TECHNICAL FIELD

This disclosure generally relates to systems and methods for addressing devices in a superconducting circuit, and, more specifically, to systems and methods for resonator-addressing of superconducting flux storage devices and digital-to-analog converters (DAC) in superconducting integrated circuits.

BACKGROUND Frequency Multiplexed Resonant (FMR) Readout

Superconducting microwave resonators have been used in a variety of fields including, but not limited to, quantum computation and astronomy. For example, in quantum computation, superconducting resonators have been used to detect the state of qubits in astronomy, superconducting microwave resonators have been used in Microwave Kinetic Inductance

Detectors ( K!Ds). In both cases, many resonators (detectors) can be coupled to a common transmission line and integrated through frequency domain multiplexing. Frequency domain multiplexing (FDM) is a technique in which a communication bandwidth is divided into a number of non-overlapping sub- bands, each sub-band used to carry a separate signal. Frequency domain multiplexing is also referred to in the present application as frequency multiplexing and frequency division multiplexing.

Using FMR technology, superconducting resonators of different resonant frequencies can be used for readout of multiple qubits. The resonators can share a common microwave transmission line by using frequency domain multiplexing. BRIEF SUMMARY

A superconducting integrated circuit may be summarized as including a microwave transmission line; a first superconducting flux storage device comprising a loop of material that is superconductive in a range of temperatures interrupted by a compound Josephson junction (CJJ), the CJJ comprising two parallel current paths each interrupted by a respective

Josephson junction; a first low-frequency address bias line; a first address interface that inductively communicatively couples the first low-frequency bias address line to the first superconducting flux storage device; and a first superconducting resonator communicatively coupled to the microwave transmission line; a first signal interface that inductively communicatively couples the first superconducting resonator to the first superconducting flux storage device, wherein the first address interface and the first signal interface are each operable to provide a respective flux bias to the first superconducting flux storage device.

The first superconducting flux storage device may be addressed by a microwave signal carried by the first superconducting resonator.

The superconducting integrated circuit may further include a second superconducting resonator communicatively coupled to the microwave transmission line; and a second signal interface that inductively

communicatively couples the second superconducting resonator to the first superconducting flux storage device, wherein the first address interface, the first signal interface and the second signal interface are each operable to provide a respective flux bias to the first superconducting flux storage device. The first superconducting resonator may extend along a first axis and the second superconducting resonator extends along a second axis, the second axis perpendicular to the first axis. The first superconducting flux storage device may be XY addressed by two microwave signals, a first one of the two microwave signals carried by the first superconducting resonator, and a second one of the two microwave signals carried by the second superconducting resonator. The first and the second superconducting resonators may have respective resonant frequencies. The respective resonant frequencies of the first and the second superconducting resonators may result in a beat signal.

The respective resonant frequencies of the first and the second

superconducting resonators may reduce an amount of cross talk therebetween.

The superconducting flux storage device may include a superconducting digitai-to-anaiog converter (DAC) that is operable to perform latching, and the first and the second superconducting resonators may be communicatively coupled to the superconducting DAC, and the

superconducting DAC may be programmed by a low-frequency bias signal carried by the first low-frequency address bias line and by two microwave signals, which combine constructively to exceed a defined address latching level.

The superconducting integrated circuit may further include a second superconducting flux storage device comprising a loop of material that is superconductive in a range of temperatures interrupted by a compound Josephson junction (CJJ), the CJJ comprising two parallel current paths each interrupted by a respective Josephson junction; a second address interface that inductively communicatively couples the first low-frequency bias address line to the second superconducting flux storage device; and a second signal interface that inductively communicatively couples the first superconducting resonator to the second superconducting flux storage device, wherein the first address interface and the first signal interface are each operable to provide a respective flux bias to the first superconducting flux storage device, and the second address interface and the second signal interface are each operable to provide a respective flux bias to the second superconducting flux storage device.

The first superconducting resonator may include at least one coupling capacitor and an inductance, and the inductance may be electrically communicatively coupled in series with the at least one coupling capacitor. The at least one coupling capacitor may include a pair of coupling capacitors, and the inductance electrically may be electrically communicatively coupled in series between the pair of coupling capacitors. The first superconducting resonator may further include a shunting capacitor that provides an electrical path to an electrical ground. The first superconducting resonator may be a distributed superconducting resonator. The first superconducting resonator may be a lumped element superconducting resonator.

In various of the above implementations, the first superconducting flux storage device may include a superconducting digita!-to-ana!og converter (DAC) that is operable to perform latching.

The superconducting integrated circuit may further include an array of superconducting resonators communicatively coupled to the microwave transmission line.

The superconducting integrated circuit may further include an array of superconducting resonators communicatively coupled to the microwave transmission line, the microwave transmission line which carries frequency domain multiplexed signals to the array of superconducting resonators.

A quantum processor may include the superconducting integrated circuit.

A control circuit for a quantum processor may include one of the various implementations of the superconducting integrated circuit described above.

A method of loading a superconducting flux storage device with a magnetic flux quantum, the superconducting flux storage device including a superconducting flux storage loop interrupted by a compound Josephson junction (CJJ), may further include setting a current bias line to a first current bias value; setting a low-frequency address line to a first address line value; setting the current bias line to a second current bias line value to cause a current bias to be applied to the superconducting flux storage device; setting the low-frequency address line to a second address line value to cause a flux bias to be applied to the CJJ of the superconducting flux storage device;

applying a first high-frequency pulse to the superconducting flux storage device by a microwave transmission line and at least a first superconducting resonator, the microwave transmission line communicatively coupled to the first superconducting resonator, and the first superconducting resonator

communicatively coupled to the superconducting flux storage device, to cause a flux quantum to be added to the CJJ of the superconducting flux storage device; setting the low-frequency address line to a third address line value to cause the flux quantum to be loaded into the superconducting flux storage loop of the superconducting flux storage device; returning the current bias line to the first current bias line value; and returning the low-frequency address line to the first address line value, wherein applying a first high-frequency pulse to the superconducting flux storage device by a microwave transmission line and a first superconducting resonator includes causing a combined low-frequency and high-frequency signal level applied to the superconducting flux storage device to exceed a predetermined upper threshold.

Setting the low-frequency address line to a third address line value to cause the flux quantum to be loaded into the superconducting flux storage loop of the superconducting flux storage device may include setting the low-frequency address line to a third address line value to cause the flux quantum to be loaded into a superconducting DAC.

Applying a first high-frequency pulse to the superconducting flux storage device by a microwave transmission line and at least a first

superconducting resonator may include applying a set of frequency-domain multiplex signals to each of the first superconducting resonator and at least a second superconducting resonator. Applying a first high-frequency pulse to the superconducting flux storage device by a microwave transmission line and at least a first superconducting resonator may include applying a set of frequency- domain multiplex signals to each of the first superconducting resonator and at least a second superconducting resonator.

The method may further include applying the first high-frequency pulse to a second superconducting flux storage device by the microwave transmission line and the at least a first superconducting resonator. Applying a first high-frequency pulse to the superconducting flux storage device by a microwave transmission line and at least a first

superconducting resonator may include applying a first high-frequency pulse to a superconducting digital-to-analog converter (DAC) by the microwave transmission line and a first and at least a second superconducting resonator, the superconducting DAC which is operable to perform latching.

The method may further include programming the

superconducting DAC by a low-frequency bias signal carried by the first low- frequency address bias line and by two microwave signals, which may combine constructively to exceed a threshold address latching level.

A method of loading a two-dimensional array of superconducting flux storage devices with a magnetic flux quantum, the superconducting flux storage devices each including a respective superconducting flux storage loop interrupted by a respective compound Josephson junction (CJJ) and arranged in rows and columns in the two-dimensional array may be summarized as including applying a bias current to the respective superconducting flux storage loop of each a number of the superconducting flux storage devices by one or more current bias lines; applying a first low-frequency flux bias to the respective CJJ of each of a number of the superconducting flux storage devices; and applying a first high-frequency flux bias to the respective CJJ of the

superconducting flux storage device in a first one of the rows of the two- dimensional array that extends along a first dimension of the two-dimensional array by a first superconducting resonator; and applying a second high- frequency flux bias to the respective CJJ of the superconducting flux storage device in a first one of the columns of the two-dimensional array that extends along a second dimension of the two-dimensional array by a second superconducting resonator that extends along a second dimension of the two- dimensional array, the second dimension different then the first dimension.

Applying a first low-frequency flux bias to the respective CJJ of each of a number of the superconducting flux storage devices may include applying the first low-frequency bias by a low-frequency address line having a frequency below a lowest threshold of a microwave frequency range, which is communicatively coupled to a biasing inductive interface. Applying a first high- frequency flux bias to the respective CJJ of the superconducting flux storage device that extends along a first one of the rows of the two-dimensional array may include applying the first high-frequency flux bias to the respective CJJ of the superconducting flux storage device by at least one microwave

transmission line, which is communicatively coupled to the first superconducting resonator. Applying a first high-frequency flux bias to the respective CJJ of the superconducting flux storage device that extends along a first one of the rows of the two-dimensional array may include applying the first high-frequency flux bias to the respective CJJ of the superconducting flux storage device by at least one coaxial microwave transmission line, which is communicatively coupled to the first superconducting resonator.

Applying a second high-frequency flux bias to the respective CJJ of the superconducting flux storage device that extends along a first one of the columns of the two-dimensional array may include applying the second high- frequency flux bias to the respective CJJ of the superconducting flux storage device by at least one microwave transmission line, which is communicatively coupled to the second superconducting resonator. Applying a second high- frequency flux bias to the respective CJJ of the superconducting flux storage device that extends along a first one of the columns of the two-dimensional array may include applying the second high-frequency flux bias to the respective CJJ of the superconducting flux storage device by at least one coaxial microwave transmission line, which is communicatively coupled to the second superconducting resonator.

Applying a first high-frequency flux bias to the respective CJJ of the superconducting flux storage device in a first one of the rows of the two- dimensional array may include applying the first high frequency flux bias by the first superconducting resonator that extends along the first dimension of the two-dimensional array and applying a second high-frequency flux bias to the respective CJJ of the superconducting flux storage device in a first one of the columns of the two-dimensional array may include applying the second high frequency flux bias by the second superconducting resonator that extends along the second dimension of the two-dimensional array. The applying a bias current, applying a first iow-frequency flux bias, applying a first high-frequency flux bias, and applying a second high-frequency flux bias in combination may cause a combined address signal level to exceed a threshold address signal latching level for one of the superconducting flux storage devices. The superconducting flux storage devices may include a respective

superconducting digita!-to-analog converter (DAC) that is operable to perform latching, each superconducting DAC having a respective pair of

superconducting resonators communicatively coupled thereto, and the applying a bias current, applying a first iow-frequency flux bias, applying a first high- frequency flux bias, and applying a second high-frequency flux bias in combination may cause a combined address signal level to exceed a threshold address latching level for one of the super conducting DACs.

