TECHNICAL FIELD

[0001] The present disclosure relates to a quantum computing, and more particularly to an arrangement for reducing qubit leakage errors and to a quantum computing system.

BACKGROUND [0002] Quantum computing is based on the idea of stor- ing information in a two-level quantum system. However, many realizations of such quantum bits have more than two energy levels. In such cases the qubit is formed by the two lowest energy levels and the excitation of the higher levels should be prevented. The situation where any of these higher energy levels are in risk of becoming excited, i.e. when their quantum states mix with those of the computational basis is called a leakage error. This can happen because of, for example, applied gate operations or system-environment interactions. Leakage errors cannot be fixed with the standard quantum error correction that only deals with errors within the com- putational basis. SUMMARY

[0003] This summary is provided to introduce a selec- tion of concepts in a simplified form that are further described below in the detailed description. This sum- mary is not intended to identify key features or essen- tial features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.

[0004] It is an objective to provide an arrangement for reducing qubit leakage errors and a quantum compu- ting system. The foregoing and other objectives are achieved by the features of the independent claims. Fur- ther implementation forms are apparent from the depend- ent claims, the description and the figures.

[0005] According to a first aspect, an arrangement for reducing qubit leakage errors comprises: at least one qubit having a ground state, and a plurality of excited states, wherein the plurality of excited states com- prises a lowest excited state, wherein an energy dif- ference between the ground state and the lowest excited state corresponds to a first frequency, and an energy difference between the lowest excited state and another excited state in the plurality of excited states corre- sponds to a second frequency; an energy dissipation structure configured to dissipate energy transferred to the energy dissipation structure; and a filter hav- ing at least one stopband and at least one passband, wherein the filter is coupled to the at least one qubit and to the energy dissipation structure, and wherein the at least one stopband comprises the first frequency and the at least one passband comprises the second fre- quency. The arrangement may, for example, reduce leakage errors in the at least one qubit by permitting transi- tions at least from the another excited state to the lowest excited state while not permitting transitions between the lowest excited state and the ground state. [0006] In an implementation form of the first aspect, the energy dissipation structure comprises at least one normal metal - insulator - superconductor, NIS, junc- tion. The arrangement may, for example, efficiently dis- sipate photon energy from the qubit via the energy dis- sipation structure.

[0007] In a further implementation form of the first aspect, the energy dissipation structure comprises a quantum circuit refrigerator, QCR, wherein the QCR com- prises a voltage-biased superconductor - insulator - normal metal - insulator - superconductor, SINIS, junc- tion. The arrangement may, for example, efficiently dis- sipate photon energy from the qubit via the energy dis- sipation structure and the dissipation may be controlled via the bias voltage. [0008] In a further implementation form of the first aspect, the energy dissipation structure is configured to dissipate photon energy transferred to the energy dissipation structure via photon-assisted electron tun- nelling in the NIS/SINIS junction. [0009] In a further implementation form of the first aspect, the filter comprises a band-stop filter or a low-pass filter. The arrangement may, for example, block unwanted state transitions of the qubit while allowing state transitions that reduce leakage errors. [0010] In a further implementation form of the first aspect, the at least one qubit comprises a superconduc- tive qubit.

[001 1] In a further implementation form of the first aspect, the another excited state comprises a second lowest excited state of the at least one qubit. The arrangement may, for example, permit relaxations from the second lowest excited state to the lowest excited states thus removing leakage errors due to occupation in the second lowest excited state.

[0012] In a further implementation form of the first aspect, the at least one passband of the filter further comprises a third frequency corresponding to an energy difference between two states in the plurality of ex- cited states. The arrangement may, for example, permit relaxations between the two states thus reducing leakage errors.

[0013] In a further implementation form of the first aspect, the at least one passband of the filter further comprises a first plurality of frequencies, wherein each frequency in the first plurality of frequencies corre- sponds to an energy difference between two consecutive excited states in the plurality of excited states of the at least one qubit. The arrangement may, for example, permit relaxations between consecutive excited states thus removing leakage errors via sequential relaxations between consecutive excited states.

[0014] In a further implementation form of the first aspect, the at least one stopband of the filter further comprises a second plurality of frequencies, wherein each frequency in the second plurality of frequencies corresponds to an energy difference between two non- consecutive states of the at least one qubit. The ar- rangement may, for example, restrict the at least one qubit to sequential relaxations thus causing more of the leakage error being retrieved in the lowest excited state instead of the ground state.

[001 5] According to a second aspect, a quantum compu- ting device comprises the arrangement according to the first aspect. The quantum computing device may, for ex- ample, perform quantum computations using the arrange- ment with reduced leakage errors.

[0016] According to a third aspect, a quantum compu- ting system comprises the arrangement according to the first aspect and a control unit coupled to the arrange- ment, wherein the control unit is configured to decouple the filter and/or the energy dissipation structure from the at least one qubit in response to a gate operation on the at least one qubit. The quantum computing system may, for example, mitigate any effects caused by the filter and/or the energy dissipation structure on the gate operation.

