On-resonance parametric gates with reduced leakage by modulation frequency selection

Abstract

Provided is a parametric gate control method, comprising: generating (S110) a flux modulation signal having a selected modulation frequency for a first qubit for tuning an on- resonance parametric gate during a gate operation having a finite gate time; and applying (S130) the flux modulation signal to a control line coupled to the first qubit, wherein the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation.

Description

[0001]271 261 s5 / sca On-resonance parametric gates with reduced leakage by modulation frequency selection Technical Field The present disclosure relates to a parametric gate control method and a system for implementing on-resonance parametric gates with reduced leakage. Background Quantum computers provide quantum physics-based computational power not available from a standard classical computer. The advantage in computational power is achieved because quantum computers can perform many calculations simultaneously due to the principles of superposition (processing a large number of states at the same time) and entanglement (leveraging and improving efficiency of specific algorithms and computations). The technical implementation of quantum computation by a quantum algorithm can be achieved by quantum gates (such as single-qubit gates, two-qubit gates) to manipulate quantum bits (qubits). Quantum gates play a central role in quantum algorithms and quantum information processing, enabling the quantum computer to perform complex calculations and solve technical problems that were previously considered intractable. The precise control and manipulation of physical qubits using these quantum gates allows quantum computers to achieve their exceptional computational power, for example in the field of cryptography, drug development, and materials science. Quantum gates may be performed, for example, by controlling a tunable physical qubit. A physical qubit may exhibit a transition frequency related to an energy difference of energy levels of quantum states, and the transition frequency may be tuned or changed by the application of an electromagnetic field. For example, a superconducting tunable qubit may be a tunable transmon qubit, a fluxonium qubit, a flux qubit, a unimon qubit or the like. Two-qubit gates may be implemented as parametrically activated gates. A parametric gate is a quantum gate for quantum computing which is defined by one or more parameters as variables in its operation. The one or more parameters (control parameters) may vary (time-dependent parameters) during the gate operation. Here, by adjusting or varying the parameter(s) of a parametric gate, the gate operation may be tuned to perform a specific quantum operation. Finding optimal parameter values for a specific quantum operations(s) may be challenging due to noise and practical imperfections of the quantum device. In general, the parameter(s) of the quantum gate may be tuned to be on (or near) or off a resonance frequency of the physical qubit or on or off a resonance frequency with a higher transition frequency of a qubit or multiple qubits (in case of a multi-qubit gate). Parameters may, for example, be a flux modulation frequency and / or amplitude of a flux modulation signal applied to a qubit (e.g. a tunable-frequency qubit), a qubit flux bias, or the like. By tuning the parameters of these gates to be on or near the resonance frequency of the qubit or quantum system, it is possible to achieve more efficient and controlled operations. The specific implementation of on-resonance parametric gates depends on the physical architecture and technology used for quantum computation. Different types of physical qubits (e.g., superconducting qubits, trapped ions, topological qubits) have different resonance frequencies, and specifying gates that exploit these resonances requires precise engineering which requires an understanding of the underlying quantum physical implementation (such as sources of noise and leakage therein). By contrast, using off-resonance parametric gates may achieve certain quantum operations while minimizing unwanted effects that might arise from resonant interactions. In particular, by operating gates off-resonance, the impact of noise, decoherence, and other sources of error can be mitigated as compared to the case when the system is near its resonance frequency. Technical Problem The skilled person understands the term infidelity as a quantitative representation of the extent to which the final state after a quantum operation deviates from the desired or expected state. This term may thus be used to assess the performance of quantum operations, gates, and algorithms by quantifying how well they maintain the integrity of quantum states and reduce errors, in particular with regard to a specific quantum computing hardware architecture. Parametric gates have been implemented since some time now. The first implementations were based on side-band driven parametric gates, where the sideband of the driven qubit was brought in resonance with the other. However, the gate speed is limited by the reduced qubit-qubit coupling and there are limitations on frequency selectivity as well [1-4]. More recently, an on-resonance parametric gate was proposed [5-6]. However, this implementation suffered from lower fidelities arising from leakage. On-resonance parametric gates, either between two qubits (qubit–qubit) or between a qubit and a resonator (qubit– resonator) can suffer from leakage into coupler states resulting in a lower fidelity, for example