The method may further include frequency domain multiplexing signals representative of the first high-frequency flux bias and the second high- frequency flux bias on a single coaxial cable that is communicatively coupled to the first and the second superconducting resonators.

A method of loading a superconducting flux storage device with a magnetic flux quantum, the superconducting flux storage device comprising a superconducting flux storage loop interrupted by a compound Josephson junction (CJJ) loop that includes a CJJ, may be summarized as including supplying a current to the superconducting flux storage loop; applying a global bias flux bias to the CJJ loop; and applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator to produce a combined flux bias of the CJJ loop that combines the applied global flux bias and the applied one or more high-frequency microwave pulses and which causes a flux quantum to be added into the CJJ loop. Supplying a current to the superconducting flux storage loop may include raising a current level of a current line that supplies the current to the superconducting flux storage loop to a first level. Applying a global bias flux bias to the CJJ loop may include applying a global bias flux bias sufficient to reduce a power of the applied one or more high-frequency microwave pulses required to cause the combined flux bias of the CJJ loop to cause the flux quantum to be added to the CJJ loop. Providing a global bias flux bias to the CJJ loop may include raising a global address bias line that supplies the global flux bias to the CJJ loop to a first calibration level, the first calibration level calibrated to cause a combined flux bias of the CJJ loop to exceed a first threshold at which the flux quantum is added into the CJJ loop.

The method may further include selecting the global bias flux bias to be sufficient to reduce a power of the applied one or more high-frequency microwave pulses required to cause the combined flux bias of the CJJ loop to exceed the first threshold.

The method may further include selecting the first threshold such that the combined flux bias of the CJJ loop from the combination of the global bias flux bias and a flux generated by the one or more high-frequency microwave pulses causes the flux quantum to be added to the CJJ loop.

Applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include applying one or more a high-frequency microwave pulses to the CJJ loop by only the first

superconducting resonator. Applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include applying one or more a high-frequency microwave pulses to the CJJ loop by the first superconducting resonator and by at least a second superconducting resonator. The first superconducting resonator may be a first distributed superconducting resonator, and applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include applying one or more a high-frequency microwave pulses to the CJJ loop by the first distributed superconducting resonator. The first superconducting resonator may be a first lumped element superconducting resonator, and applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include applying one or more a high-frequency microwave pulses to the CJJ loop by the first lumped element superconducting resonator. Applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include supplying the one or more a high-frequency microwave pulses to at least the first superconducting resonator by a transmission line. Applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include supplying the one or more a high- frequency microwave pulses to at least the first superconducting resonator via a coaxial cable. Applying one or more a high-frequency microwave pulses to the CJJ loop by at least a first superconducting resonator may include producing a frequency domain multiplexed signal, and supplying the frequency domain multiplexed signal to at least the first superconducting resonator via a transmission line.

The method may further include reducing the global bias flux bias to the CJJ loop to cause the flux quantum to move into the superconducting flux storage loop. Reducing the global bias flux bias to the CJJ loop may include lowering the global address bias line that supplies the global flux bias to the CJJ loop to a second calibration level, the second calibration level calibrated to cause a combined flux bias of the CJJ loop to fall below a second threshold at which the flux quantum is moved into the superconducting flux storage loop.

The method may further include selecting the second threshold such that the combined flux bias of the CJJ loop from the combination of the global bias flux bias and a flux generated by the one or more high-frequency microwave pulses causes the flux quantum to be moved into the

superconducting flux storage loop.

A superconducting integrated circuit may be summarized as including a microwave transmission line; a signal source; one or more electronic filters; a first superconducting resonator comprising a first superconducting resonator body that is superconductive in a range of temperatures; the first superconducting resonator communicatively coupled to the microwave transmission line; and a second superconducting resonator comprising a second superconducting resonator body that is superconductive in a range of temperatures, the second superconducting resonator

communicatively coupled to the microwave transmission line, the second superconducting resonator body communicatively coupled in series with the first superconducting resonator body; wherein the signal source is operable to provide a DC bias to the first and the second superconducting resonators by a signal path that includes the one or more electronic filters.

The microwave transmission line may be one of a microstrip transmission line, a stripline transmission line, a coaxial transmission line, and a co-pianar waveguide transmission line. The signal source may be operable to provide at least one of a low-frequency and a DC signal. The one or more electronic filters may be one of a low-pass filter, a notch filter, and a band-pass filter.

The first and the second superconducting resonators may be members of an array of three or more superconducting resonators. The first and the second superconducting resonators may be capacitively

communicatively coupled to the microwave transmission line.

The superconducting integrated circuit may be operable at a higher temperature than a cryogenic temperature to identify the presence of at least one of an electrical short, an electrical open, and an unexpected resistance in at least one of the first and the second superconducting resonators.

A superconducting integrated circuit may be summarized as including a microwave transmission line; a signal source; an electronic filter; a superconducting resonator comprising a superconducting resonator body that is superconductive in a range of temperatures, the superconducting resonator communicatively coupled to the microwave transmission line, the

superconducting resonator body communicatively coupled to the signal source by the electronic filter; and a superconducting device that is communicatively coupled to the superconducting resonator, the superconducting device operable to receive a programming signal that includes at least a DC bias from the signal source by a first signal path that includes the electronic filter.

The microwave transmission line may be one of a microstrip transmission line, a stripline transmission line, a coaxial transmission line, and a co-planar waveguide transmission line. The signal source may be operable to provide at least one of a low-frequency and a DC signal. The electronic filter is one of a low-pass filter, a notch filter, and a band-pass filter.

The superconducting resonator may be a member of an array of two or more superconducting resonators. The superconducting resonator may be capacitively communicatively coupled to the microwave transmission line. The superconducting resonator body may be galvanically communicatively coupled to the signal source by the electronic filter. The superconducting device may be galvanically communicatively coupled to the superconducting resonator. The superconducting device may be inductively communicatively coupled to the superconducting resonator.

The superconducting device may include a loop of material that is superconductive at a range of temperatures, the loop of material interrupted by a compound Josephson junction (CJJ). The low-frequency bias address line may be operable to provide a flux bias to the compound Josephson junction of the superconducting device.

The superconducting device may be a superconducting flux storage device. The superconducting flux storage device may include a digital- io-ana!og converter (DAC).

The programming signal may further include a high-frequency signal by a second signal path that includes the microwave transmission line.

In various of the implementations described above, the superconducting integrated circuit may further include a switch operable to dynamically isolate a device. The switch may comprise a plurality of cascade elements superconductlngiy electrically communicatively coupled in series, each cascade element of the plurality of cascade elements comprising a respective first plurality of N Superconducting Quantum Interference Devices (SQU!Ds) superconductingly electrically communicatively coupled in series in a first arm, each SQUID of the first plurality of SQUIDs comprising at least one Josephson Junction, a respective matching capacitor, and a respective second plurality of M SQUIDs superconductingly electrically communicatively coupled in series in a second arm, opposite the first arm with respect to the matching capacitor, each SQUID of the second plurality of SQUIDs comprising at least one Josephson Junction.

In some implementations, the total number of SQUIDS N in the first plurality of N SQUIDS is equal to the total number of SQUIDS M in the second plurality of M SQUIDS.

The switch may further include a flux bias line and at least one filtering element along the flux bias line, the filtering element operable to cause the switch to operate at an operating speed. The at least one filtering element may be selected from a group consisting of: a passive filtering elements, inductive chokes and kinetic inductors.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the drawings, identical reference numbers identify similar elements or acts. The sizes and relative positions of elements in the drawings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not necessarily drawn to scale, and some of these elements are arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn, are not necessarily intended to convey any information regarding the actual shape of the particular elements, and have been solely selected for ease of recognition in the drawings.

Figure 1 is a schematic diagram illustrating an example implementation of a microwave path to a superconducting circuit, according to the present disclosure. Figure 2 is a schematic diagram illustrating another example implementation of a microwave path to a superconducting circuit according to the present disclosure.

Figure 3A is a schematic layout of a portion of an example implementation of a superconducting circuit that includes a superconducting half-wave resonator and a transmission line, according to the present disclosure.

Figure 3B is a zoom of a coupling region of the superconducting half-wave resonator of Figure 3A.

Figure 4A is a schematic layout of a portion of an example implementation of a superconducting circuit that includes a superconducting lumped-element resonator and a transmission line, according to the present disclosure.

Figure 4B is an enlarged view of a coupling region of the superconducting lumped-element resonator of Figure 4A.

Figure 4C is a cross-section of a portion of an example implementation of a superconducting circuit that includes a superconducting lumped-element resonator and a transmission line, according to the present disclosure.

Figure 5A is a schematic diagram illustrating an example implementation of a superconducting circuit that includes a

resonator-addressed digital-to-analog converter (DAC), according to the present disclosure.

Figure 5B is a schematic diagram illustrating another example implementation of a superconducting circuit that includes a

resonator-addressed digital-to-analog converter (DAC), according to the present disclosure.

Figure 5G is a schematic diagram illustrating an example implementation of a superconducting circuit that includes a pair of resonator- addressed digital-to-analog converters (DACs), according to the present disclosure. Figure 5D is a schematic diagram illustrating another example implementation of a superconducting circuit that includes a resonator- addressed digital-to-analog converter (DAC), according to the present disclosure.

Figure 5E is a schematic diagram illustrating an example implementation of a superconducting circuit that includes a resonator- addressed digital-to-analog converter (DAC), and a resonator that is inductively coupled to a transmission line, according to the present disclosure.

Figure 5F is a schematic diagram illustrating another example implementation of a superconducting circuit that includes a resonator- addressed digital-to-analog converter (DAC), and a resonator that is inductively coupled to a transmission line, according to the present disclosure.

Figure 6 is a flow chart illustrating an example method for programming a superconducting DAC, according to the present disclosure.

Figures 7A, 7B and 7C are schematic plots illustrating an example time series for loading of a single flux quantum, according to the present disclosure.

Figure 7A is a graph showing a plot of an address signal current as a function of time.

Figure 7B is a graph showing a plot of a power signal current as a function of time.

Figure 7C is a graph showing a plot of a persistent current in a compound Josephson junction (CJJ) loop of the superconducting DAC as a function of time.

Figure 8 is a schematic diagram of an exemplary computing system including a digital computer and an analog computer in accordance with the present systems, devices, and methods.

Figure 9A is a schematic diagram illustrating an example implementation of a superconducting circuit, according to the present disclosure. Figure 9B is a schematic diagram illustrating another example implementation of a superconducting circuit, according to the present disclosure.