[001 7] In an implementation form of the third aspect, the quantum computing system further comprises a plu- rality of arrangements, wherein each arrangement in the plurality of arrangements is implemented according to the first aspect, wherein the control unit is coupled to each arrangement in the plurality of arrangements, and wherein the control unit is further configured to decouple the filter and/or the energy dissipation struc- ture of each arrangement in the plurality of arrange- ments from the at least one qubit of the arrangement in response to a gate operation on the qubit. The quantum computing system may, for example, mitigate any effects caused by the filter and/or the energy dissipation structure on the gate operation in each arrangement. [0018] According to a fourth aspect, a method for qubit leakage error reduction using at least one ar- rangement according to the first aspect comprises: de- coupling the filter and/or the energy dissipation struc- ture from the at least one qubit; and performing at least one gate operation on the qubit; recoupling the filter and/or the energy dissipation structure to the at least one qubit. The method may, for example, miti- gate any effects caused by the filter and/or the energy dissipation structure on the gate operation.

[0019] In an implementation form of the fourth aspect, the at least one arrangement comprises a plurality of arrangements, the method further comprising: decoupling the filter and/or the energy dissipation structure from the at least one qubit in each arrangement in the plu- rality of arrangements; performing at least one multi- qubit gate operation on using the at least one qubit in each arrangement in the plurality of arrangements; and recoupling the filter and/or the energy dissipation structure to the at least one qubit in each arrangement in the plurality of arrangements. The method may, for example, mitigate any effects caused by the filter and/or the energy dissipation structure on the gate op- eration in each arrangement.

[0020] Many of the attendant features will be more readily appreciated as they become better understood by reference to the following detailed description consid- ered in connection with the accompanying drawings.

DESCRIPTION OF THE DRAWINGS [0021] In the following, example embodiments are de- scribed in more detail with reference to the attached figures and drawings, in which:

[0022] Fig. 1 illustrates a schematic representation of an arrangement for reducing qubit leakage errors ac- cording to an embodiment;

[0023] Fig. 2 illustrates a schematic representation of qubit energy levels according to an embodiment;

[0024] Fig. 3 illustrates a schematic representation of a frequency response of a band-stop filter according to an embodiment;

[0025] Fig. 4 illustrates a schematic representation of a frequency response of a low-pass filter according to an embodiment;

[0026] Fig. 5 illustrates a schematic representation of an energy dissipation structure according to an em- bodiment;

[0027] Fig. 6 illustrates an effective circuit model representation of the arrangement according to an em- bodiment; [0028] Fig. 7 illustrated a table of circuit parame- ters according to an embodiment;

[0029] Fig. 8 illustrates a plot representation of qubit eigenstate occupation during gate operation ac- cording to an embodiment;

[0030] Fig. 9 illustrates a schematic representation of a quantum computing system according to an embodi- ment;

[0031] Fig. 10 illustrates a schematic representation of a control unit according to an embodiment; and

[0032] Fig. 11 illustrates a flowchart representation of a method for qubit leakage error reduction using at least one arrangement according to an embodiment.

[0033] In the following, like reference numerals are used to designate like parts in the accompanying draw- ings.

DETAILED DESCRIPTION

[0034] In the following description, reference is made to the accompanying drawings, which form part of the disclosure, and in which are shown, by way of illustra- tion, specific aspects in which the present disclosure may be placed. It is understood that other aspects may be utilised, and structural or logical changes may be made without departing from the scope of the present disclosure. The following detailed description, there- fore, is not to be taken in a limiting sense, as the scope of the present disclosure is defined be the ap- pended claims. [0035] For instance, it is understood that a disclo- sure in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if a specific method step is described, a corresponding de- vice may include a unit to perform the described method step, even if such unit is not explicitly described or illustrated in the figures. On the other hand, for ex- ample, if a specific apparatus is described based on functional units, a corresponding method may include a step performing the described functionality, even if such step is not explicitly described or illustrated in the figures. Further, it is understood that the features of the various example aspects described herein may be combined with each other, unless specifically noted oth- erwise.

[0036] Fig. 1 illustrates a schematic representation of an arrangement 100 for reducing qubit leakage errors according to an embodiment. [0037] According to an embodiment, the arrangement 100 comprises a at least one qubit 101 having a ground state and a plurality of excited states. The plurality of excited states comprises a lowest excited state. An en- ergy difference between the ground state and the lowest excited state may correspond to a first frequency. An energy difference between the lowest excited state and another excited state in the plurality of excited states may correspond to a second frequency. [0038] Herein, when referring to the order of the states of the at least one qubit 101 using phrases such as "lowest", "second lowest", and "consecutive", these terms may refer to the order of the states in terms of energy. For example, the ground state of the at least one qubit 101 may refer to a lowest state of the at least one qubit 101, meaning a state of the at least one qubit 101 with the lowest energy. Similarly, the lowest excited state of the at least one qubit 101 may refer to an excited state of the at least one qubit 101 with the lowest energy and so on.

