RELATED APPLICATIONS

[0001 ] This application claims priority from U.S. Patent Application Serial

No. 15/887524, filed 1 February 2018, which is incorporated herein in its entirety.

TECHNICAL FIELD

[0002] The present invention relates generally to quantum and classical digital superconducting circuits, and specifically to Josephson polarity and logical inverter gates.

BACKGROUND

[0003] In the field of digital logic, extensive use is made of well known and highly developed complimentary metal-oxide semiconductor (CMOS) technology. As CMOS has begun to approach maturity as a technology, there is an interest in alternatives that may lead to higher performance in terms of speed, power dissipation computational density, interconnect bandwidth, and the like. An alternative to CMOS technology comprises superconductor based single flux quantum circuitry, utilizing superconducting Josephson junctions (JJs), with typical signal power of around 4 nanowatts (nW), at a typical data rate of 20 gigabits per second (Gb/s) or greater, and operating temperatures of around 4 kelvins.

[0004] An inverter is an electrical circuit device capable of inverting an input signal into an output. A polarity inverter inverts the polarity of an input signal such that a positive input value having some magnitude is inverted to produce an output signal, or result in an output state, having a negative input value equal in magnitude to the input value but opposite in sign or polarity. In digital logic contexts having only two logical senses, a logical inverter is a gale capable of inverting a logical input into a logical output having the opposite logical sense of the logical input. Thus, an inverted "low" or "0" logical input provides a "high" or "1" logical output, and vice-versa. A polarity inverter can invert the polarity of an input signal in a manner that many involve more states than the binary states associated with logical inversion, e.g., three states or more. SUMMARY

[0005] One example provides a Josephson inverter gate circuit. The circuit includes an input to provide an input signal made up of at least one single flux quantum (SFQ) pulse, and a half-twisted Josephson transmission line (JTL) comprising at least four Josephson junctions arranged to propagate the input signal to an output and to invert the input signal into an output signal. The half-twisted JTL can have a central loop. The inverter gate circuit can be a polarity inverter, such that an output-end Josephson junction in the half-twisted JTL exhibits a -2π superconducting phase upon propagation to the output of an input signal that sets an input-end Josephson junction to a 2π superconducting phase, and the output-end Josephson junction exhibits a 0 superconducting phase upon propagation to the output of an input signal that resets the input-end Josephson junction to a 0 superconducting phase. Alternatively, the inverter gate circuit can be a logical inverter, such that, after a transient start-up period, the output-end Josephson junction exhibits a 0 superconducting phase upon propagation to the output of an input signal that sets the input-end Josephson junction to a 2π superconducting phase, and the output-end Josephson junction exhibits a 2π superconducting phase upon propagation to the output of an input signal that resets the input-end Josephson junction to a 0 superconducting phase.

[0006] Another example provides a method of logically inverting a signal value based on SFQ pulse inputs. A first positive SFQ pulse is provided to an input end of a half-twisted JTL to set an input-side Josephson junction in the half-twisted JTL (i.e., a Josephson junction that is nearer the input end of the half-twisted JTL than an output end of the half-twisted JTL) to a 2π superconducting phase. Before or after providing the first positive SFQ pulse, but before the first positive SFQ pulse can propagate through a central loop of the half-twisted JTL toward the output, one Φο of current is injected into the central loop as an initializing current, such that the first positive SFQ pulse is annihilated by the initializing current and does not propagate through the central loop toward the output end of the half-twisted JTL. This does not affect the superconducting phase of the input-side Josephson junction. Then, a negative SFQ pulse is provided to the input end of the half-twisted JTL to reset the input-side Josephson junction to a 0 superconducting phase, such that the negative SFQ pulse propagates to the output end of the half-twisted JTL to set to 2π the superconducting phase of an output-side Josephson junction in the half-twisted JTL (i.e., a Josephson junction that is nearer the output end of the half-twisted JTL than the input end of the half-twisted JTL). The method can continue by providing a second positive SFQ pulse to the input end of the half-twisted JTL to set the input-side Josephson junction to a 2π superconducting phase, such that the second positive SFQ pulse propagates to the output end of the half-twisted JTL to reset to 0 the superconducting phase of the output-side Josephson junction.

[0007] Another example provides another method of logically inverting a signal value based on SFQ pulse inputs. A first positive SFQ pulse is provided to an input end of a JTL having a floating Josephson junction in a central portion of the JTL to set an input-side

Josephson junction in the JTL to a 2π superconducting phase. Before or after providing the first positive SFQ pulse, but before the first positive SFQ pulse can propagate through the central portion of the JTL toward the output, Φο/2 of current is injected into the floating Josephson junction as an initializing current, such that the first positive SFQ pulse is annihilated by the initializing current and does not propagate through the central portion toward the output end of the JTL. This does not affect the superconducting phase of the input-side Josephson junction. Then, a negative SFQ pulse is provided to the input end of the JTL to reset the input-side Josephson junction to a 0 superconducting phase, such that the negative SFQ pulse propagates to the output end of the JTL to set to 2π the superconducting phase of an output-side Josephson junction in the JTL. The method can continue by providing a second positive SFQ pulse to the input end of the JTL to set the input-side Josephson junction to a In superconducting phase, such that the second positive SFQ pulse propagates to the output end of the JTL to reset to 0 the superconducting phase of the output-side Josephson junction.

[0008] Another example provides yet another method of logically inverting a signal value based on SFQ pulse inputs. A first negative SFQ pulse is provided to an output end of a half-twisted JTL to set an output-side Josephson junction in the half-twisted JTL to a 2π superconducting phase. Before or after providing the first negative SFQ pulse, but before the first negative SFQ pulse can propagate through a central loop of the half-twisted JTL toward the input, one Φο of current is injected into the central loop as an initializing current, such that the first negative SFQ pulse is annihilated by the initializing current and does not propagate through the central loop toward the input end of the half-twisted JTL. This does not affect the superconducting phase of the output-side Josephson junction. Then, a positive SFQ pulse is provided to the input end of the half-twisted JTL to set the input-side Josephson junction to a 2π superconducting phase, such that the positive SFQ pulse propagates to the output end of the half- twisted JTL to set to 0 the superconducting phase of the output-side Josephson junction in the half-twisted JTL. The method can continue by providing a second negative SFQ pulse to the input end of the half-twisted JTL to reset the input-side Josephson junction to a 0

superconducting phase, such that the second negative SFQ pulse propagates to the output end of the half-twisted JTL to set to 2π the superconducting phase of the output-side Josephson junction.