Figure 10 is a schematic diagram illustrating an example implementation of a superconducting circuit that includes an array of resonators, according to the present disclosure.

Figure 11 is a schematic diagram illustrating an example implementation of a superconducting circuit that includes a switch, according to the present disclosure.

Figure 12 is a schematic diagram of an example implementation of a transmission line layout with N cascade elements, according to the present disclosure.

Figure 13 is a schematic diagram of an example implementation of a single cascade element, according to the present disclosure.

DETAILED DESCRIPTION Preamble

In the following description, some specific details are included to provide a thorough understanding of various disclosed implementations and embodiments. One skilled in the relevant art, however, will recognize that embodiments may be practiced without one or more of these specific details, or with other methods, components, materials, etc. In other instances, well-known structures associated with superconductive devices and integrated

superconductive circuits have not been shown or described in detail to avoid unnecessarily obscuring descriptions of the implementations or embodiments of the present methods. Throughout this specification and the appended claims, the words“element” and“elements” are used to encompass, but are not limited to, ail such structures, systems, and devices associated with superconductive circuits and integrated superconductive circuits.

Unless the context requires otherwise, throughout the specification and claims which follow, the word“comprise” is synonymous with “including,” and is inclusive or open-ended (i.e., does not exclude additional, unrecited elements or acts).

Reference throughout this specification to“one embodiment”“an embodiment”,“another embodiment”,“one example”,“an example”,“another example”,“one implementation”,“another implementation”, or the like means that a particular referent feature, structure, or characteristic described in connection with the embodiment, example, or implementation is included in at least one embodiment, example, or implementation. Thus, the appearances of the phrases“in one embodiment”,“in an embodiment”,“another embodiment” or the like in various places throughout this specification are not necessarily all referring to the same embodiment, example, or implementation. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments, examples, or implementations.

It should be noted that, as used in this specification and the appended claims, the singular forms "a," "an," and "the" include plural referents unless the content clearly dictates otherwise. Thus, for example, reference to a readout system including "a superconducting resonator" includes a single superconducting resonator, or two or more superconducting resonators it should also be noted that the term "or" is generally employed in its sense including "and/or" unless the content clearly dictates otherwise.

The headings provided herein are for convenience only and do not interpret the scope or meaning of the embodiments.

Superconducting flux storage devices are also referred to in the present application as superconducting digital-to-anaiog converters (DACs) or flux DACs.

While a single qubit, or a handful of them, can be precisely controlled with dedicated analog lines driven by room-temperature electronics, integrating more than a few dozen qubits on a single chip can preferably be implemented using on-chip control circuitry in one implementation of a quantum processor that includes superconducting qubits and couplers, there can be six control parameters per qubit, and one control parameter per coupler. Static flux biases can be selectively applied to superconducting loops in order to realize at least some of the control parameters. The desired values of flux biases can be programmed into control devices using a relatively small number of control lines that carry signals generated at room temperature.

In some implementations, the control devices can combine the functions of persistent memory and digitai-to-ana!og conversion, and are referred to in the present application as flux DACs. In an example

implementation, each flux DAC has a size of about 10 pm. Having several of them attached to a single qubit can set a lower bound on qubit size, and can influence possible qubit shapes, hardware graph topologies, and processor architecture.

XYZ-Addressinq of Flux DACs

In an example implementation of a quantum processor, 512 qubits and associated coupling devices can be controlled by 4608 flux DACs. An XYZ-addressing approach can control the quantum processor using only 56 lines. The processor can be arranged as an 8x8 array of files, each tile having 72 flux DACs. The flux DACs of each tile can arranged in a respective 3-DAC plaquette. One of each set of three DACs in a 3-DAC piaquette can be selected using one of three lines (referred to in the present application as ADDR lines), ail three sharing another line (referred to in the present application as a TRIG line), in an arrangement that uses 15 ADDR lines and 5 TRIG lines to address DACs in the tile. The 8x8 array of files can be divided into

16 domains (referred to in the present application as PWR domains), and arranged such that 4608 flux DACs can be addressed using 30 ADDR lines,

10 TRIG lines, and 16 PWR lines in total. See for example Bunyk P. et a!., “ARCHITECTURAL CONSIDERATIONS IN THE DESIGN OF A

SUPERCONDUCTING QUANTUM ANNEALING PROCESSOR”,

arXiv:1401.55Q4v1 , 21 January 2014. A shortcoming of the DAC addressing scheme described above can be that implementation can include a large number of wires and

superconducting low-pass filters which can be costly to produce, vulnerable to failure, and a source of high-energy photons from internal filter heating in some implementations, the wires are twisted-pairs and the low-pass filters are custom-made superconducting low-pass filters. The systems and methods described in the present application include replacing wires by coaxial cable which can provide greater bandwidth and reduced complexify. A benefit of greater bandwidth can be improved processor input/output (I/O) speed. Superconducting resonators

When high-bandwidth lines are available for communicating with the processor, more efficient use of the bandwidth can be made by coupling the high-bandwidth line to a suitable on-chip element. An example on-chip element suitable for coupling to a high-bandwidth transmission line is a superconducting resonator. An advantage of using a superconducting resonator is that, similar to FMRR or kinetic-inductance detectors (KIDs), these devices can be compatible with frequency-domain multiplexing. An array of resonators can be coupled to a single transmission line using frequency-domain multiplexing by tuning a resonant frequency of each resonator. The resonant frequency can be tuned, for example, by adjusting the length of the resonator, or by adding additional capacitance. In one implementation, the resonator is a half-wave superconducting resonator. Microwave current can be excited when the resonator is driven near its resonant frequency. Addressing can be achieved by selectively transmitting a set of tones corresponding to a set of resonators in the array of resonators.

One implementation of a superconducting resonator is a distributed superconducting resonator. An example of a distributed

superconducting resonator is half-wavelength (A/2) resonator. A half- wavelength resonator can generally provide sufficient usable wiring length for coupling to another device. A shortcoming of a half-wavelength resonator can be that the presence of harmonics at integer multiples of the fundamental resonant frequency can constrain an array of superconducting resonators to one octave of bandwidth.

An additional shortcoming can be that the current distribution can vary sinusoidally along the wiring length, with a peak value occurring at the center of the device. One approach to mitigating the variation in current is to use only a section of the length of the resonator, for example a section in which the variation in current remains within, say, 90% of the peak value. Another approach is to vary the strength of coupling of the resonator to devices coupled along the length of the resonator. Yet another approach is to adjust the driving amplitudes of intersecting resonators to compensate for the resonator current distribution. The latter approach can be used in the XY~addressing scheme described below, for example in some implementations, resonators are operated individually. In some implementations, resonators are operated in groups of resonators.

Another example of a distributed superconducting resonator is quarter-wavelength (A/ 4) resonator. The peak current for a quarter-wavelength resonator is at one end of the resonator. Variation in current along the body of the resonator can be mitigated by using only a section of the length of the resonator, specifically a section at or near the end where the current is within, say 90% of the peak value.

Lumped-Element Resonators

Another Implementation of a superconducting resonator is a lumped-element resonator. An advantage of a lumped-element resonator is that the current distribution can be substantially uniform along an inductive portion of the lumped-element resonator. Furthermore, a lumped-element resonator can be compact and, in at least some implementations, is not constrained to one octave of bandwidth. In an example implementation, a lumped element resonator can operate from 5 GHz to 15 GHz and yield 10 GHz of bandwidth.

There are several constraints that can influence achievable electronic bandwidth utilization of a resonator-multiplexed programming system (also referred to in the present application as system bandwidth). The same or similar constraints can also influence the system bandwidth of implementations of superconducting resonators other than lumped-element resonators.

Firstly, system bandwidth can be constrained by cross-talk. The effective system bandwidth can be limited by as much as 30% in some implementations where resonances are spaced apart from each other to reduce cross-talk. The spacing can be several !inewidtbs. In an example

implementation, the system bandwidth is 4-8 GHz, and the per-resonator bandwidth is approximately 30 MHz centered on a resonant frequency within the system bandwidth.

Secondly, variations in fabrication, such as variations in layer thickness, can shift a resonant frequency by as much as 10% in some implementations.

Thirdly, layout of resonators and associated wiring can cause resonator-to-resonator frequency shifts. In some implementations, resonator bodies are routed in a single wiring layer to reduce resonator-to-resonator frequency shifts. In some implementations, where vias are used, an array of resonators can use the same number of vias with approximately the same distribution along the resonator length for every resonator in the array, to avoid or at least reduce differential changes in resonator lengths, resonator-to- resonator frequency shifts, and contaminant electronic cross-talk.

Fourthly, a higher-frequency resonator can be shorter than a lower-frequency resonator, and, in some instances, can become too short to span a width of a processor in which an array of resonators is used for DAC addressing. The constraint can be mitigated in some implementations by using a thin dielectric layer In the fabrication of parallel-plate capacitors in the resonators to reduce the footprint of the capacitors. See for example U.S. Provisional Patent Application Serial No 62/660,719,“SYSTEMS AND

METHODS FOR FABRICATION OF SUPERCONDUCTING DEVICES”, filed April 20, 2018

In some implementations, a kinetic-inductance transmission line is used to route microwave signals with high isolation within the processor fabric using three layers or less. See, for example, PCT Patent Application

No WO2018US016237, SYSTEMS AND METHODS FOR FABRICATION OF SUPERCONDUCTING INTEGRATED CIRCUITS, filed January 31 , 2018.

High-Bandwidth Lines

Figure 1 is a schematic diagram illustrating an example implementation of a microwave path 100 to a superconducting circuit, according to the present disclosure. Microwave path 100 is positioned in a cryogenic refrigeration system, and travels between a still 102 and a sample holder 104 The cryogenic refrigeration system includes still 102, sample holder 104, and mixing chamber 106. Microwave path 100 is a microwave path through the cryogenic refrigeration system. Still 102, sample holder 104, and mixing chamber 106 are shown as dashed lines in Figure 1 to provide context for microwave path 100.

Sample holder 104 can contain or hold the superconducting circuit in some implementations, the superconducting circuit is an integrated superconducting circuit. In some implementations, the integrated

superconducting circuit includes a quantum processor in some

implementations, the quantum processor includes a plurality of superconducting flux qubits. See, for example, PCT Patent Application No.

PGT/US20G9/G55939 (published as International patent application publication WO 2010/028183A2), SYSTEMS, METHODS AND APPARATUS FOR ACTIVE COMPENSATION OF QUANTUM PROCESSOR ELEMENTS, filed September 3, 2009.

Microwave path 100 traverses mixing chamber 106 Microwave path 100 includes a superconducting coaxial cable 108 between still 102 and mixing chamber 106. In some implementations, superconducting cable 108 is superconducting NbTi coaxial cable.