[0039] The first frequency may also be referred to as a resonance frequency of the at least one qubit 101. [0040] Herein, any energy E, such as an energy dif- ference between states of the qubit, and a corresponding frequency w may be related by

E = hω , where h is the reduced Planck constant.

[0041] The another excited state may comprise, for example, a second lowest excited state of the at least one qubit 101, a third lowest excited state of the at least one qubit 101, or any other excited state in the plurality of excites states of the at least one qubit 101. [0042] According to an embodiment, the at least one qubit 101 comprises a superconductive qubit. According to an embodiment, the at least one qubit 101 comprises at least one Josephson junction. [0043] According to an embodiment, the at least one qubit 101 comprises a transmon qubit. Alternatively, the at least one qubit 101 may comprise any other type of qubit, such as a superconductive quantum interference device (SQUID) qubit, a flux qubit, a charge qubit, or a phase qubit.

[0044] Although some embodiments may be disclosed herein with reference to a certain type of at least one qubit 101, these qubit types are only exemplary. In any embodiment disclosed herein, the at least one qubit 101 may be implemented in various ways and using various technologies.

[0045] The arrangement 100 may further comprise an energy dissipation structure 103 configured to dissipate energy transferred to the energy dissipation structure 103.

[0046] According to an embodiment, the energy dissi- pation structure 103 comprises at least one normal metal - insulator - superconductor (NIS) junction. The energy dissipation structure 103 may be configured to dissipate photon energy transferred to the energy dissipation structure 103 via photon-assisted tunnelling in the NIS junction.

[0047] According to an embodiment, the energy dissi- pation structure 103 comprises a quantum circuit re- frigerator (QCR).

[0048] The arrangement 100 may further comprise a fil- ter 102 having at least one stopband and at least one passband. The filter 102 may be coupled to the at least one qubit 101 and to the energy dissipation structure 103. The at least one stopband may comprise the first frequency and the at least one passband may comprise the second frequency. The energy dissipation structure 103 coupled to the filter 102 results in a filtered energy dissipation structure. A filtered energy dissipation structure comprises a noise spectrum that contains at least a band or passband of allowed frequencies, where said noise spectrum is non-vanishing, and at least a gap or stopband of non-allowed frequencies, where said noise spectrum substantially vanishes. When a quantum system, such as the at least one qubit 101, is coupled to the filtered energy dissipation structure, the decay of the system energy levels separated by frequencies within the stopband can be suppressed, while the decay of the sys- tem energy levels separated by frequencies within the allowed band can occur.

[0049] The filter 102 may be, for example, electri- cally, capacitively, inductively, and/or electromag- netically coupled to the at least one qubit 101 and/or to the energy dissipation structure 103.

[0050] The filter 102 may also be referred to as an electronic filter or similar. The filter 102 may com- prise, for example, discrete elements, such as capaci- tors, inductors, and resistors, and/or distributed el- ements embodied in, for example, transmission lines. Although some embodiments disclosed herein may depict the filter 102 as comprising certain components, these are only exemplary and the filter 102 may be implemented in various other ways.

[0051] The filter 102 may transfer energy from the at least one qubit 101 to the energy dissipation structure 103. The energy transfer may be selective based on the frequency response of the filter 102.

[0052] Although some examples of the filter 102 are disclosed in the embodiments herein, the filter 102 may also be implemented in various other ways and using various other components/elements. In any embodiment disclosed herein, the filter 102 may be implemented us- ing, for example, any type of band-stop filter. For example, the filter 102 may be implemented using a more complex geometry corresponding to a higher order band- stop filter. Such a higher order band-stop filter may approximate a rectangular frequency response of an ideal band-stop filter.

[0053] The arrangement 100 may be embodied in, for example, a quantum computing device. Such a quantum com- puting device may comprise a plurality of qubits for performing quantum computation. Each such qubit may be implemented using the arrangement 100.

[0054] The arrangement 100 may be realized, for exam- ple, in the superconducting circuit architecture. [0055] The arrangement 100 may be able to reduce leak- age errors of the at least one qubit 101. A leakage error may occur when the at least one qubit 101 is excited to its second lowest or higher excited state. Thus, the state of the at least one qubit 101 leaks out of the computational basis formed by the ground state and lowest excited state. The typical error correcting protocols, such as the surface code, can only fix bit and phase flip errors in the computational basis. Thus, leakage errors should be reduced in some other manner. [0056] The arrangement 100 may be able to relax the unwanted higher excited states of the at least one qubit 101 via energy dissipation to the energy dissipation structure 103 while leaving the lowest excited state untouched. This may be referred to as targeted dissipa tion. Transitions from the lowest excited state to the ground state can be protected by the at least one stop- band of the filter 102.