when compared to fast-flux gates. There is thus a need to increase the fidelity by reducing leakage when using on-resonance parametric gates in quantum computation. Solution The present inventors have realized that the leakage of on- resonance parametric gates of quantum operations can be reduced by a specific selection of the modulation frequency. According to an aspect of the present disclosure, a parametric gate control method comprises generating a flux modulation signal having a selected modulation frequency for a first qubit, in particular a first tunable-frequency qubit, for tuning an on-resonance parametric gate during a gate operation having a finite gate time; and applying the flux modulation signal to a control line coupled to the first qubit, wherein the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation. According to a preferred embodiment, the selected modulation frequency may cause the qubit population oscillation and the qubit leakage oscillation to have a predetermined phase relationship with each other. According to another preferred embodiment, the qubit population oscillation may have a first oscillation frequency being different from a second oscillation frequency of the qubit leakage oscillation. According to another preferred embodiment, the modulation frequency may be selected in conjunction with a selection of the finite gate time. According to another preferred embodiment, the selected modulation frequency may associate a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation. According to another preferred embodiment, the maximum of the qubit population oscillation may be set at the first qubit, in particular at the first tunable-frequency qubit or at a second qubit, in particular a second tunable-frequency qubit, or at a resonator circuit. According to another preferred embodiment, the selected modulation frequency may associate a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time. According to another preferred embodiment, the maximum of the qubit population oscillation may be set at the first qubit, in particular the first tunable-frequency qubit, and the on- resonance parametric gate is a CZ gate. According to another preferred embodiment, the selected modulation frequency may associate a minimum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time. According to another preferred embodiment, wherein the minimum of the qubit population oscillation may be set at the first tunable-frequency qubit and the on-resonance parametric gate is a SWAP gate. According to another preferred embodiment, the qubit leakage oscillation may be due to Landau-Zener-Stückelberg interaction. According to another preferred embodiment, the flux modulation signal may further be applied to a control line coupled to a second qubit, in particular to a second tunable-frequency qubit.According to another aspect of the present disclosure, a systemcomprises: a first qubit, in particular a first tunable- frequency qubit; and a generator circuit, wherein the generator circuit is configured to: generate a flux modulation signal having a selected modulation frequency for the first qubit for tuning an on-resonance parametric gate during a gate operation having a finite gate time; and apply the flux modulation signal to a control line coupled to the first qubit, wherein the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation. According to another preferred embodiment, the system may further comprise a second qubit, in particular a second tunable-frequency qubit, and a tunable coupler. According to another preferred embodiment, the system may further comprise a resonator and a tunable coupler. Brief Description of Drawings Fig. 1 shows a flow diagram of an embodiment of a control method for selecting a modulation frequency for an on-resonance parametric quantum gate.Fig. 2A and 2B show a simulated time-dependent population ofan on-resonance parametric two-qubit gate (SWAP gate), for a modulation frequency of 185 MHz (Fig. 2A) and a modulation frequency of 465 MHz (Fig. 2B).Fig. 3A and 3B show examples of a first oscillation (bold) anda second oscillation (dotted) with different phase relationships with each other.Fig. 4A and Fig. 4B respectively show experimental data ofsimulations using a higher modulation frequency of 465 MHz according to the present disclosure and a lower modulation frequency of 185 MHz for comparison.Fig. 5A and Fig. 5B respectively show a comparison ofexperimental data of simulations using a modulation frequency selected according to the present disclosure and a modulation frequency so that the qubit population oscillations are not synchronized with the qubit leakage oscillations. Fig. 6 shows a system for implementing on-resonance parametric gates with reduced leakage according to an embodiment. Detailed Description Where technical features in the drawings, detailed description or any claim are followed by reference signs, the reference signs have been included for the sole purpose of increasing the intelligibility of the drawings, detailed description, and claims. Accordingly, neither the reference signs nor their absence have any limiting effect on the scope of any claim elements. Parametric gates are a class of quantum gates that rely on time-varying parameters to perform quantum operations. Unlike standard, static quantum gates (such as Pauli-X, Pauli-Y, or Hadamard gates), which are applied at specific time points to manipulate qubits, parametric gates may be continuously modulated throughout a quantum computation, allowing for a dynamic and precise control over the quantum state of a