[0009] Another example provides still another method of logically inverting a signal value based on SFQ pulse inputs. A first negative SFQ pulse is provided to an output end of a JTL having a floating Josephson junction in a central portion of the JTL to set an output-side Josephson junction in the JTL to a 2π superconducting phase. Before or after providing the first negative SFQ pulse, but before the first negative SFQ pulse can propagate through the central portion of the JTL toward the input, Φο/2 of current is injected into the floating Josephson junction as an initializing current, such that the first negative SFQ pulse is annihilated by the initializing current and does not propagate through the central portion toward the input end of the JTL. This does not affect the superconducting phase of the output-side Josephson junction. Then, a positive SFQ pulse is provided to the input end of the JTL to set the input-side Josephson junction to a 2π superconducting phase, such that the positive SFQ pulse propagates to the output end of the JTL to reset to 0 the superconducting phase of the output-side Josephson junction in the JTL. The method can continue by providing a second negative SFQ pulse to the input end of the JTL to reset the input-side Josephson junction to a 0 superconducting phase, such that the second negative SFQ pulse propagates to the output end of the JTL to set to 2π the superconducting phase of the output-side Josephson junction.

[0010] Yet another example provides a superconducting reciprocal quantum logic (RQL) inverter circuit made up of an input end, an output end, and a central portion connecting the input end and the output end. The central portion includes at least one of a central loop comprising at least two JJs, and/or a floating Josephson junction in series with a transformer-coupled DC flux bias injection source configured to inject one Φο of current. The input end includes a first inductor connected between an input node and a first node, a first Josephson junction connected between the first node and a circuit ground, a second inductor connected between the first node and a second node, a second Josephson junction connected between the second node and a third node, and a third inductor connected between the third node and the circuit ground. The output end includes a fourth inductor connected between a fourth node and the circuit ground, a third Josephson junction connected between the fourth node and a fifth node, an fifth inductor connected between the fifth node and a sixth node, a fourth Josephson junction connected between the sixth node and the circuit ground, and a sixth inductor connected between the sixth node and an output node.

[0011] Still another example provides a Josephson inverter gate circuit. The circuit includes an input to provide an input signal made up of at least one single flux quantum (SFQ) pulse, and a JTL comprising at least five Josephson junctions arranged to propagate the input signal to an output and to invert the input signal into an output signal, one of them being a floating Josephson junction located centrally in the JTL. The inverter gate circuit can have a single DC flux bias input to provide an initializing current to the floating Josephson junction. The inverter gate circuit can be a polarity inverter or logical inverter, as defined with regard to the first example.

[0012] Still yet another example provides a superconducting reciprocal quantum logic (RQL) inverter circuit made up of an input end, an output end, and a central portion connecting the input end and the output end, the central portion comprising a floating Josephson junction and a transformer-coupled DC flux bias injection source configured to inject one Φο of current. The input end includes a first inductor connected between an input node and a first node, a first Josephson junction connected between the first node and a circuit ground, a second inductor connected between the first node and a second node, a third inductor connected between the second node and a third node, and a second Josephson junction connected between the third node and the circuit ground. The output end includes a third Josephson junction connected between a fifth node and the circuit ground, a seventh inductor connected between the fifth node and a sixth node, an eighth inductor connected between the sixth node and a seventh node, a fourth

Josephson junction connected between the seventh node and the circuit ground, and a tenth inductor connected between the seventh node and an output node. The circuit can further include two bias inputs each providing a bias signal having an AC component, the first bias input connected via a fourth inductor connected to the circuit at the second node, and the second bias input connected via a ninth inductor connected to the circuit at the sixth node.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] FIG. 1 A is a block diagram of an example Josephson inverter gate having a half- twisted Josephson transmission line (JTL).

[0014] FIG. IB is a block diagram of another example Josephson inverter gale having a JTL that includes a floating Josephson junction.

[0015] FIG. 3 A is a schematic of an example Josephson polarity inverter gate using a half-twisted JTL.

[0016] FIG. 3B is a graph of simulation results for the example gate of FIG. 3 A.

[0017] FIGS. 3C-3G are annotated schematics of the example Josephson polarity inverter gate of FIG. 3A showing an example functioning of the circuit.

[0018] FIG. 4A is a schematic of an example Josephson logical inverter gate using a half- twisted JTL.

[0019] FIG. 4B is a graph of simulation results for the example gate of FIG. 4A.

[0020] FIGS. 4C-4J are annotated schematics of the example Josephson logical inverter gate of FIG. 4A showing a first example functioning of the circuit. [0021 ] FIGS. 4K-4R are annotated schematics of the example Josephson logical inverter gate of FIG. 4A showing a second example functioning of the circuit.

[0022] FIG. 5 is a schematic of an example Josephson logical inverter gate using a half- twisted JTL and direct coupling.

[0023] FIG. 6 is a schematic of another example Josephson logical inverter gate using a half-twisted JTL.

[0024] FIG. 7 is a schematic of a Josephson polarity inverter gate using a JTL that includes a floating Josephson junction.

[0025] FIG. 8 is a schematic of a Josephson logical inverter gate using a JTL that includes a floating Josephson junction.

[0026] FIG. 9 is a graph of simulation results for the example logical inverter gate of FIG. 8 having positive central DC flux bias.

[0027] FIG. 10 is a graph of simulation results for the example logical inverter gate of FIG. 8 having negative central DC flux bias.

[0028] FIGS. 11A and 1 IB are flow charts illustrating methods of logically inverting a signal value based on single flux quantum (SFQ) pulse inputs.

DETAILED DESCRIPTION

[0029] Inversion in CMOS technology generally involves conversion of a low voltage to a high voltage or vice versa. Inversion of signals in circuits that use phase mode logic (PML) poses a more difficult problem, because in PML circuits, logical states are encoded as superconducting phases of, for example, Josephson junctions, such phases being set or reset with positive or negative pulses, e.g., single flux quantum (SFQ) pulses, that propagate through the circuits. The difference in encoding paradigm means that the techniques and structures of CMOS inversion methods are not assistive in achieving PML inversion, and new techniques and structures must be devised in order to implement a simple and effective PML inverter. Existing techniques for achieving signal inversion in phase-mode circuits, such as circuits from the family of reciprocal quantum logic (RQL) superconducting logic circuits, rely upon the use of a polarity-inverting transformer followed by a JTL that initializes to logical "high." However, the transformer is required to be physically large and high-efficiency.