Microwave path 100 includes a bandpass filter 110, a powder filter 112, and a switch 114 for the operating band in some implementations, powder filter 112 is a superconducting powder filter. Bandpass filter 110 is communicatively coupled to powder filter 112 by coaxial cable 116. In some implementations, bandpass filter 110 is a GHz bandpass filter with greater than 90 dB out-of-band suppression of frequencies below 60 GHz. Bandpass filter 110 and powder filter 112 in combination can determine at least in part the filtering of signals on microwave path 100. The use of one or both of bandpass filter 110 and powder filter 112 can be optional in some implementations. One reason for including filters is to reduce noise on high-bandwidth lines used for programming devices such as devices in a quantum processor.

In some implementations operating at a temperature of approximately 4 K, switch 114 is a CMOS switch. In other implementations operating at a temperature of approximately 4 K, switch 114 is a

superconducting switch, e.g., a cryotron in some implementations operating at a temperature of approximately 10 mK, switch 114 is a superconducting switch, e.g., a cryotron. In yet other implementations, switch 114 is a broadband switch as described in the present application with reference to Figures 11 , 12, and 13. The broadband switch of Figures 11 , 12, and 13 is further described in US Patent Application No. 16/397,790, DYNAMICAL ISOLATION OF A

CRYOGENIC PROCESSOR, filed April 29, 2019.

In some implementations, coaxial cable 116 is a copper coaxial cable. Powder filter 112 is communicatively coupled to switch 114 by coaxial cable 118. In some implementations, coaxial cable 118 is a copper coaxial cable. Switch 114 is communicatively coupled to sample holder 104 by coaxial cable 120. in some implementations, coaxial cable 120 is a copper coaxial cable.

Figure 2 is a schematic diagram illustrating another example implementation of a microwave path to a superconducting circuit according to the present disclosure. Microwave path 200 is positioned in a cryogenic refrigeration system, and travels between a still 202 and a sample holder 204. The cryogenic refrigeration system includes still 202, sample holder 204, and mixing chamber 206. Microwave path 200 is a microwave path through the cryogenic refrigeration system. Still 202, sample holder 204, and mixing chamber 206 are shown as dashed lines in Figure 2 to provide context for microwave path 200.

Sample holder 204 can contain or hold the superconducting circuit in some implementations, the superconducting circuit is an integrated superconducting circuit. The integrated superconducting circuit may comprise or include a quantum processor with one or more other superconducting processors or devices, or may consist of a quantum processor. In some implementations, the quantum processor includes a plurality of superconducting flux qubits.

Microwave path 200 traverses a mixing chamber 206. Microwave path 200 includes a coaxial cable 208 between still 202 and mixing chamber 206. In some implementations, coaxial cable 108 is a superconducting coaxial cable. In some implementations, coaxial cable is a stainless steel coaxial cable.

Microwave path 200 includes a bandpass filter 210, and a switch 212. Bandpass filter 210 is communicatively coupled to switch 212 by coaxial cable 214. in some implementations, bandpass filter 210 is a GHz bandpass filter with greater than 90 dB out-of-band suppression of frequencies below 60 GHz. in some implementations, coaxial cable 214 is a copper coaxial cable. Switch 212 is communicatively coupled to sample holder 204 by coaxial cable 216. In some implementations, coaxial cable 216 is a copper coaxial cable.

In some implementations operating at a temperature of approximately 4 K, switch 212 is a CMOS switch. In other implementations operating at a temperature of approximately 4 K, switch 212 is a

superconducting switch, e.g., a cryotron. In some implementations operating at a temperature of approximately 10 mK, switch 212 is a superconducting switch, e.g., a cryotron.

Example Lumped-Element Resonators

Figure 3A is a schematic layout of a portion of an example implementation of a superconducting circuit 300 that includes a

superconducting half-wave resonator 302 and a transmission line 304, according to the present disclosure. For example, superconducting half-wave resonator 302 can be a microstrip resonator, a coplanar waveguide (CPW) resonator, or a lumped-element resonator. For example, transmission line 304 can be a microstrip transmission line, a CPW transmission line, or a coaxial transmission line.

Superconducting half-wave resonator 302 includes a resonator body 306. Figure 3A illustrates a portion of resonator body 306. In one implementation, resonator body 306 is 7 mm in length.

Region 308 (shown in Figure 3A by hatching) is a region over which the current is within a suitable threshold of the maximum current in some implementations, the suitable threshold is 90% of the maximum current in some implementations, a length of resonator body 306 that is not used for coupling to a DAC is routed in a serpentine shape, e.g., to increase

compactness of the superconducting half-wave resonator 302.

Figure 3B is an enlarged view of a coupling region 310 of superconducting half-wave resonator 302 of Figure 3A. Superconducting half wave resonator 302 includes wiring 312 and a coupling capacitor 314. In some implementations, coupling capacitor 314 is a parallel-plate capacitor. In some implementations, coupling capacitor 314 includes a thin-film metal-oxide layer for a dielectric in the parallel-plate capacitor. See for example U.S. Provisional Patent Application Serial No. 62/660,719,“SYSTEMS AND METHODS FOR FABRICATION OF SUPERCONDUCTING DEVICES”, filed April 20, 2018. In one implementation, coupling capacitor 314 has dimensions of 30 pm x 20 pm. Superconducting circuit 300 includes a via 316, for example to provide a superconducting communicative coupling between transmission line 304 and a lower capacitor plate (not shown in Figure 3B) of coupling capacitor 314.

Superconducting half-wave resonator 302 can have a symmetric coupling to ground (also referred to in the present application as an electrical ground) at the other end (not shown in Figures 3A and 3B). in some implementations, wiring 312 and/or transmission line 304 include 2 pm lines over a 1 pm silicon dioxide (Si02) dielectric, which can provide an impedance close to 50 W.

In one implementation, superconducting half-wave resonator 302 has a resonance at 9.58 GHz with approximately 35 MHz of bandwidth.

Figure 4A is a schematic layout of a portion of an example implementation of a superconducting circuit 400a that includes a

superconducting lumped-element resonator 402 and a transmission line 404, according to the present disclosure. Superconducting lumped-element resonator 402 includes a resonator body 406.

Region 408 (shown in Figure 4A by hatching) is a region over which the current is within a suitable threshold of the maximum current in some implementations, the suitable threshold is 90% of the maximum current.

Figure 4B is an enlarged view of a coupling region 410 of superconducting lumped-element resonator 402 of Figure 4A. Superconducting lumped-element resonator 402 includes wiring 412, a coupling capacitor 414, and a shunting capacitor 416. in some implementations, coupling capacitor 414 is a parallel-plate capacitor. In some implementations, coupling capacitor 414 includes a thin-film metal-oxide layer for a dielectric in the parallel-plate capacitor. In one implementation, coupling capacitor 414 has dimensions of 30 pm x 30 pm. In one implementation, shunting capacitor 416 has dimensions 100 pm x 120 pm.

Superconducting lumped-element resonator 302 can have a symmetric coupling to ground at the other end. In some implementations, wiring 412 and/or transmission line 404 include 2 lines over a 1 pm silicon dioxide (Si02) dielectric, which can provide an impedance dose to 50 W.

In one implementation, superconducting lumped-element resonator 402 has a resonance at 9.82 GHz with approximately 30 MHz of bandwidth.

Figure 4C is a schematic drawing of a cross-section of a portion of an example implementation of a superconducting circuit 400c that includes a superconducting lumped-element resonator 418 and a transmission line 420, according to the present disclosure.

Superconducting circuit 400c is implemented in a multilayer superconducting integrated circuit. Transmission line 420 is coupled to a ground plane 422. Transmission line 420 is superconductingiy

communicatively coupled by a superconducting via 424 and superconducting wiring 426 to a lower plate 428 of a coupling capacitor (such as coupling capacitor 414 of Figure 4B).

An upper plate 430 of the coupling capacitor is superconductingiy communicatively coupled to a resonator body 432 and an upper plate 434 of a capacitor coupling superconducting lumped-element resonator 418 to a ground plane 436.

Resonator-addressed DAC

Resonator-addressing of a superconducting digital-to-analog converter (DAC) can use the DAC in operation to perform latching. Resonator addressing is also referred to in the present application as resonator-loading of a DAC. A resonator can be integrated with a DAC, and can, for example, be used to replace an ADDR line and/or a TRIG line.

The superconducting DAC is a flux memory or flux storage device. The DAC can convert a digital amount of flux Into flux stored in an analog device.

In some implementations, the DAC includes a loop of material that is superconducting in a range of temperatures, the loop interrupted by one or more Josephson junctions. In one implementation, the DAC is an rf-SQU!D and includes a superconducting loop interrupted by a single josephson junction. In another implementation, the DAC is a superconducting loop interrupted by a compound Josephson junction (CJJ). The DAC CJJ loop can behave as a summing element for magnetic flux. The loop can combine a DC current provided by a wire (e.g., a twisted-pair) with a current induced by a resonator. In some implementations, the wire is shared globally, i.e., shared by ail the DACs in the processor, and the wire can provide a global bias signal.

Storing flux in the DAC includes adding flux to the CJJ loop of the DAC, and moving flux in the CJJ loop of the DAC into the storage loop of the DAC (i.e., into the superconducting loop of the DAC). Flux can be added to the CJJ loop of the superconducting DAC by suppressing a Josephson junction in the CJJ using one or more control signals or biases. Flux can be moved into the DAC storage loop by suppressing another josephson junction in the CJJ loop using one or more control signals or biases. Multiple flux quanta can be stored in a superconducting DAC implemented using a CJJ loop.

See, for example, Johnson M.W. et al.“A scalable control system for a superconducting adiabatic quantum optimization processor”,

arXiv:0907.3757v2, 24 March 2010 for a description of flux DACs operable to control superconducting devices in an integrated circuit

An array of resonators can be used to provide one or more signals that can be combined with the global bias signal to enable addressing.

Figure 5A is a schematic diagram illustrating an example implementation of a superconducting circuit 500a that includes a resonator- addressed DAC 502, according to the present disclosure.

In superconducting circuit 500a, resonator-addressed DAC 502 includes a loop 504 of material that is superconductive in a range of temperatures, and a CJJ 506. CJJ 506 includes two parallel current paths each Interrupted by a Josephson junction, Josephson junctions 508 and 510, respectively. Superconducting circuit 500a further includes a resonator 512, a transmission line 514, and interface 516 operable to provide a global flux bias by a global flux bias line 516-1. Resonator 512 includes coupling capacitors 518 and 520, an inductive interface 522 to resonator-addressed DAC 502, and an optional shunting capacitor 524. The term“inductive interface” is also referred to in the present application as a coupling inductance. Coupling inductance 522, for example, inductively communicatively couples resonator 512 and resonator-addressed DAC 502.