[0057] Herein, decay rates from the second lowest to the lowest excited state and from the lowest excited state to the ground state in the ar- rangement 100 are derived. The latter decay can be sup- pressed by at least one stopband of the filter 102 while the leakage-removing transition rate remains high. Fur- thermore, the effect of this method to three-level trun- cated qubit subjected to a X-gate operation pulse are studied. According to numerical results based on the Lindblad equation, the targeted dissipation reduces gate fidelity when the gate is applied to qubit in the lowest excited state |e) . To overcome this problem, it is pos- sible to use of tuneable energy dissipation structure, such as QCR, that allow to switch off its connection to the at least one qubit 101 during gate operations. [0058] Fig. 2 illustrates a schematic representation of qubit energy levels according to an embodiment.

[0059] The at least one qubit 101 may have a ground state I g) 201. Herein, the ground state 201 may refer to a quantum state of the at least one qubit 101 with the lowest energy.

[0060] The at least one qubit 101 may further have a plurality of excited states 204. The plurality of ex- cited states 204 may comprise a lowest excited state |e) 202. Herein, the lowest excited state 202 may refer to a quantum state of the at least one qubit 101 with the second lowest energy.

[0061] The ground state 201 and the lowest excited state 202 of the at least one qubit 101 may correspond to the computational basis of the at least one qubit 101. For example, the ground state 201 may correspond to the |0) state of the at least one qubit 101 and the lowest excited state 202 may correspond to the |1) state of the at least one qubit 101 or vice versa. [0062] The energy difference 211 between the ground state 201 and the lowest excited state 202 may corre- spond to a first frequency _ge . The first frequency ω _ge may also be referred to as the resonance frequency of the at least one qubit 101. [0063] The plurality of excited states 204 may further comprise a second lowest excited state \f) 203. The sec ond lowest excited state 203 has a higher energy than the ground state 201 and the lowest excited state 202. [0064] Herein, any excited state above the second low est excited state \f) may be denoted by | f + k), where k refers to the position of the state above the second lowest excited state \f). For example, the third lowest excited state may be denoted by \f + 1) and so on.

[0065] The plurality of excited states 204 may com- prise any number of excited states. In the embodiment of Fig. 2, four excited states are illustrated. However, as is denoted in Fig. 2, there may be any number of excited states between the state \f + 1) and the state

I f + n) .

[0066] According to an embodiment, the energy differ- ence 212 between the lowest excited state 202 and the second lowest excited state 203 corresponds to the sec- ond frequency ω _ef . Alternatively, the second frequency may correspond to an energy difference 213, 214 between the lowest excited state 202 any other excited state in the plurality of excited states 204.

[0067] The at least one passband of the filter 102 may comprise the second frequency. An energy difference be- tween the lowest excited state 202 and the another ex- cited state in the plurality of excited states 204 may correspond to the second frequency.

[0068] According to an embodiment, the another excited state comprises a second lowest excited state 203 of the at least one qubit 101. Thus, the filter 102 may permit the at least one qubit 101 to relax from the second lowest excited state 203 to the lowest excited state 202 by transferring the corresponding energy to the energy dissipation structure 103.

[0069] According to an embodiment, the at least one passband of the filter 102 further comprises a third frequency corresponding to an energy difference between two states in the plurality of excited states 204. The third frequency may be different from the second fre- quency. Thus, the filter 102 may permit a relaxation between two states in the plurality of excited state 204 by transferring the corresponding energy to the energy dissipation structure 103.

[0070] According to an embodiment, the at least one passband of the filter 102 further comprises a first plurality of frequencies. Each frequency in the first plurality of frequencies may correspond to an energy difference between two consecutive or non-consecutive excited states in the plurality of excited states 204 of the at least one qubit 101. Thus, the filter 102 may permit transitions between two consecutive states in the plurality of excited states 204. Thus, the at least one qubit 101 may relax from an excited state in the plu- rality of excited state 204 to the computational basis via a plurality of transitions between consecutive ex- cited states. [0071] Since the at least one stopband of the filter

102 may comprise the first frequency ω _ge , the filter 102 and/or the energy dissipation structure 103 may not sig- nificantly affect the operation of at least one qubit 101 when the at least one qubit 101 is in the ground state 201, in the lowest excited state 202, or in a superposition thereof.

[0072] On the other hand, since the at least one pass- band of the filter 102 may comprise the second fre- quency, when the at least one qubit 101 is in the another excited state, such as the second lowest excited state 203, energy may be transferred from the at least one qubit 101 to the energy dissipation structure 103 via the filter 102. Thus, the at least one qubit 101 may relax from the another excited state to the lowest ex- cited state 202, thus returning to the computational basis and reducing leakage errors.