[0030] This disclosure therefore relates generally to logical gate circuits for use in superconducting systems. In some examples, a one-input, one-output superconducting inverter gate can provide polarity inversion of phase mode logic inputs. In other examples, a one-input, one-output superconducting inverter gate can provide logical inversion of phase mode logic inputs. Thus, for example, when "low" and "high" logic states are encoded as 0 and 2π superconducting phases of Josephson junctions, respectively, a gate in a superconducting circuit, such as in an RQL superconducting circuit, can be configured to deliver a negative SFQ pulse on a gate output in response to a positive SFQ pulse on a gale input, and to deliver a positive SFQ pulse on a gate output in response to a negative SFQ pulse on a gate input. These pulses can set or reset the phases of Josephson junctions encoding the corresponding logical "low" or "high" states. The inverters described herein eliminate the need for physically-large, high-efficiency transformers on the signal path.

[0031 ] FIG. 1 A is a block diagram of an example Josephson inverter gate 100 having input IN 102 and output OUT 104 corresponding to an inversion of input IN 102. Depending on (he configuration of Josephson inverter gate 100, output OUT 104 can provide a polarity inversion or a logical inversion of input IN 102. Inverter gate 100 includes half-twisted

Josephson transmission line (JTL) 106, which includes at least four Josephson junctions (JJs) 108-1, 108-2, 108-3, 108-4, and which receives inputs from two AC bias lines 110, 112 opposite in phase from each other. By "half-twisted JTL," it is meant that the structure of a conventional JTL has had a half-twist applied to it, such that the ground reference at the output is on the opposite side of the JTL relative to the input, and such that the half-twisted JTL inverts the polarity of an applied SFQ voltage pulse. The inverter gate 100 can include more than four Josephson junctions. Inverter gate 100 can also include one or more DC inputs 114, 116, which can be provided to establish initializing conditions on half-twisted JTL 106 at system start-up. For example, DC inputs 114, 116 can each inject Φο/2 worth of current into a central loop of half-twisted JTL 106. In some examples, the two Φο/2 currents can be provided to cancel each other out. In other examples, the two Φο/2 currents can be provided to sum to a full Φο and thus to place one whole Φο worth of current into the central loop. Signals provided to and by input IN 102 and output OUT 104 can consist of positive or negative single flux quantum (SFQ) pulses corresponding to asserted or de-asserted logic states, respectively. Corresponding input and output logic stales can be stored on, i.e., encoded by, the superconducting phases of JTL

Josephson junctions 108-1 through 108-4.

[0032] FIG. IB is a block diagram of another example Josephson inverter gate ISO having input IN 152 and output OUT 154 corresponding to an inversion of input IN 152. As with gate 100, depending on the configuration of Josephson inverter gate 150, output OUT 154 can provide a polarity inversion or a logical inversion of input IN 152. Inverter gate 150 includes JTL 156, which includes at least four Josephson junctions (JJs) 108-1, 108-2, 108-3, 108-4, and, in addition, underdamped floating Josephson junction 158-5. "Floating" in this context means that neither of the Josephson junction terminals is grounded. JTL 156 receives inputs from two AC bias lines 160, 162, which are functionally, although not necessarily precisely, opposite in phase from each other. The inverter gate 150 can include more than four Josephson junctions. Inverter gate 150 can also include a DC input 164, which can be provided to establish initializing conditions on JTL 156 at system start-up. For example, DC input 164 can inject Φο/2 of current into floating JJ 158-5. Signals provided to and by input IN 152 and output OUT 154 can consist of positive or negative single flux quantum (SFQ) pulses corresponding to asserted or de-asserted logic states, respectively. Corresponding input and output logic slates can be stored on, i.e., encoded by, the superconducting phases of JTL

Josephson junctions 158-1 through 158-4.

[0033] FIG. 2 illustrates the topology 200 of a JTL with a "half-twist," as discussed previously, consisting of AC-biased Josephson junctions with inductive interconnect shown as upper and lower bold lines that cross at a twist point 202 without electrically connecting at the twist point 202. The topology 200 can accommodate RQL data encoding, wherein every positive SFQ pulse is followed by a negative one. [0034] Each end of the JTL 200 has an upper signal propagation side and a lower ground side, which are not voltage nodes but rather are inductive interconnects, such that there is appropriate isolation and gain between input and output. The half-twist 202 of the JTL 200 means that the signal propagation side becomes the ground side as the JTL 200 progresses from input to output. Because the connection between input RQL In and output RQL Out is inductive, the signal propagation side at the input can be grounded at the output without shorting out the topology 200. Each Josephson junction triggers locally, at times dictated in part by biasing provided by AC biases 204, 206, 208, 210, and by the time a signal propagates to the output, it has been inverted in voltage polarity. Because of half -twist 202, AC biases 208 and 210 are opposite in direction (i.e., inverted in AC phase) from AC biases 204, 206. Initialization of the output to logical high can be accomplished within the signal inversion stage with flux biasing (not shown in FIG. 2).

[0035] RQL circuits propagate logical changes as SFQ pulses or trains of such pulses.

Inversion of signals therefore might be conceptualized to entail creation or annihilation of SFQ pulses to invert a signal train, but implementation of such functionality proves physically difficult. Therefore, rather than conceptualizing logic signals in terms of SFQ pulses, logic states can be conceptualized as superconducting phases of Josephson junctions used as logic elements, where phase is defined as the time-integral of voltage at every node. The half-twisted JTL topology 200 can convert high phase to low phase and vice- versa and thereby invert the polarity of an incoming SFQ voltage pulse provided at terminal RQL In, because instead of attempting to create or annihilate pulses, topology 200 flips Josephson junction phase polarities upside down between input and output. Although FIG. 2 shows half-twisted JTL 200 as having eight

Josephson junctions, an inverter in accordance with the present disclosure can be made with fewer Josephson junctions.

[0036] FIG. 3A shows a schematic, with accompanying simulation result plots shown in FIG. 3B, of an RQL signal polarity inverter 300 with direct coupling, that follows the half-twist topology model of FIG. 2. With reference to dots placed near Josephson junctions Ji, J ₂ , J3, J4, circuit 300 is illustrated in a final state after a positive input pulse has propagated to the output. From an initial state (at the 0 picosecond mark in FIG. 3B) wherein all of the Josephson junctions Ji, J ₂ , J3, J ₄ are at 0 superconducting phase, an input signal provided as a positive SFQ pulse at the INPUT line causes the first Josephson junction Ji to trigger (placing it in the 2π superconducting phase), which in turn triggers second Josephson junction J ₂ , which in turn triggers third Josephson junction J3, which in turn triggers fourth Josephson junction J ₄ (placing it in the -2π superconducting phase). Fourth Josephson junction J ₄ triggers with an "opposite" polarity as compared to the triggering of first Josephson junction Ji, as is indicated in FIG. 3A by the relative placement of Josephson junction superconducting phase dots near each Josephson junction. Thus, the OUTPUT line transmits a negative SFQ pulse in response to a positive SFQ pulse at the INPUT, and vice versa.