qubit. Parametric gates can be especially useful in certain quantum algorithms and error correction schemes. By carefully tuning or adjusting the parameters of these gates, quantum processors can perform operations with enhanced precision and resilience against noise and decoherence, making them a valuable tool in the pursuit of fault-tolerant quantum computing. An on-resonance parametric gate is a specific type of parametric gate that operates at a frequency matching the natural frequency (or resonance frequency) of the qubit it is applied to. In quantum systems like superconducting qubits, qubits have specific resonance frequencies associated with their energy levels. By driving a parametric gate at this resonance frequency, a desired quantum operation on the qubit can be achieved. The term "on-resonance" implies that the parametric gate's modulation amplitude may be determined by a transition frequency of a qubit. In addition, an amplitude may have a pre-determined (sufficient) value so that the qubit frequency is tuned or adjusted to a required transition frequency, e.g. corresponding to the frequency of another qubit (in case of qubit-qubit interaction) and / or resonator (in case of qubit-resonator interaction). For qubit-qubit interaction, the skilled person understands that one qubit may be modulated by the modulation signal while the corresponding transition frequency refers to the other qubit. When a parametric gate is applied on-resonance, it can induce precise and controlled quantum operations, making it an efficient tool for various quantum computing tasks. On-resonance parametric gates are particularly relevant in superconducting qubit architectures, where the qubit's energy levels are highly sensitive to external fields, like microwave signals. By carefully tuning the parameters and frequencies of these gates to match the qubit's resonance frequency, quantum circuits can be engineered that maximize the fidelity and accuracy of quantum computations while minimizing errors and unwanted effects. Examples of such on-resonance parametric gates are parametric X- and Y-gates, two-qubit CNOT gates, two-qubit iSWAP gates, two-qubit CZ (Controlled-Z) gates, two-qubit cross-resonance gates, and the like. More specifically, in superconducting qubit systems, the energy levels of qubits are sensitive to the magnetic flux passing through a superconducting loop. By applying time- varying magnetic fields, known as flux modulation or flux pulses, the energy levels of the qubit can be manipulated. This manipulation may change the state of the qubit, and these altered qubit states can be used as inputs to quantum gates to perform specific quantum operations on that qubit. The combination of flux modulation and quantum gates thus allows for the implementation of quantum algorithms and quantum computation. Fig. 1 shows a flow diagram of an embodiment of a parametric gate control method which may be applied for selecting a modulation frequency for an on-resonance parametric quantumgate. According to step S110, the control method generates aflux modulation signal having a selected modulation frequency, i.e. having a frequency value causing an oscillating magnetic flux. Here, the selected modulation frequency may cause an oscillation at a time scale that is 10 to 100 times smaller than the gate duration (gate time). In addition, the selected modulation frequency may also depend on or limited by gate fidelity and / or time constants in filters in the control line (e.g. flux line) supplying the flux modulation signal. The flux modulation signal having the selected modulation frequency may control a (first) physical qubit, in particular a tunable-frequency physical qubit, over a (finite) duration of the flux modulation signal. The flux modulation signal may thus be used to tune or adjust the physical qubit which can then be used to implement an on-resonance parametric gate during a gate operation having a finite gate time. Gate times (also referred to as gate durations) for superconducting qubits can be tens of nanoseconds, and preferably less than 10 nanoseconds. Further, a typical flux modulation frequency may be in the MHz range. The skilled person understands that a flux modulation frequency and a flux modulation amplitude cause a modulation of a qubit flux bias which in turn causes the transition frequency of the qubit to be modulated. Preferably, an amplitude of the flux modulation signal should have a pre-determined (sufficient) value so that the qubit frequency is tuned, for example tuned down or lowered to a required transition frequency (e.g. corresponding to the gate, to the frequency of another qubit or resonator, or the like) which may be in the GHz range.Further, according to step S130 in Fig. 1, the generated fluxmodulation signal having the selected flux modulation frequency may be applied to at least one control line. A control line may be electromagnetically coupled to the (first) tunable-frequency physical qubit (of a quantum device or quantum computer). In general, each qubit may have a dedicated control line. The control line may be a physical hardware connection, such as a microwave or radiofrequency transmission line configured to apply the control signal (flux modulation signal) as a microwave pulse, a voltage pulse, or the like. Such control lines may be implemented using microwave cables, coaxial cables or other electronic components for carrying the control signal from the flux modulation signal generator to the physical qubit. In general, the control line may be implemented as a transmission line between classical control