Superconducting circuit 500a further includes a current bias line 526 operable to provide a current bias to resonator-addressed DAC 502.

In some implementations, multiple resonators are coupled to a single DAC CJJ loop. Figure 5B is a schematic diagram illustrating another example implementation of a superconducting circuit 500b that includes a resonator-addressed digital-to-analog converter (DAC) 502, according to the present disclosure.

In Figure 5B, the same reference numbers as used in Figure 5A are used in Figure 5B to denote the same or similar elements. In instances where there is more than one of a given element in the implementation of Figure 5B, the reference number has a suffix, for example“a” or“b”.

In superconducting circuit 500b, resonator-addressed DAC 502 includes a loop 504 of material that is superconductive in a range of temperatures, and a CJJ 506. CJJ 506 includes two parallel current paths each interrupted by a Josephson junction, Josephson junctions 508 and 510, respectively.

Superconducting circuit 500b further includes a pair of resonators 512a and 512b, a transmission line 514, and interface 516 operable to provide a global flux bias by a global flux bias line 516-1. Resonator 512a includes coupling capacitors 518a and 520a, an inductive interface 522a to resonator- addressed DAC 502, and an optional shunting capacitor (not shown in Figure 5B for clarity) like optional shunt capacitor 524 of Figure 5A Resonator 512b includes coupling capacitors 518b and 520b, an inductive interface 522b to resonator-addressed DAC 502, and an optional shunting capacitor (not shown in Figure 5B for clarity) like optional shunt capacitor 524 of Figure 5A.

Superconducting circuit 500b further includes a current bias line 528 operable to provide a current bias to resonator-addressed DAC 502.

In one implementation, resonators 512a and 512b are both coupled to transmission line 514. In another implementation, resonator 512a is coupled to transmission line 514, and resonator 512b is coupled to another transmission line (not shown in Figure 5B).

In an example where two resonators are coupled to the same DAC CJJ loop, programming of the DAC can be achieved by having a low- frequency bias signal and two microwave signals combine constructively to exceed a determined address latching level. The arrangement can be used to construct an XY-addressing scheme for a set of DACs, for example as follows. Firstly, a DC current bias is applied to the DAC by a line referred to in the present application as a power line (PWR). Next, a global low-frequency bias address line (such global flux bias line 516-1 of Figures 5A and 5B) is coupled to each DAC CJJ loop of the set of DAC CJJ loops. Then, an array of resonators is laid out across the processor in a two-dimensional X-Y grid. Programming a selected DAC can be achieved by pre-biasing the set of DAC CJJ loops, and activating the two resonators (also referred to in the present application as the address and trigger resonators) that intersect at the selected DAC.

The selected DAC can be programmed when the two microwave tones (at frequencies f _l0 and f _hi respectively) add constructively at the selected DAC. A relative phase between the two microwave tones can evolve at an angular frequency w given by the following expression:

w = 2n(f _hi - f _l0 )

The higher frequency tone can advance by dq = w// _ί0 radians per cycle relative to the low frequency tone. Each time the low frequency tone is at a maximum, the high frequency tone can have advanced by an angle of _< 50. The angle dq can represent the closest the maxima can approach within a cycle at the beat frequency, and, since the angle dq can be linearly proportional to the beat frequency w, it can be desirable to select a smaller beat frequency w. On the other hand, the smaller the beat frequency w, the longer it will take to complete a full 2p rotation.

It can be desirable to leave the resonator charged no more than a few multiples of r = ΐ/(2pDί) where Af is the bandwidth per tone. It can typically take a few multiples of t to charge and discharge the resonator.

Programming can be completed roughly on the same timescale for efficient bandwidth utilization. This suggests that it can be desirable to match the beat frequency to the linewidth of the resonator.

This leads to a criterion for completing the programming in a reasonable time - that the beat frequency should be approximately matched to a resonator linewidth (w~2pA†). This is consistent with a preference for resonances within a single array to be separated by a few linewidths Af to avoid, or at least reduce, cross-talk between channels. With the beat frequency determined, the resonant frequencies can, in general, be selected to be at higher frequencies consistent with the determined beat frequency.

Figure 5C is a schematic diagram illustrating an example implementation of a superconducting circuit 500c that includes a pair of resonator-addressed digital-to-analog converters (DACs) 502a and 502b, according to the present disclosure.

The same reference numbers as used in Figure 5A are used in Figure 5C to denote the same or similar elements. In instances where there is more than one of a given element in the implementation of Figure 5C, the reference number has a suffix, for example“a” or“b”.

In superconducting circuit 500c, resonator-addressed DAC 502a includes a loop 504a of material that is superconductive in a range of temperatures, and a CJJ 506a. CJJ 506a includes two parallel current paths each interrupted by a Josephson junction, Josephson junctions 508a and 510a, respectively. Resonator-addressed DAC 502b includes a loop 504b of material that is superconductive in a range of temperatures, and a CJJ 506b. CJJ 506b includes two parallel current paths each interrupted by a Josephson junction, Josepbson junctions 508b and 510b, respectively.

Superconducting circuit 500c further includes a resonator 512, a transmission line 514, and interfaces 516a, 516b operable to provide a global flux bias by a global flux bias line 516-1. Resonator 512 includes coupling capacitors 518 and 520, an inductive interface 522a to resonator-addressed DAC 502a, an inductive interface 522b to resonator-addressed DAC 502b, and an optional shunting capacitor 524.

Superconducting circuit 500c further includes a current bias line 526a operable to provide a current bias to resonator-addressed DAC 502a, and a current bias line 526b operable to provide a current bias to resonator- addressed DAC 502b.

A variation in current along the body of resonator 512 of

Figure 5C can be at least partially compensated for by tuning each coupling of resonator 512 to DAC CJJ loops 502a and 502b. In some implementations, resonator 512 can be coupled to more than two DAC CJJ loops (only two DAC CJJ loops are shown in Figure 5C), and a variation in current along the body of resonator 512 can be at least partially compensated for by tuning the coupling of resonator 512 to a respective DAC CJJ loop.

Figure 5D is a schematic diagram illustrating another example implementation of a superconducting circuit 5GQd that includes a resonator- addressed digital-to-analog converter (DAC) 502, according to the present disclosure. Resonator 512 of Figure 5D includes a DC-SQUID 528. DC- SQUID 528 includes a pair of Josephson junctions 530 and 532.

Superconducting circuit 500d includes an interface 534. interface 534 can be provided by a DAC or an analog line, for example. A resonant frequency of resonator 512 of Figure 5D can be tuned by applying a flux bias to DC-SQUID 532 by interface 534.

In some implementations, a critical current of at least one of Josephson junctions 530 and 532 is comparable in magnitude to a current in resonator 512. Instead of capacitively communicatively coupling resonator 512 of Figures 5A, 5C, and 5D to transmission line 514 by coupling capacitor 518, resonator 512 can be communicatively coupled to transmission line 514 by an inductive coupling. For example, resonator 512 can be inductively

communicatively coupled to transmission line 514 by an inductive coupling to a portion of the body of resonator 512.

Similarly, instead of capacitively communicatively coupling each of resonators 512a and 512b of Figure 5B to a respective transmission line by a respective coupling capacitor, at least one of resonators 512a and 512b can be inductively communicatively coupled to the respective transmission line by an inductive coupling to a portion of the body of the resonator.

Figure 5E is a schematic diagram illustrating an example implementation of a superconducting circuit 5GQe that includes a resonator- addressed digital-to-analog converter (DAC) 502, and a resonator 512 that is inductively coupled to transmission line 514, according to the present disclosure. The same reference numbers as used in Figure 5A are used in Figure 5E to denote the same or similar elements.

Superconducting circuit 500e includes a transmission line inductance 536, and matching capacitors 538 and 540. Matching capacitors 538 and 540 can at least partially tune an impedance of transmission line 514 (e.g. to cause the impedance of transmission line 514 to be approximately 50 W).

Resonator 512 can be inductively communicatively coupled to transmission line 514 by transmission line inductance 536. Resonator 512 can be used to address one or more DACs or other superconducting devices.

Resonator 512 includes a first part 542 that includes an inductive interface 522 to resonator-addressed DAC 502, and a second part 544 that includes an inductive interface 546 to another resonator-addressed DAC (not shown in Figure 5E). First part 542 includes a capacitor 520, and second part 544 includes a capacitor 548 Figure 5F is a schematic diagram illustrating another example implementation of a superconducting circuit 5G0f that includes a resonator- addressed digital-to-analog converter (DAC) 502, and a resonator 512 that is inductively coupled to transmission line 514, according to the present disclosure. The same reference numbers as used in Figure 5A are used in Figure 5F to denote the same or similar elements. Superconducting circuit 50Qf includes a coupling inductance 550 that can inductively communicatively couple resonator 512 to transmission line 514 by transmission line inductance 536.

Figure 6 is a flow chart illustrating an example method 600 for programming a DAC according to the present disclosure. Figure 6 illustrates a sequence of currents used to load a single flux quantum into a DAC storage loop using standard 30 MHz lines. Method 600 may be implemented by a programming system. For example, method 600 may be implemented in electronics operating at room temperature. In some implementations, the phases and amplitudes of the high-frequency tones are controllable, for example by electronics operating at room temperature.

Method 600 includes acts 602 to 618, though those of skill in the art will appreciate that in alternative embodiments certain acts may be omitted and/or additional acts may be added. Those of skill In the art will appreciate that the illustrated order of the acts is shown for exemplary purposes only and may change in alterative embodiments.

Method 600 starts at 602, for example in response to an initiation of the programming of a DAC. At 604, the DAC power level in the current line to the resonator-addressed DAC (such as resonator-addressed DAC 502 of Figure 5A) is raised to a pre-call brated level or defined calibration level. At 606, the global address bias line is raised to a pre-calibrated level or defined calibration level, thereby applying a flux bias to one or more resonator- addressed DACs The level is calibrated to cause the level to cross a predetermined or defined upper threshold when high-frequency microwave pulses are applied by a transmission line (such as transmission line 514 of Figure 5A) and a resonator (such as resonator 512 of Figure 5A). The upper threshold is selected to cause a flux quantum to be added into the CJJ loop when the upper threshold is exceeded by applying and combining the global bias and the high-frequency microwave pulses.

At 608, the high-frequency ADDR and/or TRIG (X and/or Y) microwave pulses are applied to selected devices, e.g., to selected resonator- addressed DACs. in an XY-addressing scheme, the flux bias injected into the DAC by the ADDR (or X) line is one-half of the flux needed to cause the combined address signal current (see Figure 7) to exceed a threshold for programming the DAC, and the flux bias injected into the DAC by the TRIG (or Y) line is similarly one-half of the flux needed to cause the combined address signal current to exceed a threshold for programming the DAC. A DAC is programmed only when a threshold is exceeded in the combined address signal current (see Figures 7A-7C and accompanying description). The systems and methods described in the present application cause the threshold to be exceeded when a) a sufficient flux bias is applied by a global address bias line, and b) there are pulses on the ADDR (X) and TRIG (Y) address lines.