[0037] The superconducting phases at the input (e.g., as measured at first Josephson junction Ji) and the output (e.g., as measured at fourth Josephson junction J ₄ ) are plotted in the graph of FIG. 3B. The input superconducting phase is plotted as a broken line and the output superconducting phase is plotted as a solid line. As can be seen in the graph, some short lag time after the input superconducting phase transitions from 0 to 2π (at around the 200 picosecond mark), the output superconducting phase transitions from 0 to -2π. Subsequently, when a negative pulse arrives at the input to restore the input superconducting phase to 0 (at around the 350 picosecond mark), the output superconducting phase also returns to 0. Then, when another positive pulse arrives at the input to again raise the input superconducting phase to 2π (at around the 400 picosecond mark), the output superconducting phase again presents the polar inversion of -2π phase after a short propagation time. After that, when another negative pulse arrives at the input to again bring the input superconducting phase to 0 (at around the 4S0 picosecond mark), the output superconducting phase also returns to 0. The graph of FIG. 3B thus accurately characterizes the behavior of the polarity inverter 300 of FIG. 3 A.

[0038] Still with regard to FIG. 3 A, second and third junctions J ₂ and J3 are part of a superconducting loop in the center of circuit 300. AC bias signals AC IN and ACour can be, for example, AC sine wave signals that are equal in magnitude and are functionally, although not necessarily precisely, opposite in AC phase, as indicated by the relative pointing of the arrows in the symbols. Other relative AC phase assignments can also result in an operative circuit 300. In order to provide appropriate biasing, DC offset sources DC IN and DCOUT can each put Φο/2 of current into the central loop shared by Josephson junctions h and J ₃ via transformer couplings IVL ₁₀ and L11/L12, where Φο is approximately equal to 2.07 milliamps-picohenries. These Φο/2 flux biases help maintain symmetry of the double- well potential in view of inductors to ground L3 and L*; the circuit would be loaded hard without the DC flux biases to compensate for the sending of the signal directly to ground via inductors L3 and L*. However, in circuit 300, with respect to any initialization current provided to the central loop, the functionally equal and opposite currents provided by the two DC sources DC IN and DCOUT cancel each other out. First junction Ji is loaded to ground not through the inductor L3 to ground at the bottom of circuit 300, but through the inductor L4 to ground at the top of circuit 300. It may further be noted that in the polarity inverter of FIG. 3A, there is no high-efficiency transformer in the signal path. (As used here, a "high efficiency" transformer is one having a coefficient of coupling k greater than 0.5, i.e., k = Lm/V(LpLs) > 0.S, where L _m is mutual inductance and Lp and Ls are the respective self- inductances of the primary and secondary inductors. In circuit 300, the L9/L10 and Ln/Li ₂ transformers are not in the signal path in that the primary inductors L10 and L12 transmit DC biases that can have arbitrary amplitude irrespective of signal amplitude.) Thus, coupling can be arbitrarily small with proportionate scaling of the DC bias current.

[0039] FIGS. 3C through 3G illustrate an example functioning of the polarity inverter circuit 300 of FIG. 3A. FIG. 3C shows a positive input pulse being introduced to the input of the polarity inverter 300 to cause current 302. This causes first Josephson junction Ji to trigger, raising its superconducting phase from 0 to 2π, as indicated with the dot placed above first Josephson junction Ji in FIG. 3D. The triggering of first Josephson junction Ji causes a functionally equal and opposite current 304 to annihilate the initial input pulse 302, and also propagates the initial pulse forward through the circuit 300 via current 306, which in turn causes second Josephson junction J ₂ to trigger. As shown in FIG. 3E, the triggering of second

Josephson junction J2 results in an another annihilating current 308 and a propagating current 310, which causes third Josephson junction J3 to trigger. FIG. 3F, shows this third triggering, annihilating current 310 with functionally equal and opposite current 312, and also causing propagating current 314. Finally, in the same fashion, fourth Josephson junction J ₄ triggers, annihilating 316 current 314 and propagating a negative pulse 318 out of the output in FIG. 3G.

[0040] Thus, it is that positive input pulse 302 results in negative output pulse 318. As noted by the dot on fourth Josephson junction JU at the far side of the signal propagation line in FIG. 3G, fourth Josephson junction J ₄ is in a -2π superconducting phase at the conclusion, consistent with the polarity inverter function illustrated in FIG. 3B. That is, the output is at -2π when the input is at 2π. In similar fashion, a subsequent negative pulse introduced to the input of circuit 300 will result in a positive pulse issuing from the output of circuit 300, and will return all Josephson junctions J1-J4 to the 0 superconducting phase.

[0041 ] FIG. 4A shows a schematic, with accompanying simulation result plots shown in FIG. 4B, of an RQL logical inverter with direct coupling. The logical inverter of FIG. 4A is similar to the polarity inverter 300 of FIG. 3A, except that the input and output can be initialized to opposite logical states, e.g., the input can be initialized to high. This initialization can be accomplished in part by reversing the polarity of one of the two DC flux biases, in the illustrated case, DCOUT, as compared to circuit 300 of FIG. 3 A.

[0042] As one example, at system start-up, an introductory positive input SFQ pulse can be introduced to the INPUT line, and shortly thereafter, before the input signal can propagate through the circuit 400 to the OUTPUT line, the DCEN and DCOUT biases are applied, together injecting one full Φο of current into the central loop that includes Josephson junctions J2 and J3. Absent the earlier introduction of the first positive input pulse, the injected central loop current could create an unstable state, because, as in any JTL, a Josephson junction in receipt of a Φο of current wants to pass it on, but it would be uncertain whether it would be passed back to the input (i.e., from second Josephson junction J ₂ back to first Josephson junction Ji) or on to the output (i.e., from third Josephson junction J3 to fourth Josephson junction J ₄ ). By providing a first positive input SFQ pulse just before injecting the full Φο of current into the central loop, the positive input pulse is "eaten" (annihilated) by the functionally equal and opposite central loop current, retaining input-side Josephson junctions Jj and J ₂ in the 2π superconducting phase (e.g., a logical "high" state) without any change to the superconducting phase of the output-side Josephson junctions J3 and U from their initial 0 superconducting phase (e.g., a logical "low" state). Thus, the possibility of the initialization DC injection pulse being propagated to the output can be avoided by turning on the DC IN and DCour initialization currents timely after supplying the first input pulse to circuit 400.