[0073] Fig. 3 illustrates a schematic representation of a frequency response of a band-stop filter according to an embodiment. The frequency response illustrates in the embodiment of Fig. 3 is only exemplary and may not represent a physically realisable frequency response. [0074] According to an embodiment, the filter 102 com- prises a band-stop filter. [0075] In the embodiment of Fig. 3, the at least one stopband 301 of the filter 102 comprises the first fre- quency ω _ge ,. Thus, the filter 102 may prevent energy transfer from the at least one qubit 101 to the energy dissipation structure 103 at the first frequency ω _ge ,. Therefore, the filter 102 does not cause transitions from the lowest excited state 202 to the ground state 201 and the filter 102 does not therefore substantially affect the computational basis of the at least one qubit 101. [0076] The at least one passband 302 of the filter 102, on the other hand, may comprise frequencies corre- sponding to various other transitions between the states of the at least one qubit 101. In the embodiment of Fig. 3, two frequencies corresponding to energy differences between excited states of the at least one qubit 101 are illustrated. Since these energy differ- ences are smaller than the energy difference between |g) and |e), the corresponding frequencies are lower than the first frequency _ge . Since the at least one passband 302 of the filter 102 comprises these frequencies, the cor- responding relaxations may occur in the at least one qubit 101, which can reduce leakage errors.

[0077] The at least one passband 302 of the filter 102 further comprises frequencies that correspond to larger energy differences than the gap between |g) and |e) . Three such frequencies, and are il- lustrated in the embodiment of Fig. 3. Thus, the filter 102 may also allow direct transitions from the higher excited states to the ground state 201, which may fur- ther reduce leakage errors in the at least one qubit 101.

[0078] Fig. 4 illustrates a schematic representation of a frequency response of a low-pass filter according to an embodiment. The frequency response illustrates in the embodiment of Fig. 4 is only exemplary and may not represent a physically realisable frequency response. [0079] According to an embodiment, the filter 102 com- prises a low-pass filter. [0080] Since the anharmonicity of the at least one qubit 101 may render the gaps between successive excited states smaller than the first frequency , it may suf- fice to implement the filter 102 using a low-pass filter instead of a band-stop filter.

[0081] Similarly to the band-stop filter of the em- bodiment of Fig. 3, a low-pass filter may allow transi- tions between the excited states of the at least one qubit 101, since the frequencies are in the at least one passband of the filter 102. On the other hand, since the higher frequency transitions, such as are in the at least one stopband 301 of the filter 102, these transitions may not occur and cannot be therefore used to remove leakage errors. [0082] On the other hand, by restricting the at least one qubit 101 to sequential transitions, the low-pass filter could guarantee that all the leakage error is retrieved in the |e) state instead of the |g) state.

[0083] According to an embodiment, the at least one stopband 301 of the filter 102 further comprises a sec- ond plurality of frequencies, wherein each frequency in the second plurality of frequencies corresponds to an energy difference between two non-consecutive states of the at least one qubit 101. [0084] Fig. 5 illustrates a schematic representation of an energy dissipation structure 103 according to an embodiment.

[0085] According to an embodiment, the energy dissi- pation structure 103 comprises at least one normal metal 501 - insulator 502 - superconductor 503 (NIS) junction. The NIS junction may be voltage-biased.

[0086] For example, in the embodiment of Fig. 5, the energy dissipation structure 103 comprises two NIS junc- tions. These two NIS junctions form a superconductor - insulator - normal metal - insulator - superconductor (SINIS) junction. The SINIS junction may be biased by a bias voltage V _B . The metallic island 501 may be charged and discharged by controlling the bias voltage V _B due to electron tunnelling across the insulating barriers 502. These tunnelling transitions may also involve ab- sorption or emission of photons. The SINIS junction may be referred to as a quantum circuit refrigerator (QCR). [0087] The filter 102 may be coupled to the normal metal 501 of the NIS/SINIS junction. Thus, the energy dissipation structure 103 may absorb photon energy from the at least one qubit 101 via the filter 102.

[0088] According to an embodiment, the energy dissi- pation structure 103 comprises an QCR, wherein the QCR comprises a SINIS junction.

[0089] According to an embodiment, the energy dissi- pation structure 103 is configured to dissipate photon energy transferred to the energy dissipation structure via photon-assisted electron tunnelling in the NIS/SINIS junction.

[0090] Fig. 6 illustrates an effective circuit model representation of the arrangement 100 according to an embodiment. It should be appreciated that the arrange- ment 100 may not comprise the circuit elements illus trated in the embodiment of Fig. 6. Rather, the circuit elements illustrated in the embodiment of Fig. 6 may represented the electrical properties of the at least one qubit 101, of the filter 102, and/or of the energy dissipation structure 103 and can be used to analyse the behaviour of the arrangement 100.

[0091] In the embodiment of Fig. 6, the energy dissi- pation structure 103 comprises a QCR. The QCR is mod- elled by a NIS junction with capacitance C _j and voltage bias V = V _b /2 . Tunnelling through this NIS junction is modelled by the tunnel junction in parallel with capac- itance C _j . The second NIS junction of the QCR is omitted and its contribution to the tunnelling through the NIS junction is included in the capacitance C _m of the me- tallic island to the ground.

[0092] In the embodiment of Fig. 6, the filter 102 comprises an LC band-stop filter. The filter is modelled by the capacitance C _c connecting it to the QCR and the series capacitance C _f and inductance L _f connected to the ground.