electronics and the quantum computer architecture (involving qubits and coupling elements between the qubits). In case of a flux modulation signal for a superconducting qubit, the control line is a flux line adapted to control the magnetic flux permeating the superconducting loop of the qubit. For the flux line, the coupling is an inductive coupling. Filters in the flux line may be configured (e.g. based on a LC resonator) to stabilize the flux modulation signal and reduce noise and fluctuations in the flux signal. In addition, the flux modulation signal may be applied to at least one control line, in particular a first control line (flux line) coupled to the first qubit (e.g. unable-frequency qubit) and / or a second control line (flux line) coupled to the second qubit (e.g. tunable-frequency qubit). The first and second control lines are preferably separate flux lines. In the context of a quantum qubit being driven by a selected flux modulation signal, several dynamic processes may come into play. The state population of the qubit may undergo temporal changes primarily as a result of two fundamental mechanisms. First, internal oscillations between the qubit's different quantum states, such as the ground and excited states, may occur due to the time-varying magnetic fields induced by the flux modulation signal. Simultaneously, the qubit can experience leakage oscillations, which can result from imperfections in the quantum hardware and / or environmental interactions and couplings. These leakage oscillations can lead to the undesired excitation of higher energy states or transitions to non-computational states. Therefore, achieving higher fidelity quantum computations may involve not only effectively driving the qubit with the flux modulation signal but also mitigating and controlling the impact of leakage oscillations to ensure the accurate execution of quantum operations. The first dynamic process leading to qubit population oscillations having a first frequency Ω1is related to Rabi oscillations, which may be oscillations between energy states of a single or multiple qubits (e.g. between a first excited state of a first qubit and a first excited state of a second qubit, or between a first excited state of a first qubit and a resonator) in the presence of the flux modulation signal which turn on an effective coupling between a first qubit and second qubit, a first qubit and a resonator, or the like. That is, the qubit population oscillations Ω1may depend on the flux modulation frequency. The second dynamic process (different from the first dynamic process) leads to qubit leakage oscillations out of the first and / or second energy state of the qubit and has a second, different frequency Ω2. In particular, a leakage mechanism in parametric gates may arise from Landau-Zener-Stückelberg (LZS) interactions [7] that occur when the physical qubit is subject to frequency tuning by the flux modulation signal, e.g. driven with a parametric drive (for example, cosine-modulated flux pulses of qubit frequency). The LZS interaction can lead to leakage for several interrelated reasons: (i) Non-Adiabatic Transitions: Here, the Landau-Zener effect, a component of the LZS interaction, causes non-adiabatic transitions in the sense of a non-Markovian, stochastic effect between qubit energy levels when the flux modulation signal changes. If the change in parameters is too rapid compared to the energy gap between the qubit's states, non-adiabatic transitions can occur, causing the qubit to end up in unintended states. This may be a primary mechanism of leakage. (ii) Stückelberg Interference: The Stückelberg oscillations can lead to qubit leakage because they introduce interference patterns in the qubit's dynamics. These oscillations may not be perfectly controllable, leading to unintended energy state population changes. The interference can result in some population leaking into states other than the intended ground and excited states of the physical qubit. (iii) Off-Resonant Driving: In the LZS interaction, precise tuning of the modulation frequency to match the qubit's energy splitting (the resonant condition) may be required for coherent operations. If the driving frequency is not perfectly aligned with the qubit's energy levels, off-resonant driving can also lead to leakage, as the qubit may respond to frequencies that cause it to deviate from its desired state. (iv) Imperfections and Noise: In practical quantum systems, there are always imperfections and external noise sources that may also affect the LZS interaction. These imperfections and noise can lead to further deviations from the ideal LZS dynamics and result in leakage. In other words, when a physical qubit is driven by an external flux modulation signal that changes with time, the qubit's energy levels can shift as well, leading to transitions between its states. These transitions are non-adiabatic if the external driving is changed rapidly compared to the characteristic timescale of the qubit dynamics. The non-adiabatic transitions may be governed by the Landau-Zener formula, P= e−2πΓDwhich is based on the speed at which the energy levels of the (transmon) qubits are changed upon flux-modulation describedabove and provides the probability PD of diabatic transitioningbetween the qubit states as a function of energy separation (distance) between the qubit states which determines Γ. Leakage thus occurs because a sufficiently large amplitude modulation of the qubit flux is needed to obtain an average flux amplitude required for the gate operation. This large, fast and cyclic detuning of the