[0043] The above-described initialization example, which is illustrated more fully in FIGS. 4C-4J, manifests in the plots of FIG. 4B as a transient 440 in the input Josephson junction phase triggered by the provision of the initial input signal (i.e., a positive SFQ pulse) in conjunction with the DC flux-bias turn-on, without any change to the output during the same time period. Later, when a subsequent negative SFQ pulse is provided to the input (around the 50 picosecond mark), the output goes to logical "high" for the first time. The transient 2π superconducting phases of input-side Josephson junctions Ji and J ₂ during initialization are illustrated in FIG. 4A by stipple-filled superconducting phase dots at Josephson junctions Ji and J ₂ . The transition (at about the 50 picosecond mark) from high to low on the input and from low to high on the output is indicated by the solid dots near Josephson junctions Ji and J ₂ in FIG. 4A. Although these dots are placed on the opposite side of the Josephson junctions from the stipple-filled superconducting phase dots, they are meant only to indicate a return to 0 superconducting phase and not that the Josephson junctions have transitioned to a -2π superconducting phase.

[0044] Upon the introduction of a positive pulse to the input (at around the 200 picosecond mark), raising the input Josephson junction superconducting phase from 0 to In, the output Josephson junction superconducting phase falls from 2π to 0 after some short propagation time. A negative pulse at the input (at around the 350 picosecond mark) causes the input Josephson junction superconducting phase to fall from 2π to 0 and, conversely, the output Josephson junction superconducting phase to rise from 0 to 2π. A second positive pulse introduced to the input (at around the 400 picosecond mark) causes the input Josephson junction superconducting phase to again rise from 0 to 2π and, conversely, the output Josephson junction superconducting phase to again fall from 2π to 0. A second negative pulse arriving at the input (at around the 450 picosecond mark) causes the input Josephson junction superconducting phase to again fall from 2π to 0 and, conversely, the output Josephson junction superconducting phase to again rise to 2π from 0. The plots of FIG. 4B thus accurately characterize the behavior of the logical inverter 400 of FIG. 4A.

[0045] As another initialization example for the circuit of FIG. 4A, not illustrated in FIG. 4B, an introductory negative SFQ pulse can be introduced via the OUTPUT line at system start-up, and shortly thereafter, before the signal can propagate through the circuit 400 to the INPUT line, the DC IN and DCOUT biases are applied, together injecting one full Φο of current into the central loop that includes Josephson junctions J ₂ and J ₃ . The negative pulse is "eaten" (annihilated) by the functionally equal and opposite central loop current, retaining output-side Josephson junctions J ₃ and J* in the 2π superconducting phase (e.g., a logical "high" state) without any change to the superconducting phase of the input-side Josephson junctions Ji and J ₂ from their initial 0 superconducting phase (e.g., a logical "low" state). This initialization example is illustrated more fully in FIGS. 4K-4R. In either initialization example, the DC biases can be applied before or after the initializing pulses are applied.

[0046] FIGS. 4C through 4 J illustrate the first-described example functioning of the logical inverter circuit 400 of FIG. 4A, wherein an initializing pulse is provided through the INPUT and "eaten" in the central loop. FIG. 4C shows a positive input pulse being introduced to the INPUT of the logical inverter 400 to cause current 402. This causes first Josephson junction Ji to trigger, raising its superconducting phase from 0 to 2π, as indicated with the dot placed above first Josephson junction Ji in FIG. 4D. The triggering of first Josephson junction Ji causes a functionally equal and opposite current 404 to annihilate the initial current 402, and also propagates the initial pulse forward through the circuit 400 via current 406, which in turn causes second Josephson junction J ₂ to trigger. As shown in FIG. 4E, the triggering of second

Josephson junction J ₂ results in another annihilating current 408 and a propagating current 412.

[0047] In contrast, however, to the functioning of polarity inverter 300 of FIG. 3 A, before the propagating current 412 can propagate on through the circuit, a functionally equal and opposite current 410 is induced in the central loop of circuit 400 by DC current sources DCIN and DC OUT, annihilating current 412 before it can cause third Josephson junction J ₃ to trigger. The superconducting phase of fourth Josephson junction J ₄ thus stays at 0 despite the

superconducting phase of first Josephson junction Ji being at 2π. As mentioned previously, as an alternative, DC-source-induced central loop current 410 can be introduced prior to the input of initializing current 402.

[0048] Following from FIG. 4E, FIG. 4F shows the subsequent introduction of a negative input pulse applied to the INPUT of circuit 400 to induce current 414. Thus, in FIG. 4G, first Josephson junction Ji untriggers, annihilating current 414 with functionally equal and opposite current 416. The negative input pulse propagates via negative current 418, which causes second Josephson junction J ₂ to untrigger, as shown in FIG. 4H. Concomitantly, annihilating current 420 and propagating current 422 are produced by the untriggering of second Josephson junction J ₂ , and at this point both first and second Josephson junctions Ji and J ₂ again exhibit the initial superconducting phase of 0.

[0049] The untriggering of second Josephson J ₂ and propagation of negative pulse 422 through the central loop of circuit 400 then cause third Josephson junctionh to trigger, as shown in FIG. 41, annihilating current 422 with functionally equal and opposite current 424, and also causing propagating current 426. Comparing FIG. 41 with FIG. 3F, that third Josephson junctionh has triggered in the opposite direction in the logical inverter configuration 400 as compared to the polarity inverter configuration 300, and thus, as shown by the relative placement of the dots around third Josephson junction J ₃ in the respective drawings, third Josephson junctionh has obtained a 2π superconducting phase in FIG. 41 as compared to the -2π superconducting phase obtained in FIG. 3F. Finally, in the same fashion, fourth Josephson junction J ₄ triggers to obtain a 2π superconducting phase of its own, annihilating 428 current 426 and propagating a positive pulse via current 430 out of the OUTPUT in FIG. 4J.

[0050] Thus, it is that negative input pulse 414 results in positive output pulse 430.

Moreover, as noted by the dot on fourth Josephson junction J ₄ at the near side the signal propagation line in FIG. 4J, fourth Josephson junction J ₄ is in a 2π superconducting phase at the conclusion of this sequence, which corresponds to the end of transient 440 illustrated in FIG. 4B. The above-described functioning is consistent with the polarity inverter function illustrated in FIG. 4B. That is, the output is at 2π when the input is at 0. In similar fashion, a subsequent positive pulse introduced to the INPUT of circuit 400 will result in a negative pulse issuing from the OUTPUT of circuit 400, and will place Josephson junctions h andh in a 2π superconducting phase and will return Josephson junctions J ₃ and J ₄ to a 0 superconducting phase, again consistent with FIG. 4B and the desired polarity inverter function.