[0093] In the embodiment of Fig. 6, the at least one qubit 101 comprises a transmon qubit. The at least one qubit 101 comprises capacitances C _B and C _j , and Joseph- son energy E _j is capacitively coupled to the filter 102 with capacitance C _g .

[0094] The node fluxes of the normal metal island of the QCR, of the two nodes on the filter 102 and of the at least one qubit 101 are denoted by and , respectively, along with their conjugate node charges Q _N , Q, Q _f , and Q _q .

[0095] As a first approximation, the band-stop filter may be modelled by a first order lumped-element LC fil- ter. The arrangement 100 comprises a QCR connected to an LC band-stop filter that is capacitively connected to the at least one qubit 101. Thus, the band-stop filter and the QCR form a band-stop filtered QCR. The band- stop filtered QCR is an example of a filtered energy dissipation structure. In the following, the canonical approach is followed by writing down the system Lagran- gian and deriving from it a Hamiltonian expressed in terms of node fluxes and charges that are elevated to quantum operators.

[0096] The transition rates from \f) to |e) and from |e) to I g) are calculated as predicted by the Hamiltonian when the QCR and the band-stop filter are considered as an environment for the at least one qubit 101. This involves solving the noise spectral density of the en- vironment formed by QCR and the filter 102, which is done using classical circuit analysis.

[0097] The Lagrangian L of the system may be written as where is the flux quantum and Φ _i denotes the time derivative of the node flux Φ _ί . The quantum mechanical Hamiltonian of the system can be obtained from the La- grangian by defining the node charges Qi as generalised momentum of the node fluxes Φ _ί , performing a Legendre transformation and elevating the degrees of freedoms to operators and . This yields the system Ham- iltonian and The node variables fi and Qi do not appear in the Ham- iltonian, since Q is a constant of motion.

[0098] Fermi 's golden rule may be used to calculate the environment-induced decay rates of the three-level transmon. where Q _int is the charge-transfer operator taking charge from the initial eigenstate i of the qubit to another eigenstate the voltage noise spectrum of band-stop filtered QCR and is the frequency dif- ference of the final state and initial state. In view of this, the Hamiltonian can be divided into system H _sysr environment H _env and interaction H _int parts:

The system Hamiltonian can be written in its eigenbasis I k) up to third level as where the frequency of the ground state is set to zero, ω _ge is the resonance frequency of the at least one qubit 101 and a is the anharmonicity of the f energy level. [0099] The charge transfer operator Q _int may be ex- pressed as and Q _q can be expressed in the transmon eigenbasis as where the "ket-bras" \f){i\ cause sequential transitions between adjacent energy levels and 2e is the charge of a Cooper pair.

[0100] The voltage noise spectrum of the band-stop filtered QCR can be obtained by solving the voltage response of the filter 102 , and using this relation to express the voltage noise after the filter 102 as is the noise voltage of the QCR. The noise of the QCR alone is ob- tained from the fluctuation-dissipation theorem. Hence, the voltage noise spectrum of band-stop filtered QCR is where in the last line the product of transfer functions is written out explicitly and the result of fluctuation- dissipation theorem is used. is the central rejected frequency, R _f and R _c are the series resistances of the filter and QCR-filter coupling conductor. They arise e.g. from losses of dielectrics and are needed to keep the Q-factor of LC-circuit finite. The approximate equality follows from approximations based on relation .

[0101] Fig. 7 illustrates a table of circuit parame- ters according to an embodiment. The circuit parameters are only exemplary and are intended to illustrate whether the filter 102 can protect the e g transition while the leakage error at the /-state is decaying in f→ e transition. The effective resistance of the QCR is set such that the QCR-induced relaxation rate is in nanoseconds.

[0102] Inserting the expressions for Q _q and S _QCR+BSF (ω) ) and frequencies ω _ge and ω _ef = 2ω _{ge —} a to the Fermi's golden rule yields decay rates and [0103] The decay rates of above indicate that a band- stop filter can efficiently protect the computational basis from decaying while allowing QCR to quickly de- plete the second lowest excited state. The decay process f g is ignored in the analysis based on the fact that transitions with even difference between the states are exponentially suppressed because of parity. [0104] In the following, the effect of the band-stop filtered QCR to the time-evolution of a driven three- level truncated transmon is studied. The Hamiltonian of the system includes the transmon and drive terms and is represented in the ordered basis as a matrix where the ground state energy is set to zero and the energy of the \f) state is, in terms of the first excited state energy and an anharmonicity ha. The duration of coherent transitions between the basis states are dictated by the Rabi frequency W, se- lective driving is made possible by choosing the fre- quency of the driving field ω _ά to match the energy dif- ference between the target states and the duration of the drive is given by the time parameter t .

[0105] The dissipative time-evolution is modelled us- ing the Lindblad master equation, where the amplitudes of the dissipation terms are given by the decay rates and calculated above. For a vectorized form of the density matrices, the Lindblad equation yields a following time-evolution superoperator where is a superoperator corresponding to the coherent von Neumann part and is a Lindblad generator with Lindblad operators the 3x3 identity matrix and signifies the tensor (Kronecker) product of matrices.