Use of a global flux bias can be beneficial for at least the following two reasons. Firstly, the bias breaks the symmetry, and eliminates, or at least reduces, the possibility that the method described above causes the DAC to be “de-programmed”. Secondly, the bias reduces the power requirements for the high-frequency microwave pulses.

At 610, the programming system determines whether it is ready to lower the address line. If, the programming system determines it is ready to lower the address line, then method 600 proceeds to 612, otherwise it continues to wait. At 612, the global address bias line level is lowered below a predetermined threshold, thereby reducing the global flux bias applied to the DACs. Lowering the address line below the predetermined lower threshold causes the flux quantum to move into the DAC storage loop.

At 614, the programming system determines whether programming is complete. If, the programming system determines that programming is complete, then method 600 proceeds to 616, otherwise it continues to wait. At 616, the power iine is returned to its initial level in some implementations, the power line is returned to zero current. At 618, the address line is returned to its initial level. In some implementations, the address line is returned to zero current.

Figures 7A, 7B and 7C are graphs showing plots illustrating an example time series for loading of a single flux quantum according to the present disclosure. Figure 7A is a plot of the combined address signal current 702 as a function of time. Combined address signal current 702 includes a global bias 704 and high-frequency pulses 706. Combined address signal current is raised above threshold 708 to cause loading of a single flux quantum into a DAC CJJ loop.

Figure 7B is a plot of a power signal current 710 as a function of time. At time to, power signal current 710 is raised. At time ti, power signal current 710 is raised.

Referring again to Figure combined address signal current exceeds threshold 708 at time t ₂ . At time fe, global bias 704 is lowered thereby lowering combined address signal current 702.

Figure 7C is a plot of the persistent current 712 in the DAC storage loop as a function of time. Persistent current 712 steps down at time t ₂ , and steps down again at time t3. At time t ₃ , persistent current 712 steps down below a predetermined lower threshold 714.

In the example illustrated In Figure 7C, the DAC persistent current at to is zero. After t ₂ , persistent current 712 includes a current induced by magnetic flux added to the DAC CJJ loop. After t ₃ , persistent current 712 includes a current induced by magnetic flux moved from the DAC CJJ loop into the DAC storage loop. After persistent current 712 includes a current induced by the one or more flux quanta stored in the DAC.

Loading one or more flux quanta into the DAC is also referred to in the present application as programming the DAC. Programming the DAC can also Include removing one or more flux quanta from the DAC storage loop. In one implementation, removing one or more flux quanta from the DAC storage loop includes reversing a signal on an address line.

Two additional control interfaces can be included to ensure programming can be completed by intersecting resonators. The two additional control interfaces are to control the phase and the amplitude of the driving tones (also referred to in the present application and resonant frequencies). Calibration can be used determine a range over which the phase and the amplitude of the driving tones can be adjusted consistent with programming for a target DAC. Depending on the size of the range, it can be possible to program multiple devices at the same time. The size of the range can at least partially depend on: a) a variation in the current amplitude over the length of the resonator, and, or b) a degree of homogeneity in a coupling between each resonator and a respective CJJ loop.

Addressing Technology for Superconducting Devices

Figure 8 shows a hybrid computing system 800 according to at least one exemplary implementation, including a digital computer 802 and a quantum computer 804, that may incorporate systems and methods described in the present application, including resonator-addressed DACs.

Digital computer 802 comprises CPU 806, user interface elements 808, 810, 812 and 814, disk 816, controller 818, bus 820 and memory 822. Memory 822 comprises modules 824, 826, 828, 830, 832, and 834.

Quantum computer 804 comprises quantum processor 836, readout control system 838, qubit control system 840 and coupler control system 842. Quantum computer 804 can incorporate FMR technology comprising superconducting resonators (such as superconducting resonator 300 of Figure 3A). Computing system 800 can comprise a resonator- addressed DAC such as resonator-addressed DAC 502 of Figures 5A-5F. Power line (Z) frequency-multiplexed flux-DAC addressing

As described above, a shortcoming of a conventional scheme for addressing a superconducting device, for example a flux storage device such as DAC, can be that implementation may include a large number of wires and superconducting low-pass filters which can be costly to produce, vulnerable to failure, and a source of high-energy photons from internal filter heating. In some implementations, the wires are twisted-pairs and the low-pass filters are custom-made superconducting low-pass filters. The systems and methods described in the present application include replacing wires by coaxial cable which can provide greater bandwidth and reduced complexity. A benefit of greater bandwidth can be improved processor input/outpuf (I/O) speed.

The systems and methods described in the present application include frequency-domain multiplexing of signals on microwave lines. The systems and methods described in the present application include

programming a DAC, or other suitable superconducting device on a superconducting circuit, by using signals to activate power line resonators with calibrated resonator amplitudes, and by using signals to implement an XY- addressing scheme (such as trigger and address signals described in the present application).

In some implementations of addressing schemes, superconducting devices are addressable on a three-dimensional grid using combinations of three external bias signals. One of these external bias signals is typically a common current bias provided to a set of one or more

superconducting devices. The line that provides the common current bias is referred to in the present application as a power (PWR) line. Each PWR line can provide a current bias to a respective set of one or more superconducting devices.

The other two bias signals are referred to in the present application as trigger (TRIG) and address (ADDR) biases. See, for example, a description of XYZ-addressing of flux DACs earlier in the present application. As described above with reference to Figures 5A through 5F, Figure 6, and Figure 7A through 7C, programming of a DAC coupied to two superconducting resonators can be achieved by having a low-frequency bias signal and two microwave signals combine constructively to exceed a determined address latching level. The arrangement can be used to construct an XY-addressing scheme for a set of DACs. In one implementation, an array of superconducting resonators is laid out across a superconducting integrated circuit (e.g , a quantum processor) in a two-dimensional grid. A global low- frequency bias address line is coupled to each DAC CJJ loop of the set of DAC CJJ loops. Programming a selected DAC is achieved by pre-biasing the set of DAC CJJ loops, and activating the two resonators (also referred to in the present application as the address and trigger resonators) that intersect at the selected DAC.

Some implementations employ an XYZ-addressing scheme. A low-frequency or DC current bias is applied to a DAC by a line referred to in the present application as a power (PWR) line. Current bias line 526 of Figures 5A and 5B, and current bias lines 526a and 526b of Figure 5C are examples of PWR lines.

In a conventional arrangement, the PWR lines provide a DC bias and are generally galvanically coupied to a body of the DAC. The DC bias can be applied to an array of DACs referred to in the present application as a domain or power domain. The domains are the Z-dimension in the XYZ- addressing scheme. In an example implementation of a quantum processor, the quantum processor includes 24 domains supplied by 24 twisted pair lines. Programming a DAC in a domain can include applying a DC bias to the domain, and applying XY signals (as described above) to a CJJ loop of the DAC.

The present application describes a technology for providing a PWR bias by a superconducting resonator. The domain can include one or more DACs coupled to a superconducting resonator. The superconductor resonator can be supplied with a DC bias as a DC current through a body of the superconducting resonator. The DC current can be common to multiple resonators and can be supplied, for example, by a single twisted pair. The domain can be activated (to provide Z-addressing of the domain) by exciting microwave current in the resonator.

The PWR lines can be frequency multiplexed. Frequency multiplexing on the transmission line can be used to activate one of an array of resonators coupled to the transmission line.

In some implementations of the technology described in the present application, a superconducting resonator used to provide a PWR bias can be a superconducting resonator that is communicatively coupled to a microwave transmission line.

The technology described in the present application can cause a low-frequency (or DC) PWR bias to be combined with a high-frequency (e.g., microwave) current along the resonator body to provide PWR addressing. The low-frequency bias can be applied to more than one resonator. The low- frequency bias can be a global bias applied to a set of resonators.

In some implementations (e.g , implementations described below with reference to Figures 9A and 9B), programming the DAC (or other suitable superconducting device or flux storage device) includes summing the low- frequency (or DC) and high-frequency (or microwave) currents in the resonator body. In other implementations (e.g., implementations described with references to Figures 5A through 5F, Figure 6, and Figures 7 A through 7C), low-frequency and high-frequency signals are combined by summing of flux in a CJJ (such as CJJ 506 of Figure 5A).

The technology described in the present application integrates a superconducting resonator with a DC line. In some implementations, the DC line is integrated with an electronic filter. In some implementations, a superconducting resonator (or an array of superconducting resonators) and a DC line are integrated on-chip. The integrated component can sum currents in at least two frequency bands (e.g., microwave and DC).

It is generally desirable for the low-frequency signal path to the superconducting resonator to have a high impedance at microwave frequencies. One reason for having a high impedance at microwave frequencies is to reduce a load on the superconducting resonator. The inclusion of an electronic filter in the low-frequency signal path can cause the low-frequency (or DC) line to appear as a high impedance at a microwave frequency e.g., a resonance frequency of the superconducting resonator. In some implementations (including those discussed with reference to Figure 10, and those that include an array of superconducting resonators), the inclusion of an electronic filter in the low-frequency signal path can cause a decoupling of two superconducting resonators.

Figure 9A is a schematic diagram illustrating an example implementation of a superconducting circuit 900a, according to the present disclosure. Superconducting circuit 900a includes a superconducting device addressed by a superconducting resonator. The superconducting device can be a superconducting flux storage device such as resonator-addressed DAC 902 of Figure 9A, according to the present disclosure.

Superconducting circuit 900a can provide Z (power line) addressing of resonator-addressed DAC 902. As described above, a DC bias can be supplied to a body of a resonator, and summed with a high-frequency current from a microwave transmission line. Z (power line) addressing can be achieved by selectively activating a domain using a high-frequency (microwave) current.

In superconducting circuit 900a, resonator-addressed DAC 902 includes a loop 904 of material that is superconductive in a range of

temperatures, and a compound Josephson junction (CJJ) 906. CJJ 906 includes two parallel current paths each interrupted by a Josephson junction, Josephson junctions 908 and 910, respectively.

Superconducting circuit 900a further includes a resonator 912, and a transmission line 914. Resonator 912 of Figure 9A is referred to in the present application as a power (PWR) line resonator. Resonator 912 includes coupling capacitors 916 and 918, and lines 920 and 922 to resonator- addressed DAC 902. Lines 920 and 922 are communicatively coupled to a low- frequency (or DC) signal source 924 by an electronic filter 926. Electronic filter 926 is operable to remove selected frequency components. Selected frequency components can include a high-frequency component. Selected frequency components can include a resonance frequency of a

superconducting resonator. Electronic filter 926 can be a low-pass filter, a notch filter, or a band-pass filter.