qubit during the finite gate time may cause the energy level of the qubit to pass / come close to a coupler state of a multiple times. Each time the qubit is pushed close to this state, there is a finite probability of leakage into a coupler state (a state related to a second qubit, a state related to a resonator, or the like). Although this issue is not similarly significant in a single occurrence of non-parametric gate, there have been works to mitigate it [8, 9]. However, no technical solution exists for parametrically driven gates. The present inventors have simulated the time-dependent population of an on-resonance parametric two-qubit gate (SWAP gate), as shown for modulation frequency of 185 MHz (Fig. 2A) and a modulation frequency of 465 MHz (Fig. 2B). As indicatedin Fig. 2A and 2B, two fundamental oscillations are present in the system. In particular, first, a frequency dependent (havinga frequency Ω2) leakage in the qubit population spectrum canbe observed in the form of vertical stripes. These leakage oscillations are attributed to LZS interference. Althoughsecond, Rabi oscillations (having a frequency Ω1) between thequbit and coupler exist as well, the LZS interference may even be a dominant effect in case of parametric gates due to thespeed of interaction. The on-resonance parametric gatesimplemented in the prior art suffer from leakage, as LZS interference mechanisms are not taken into consideration. However, the present simulations show that this is also a non- negligible source of infidelity. While these two different oscillations (Ω1, Ω2) are a result of operating the parametric gate, and the leakage oscillations (due to LZS interaction) themselves may not be perfectly controllable, according to an embodiment, the selection of the modulation frequency (and in consequence a control of the qubit population oscillation) may be performed in conjunction with the finite gate time reduce to reduce leakage when operating an on-resonance parametric gate. Based on this, in the qubit control method according to theembodiment of Fig. 1, the modulation frequency is selectedbased on the finite gate time, the qubit population oscillation and the qubit leakage oscillation, that is three independent parameters. Preferably, the two different oscillationfrequencies (Ω1, Ω2) related to the two different technicalprocesses that involve, on the one hand, internal quantum state population oscillations and, on the other hand, leakage oscillations from the qubit state in the qubit-environment system, are brought to a predetermined phase relationship with regard to the finite gate time. That is, the modulation frequency may be selected so that the phase φ1of the internal quantum state population oscillations which may generally be considered to follow: Cos(Ω1t + φ1) and the phase φ2of the leakage oscillations which may generally be considered to follow: Cos(Ω2t + φ2)preferably have a predefined phase relationship between φ1 andφ2regarding the finite gate time. The skilled person understands that the selected modulation frequency causes, in particular, a change of the qubit population oscillation frequency Ω1which also causes a change in the phaserelationship to the qubit leakage oscillation.Fig. 3A and 3B show examples of a first oscillation (bold) anda second oscillation (dotted) with different phaserelationships with each other. In Fig. 3A, the phaserelationship is set so that a maximum value of the first oscillation (bold) may coincide with a minimum value of the second oscillation (dotted). That is, the selected modulation frequency associates a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation.In Fig. 3B the phase relationship is set so that a maximumvalue of the first oscillation (bold) may coincide with a maximum value of the second oscillation (dotted). Preferably, such coinciding time points may be selected in consideration of the finite gate time, for example by having the finite gate time also coinciding. Accordingly, setting such a predetermined phase relationships between the qubit population oscillation (first oscillation) and the qubit leakage oscillation (second oscillation) allows for an improved control over state population and leakage. In a preferred embodiment, to reduce the leakage due to LZS interference, a flux modulation frequency may be selected so that the two specific oscillations (Ω1, Ω2) match the finite gate time. In other words, the predefined phase relationship may be set so that the qubit population oscillation (for example, a maximum thereof) and the qubit leakage oscillation (for example, a minimum thereof) are co-incident at one or more time points during the finite gate time. Namely, a modulation frequency tracking of the qubit population (for example, returned completely back to the initial qubit in case of CZ; to the other in case of SWAP) selected such that leakage due to LZS interference is minimized. By systematically tuning the system to controllably match these phenomena, the leakage in on-resonance parametric gates can be reduced.As explained, the qubit population oscillation (based on Rabioscillations) has a first frequency that depends on the selected modulation frequency of the flux modulation signal, while the qubit leakage oscillation (based on LZS interaction) has a second frequency that is different from the first frequency. Typical values of the first frequency, for example observed in the simulations described above, may be in the range of 10 MHz, and typical values of the second frequency may be in the range of 600 MHz. The selection of the modulation frequency is thus based on