[0051 ] FIGS. 4K through 4R illustrate the second-described example functioning of the logical inverter circuit 400 of FIG. 4A, wherein an initializing pulse is provided through the OUTPUT and "eaten" in the central loop. From an initial state, FIG. 4K shows a negative pulse being introduced to the OUTPUT of the logical inverter 400 to induce current 450. This causes fourth Josephson junction J ₄ to trigger, raising its superconducting phase from 0 to 2π, as indicated with the dot placed below fourth Josephson junction J ₄ in FIG. 4L. The triggering of fourth Josephson junction J ₄ causes a functionally equal and opposite current 4S2 to annihilate the initial input current 450, and also propagates the initial pulse backward through the circuit 400 via current 454, which in turn causes third Josephson junction J ₃ to trigger. As shown in FIG. 4M, the triggering of third Josephson junction J ₃ results in another annihilating current 456 and a propagating current 460.

[0052] Similarly, to the previously described functioning and as shown earlier in

FIG. 4E, current 460 is not permitted to propagate further but is "eaten" in the central loop of circuit 400 by a functionally equal and opposite current 458 induced in the central loop by of circuit 400 by DC current sources DC IN and DCOUT. The superconducting phase of second Josephson junction J ₂ thus stays at 0 despite the superconducting phase of fourth Josephson junction J ₄ being at 2π.

[0053] Following from FIG. 4M, FIG. 4N shows the subsequent introduction of a positive input pulse applied to the INPUT of circuit 400 to induce current 462. Thus, in FIG. 40, first Josephson junction Ji triggers, annihilating current 462 with functionally equal and opposite current 464. The positive input pulse propagates via positive current 466, which causes second Josephson junctionh to trigger, as shown in FIG. 4P. Concomitantly, annihilating current 468 and propagating current 470 are produced by the triggering of second Josephson junction J ₂ , and at this point all four Josephson junctions J1-J4 exhibit the 2π superconducting phase.

[0054] The triggering of second Josephson J ₂ and propagation of positive pulse 470 through the central loop of circuit 400 then cause third Josephson junction J3 to untrigger, as shown in FIG. 4Q, annihilating current 470 with functionally equal and opposite current 472, and also causing propagating current 474. Comparing FIG. 4Q with FIG. 41, third Josephson junction J3 has triggered in the opposite direction by this sequence of operation (illustrated in FIGS. 4K-4R) as compared to the previously described sequence of operation (illustrated in FIGS. 4C-4J), and thus, as shown by the relative placement of the dots around third Josephson junction J3 in the respective drawings, Josephson junction J3 exhibits a 0 superconducting phase in FIG. 4Q as compared to the 2π superconducting phase exhibited in FIG. 41. Finally, in the same fashion, fourth Josephson junction J ₄ untriggers to exhibit a 0 superconducting phase of its own, annihilating 476 current 474 and propagating a negative pulse 478 out of the OUTPUT of logical inverter circuit 400 in FIG. 4R.

[0055] Thus, it is that positive input pulse 462 results in negative output pulse 478. Moreover, as noted by the absence of any dot near fourth Josephson junction J ₄ in FIG. 4R, fourth Josephson junction J ₄ is in a 0 superconducting phase at the conclusion of this sequence. The above-described functioning is consistent with the desired polarity inverter function. That is, the output is at 0 when the input is at 2π. In similar fashion, a subsequent negative pulse introduced to the INPUT of circuit 400 will result in a positive pulse issuing from the OUTPUT of circuit 400, and will return Josephson junctions Ji andh to a 0 superconducting phase and will place Josephson junctions J ₃ and J ₄ in a 2π superconducting phase, again consistent with the desired polarity inverter function.

[0056] FIG. S shows an example schematic of an RQL logical inverter 500 with direct coupling that has the implantation of the flux bias using an extra junction on each side, i.e., Josephson junctions Js and Je. This implementation may also be referred to as a "digital flux bias" implementation. In this configuration 500, the biases DC IN and DCour trigger respective 2π phase advances in Josephson junctions J 5 and J ₆ . On the left side of the central loop of circuit 500, when the inductances in the L9 branch and the J5/L10 branch are similar, then the desired Φο/2 flux bias obtains with reduced sensitivity to the value of DCIN. Similarly, on the right side of the central loop of circuit 500, when the inductances in the L12 branch and the Je/Ln branch are similar, then the desired Φο/2 flux bias obtains with reduced sensitivity to the value of DCOUT. Circuit 500 otherwise operates similarly to the previously described logical inverter 400 of FIG. 4A and the opposing superconducting phases in the input Josephson junctions Ji, J ₂ versus the output Josephson junctions J3, h can be achieved in circuit 500 as already described with respect to circuit 400 in order to achieve the desired logical inversion functionality.

[0057] FIG. 6 shows an example schematic of an RQL logical inverter 600 that produces higher output drive as compared to the previously described examples. The input and output are more isolated with respect to ground as compared to the previous examples, but with the tradeoff that inverter 600 has a higher component count as compared to the previous examples.

Circuit 600 otherwise operates similarly to the previously described logical inverter 400 of FIG. 4A and the opposing superconducting phases in the input Josephson junctions Ji, J ₂ versus the output Josephson junctions J3, J ₄ can be achieved in circuit 600 as already described with respect to circuit 400 in order to achieve the desired logical inversion functionality.

[0058] FIG. 7 is a schematic of an example RQL polarity inverter circuit 700 that uses a single floating junction JF that produces two SFQ pulses when triggered, which then triggers the output with negative polarity, i.e., such that output Josephson junction J ₄ exhibits -2π

superconducting phase after a positive input signal brings input Josephson junction Ji to 2π superconducting phase. Similarly, FIG. 8 is a schematic of an example RQL logical inverter circuit 800 that operates in a similar fashion, except that output Josephson junction J ₄ exhibits 2π superconducting phase after a negative input signal brings input Josephson junction Ji to 0 superconducting phase. Thus, circuits 700 and 800 do not rely on the half-twisted JTL structure conceptually illustrated in FIG. 2. Logical inverter of FIG. 8 can function irrespective of the direction of the DC current source in the middle of circuit provided through transformer coupling L10/L9, so it is actually the polarity of a DC offset associated with ACOUT that distinguishes logical inverter 800 from polarity inverter 700. In polarity inverter 700, said DC offset is -Φο/2, whereas in logical inverter 800, said DC offset is +Φο/2. This negative DC offset in ACOUT in the polarity inverter 700 is so that the first transition on the OUTPUT of circuit 700 is negative (from 0 to -2π). In either circuit 700 or 800, ACIN has a DC offset of +Φ</2.