Instead of capacitively communicatively coupling resonator 912 to transmission line 914 by coupling capacitor 916, resonator 912 can be communicatively coupled to transmission line 914 by an inductive coupling. For example, resonator 912 can be inductively communicatively coupled to transmission line 914 by an inductive coupling to a portion of the body of resonator 912. See, for example, Figures 5E and 5F.

In some implementations, multiple resonators are coupled to a single DAC CJJ loop. See, for example, Figure 5B.

Figure 9B is a schematic diagram illustrating another example implementation of a superconducting circuit 900b, according to the present disclosure. Superconducting circuit 900b includes a superconducting device addressed by a superconducting resonator. The superconducting device can be a superconducting flux storage device such as resonator-addressed DAC 902 of Figure 9B. The same reference numbers as used in Figure 9A are used in Figure 9B to denote the same or similar elements.

In superconducting circuit 900b, resonator-addressed DAC 902 includes a loop 904 of material that is superconductive in a range of

temperatures, and a CJJ 906. CJJ 906 includes two parallel current paths each interrupted by a Josephson junction, Josephson junctions 908 and 910, respectively.

Superconducting circuit 900a further includes a resonator 912, and a transmission line 914. Resonator 912 of Figure 9B is referred to in the present application as an XY-resonator. Resonator 912 includes coupling capacitors 916 and 918, and lines 920 and 922 to resonator 912. Lines 920 and 922 are communicatively coupled to a low- frequency (or DC) signal source 924 by an electronic filter 926. Electronic filter 926 is operable to remove selected frequency components. Selected frequency components can include a high-frequency component. Selected frequency components can include a resonance frequency of a

superconducting resonator. Electronic filter 926 can be a low-pass filter, a notch filter, or a band-pass filter.

Referring again to Figure 5A, superconducting circuit 500a includes resonator 512 (referred to in the present application as an XY resonator - for its role in XY addressing described in the present application), and interface 516 and global flux bias line 516-1. in the implementation of superconducting circuit 900b of Figure 9B, interface 516 and global flux bias line 516-1 are replaced by a DC current bias supplied to the body of resonator 912 of Figure 9B by signal source 924 on a signal path that includes electronic filter 926.

A benefit of superconducting circuit 900b over superconducting circuit 500a is that interface 516 and global flux bias line 516-1 can be eliminated in some implementations, thereby reducing a challenge of laying out superconducting circuit 900b, e.g., reducing an area occupied by

superconducting circuit 900b on a superconducting integrated circuit.

Instead of capacitively communicatively coupling resonator 912 of Figure 9B to transmission line 914 by coupling capacitor 916, resonator 912 can be communicatively coupled to transmission line 914 by an inductive coupling. For example, resonator 912 can be inductively communicatively coupled to transmission line 914 by an inductive coupling to a portion of the body of resonator 912 See, for example, Figures 5E and 5F !n some implementations, multiple resonators are coupled to a single DAC CJJ loop. See, for example, Figure 5B

Resonator 912 further includes an inductive interface 928 to resonator-addressed DAC 902 The term“inductive interface” is also referred to in the present application as a coupling inductance. Coupling inductance 928, for example, inductively communicatively couples resonator 912 and resonator-addressed DAC 902.

Resonator 912 further includes an optional shunting capacitor

930.

Process control monitoring (PCM) line for an array of capacitively-coupled superconducting resonators.

As described in the present application, an array of superconducting resonators can be coupled to a single transmission line, and used, for example, to enable frequency-domain multiplexing in a variety of applications (e.g., in applications involving qubits, kinetic-inductance defectors (K!Ds), transition-edge sensors and in other applications which can be integrated with resonant devices).

It can be desirable to probe a superconducting device or a superconducting integrated circuit using a low-frequency or DC signal at room temperature (or at least at a higher temperature than a cryogenic temperature) to determine whether a resonator body in the device, or circuit, includes an electrical short or open, or has an unexpected resistance. An electrical short, open, or unexpected resistance can be caused by, for example, shrinkage of a critical dimension, variations in the thickness of a layer in a multilayer integrated circuit, material quality issues, and/or other fabrication issues.

One approach to dealing with fabrication failures is to simply accept a loss in yield in some applications, for example, KIDs are used for imaging, and the loss of a few resonators corresponds to the loss of a few pixels in an imaging array. This can be acceptable as the array can be scanned over the image area in order to fili-in the missing data arising from the loss of pixels. In other applications, a frequency-multiplexed resonant readout uses an array of resonators for readout of qubit values in a quantum processor. The loss of a few resonators results in a commensurate reduction in readout speed, and may be acceptable. In other applications, it can be important to know that at least an acceptable number of resonators in an array of resonators have yielded in an application where an array of resonators is used to address an array of superconducting fiux-DACs in a quantum processor, loss of a resonator can result in an inoperable processor. In another application where a large-scale array of superconducting transmon qubits is used for gate-model quantum computing, loss of a resonator can similarly result in an inoperable system.

For at least these reasons, it can be advantageous to measure the health of an array of resonators at room temperature (or at least at a higher temperature than a cryogenic temperature) to reduce the likelihood of encountering an inoperable resonator after a system has been cooled to a cryogenic temperature.

The systems and methods described in the present application include the use of a low-frequency or DC line at a temperature higher than a cryogenic temperature to determine the health of a resonator in an array of superconducting resonators or the health of a qubit in an array of

superconducting qubits in some implementations, the resonators in the array of superconducting resonators is capacitiveiy communicatively coupled to one or more transmission lines (e.g , microwave transmission lines) in other implementations, the resonators in the array of superconducting resonators are inductively coupled to one or more transmission lines.

In some implementations, the PCM line is also used, during operation of a superconducting circuit, as a DC bias line. The DC bias line may be global DC bias line. The superconducting circuit may include an array of capacitively-coupled superconducting resonators.

As described above with reference to Figures 9A and 9B, a DC bias can be added to a resonator body. The DC bias of Figures 9A and 9B can be used to probe the health of a resonator at a temperature higher than a cryogenic temperature, for example at room temperature (approximately 20°C). For example, the DC bias can be used to probe for an electrical short, an open, or an unexpected resistance. Figure 10 is a schematic diagram illustrating an example implementation of a superconducting circuit 1000 that includes an array of N resonators 1002-1 , 1002-2, and 1GG2-N (collectively referred to in the present application as 1002), according to the present disclosure. Though at least three resonators are shown In Figure 10, the number N of resonators in array of N resonators 1002 can be less than three or more than three.

Array of N resonators 1002 can be XY resonators or Z resonators (as described in the present application). For example, resonator 912 of Figure 9A (a Z resonator) can be a member of an array of resonators communicatively coupled to one another and to a transmission line as illustrated by array of N resonators 1002 of Figure 10. Similarly, resonator 912 (an XY resonator) of Figure 9B can be a member of an array of resonators communicatively coupled to one another and to a transmission line as illustrated by array of N resonators 1002 of Figure 10.

Figure 10 is an example implementation of a superconducting circuit that can be used to assess whether a DC short or open is present in a body of a resonator in an array of resonators. The array of resonators can be superconducting resonators. A superconducting resonator is a resonator with a resonator body that is superconducting in a range of temperatures or below a critical temperature. The superconducting resonator can be capacitive!y coupled or inductively coupled to a transmission line, for example a microwave transmission line. The microwave transmission line can be, for example, one of a microstrip transmission line, a stripline transmission line, a coaxial transmission line, and a co-planar waveguide transmission line.

Superconducting circuit 1000 of Figure 10 includes a DC wire that communicatively couples one or more resonators in an array of resonators to a signal source. In this configuration, the resonator bodies can be wired in series at DC. As described with reference to Figure 10 below, one or more electronic filters can be used to at least reduce cross-talk between resonators.

The example implementation of Figure 10 includes a DC bias for DAC power line resonators (see Figures 9A and 9B above). In other implementations, a DC bias can be included for address and trigger line resonators (e.g., address and trigger line resonators described in the present application).

Each resonator of array of N resonators 1002 is capadtiveiy communicatively coupled to transmission line 1004 by a respective coupling capacitance 1006-1 , 1006-2, and 1006-N (collectively referred to in the present application as 1006) and a respective coupling capacitance 1008-1 , 1008-2, and 1008N (collectively referred to in the present application as 1008) While Figure 10 shows each resonator of array of N resonators 1002 coupled to transmission line 1004, at least one resonator of array of N resonators can be coupled to another transmission line (not shown in Figure 10).

Each resonator of array of N resonators 1002 is communicatively coupled by a respective pair of lines and at least one electronic filter to a signal source 1010. The at least one electronic filter belongs to an array of electronic filters 1012-1 , 1012-2, 1012-3, and 1012-M (collectively referred to in the present application as 1012). For example, resonator 1002-1 is

communicatively coupled by lines 1014-1a and 1014-1 b and electronic filters 1012-1 and 1012-2. Resonator 1002-2 is communicatively coupled by lines 1014-2a and 1014-2b and electronic filters 1012-2 and 1012-3. Resonator 1002-N is communicatively coupled by lines 1Q14~Na and 1014-Nb and electronic filter 1012-M

Each filter of electronic filters 1012 can be a low-pass filter, a notch filter, or a band-pass filter. It is generally desirable that each filter of electronic filters 1012 at least reduces the magnitude of components of a signal from signal source 1010 at one or more resonant frequency bands of resonators 1002.

Figure 11 is a schematic diagram illustrating an example implementation of a superconducting circuit 1100 that includes a first broadband switch 1102a, according to the present disclosure. In some implementations, a second broadband switch 1102b may be employed together with first broadband switch 1102a. First and second broadband switches 1102a and 1102b may be implemented as broadband switch 1202 of Figure 12 and may be used as part of a superconducting circuit to address a device. Superconducting circuit 1100 includes a resonator addressed DAC 1104.

In superconducting circuit 1100, resonator-addressed DAC 1104 includes a loop 1106 of material that is superconductive in a range of temperatures, and a compound Josephson junction (CJJ) 1108. CJJ 1108 includes two parallel current paths each interrupted by a Josephson junction, Josephson junctions 1110 and 1112, respectively.

Superconducting circuit 1100 further includes a resonator 1114, a transmission line 1116, and interface 1118 operable to provide a global flux bias by a global flux bias line 1118-1. Resonator 1114 includes coupling capacitors 1120 and 1122, an inductive interface 1124 to resonator-addressed DAC 1104, and an optional shunting capacitor 1126. The term“inductive interface” is also referred to in the present application as a coupling inductance. Coupling inductance 1124, for example, inductively communicatively couples resonator 1114 and resonator-addressed DAC 1104.