the gate time to ensure that the gate operation aligns with this finite time window of the gate time and has reduced leakage.In addition, as the modulation frequency influences the qubitpopulation oscillation (based on Rabi oscillations), the modulation frequency is further selected in such a way that the finite gate time (or another specific time during the finite gate time) is associated or matches with a specific value of the qubit population oscillation (e.g. being related to a maximum or minimum of the ground or excited state of the first qubit (e.g. tunable-frequency qubit), a second qubit (e.g. a fixed-frequency qubit) and also is associated or matches with a specific value of the qubit leakage oscillation (e.g. being related to a minimum of the leakage oscillation). That is, by selecting and fine-tuning the modulation frequency, a gate operation may be performed at a point in time that corresponds with a specific optimized qubit population while minimizing leakage, which thus leads to higher fidelities. This specific choice of the modulation frequency for matching the gate operation time with intended values of the qubit population and minimized leakage values improves the control and achieve a proper balance between these factors. As such, the performance and fidelity of quantum gates in qubit-based quantum processors can directly be optimized and leads to more reliable and accurate quantum computations. According to another preferred embodiment, the selection of the modulation frequency selection may be performed together with a selection of the gate time. That is, by selecting the modulation frequency in conjunction with a selection of the finite gate time, more flexibility may be provided to achieve a fine-tuned balance between qubit population and minimal leakage at the time of gate operation. In other words, the additional flexibility to select values of the gate time in addition to the selection of the modulation frequency allows an improved control to set the predetermined phase relationship. By contrast, while US 10,483,980 B2 and WO 2022 / 140674 A1 refer to leakage state transitions (for example, in scenarios in which qubit devices are coupled to each other in physical proximity and define overlapping quantum states) and using a bipolar flux pulse for improving dephasing time and performance of two-qubit gates, respectively, the present disclosure of modulation frequency selection, preferably together with the gate time selection, in view of qubit population oscillation frequencies and leakage frequencies (due to LZS interference) is not considered therein. According to a preferred embodiment, the predetermined phase relationship may be one in which a maximum of the qubit population oscillation (e.g. measured back on the first tunable-frequency qubit, the second tunable-frequency qubit, or a resonator circuit) is associated with or matches a minimumof the qubit leakage oscillation (see also Fig. 3A). That is,by selecting the modulation frequency, the maximum of the qubitpopulation oscillation may be set at the first tunable- frequency qubit or at a second tunable-frequency qubit or ata resonator circuit. The skilled person understands that thissetting of the predetermined phase relationship achieves proper gate interaction while minimizing leakage and thusincreases fidelity. Such a selection is indicated in Fig. 2left: Here, an operation point (indicated by “O”) is shown at which the qubit population is maximized while the leakage oscillations are minimized. According to another preferred embodiment, the predeterminedphase relationship may be one in which the selected modulationfrequency associates or matches a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time. For example, the maximum of thequbit population oscillation can preferably be set or measuredback on the first qubit, the first tunable-frequency qubit. This setting of the phase relationship at the gate time improves high-fidelity operations of a CZ-gate, for example. According to another preferred embodiment, the predeterminedphase relationship may be one in which the selected transitionfrequency associates or matches a minimum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time. For example, the minimum of thequbit population oscillation can preferably be set or measuredback on the first tunable-frequency qubit. This setting of the phase relationship at the gate time improves high-fidelity operations of an iSWAP gate, for example. According to another preferred embodiment, the predeterminedphase relationship may be one in which the selected modulation frequency associates or matches a maximum of the qubit population oscillation with a maximum of the qubit leakageoscillation at the gate time, see Fig. 3B.By restricting and setting the modulation frequency of the gates as mentioned above, the leakage may suitably be limited and an increase of the 2-qubit gate fidelity can be achieved.Fig. 4A and Fig. 4B respectively show experimental data ofsimulations using a higher modulation frequency of 465 MHz according to the present disclosure and a lower modulation frequency of 185 MHz for comparison. In other words, experimental data of two different simulations corresponding to these modulation frequencies are shown.As shown in Fig. 4A, a higher modulation frequency may providebetter fidelity, in particular due to the selection of the modulation frequency which is seen to lower the LZS leakage to coupler state. Here, the leakage population (identified by the[0, 0, 1] state) is shown in Fig. 4A by the solid black curve,and at 70 ns time, the leakage is