[0059] The operation of floating Josephson junction JF in circuits 700 and 800 can be described with reference to the pendulum mechanical analogy for the Josephson junction device. The equations of motion of a Josephson junction are isomorphic to equation of motion of a physical pendulum suspended at a central swinging point, and with nothing to prevent the pendulum from swinging all the way around this central swinging point once or even multiple times. In the analogy, superconducting phase in the Josephson junction can be likened to mechanical phase of the pendulum; current in the Josephson junction is equivalent to torque in the pendulum; voltage on the Josephson junction is analogous to angular velocity on the pendulum; and the inductors associated with the Josephson junction in a circuit would be torsion springs in the analogy.

[0060] A Josephson junction, like a pendulum, can function as an oscillator. In many circuit implementations, a Josephson junction is provided with a damping resistor so that it is close to critically damped. The Josephson junction then does not swing back and forth like a pendulum in a grandfather clock, but instead, when made to trigger, goes "all the way over the top," does a 2π superconducting phase rotation, and then settles. If underdamped, by, for example, increasing the value of the damping resistor or removing said resistor completely (i.e., to create an open circuit), then upon triggering, the underdamped Josephson junction may roll around "over the top" and begin oscillating like a grandfather clock, and may even roll over twice, i.e., to the 4π superconducting phase.

[0061 ] With reference to FIG. 7, the introduction of a positive SFQ pulse on the INPUT line triggers the first Josephson junction Ji, which subsequently triggers the second Josephson junction J ₂ , which in turn triggers floating Josephson junction JF. Floating Josephson junction JF is arranged so as to be underdamped, like a pendulum that rolls over once and then once again, e.g., by configuring the Josephson junction to be without its shunt resistor, and by configuring the flux bias provided by the DC current source in the middle of circuit 700 to inject +Φο/2 of current into floating Josephson junction JF, i.e., into the central loop formed by second Josephson junctionh, floating Josephson junction JF, inductor Ls, and third Josephson junction J3. When floating Josephson junction JF triggers the first time, the current in the central loop goes to -Φο/2, putting floating Josephson junction JF in a different potential well without "loading" it— floating Josephson junction JF remains at the same energy level where it started.

[0062] Therefore, the "momentum" of floating Josephson junction JF is able to carry it "over the top" a second time, i.e., to a 4π superconducting phase, as indicated by the double dots near floating Josephson junction JF in FIG. 7. The resultant state is not stable, and floating Josephson junction JF negatively triggers. The resultant negative pulse propagates through the circuit 700, such that the positive input pulse and 2π superconducting phase at input-side Josephson junctions Ji and J2 result in output-side Josephson junctions J ₃ and J ₄ being in a -2π superconducting phase, and a negative pulse propagating out the OUTPUT line. As illustrated in FIG. 7, output AC bias signal ACOUT is configured to be functionally if not exactly opposite in polarity (e.g., 180° different in AC phase) from the input AC bias signal ACm.

[0063] Logical inverter 800 of FIG. 8 works similarly to polarity inverter 700 of FIG. 8, leveraging floating Josephson junction JF to provide inversion, except that, just as with the logical inverter circuit 400 of FIG. 4A as compared to the polarity inverter circuit 300 of FIG. 3A, the logical inverter circuit 800 of FIG. 8 performs logical inversion rather than polarity inversion. Circuit 800 differs from circuit 700 both structurally and functionally. Structurally, ACOUT is configured to have a DC offset of is +Φ0/2, as opposed to -Φο/2 in polarity

inverter 700. Functionally, circuit 800 works by an initialization process similar to those described with regard to the functioning of circuit 400 in FIG. 4A.

[0064] As one example, shortly after a positive SFQ pulse is introduced to the INPUT of circuit 800, but before it can propagate through the floating Josephson junction JF, ΦΟ/2 worth of initializing current is introduced into floating Josephson junction JF via source DC and transformer coupling L10/L9, which initializing current annihilates the incoming positive input SFQ pulse as it propagates from input to output, but while maintaining the input-side Josephson junctions Ji and h in the 2π superconducting phase, as indicated by stipple-filled superconducting phase dots at Josephson junctions Ji and J ₂ in FIG. 8. A subsequent negative input SFQ pulse introduced at the INPUT resets input-side Josephson junctions Ji and J ₂ to the 0 superconducting phase and, propagating through to the output, sets output-side Josephson junctions J ₃ and U to the 2π superconducting phase. Subject to propagation delay, subsequent alternating positive and negative input SFQ pulses will each cause the circuit to exhibit logical inversion, i.e., cause the circuit to exhibit 0 superconducting phase at the output junctions when the input junctions exhibit 2π superconducting phase and vice-versa. The transition (at about the SO picosecond mark) from high to low on the input and from low to high on the output is indicated by the solid dots near Josephson junctions JI and J2 in FIG. 4A. The solid dots placed on the opposite side of Josephson junctions Ji and J ₂ from the stipple-filled superconducting phase dots are meant only to indicate a return to 0 superconducting phase and not that the Josephson junctions have transitioned to a -2π superconducting phase.

[0065] As another initialization example, shortly after a negative SFQ pulse is introduced to the OUTPUT of circuit 800, but before it can propagate through the floating Josephson junction JF, Φο/2 worth of initializing current is introduced into floating Josephson junction JF via source DC and transformer coupling L10/L9, which initializing current annihilates the incoming negative SFQ pulse as it propagates from output to input, but while maintaining the output-side Josephson junctions J3 and J4 in the 2π superconducting phase. A subsequent positive SFQ pulse introduced at the INPUT sets input-side Josephson junctions Ji and J ₂ to the 2π superconducting phase and, propagating through to the output, resets output-side

Josephson junctions h and J* to the 0 superconducting phase. Subject to propagation delay, subsequent alternating negative and positive input SFQ pulses will each cause the circuit to exhibit logical inversion, i.e., cause the circuit to exhibit 2π superconducting phase at the output junctions when the input junctions exhibit 0 superconducting phase and vice- versa.

[0066] FIG. 9 is a graph of simulation results for the example logical inverter gale of FIG. 8 having positive central DC flux bias, while FIG. 10 is a graph of simulation results for the example logical inverter gate of FIG. 8 having negative central DC flux bias. Logical inverter 800 works whether the central DC flux bias introduces a positive or negative Φο/2. The only difference is in the turn-on transient 1000.