[0106] Being superoperator itself, the propagator of the superoperator equation above gives the (vectorized) density matrix p(t) of the system at a time t as a matrix multiplication of the vectorized initial state p(t ₀ )

[0107] The cumbersome Dyson series of the superoper- ator equation may be approximated by the following prod- uct of matrix exponentials (each corresponding to an evolution lasting for a short time dt): where N = t/dt — 1.

[0108] Fig. 8 illustrates a plot representation of qubit eigenstate occupation during gate operation ac- cording to an embodiment. In addition to the excitations induced by the environment, leakage due to driving fields may be a major cause of leakage errors. Fig. 8 illustrates the effect of a band-stop filtered QCR to a qubit undergoing a X-gate operation i.e. being subjected to a p-pulse. [0109] The plots of Fig. 8 depict the time-evolution of a three-level transmon qubit undergoing a p-pulse in resonance transitions and then idling. Fig. 8 illustrates the results for a typical Rabi frequency of corresponding to a pulse duration (approximately 67 ns). Curve 801 corresponds to the ground state |g), curve 802 cor- responds to the lowest excited state |e), and curve 803 corresponds to the second lowest excited state \f). [01 10] The left column plots 810 show the situation with a band-stop filtered QCR and the right column plots 820 show the evolution without a band-stop filtered QCR. [01 1 1] If one considers a maximal fidelity, a p-pulse applied to a qubit initially in the ground state, like in the upper panels of Fig. 8, or the first excited state, like in the lower panels of Fig. 8, the qubit is flipped to the excited state and ground state, respec- tively. Comparison of the plots with and without dis- sipation reveals that the dissipation increases the in- fidelity of the gate by an order of magnitude. However, when the pulse has ended, the leakage occupation of the I/) state starts to relax, particularly when the initial state is the excited state |e), while the relaxation of the excited state |e) is impeded by the band-stop filter. [01 12] Fig. 9 illustrates a schematic representation of a quantum computing system 1000 according to an em- bodiment. [01 13] According to an embodiment, a quantum computing system 1000 comprises the arrangement 100 and a control unit 1001 coupled to the arrangement 100. The control unit 1001 may be configured to decouple the filter 102 and/or the energy dissipation structure 103 from the at least one qubit 101 in response to a gate operation on the at least one qubit 101.

[01 14] The gate operation may be performed by the con- trol unit 1001 or by some other device in the quantum computing system 1000.

[01 1 5] The filter 102 and/or the energy dissipation structure 103 may be coupled to the at least one qubit 101 via a controllable coupling. The control unit 1001 may be coupled to the controllable coupling and the control unit 1001 may be configured to decouple the filter 102 and/or the energy dissipation structure 103 from the at least one qubit 101 via the controllable coupling in response to a gate operation on the at least one qubit 101. For example, in embodiments where the energy dissipation structure 103 comprises a NIS/SINIS junction, the control unit 1001 may be configured to decouple the energy dissipation structure 103 by ad- justing the bias voltage V _B .

[01 16] A gate operation may refer to a quantum logic gate operation, such as X-gate, Y-gate, Z-gate, Hadamard gate, or phase gate, performed on the at least one qubit 101.

[01 1 7] Decoupling the filter 102 and/or the energy dissipation structure 103 from the at least one qubit 101 may prevent the filter 102 and/or the energy dissi- pation structure 103 from affecting the computational basis of the at least one qubit 101. Although, according to the analysis above, this may not be required in some cases, such as with slow decay rates, it may be benefi- cial in some other situations.

[01 18] When the system 1000 is operational, the ar- rangement 100 may be physically located in a cryostat or similar. The cryostat may cool the at least one qubit 101 and other components of the arrangement 100, such as the energy dissipation structure 103, to cryogenic temperatures. This may be required if the at least one qubit 101 comprises, for example, a superconductive qubit. The control unit 1001 may be located outside the cryostat.

[01 19] Although the connection between the arrange- ment 100 and the control unit 1001 is schematically illustrated as a single connection in the embodiment of Fig. 10, the connection may comprise any number of con- nections. For example, the control unit 1001 may be coupled to the at least one qubit 101, to the filter 102, and/or to the energy dissipation structure 103. [0120] Fig. 10 illustrates a schematic representation of a control unit 1001 according to an embodiment. [0121] The control unit 1001 may comprise at least one processor 1101. The at least one processor 1101 may comprise, for example, one or more of various processing devices, such as a co-processor, a microprocessor, a control unit 1001, a digital signal processor (DSP), a processing circuitry with or without an accompanying DSP, or various other processing devices including in- -egrated circuits such as, for example, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a microprocessor unit (MCU), a hard- ware accelerator, a special-purpose computer chip, or the like.