almost zero, which gives higher fidelity. For the case of the lower modulation frequencyin Fig. 4B, the fidelity is lower because this choice ofmodulation frequency allows for larger LZS leakage. This can again be seen in the plot at around 100 ns - the solid black curve does not end in zero but rather at some finitepopulation. Fig. 4A and Fig. 4B clearly illustrate thedifference, and the higher modulation frequency of 465 MHz has lower leakage (that is, higher fidelity) and the lower modulation frequency of 185 MHz has higher leakage (that is, lower fidelity). As such, an improved selection of the modulation frequency allows for lower LZS leakage.Fig. 5A and Fig. 5B show another comparison of experimentaldata of simulations using, as shown in Fig. 5A, a modulationfrequency selected according to the present disclosure so that the qubit population oscillations are synchronized with thequbit leakage oscillations, and using, as shown in Fig. 5B, amodulation frequency so that the qubit population oscillations are not synchronized with the qubit leakage oscillations. In this case, the population on the initial qubit state is doubled. This is to see how the population in the qubit state and leakage states develop over time when 100% of the initialpopulation is in the qubit. As shown in Fig. 5A, at timesapproaching 70 ns (indicated by the dotted line), the coupler state population (for state [0, 0, 1]) as well as the qubit state (for state [0,1,0]) overlap at a value that ispractically zero. Fig. 5A indicates a phase relationshipbetween the respective states in which a maximum of the qubit population is synchronized with a minimum of the leakage oscillations (into the coupler state). Further, as shown in Fig. 5B, at times approaching 70 ns, in case of not synchronizing the two oscillations, the curves for the coupler state population at state [0, 0, 1] and the qubit statepopulation for state [0,1,0] do not overlap. Fig. 5B, bycomparison, indicates that a maximum of the qubit population is not synchronized with a minimum of the leakage oscillations(into the coupler state). The results in Fig. 5B at timesapproaching 70 ns and beyond indicate that a leakage into the coupler state is observed and the skilled person understands that such a leakage can be avoided by synchronizing the qubit population oscillations with the qubit leakage oscillations according to the present disclosure. Fig. 6 shows a system 100 for implementing on-resonance parametric gates with reduced leakage according to an embodiment. As illustrated, the system 100 comprises at least a first qubit (e.g. tunable-frequency qubit) 110 and a generator circuit 130. The system may further include at least a second qubit (e.g. tunable-frequency qubit) 150 and a tunable coupler 170 (e.g. between the first and second qubit). Such a qubit-coupler-qubit structure may be implemented by two grounded transmon qubits interacting through a floating coupler, including a capacitive coupling between the qubits and the coupler. Alternatively, the tunable coupler may be provided between the first qubit and a resonator. In such a quantum computing architecture, the resonator may be capacitively coupled to the qubit(s) and may be associated with individual filters (such as Purcell filters) for high- fidelity readout. The generator circuit 130 may include a generator for generating a flux modulation signal at a selected modulation frequency, as well as a control line (flux line), as explained above, for supplying the generated flux modulation signal to the qubit(s). The same flux modulation signal may be supplied to two qubits (or a plurality of qubits), or a first flux modulation signals may be supplied to a first qubit and a second flux modulation signals may be supplied to a second qubit, whereby the first and second flux modulation signals are different from each other, for example in terms of one or more of amplitude, selected modulation frequency, phase relationship to each other, etc. A flux modulation signal may also be supplied to the coupler 170. In particular, the generator circuit 130 may be configured to generate a flux modulation signal having a selected modulation frequency for a first qubit (e.g. tunable-frequency qubit) for tuning an on- resonance parametric gate during a gate operation having a finite gate time, and to apply the flux modulation signal to the control line coupled to the first qubit. As described above, the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation. The system 100 has a hardware configuration (including suitable electrical components and control components) for implementing the aspects and embodiments of the method described above. List of References [1] Beaudoin, F. et al. “First-order sidebands in circuit QED using qubit frequency modulation,” arxiv 1208.1946, 2012. [2] Strand, J. D. et al. "First-order sideband transitions with flux-driven asymmetric transmon qubits,” arxiv 1301.0535, 2013. [3] Caldwell, S. et al. “Parametrically activated entangling gates using transmon qubits,” arxiv 1706.06562, 2017. [4] WO 2018 / 236922 A1. [5] Sete, E. A. et al. "Parametric-resonance entangling gates with a tunable coupler,” arxiv 2104.03511, 2021. [6] WO 2022 / 140674 A1. [7] Shevchenko, S. N. et al., "Landau-Zener-Stückelberg interferometry,“ arxiv 0911.1917, 2010. [8] Rol, M. A., et al. “Fast, high-fidelity conditional-phase gate exploiting interference in weakly anharmonic superconducting qubits,” Phys. Rev. Lett. 123, 120502, 2019. [9] Garcia-Ripoll, J. J. et al. “Quantum control of frequency tunable transmon superconducting qubits,” arxiv 2002.10320, 2020.