[0067] The flow charts of FIGS. 11A and 1 IB illustrate methods 1100, 1150 of logically inverting a signal value based on single flux quantum (SFQ) pulse inputs. The

methods 1100, 1150 can be used, for example, with either of the circuits 400 or 800 of FIGS. 4A or 8, respectively. In method 1100 of FIG. 11 A, a first positive SFQ pulse is provided 1102 to an input end of a JTL to set an input-side Josephson junction (JJ) in the JTL (i.e., a Josephson junction that is nearer the input end of the JTL than an output end of the JTL) to a 2n

superconducting phase. The JTL can be either a half-twisted JTL as shown in circuit 400 of FIG. 4A or a JTL having a floating Josephson junction in the middle, as in circuit 800 of FIG. 8. Before the input SFQ pulse can propagate through a central loop of the half-twisted JTL to the output, or before the input SFQ pulse can propagate through a central floating Josephson junction of the JTL, as applicable, one Φο of current is injected 1104 into the central loop or Φο/2 of current is injected 1106 into the floating Josephson junction as an initializing current, e.g., by turning on one or more DC bias currents. This causes the first positive SFQ pulse to be annihilated by the initializing current, and the first positive SFQ pulse therefore does not propagate through to the output end of the JTL. However, the superconducting phase of the input-side Josephson junction is unaffected, and remains at 2π. Alternatively, turning on a DC bias current to inject 1104, 1106 the applicable amount of current can be performed before providing 1102 the first SFQ pulse, thus reordering actions 1102 and 1104/1106.

[0068] Subsequently, a negative SFQ pulse is provided 1108 to the input end of the JTL (half-twisted or having a central floating Josephson junction, as applicable) to reset the input-side Josephson junction to a 0 superconducting phase, such that the negative input SFQ pulse propagates 1110 to the output end of the JTL to set to 2π the superconducting phase of an output- side Josephson junction (i.e., a Josephson junction in the JTL that is nearer the output end of the JTL than the input end of the JTL). Thus, logical inversion is provided.

[0069] The method 1100 of FIG. 11 A can further include providing 1112 a second positive SFQ pulse to the input end of the JTL (half-twisted or otherwise) to set the input-side Josephson junction to a 2π superconducting phase, such that the second positive SFQ pulse propagates 1114 to the output end of the JTL to reset to 0 the superconducting phase of the output-side Josephson junction. This again demonstrates that the logical inversion function is correctly implemented.

[0070] In method 1150 of FIG. 1 IB, a first negative SFQ pulse is provided 1152 to an output end of a JTL to set an output-side Josephson junction in the JTL to a 2π superconducting phase. The JTL can be either a half-twisted JTL as shown in circuit 400 of FIG. 4A or a JTL having a floating Josephson junction in the middle, as in circuit 800 of FIG. 8. Before the output SFQ pulse can propagate through a central loop of the half-twisted JTL to the input, or before the output SFQ pulse can propagate through a central floating Josephson junction of the JTL, as applicable, one Φο of current is injected 1154 into the central loop or Φο/2 of current is injected 1156 into the floating Josephson junction as an initializing current, e.g., by turning on one or more DC bias currents. This causes the first negative SFQ pulse to be annihilated by the initializing current, and the first negative SFQ pulse therefore does not propagate through to the input end of the JTL. However, the superconducting phase of the output-side Josephson junction is unaffected, and remains at 2π. Alternatively, turning on a DC bias current to inject 1154, 1156 the applicable amount of current can be performed before providing 1152 the first SFQ pulse, thus reordering actions 1152 and 1154/1156.

[0071 ] Subsequently, a positive SFQ pulse is provided 1158 to the input end of the JTL (half-twisted or having a central floating Josephson junction, as applicable) to set the input-side Josephson junction to a 2π superconducting phase, such that the positive input SFQ pulse propagates 1160 to the output end of the JTL to reset to 0 the superconducting phase of an output-side Josephson junction. Thus, logical inversion is provided. Alternatively, rather than providing 1152 a first positive pulse to the input end of the JTL, the first pulse applied 1102 can be a negative pulse applied to the output of a first positive pulse to the input— anything that gets to input and output initialized opposite each other is good.

[0072] The method 1150 of FIG. 1 IB can further include providing 1162 a second negative SFQ pulse to the input end of the JTL (half-twisted or otherwise) to reset the input-side Josephson junction to a 0 superconducting phase, such that the second negative SFQ pulse propagates 1164 to the output end of the JTL to set to 2π the superconducting phase of the output-side Josephson junction. This again demonstrates that the logical inversion function is correctly implemented. It will be appreciated that either method 1100 or 1 ISO can be modified in various ways. Any reordering of the actions or modification of the actions that still results in an input Josephson junction and an output Josephson junction to be initialized to have superconducting phases opposite each other (i.e., one being 0 when the other is 2π, or vice-versa) will thereafter result in the desired logical inversion.

[0073] The methods 1100, 1150 described above can also include the actions of providing appropriate AC biasing as discussed previously in this disclosure to induce the timely triggering of Josephson junctions in the JTL and thus to cause signal propagation from input to output.

[0074] The example inverter gates described herein by the gate schematics and accompanying description can perform logical inversion for Josephson circuits that use RQL data encoding. They achieve an efficient implementation of logical inversion while eliminating the need for magnetic transformers in the signal path.

[0075] The Josephson inverter gates described herein have very good parametric operating margins, a low component count, and provide efficiency and cost advantages as compared to other inverter implementations. By eliminating high-efficiency transformers in their designs, the Josephson inverter gates described herein can save a number of metal layers, e.g., two metal layers, in the fabrication process, which sets the number of process steps and yield and thereby determines cost. The Josephson inverter gates described herein can be fabricated according to either a half -twist JTL signal path approach that involves switching the location of signal and ground at the output to produces signal inversion, or an approach involving an unshunted floating Josephson junction in the signal line to produces two SFQ pulses when triggered by the SFQ input signal, which results in an output SFQ signal of reversed polarity. This latter implementation, as shown in FIGS. 7 and 8, is schematically simpler but has narrower parametric operating margins than the half-twisted JTL implementation shown in FIGS. 3 A and 4A.

[0076] What have been described above are examples of the invention. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the invention, but one of ordinary skill in the art will recognize that many further combinations and permutations of the invention are possible. Accordingly, the invention is intended to embrace all such alterations, modifications, and variations that fall within the scope of this application, including the appended claims. Additionally, where the disclosure or claims recite "a," "an," "a first," or "another" element, or the equivalent thereof, it should be interpreted to include one or more than one such element, neither requiring nor excluding two or more such elements. As used herein, the term "includes" means includes but not limited to, and the term "including" means including but not limited to. The term "based on" means based at least in part on. The term "functionally equal" as used herein means sufficiently equal such that the described inverter functioning is achieved, and not necessarily exactly equal.