[0122] The control unit 1001 may further comprise a memory 1102. The memory 1102 may be configured to store, for example, computer programs and the like. The memory 1102 may comprise one or more volatile memory devices, one or more non-volatile memory devices, and/or a com- bination of one or more volatile memory devices and non- volatile memory devices. For example, the memory 1102 may be embodied as magnetic storage devices (such as hard disk drives, floppy disks, magnetic tapes, etc.), optical magnetic storage devices, and semiconductor mem- ories (such as mask ROM, PROM (programmable ROM), EPROM (erasable PROM), flash ROM, RAM (random access memory), etc.).

[0123] The control unit 1001 may further comprise other components not illustrated in the embodiment of Fig. 10. The control unit 1001 may comprise, for exam- ple, an input/output bus for connecting the control unit 1001 to the arrangement 100. Further, a user may control the control unit 1001 via the input/output bus. The user may, for example, control quantum computation operations performed by the arrangement 100 via the control unit 1001 and the input/output bus. [0124] When the control unit 1001 is configured to implement some functionality, some component and/or com- ponents of the control unit 1001, such as the at least one processor 1102 and/or the memory 1102, may be con- figured to implement this functionality. Furthermore, when the at least one processor 1101 is configured to implement some functionality, this functionality may be implemented using program code comprised, for example, in the memory 1102. [0125] The control unit 1001 may be implemented using, for example, a computer, some other computing device, or similar.

[0126] Fig. 11 illustrates a flowchart representation of a method 1200 for qubit leakage error reduction using at least one arrangement according to an embodiment.

[0127] The method 1200 may comprise decoupling 1201 the filter and/or the energy dissipation structure from the qubit.

[0128] The method 1200 may further comprise performing 1202 at least one gate operation on the qubit.

[0129] The method 1200 may further comprise recoupling 1203 the filter and/or the energy dissipation structure to the qubit.

[0130] The at least one qubit 101 may be decoupled from the filter and/or the energy dissipation structure all the time except at specific "leakage removal times" when the coupling to the energy dissipation structure is turned on for a time sufficient to remove the leakage. After the leakage removal, the at least one qubit 101 can again be decoupled. The frequency at which these "leakage removal times" occur can depend on the amount of leakage that is accumulated in the at least one qubit 101 during quantum computation. Thus, the leakage re- moval maybe performed, for example, after every gate, after two gates, after three gates etc.

[0131] The at least one arrangement may comprise a plurality of arrangements. This may be the case, for example, in a quantum computing system comprising a plu- rality of qubits where each qubit can be implemented using the arrangement 100. In such a case, the method 1200 may be performed on each such arrangement in order to remove leakage error from each qubit in the plurality of qubits. [0132] The method 1200 may comprise decoupling 1201 the filter and/or the energy dissipation structure from the qubit in each arrangement in the plurality of ar- rangements.

[0133] The method 1200 may further comprise performing 1202 at least one of a single-qubit gate operation or multi-qubit gate operation using the qubit in each ar- rangement in the plurality of arrangements.

[0134] The method 1200 may further comprise recoupling 1203 the filter and/or the energy dissipation structure to the qubit in each arrangement in the plurality of arrangements.

[0135] The method 1200 may be performed by, for exam- ple, the control unit 1001 or any other device capable of controlling the at least one arrangement. [0136] In a different embodiment only one or more qubits within the plurality of qubits is coupled to their filter and/or dissipation structure for leakage error removal, while other qubits within the plurality of qubits remain decoupled. In a yet different embodi- ment, the plurality of qubits may be coupled or decou- pled to a common filter and/or dissipation structure. [0137] Any range or device value given herein may be extended or altered without losing the effect sought. Also any embodiment may be combined with another embod- iment unless explicitly disallowed.

[0138] Although the subject matter has been described in language specific to structural features and/or acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as examples of implementing the claims and other equiv- alent features and acts are intended to be within the scope of the claims.

[0139] It will be understood that the benefits and advantages described above may relate to one embodiment or may relate to several embodiments. The embodiments are not limited to those that solve any or all of the stated problems or those that have any or all of the stated benefits and advantages. It will further be un- derstood that reference to 'an' item may refer to one or more of those items. [0140] The steps of the methods described herein may be carried out in any suitable order, or simultaneously where appropriate. Additionally, individual blocks may be deleted from any of the methods without departing from the spirit and scope of the subject matter de- scribed herein. Aspects of any of the embodiments de- scribed above may be combined with aspects of any of the other embodiments described to form further embodiments without losing the effect sought. [0141] The term 'comprising' is used herein to mean including the method, blocks or elements identified, but that such blocks or elements do not comprise an exclu- sive list and a method or apparatus may contain addi- tional blocks or elements. [0142] It will be understood that the above descrip- tion is given by way of example only and that various modifications may be made by those skilled in the art. The above specification, examples and data provide a complete description of the structure and use of exem- plary embodiments. Although various embodiments have been described above with a certain degree of particu- larity, or with reference to one or more individual embodiments, those skilled in the art could make numer- ous alterations to the disclosed embodiments without departing from the spirit or scope of this specifica- tion.