Claims

CLAIMS 1. A parametric gate control method, comprising: generating (S110) a flux modulation signal having a selected modulation frequency for a first qubit for tuning an on-resonance parametric gate during a gate operation having a finite gate time; and applying (S130) the flux modulation signal to a control line coupled to the first qubit, wherein the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation.

2. The parametric gate control method of claim 1, wherein the selected modulation frequency causes the qubit population oscillation and the qubit leakage oscillation to have a predetermined phase relationship with each other.

3. The parametric gate control method of one of claims 1 - 2, wherein the qubit population oscillation has a first oscillation frequency being different from a second oscillation frequency of the qubit leakage oscillation.

4. The parametric gate control method of one of claims 1 – 3, wherein the modulation frequency is selected in conjunction with a selection of the finite gate time.

5. The parametric gate control method of one of claims 1 - 4, wherein the selected modulation frequency associates a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation.

6. The parametric gate control method of claim 5, wherein the maximum of the qubit population oscillation is set at the first qubit or at a second qubit or at a resonator circuit.

7. The parametric gate control method of one of claims 1 - 4, wherein the selected modulation frequency associates a maximum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time.

8. The parametric gate control method of claim 7, wherein the maximum of the qubit population oscillation is set at the first qubit and the on-resonance parametric gate is a CZ gate.

9. The parametric gate control method of one of claims 1 - 4, wherein the selected modulation frequency associates a minimum of the qubit population oscillation with a minimum of the qubit leakage oscillation at the gate time.

10. The parametric gate control method of claim 9, wherein the minimum of the qubit population oscillation is set at the first qubit and the on-resonance parametric gate is a SWAP gate.

11. The parametric gate control method of any of claims 1 - 10, wherein the qubit leakage oscillation is due to Landau-Zener-Stückelberg interaction.

12. The parametric gate control method of any of claims 1 - 11, wherein the flux modulation signal is further applied to a control line coupled to a second qubit.

13. A system, comprising: a first qubit (110); anda generator circuit (130), wherein the generator circuit (130) is configured to: generate a flux modulation signal having a selected modulation frequency for a first qubit for tuning an on- resonance parametric gate during a gate operation having a finite gate time; and apply the flux modulation signal to a control line coupled to the first qubit, wherein the modulation frequency is selected based on the finite gate time, a qubit population oscillation and a qubit leakage oscillation.

14. The system of claim 13, further comprising a second qubit (150) and a tunable coupler (170).

15. The system of claim 13, further comprising a resonator and a tunable coupler (170).

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