Instead of capacitively communicatively coupling resonator 1114 to transmission line 1116 by coupling capacitor 1120, resonator 1114 can be communicatively coupled to transmission line 1116 by an inductive coupling. For example, resonator 1114 can be inductively communicatively coupled to transmission line 1116 by an inductive coupling to a portion of the body of resonator 1114.

Superconducting circuit 1100 further includes a current bias line 1128 operable to provide a current bias to resonator-addressed DAC 1104.

Figure 12 is a schematic diagram of an example implementation of a transmission line layout 1200 according to the present systems, methods and apparatus. Transmission line layout 1200 may be used for dynamically isolating a device, such as a superconducting processor. Transmission line 1200 comprises a broadband switch 1202 with a number N of cascade elements 1204_1 through 1204_N (collectively 1204) electrically coupled in series.

Each cascade element 1204 may be Implemented as cascade element 1300 of Figure 13. It is understood by those skilled in the art that transmission line layout 1200 may be of different construction, and the description of Figure 12 is specific to one implementation in particular, the number N of cascade elements in broadband switch 1202 may be different from the number of DC-SQU!Ds 1310a and 1310b.

Transmission line 1200 comprises transmission lines 1206, 1208 and 1210. Transmission line 1210 passes through each cascade element 1204. A first end of transmission lines 1206, 1208 and 1210 (e.g., the right-hand sides in the plane of the drawing sheet of Figure 12) is electrically communicatively coupled to a device (e.g., a superconducting processor, not shown in Figure 12), and a second end of transmission lines 1206, 1208 and 1210 (e.g., the left- hand sides) is electrically communicatively coupled to signal electronics (e.g., readout lines, not shown in Figure 12).

Transmission lines 1206, 1208 and 1210 may comprise vias 1212 and 1214 to connect to ground. Vias 1212 and 1214 can cause transmission lines 1206, 1208 and 1210 to stay at approximately the same electrical potential during an operation cycle of the device.

Each cascade element 1204 in broadband switch 1202 is electrically connected by lines 1216_1 through 1216_N (collectively 1216) to transmission line 1208 to ground.

Broadband switch 1202 comprises activation lines 1218 and 1220 and inductances 1218__a and 1220_a that are operable to cause the state of each cascade element 1204 to change from a suppression state to a passing state, and vice versa. Activation lines 1218 and 1220 are electrically coupled to cascade elements 12G4__1 and 1204_N, respectively. In at least one

implementation, activation lines 1218 and 1220 are poor electrical conductors over the frequency range used for transmitting electrical signal to the device, and receiving electric signals from the device (e.g., 4GHz -8GHz), for example activation lines 1218 and 1220 can provide >4kQ impedance at 4GHz. in the passing phase, broadband switch 1202 allows the transmission of signals, and, in the suppression phase, broadband switch 1202 suppresses blackbody radiation in one implementation, activation lines 1218 and 1220 may be implemented as a low-frequency twisted pair. Line 1222 and inductances

1222__a through 1222_n electrically couple successive cascade elements 1204 (e.g., cascade element 1204_1 to cascade element 1204_2) to provide high impedance to photons, thus at least partially obstructing a pathway to the device.

When broadband switch 1202 is implemented with cascade elements such as cascade elements 1300 of Figure 13, activation line 1218 corresponds to activation line 1318 for cascade element 1204__1 , and line 1222 to activation line 1320. Similarly, line 1222 corresponds to activation line 1318, and activation line 1220 to activation line 1320 for cascade element 1204__N

In at least one implementation, filtering elements may be integrated into activation lines 1218 and 1220 to at least limit the introduction of noise while allowing broadband switch 1202 to operate at a desired operating speed. For example, passive filtering elements or inductive chokes can be used in addition, filtering elements may be made of superconducting material or elements, such as kinetic inductors. Examples of kinetic inductors are discussed in detail in International Patent Publication No. WO2017192733A2.

In one implementation, broadband switch 1202 may be used as part of a superconducting circuit comprising microwave paths, as described above with reference to Figure 11.

Figure 13 is a schematic diagram of an example implementation of a single cascade element 1300 according to the present systems, methods and apparatus. Cascade element 1300 may be used as one of a plurality of cascade of elements as part of a broadband switch for dynamically isolating a device, such as a superconducting processor. See, for example, Figure 12 and its accompanying description. Cascade element 1300 comprises a transmission line 1302. A first end of transmission line 1302 (e.g., the right-hand side in the plane of the page of Figure 13) is electrically connected to a device (e.g., a superconducting processor, not shown in Figure 13) and a second end of transmission line 1302 (e.g., the left-hand side in the plane of the page of Figure 13) is electrically connected to signal electronics (e.g., readout lines, not shown in Figure 13). Readout electronics may be in an exterior environment, at a different temperature than the operating temperature of the device. Readout electronics may be in an exterior environment, at a different temperature than the operating temperature of the device.

Cascade element 1300 comprises a first segment 1304 and a second segment 1306, where first segment 1304 is on the left-hand side of a matching capacitor 1308 in the plane of the drawing sheet of Figure 13, and second segment 1306 is on the right-hand side of matching capacitor 1308 in the plane of the drawing sheet of Figure 13. First segment 1304 comprises a number N of DC-SQUIDs 1310a__1 through 131Ga__N (collectively 1310a) in series, and second segment 1306 comprises a number M of DC-SQUIDs 1310b__1 through 1310b_M (collectively 1310b) in series. In at least one implementation, the number of DC-SQUIDs 1310a is equal to the number of DC-SQUIDs 1310b. In some implementations, cascade element 1300 may comprise RF-SQUIDs.

Each DC-SQUID 1310a and 1310b comprises a pair Josephson junctions 1312a and 1312b (collectively 1312, only one pair called out to reduce clutter). Each Josephson junction of the pair of josephson junctions 1312 has a respective critical current l_c. Each DC-SQUID 1310a and 1310b is inductively coupled by inductance 1314a_1 through 1314a_N (collectively 1314a, only one called out in Figure 13 to reduce clutter) and 1314b__1 through 1314b_M (collectively 1314b, only one called out in Figure 13 to reduce clutter), respectively, to activation line loops 1318a__1 through 1316a__N (collectively 1316a) and 1316b_1 through 1316b_M (collectively 1316b), respectively. Loops 1316a are electrically coupled to activation line 1318, and loops 1316b are electrically coupled to activation line 1320. Activation lines 1318 and 1320 are operable to cause the state of cascade element 1300 to change from a suppression state to a passing state, and vice versa, when used as part of a broadband switch for dynamic isolation of a device, such as a superconducting processor.

Cascade element 1300 is symmetric with respect to matching capacitor 1308. Inductance 1314a and 1314b are approximately the same for DC-SQUiDs 1310a and 1310b, respectively, and junctions 1312 have approximately the same critical current 7 _C for DCJSGUIDs 1310a and 1310b. In some implementations, small variations during building and assembly may lead to values of inductance 1314a and 1314b and of critical current I _c that are not identical for each DC-SGUID of DC-SQUIDs 1310a and 1310b.

The various embodiments and implementations of the present application describe addressing schemes that include the following:

a low-frequency current bias in combination with a low-frequency flux bias (also referred to in the present application as YZ addressing and/or XYZ addressing);

a iow-frequency current bias in combination with a low-frequency flux bias and a high-frequency flux bias (also referred to in the present application as YZ addressing and/or XYZ addressing where Y and/or X can be frequency-domain multiplexed);

a low-frequency current bias in combination with a high-frequency current bias and a low-frequency flux bias (also referred to in the present application as YZ addressing and/or XYZ addressing where Z can be frequency-domain multiplexed); and

a iow-frequency current bias in combination with a high-frequency current bias, a iow-frequency flux bias, and a high-frequency flux bias (also referred to in the present application as YZ and/or XYZ addressing where X, Y, and/or Z can be frequency-domain multiplexed. In each of the above addressing schemes, the high-frequency bias (which can be a current bias across a CJJ loop and/or a flux bias into a CJJ loop) can be provided by a respective superconducting resonator. The superconducting resonator may enable frequency-domain multiplexing.

The various embodiments described above can be combined to provide further embodiments. To the extent that they are not inconsistent with the specific teachings and definitions herein, all of the U.S. patents, U.S. patent application publications, U.S. patent applications, foreign patents, foreign patent applications and non-patent publications referred to in this specification and/or listed in the Application Data Sheet, including but not limited to the following: PCT Patent Application No. WO2016US031885 (published as International patent application publication WG2016183213A1), FREQUENCY

MULTIPLEXED RESONATOR INPUT AND/OR OUTPUT FOR A

SUPERCONDUCTING DEVICE, filed May 11 , 2018; PCT Patent Application No. WO2018G16237, SYSTEMS AND METHODS FOR FABRICATION OF SUPERCONDUCTING INTEGRATED CIRCUITS, filed January 31 , 2018;US Provisional Patent Application No. 62/638,043, SYSTEMS AND METHODS FOR COUPLING A SUPERCONDUCTING TRANSMISSION LINE TO AN ARRAY OF RESONATORS, filed February 27, 2018; PCT Patent Application US2019/18792, SYSTEMS AND METHODS FOR COUPLING A

SUPERCONDUCTING TRANSMISSION LINE TO AN ARRAY OF RESONATORS, filed February 20, 2019; US Patent Application

No. 16/397,790, DYNAMICAL ISOLATION OF A CRYOGENIC PROCESSOR, filed April 29, 2019; US Provisional Patent Application No. 82/778,643,

DYNAMICAL ISOLATION OF A CRYOGENIC PROCESSOR, filed December 12, 2018; US Provisional Patent Application No. 62/680,824, DYNAMICAL ISOLATION OF A CRYOGENIC PROCESSOR, filed June 5, 2018; US

Provisional Patent Application No. 62/679079, SYSTEMS AND METHODS FOR ADDRESSING DEVICES IN A SUPERCONDUCTING CIRCUIT, filed June 1 , 2018; ; US Provisional Patent Application No. 62/672,392, SYSTEMS AND METHODS FOR ADDRESSING DEVICES IN A SUPERCONDUCTING CIRCUIT, filed May 18, 2018; and US Provisional Patent Application No.

82/883,253, SYSTEMS AND METHODS FOR ADDRESSING DEVICES IN A

SUPERCONDUCTING CIRCUIT, filed June 11 , 2018 are incorporated herein by reference, in their entireties. Aspects of the embodiments can be modified, if necessary, to employ systems, circuits and concepts of the various patents, applications and publications to provide yet further embodiments.

These and other changes can be made to the embodiments in light of the above-detailed description. In general, In the following claims, the terms used should not be construed to limit the claims to the specific

embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.