Dynamically decoupled driven controlled Z-gate

Abstract

The present disclosure provides a system and method for performing a dynamically decoupled controlled-Z gate operation. The superconducting circuit of the system includes a first qubit and a second qubit laterally coupled to the first qubit. The system can apply an external magnetic flux to the second qubit to cause a frequency of the second qubit to resonate with a frequency of the first qubit. The system can also apply a continuous alternating drive having a continuous phase to the second qubit, a duration and an amplitude of the continuous alternating drive configured to synchronize a gate duration of the dynamically decoupled controlled-Z gate operation with an integer number of Rabi oscillation periods. After providing the continuous alternating drive, the system can read out a state of the quantum computing system.

Description

TECHNICAL FIELD

[0001] The present disclosure relates to the field of quantum computing, and more specifically, to the operation of a dynamically decoupled driven controlled-Z gate. BACKGROUND

[0002] Quantum computing can be performed on qubits or quantum bits through quantum gates, which are analogous to logical gates for digital computing. One quantum gate can be a Z gate, which can perform a phase shift on a qubit, rotating the phase of the qubit by π radians when the qubit is in an excited state. The Z gate is not affected when the qubit is not in an excited state. A controlled-Z gate is an extension of the Z gate to two qubits. The controlled-Z gate can perform a Z gate operation on a second qubit of the two qubits when a first qubit of the two qubits is in an excited state. The state of the second qubit is not affected when the first qubit of the two qubits is not in an excited state.

[0003] Quantum computing can be limited by decoherence that arises from the interaction of qubits with their environment. Dynamically decoupled driving can be used to suppress this decoherence to extend the time over which quantum operations are performed. SUMMARY

[0004] The disclosed systems and methods relate to a controlled-Z gate configured to use dynamically decoupled (DD) driving that is synchronized to an integer number of Rabi oscillations. This DD driving can support reduced gate durations and increased gate fidelities, constituting a technical improvement in quantum computing.

[0005] A disclosed embodiment includes a quantum computing system for performing a dynamically decoupled controlled-Z gate operation. The system can include a superconducting circuit and at least one computing device. The superconducting circuit can include a first qubit and a second qubit laterally coupled to the first qubit. The at least one computing device can be configured to provide first, second, and third instructions. The first instruction can be provided to a first drive source. The first instruction can cause the first drive source to apply an external magnetic flux to the second qubit to cause a frequency of the second qubit to resonate with a frequency of the first qubit. The second instruction can be applied to a second drive source. The second instruction can cause the second drive source to apply a continuous alternating drive having a continuous phase to the second qubit, the continuous alternating drive having a duration and an amplitude configured to synchronize a gate duration of the dynamically decoupled controlled-Z gate operation to an integer number of Rabi oscillation periods. The third instruction can be provided after the second instruction is provided to the second drive source. The third instruction can be to read out a state of the quantum computing system.

[0006] The disclosed embodiments include a method for performing a dynamically decoupled controlled-Z gate operation. The method can include providing, by a first drive source, an external magnetic flux to a second qubit of a superconducting circuit of a quantum computing system to tune a frequency of the second qubit to a frequency of a first qubit, the first qubit being transversely coupled to the second qubit. The method can further include providing, by a second drive source, a continuous alternating drive to the second qubit, an amplitude of the continuous alternating drive corresponding to a peak in a relationship between an amplitude and a fidelity of the dynamically decoupled controlled-Z gate operation. The method can further include reading out a state of the quantum computing system after providing the second drive source.

[0007] The disclosed embodiments can include a non-transitory computer readable medium comprising instructions that, when processed by a quantum computing system, cause the quantum computing system to perform a first operation for implementing a dynamically decoupled controlled-Z gate operation. The first operation can include providing, by a first drive source, an external magnetic flux to a second qubit of a superconducting circuit to tune a frequency of the second qubit to a frequency of a first qubit, the first qubit being transversely coupled to the second qubit. The first operation can further include providing, by a second drive source, a continuous alternating drive to the second qubit. A duration of the continuous alternating drive can be inversely proportional to an amplitude of the continuous alternating drive and can be within 10% of a quotient of π divided by the amplitude of the continuous alternating drive. The first operation can include reading out a state of the quantum computing system after providing the second drive source.

[0008] It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosed embodiments. BRIEF DESCRIPTION OF DRAWINGS

[0009] The accompanying drawings, which constitute a part of this specification, illustrate several embodiments and together with the description help explain the principles and features of the disclosed embodiments. In the drawings:

[0010] Figure 1A shows a schematic diagram of an exemplary circuit for implementing a controlled-Z gate according to embodiments of the present disclosure;

[0011] Figure 1B shows a table of values for an example of the circuit shown in Figure 1A

[0012] Figure 1C shows a Hamiltonian coefficient matrix for the circuit shown in Figure 1A

[0013] Figure 1D shows a matrix representing a dynamically decoupled drive applied to the circuit shown in Figure 1A ​​​

[0014] Figure 1E A unitary matrix showing the dynamics decoupled driving CZ gate (DDCZ gate) locally equivalent to a controlled-Z gate is shown, according to embodiments of the present disclosure;

[0015] Figure 2A Gate fidelity as a function of DD driving amplitude for regular DD driving, excluding leakage effects, is shown;

[0016] Figure 2B Gate fidelity as a function of DD driving amplitude, excluding leakage effects, is shown, according to embodiments of the present disclosure;

[0017] Figure 3A Gate fidelity as a function of DD driving amplitude for regular DD driving, including leakage effects, is shown;

[0018] Figure 3B Gate fidelity as a function of DD driving amplitude, including leakage effects, is shown, according to embodiments of the present disclosure;

[0019] Figure 4A A state evolution process for a controlled-Z gate is shown, according to embodiments of the present disclosure;

[0020] Figure 4B A state evolution process for a controlled-Z gate for regular DD driving when applying a first DD amplitude is shown;

[0021] Figure 4C A state evolution process for a controlled-Z gate for regular DD driving when applying a second DD amplitude is shown;

[0022] Figure 5 A flowchart of an exemplary method for operating a controlled-Z gate is shown, according to embodiments of the present disclosure. DETAILED DESCRIPTION

[0023] Reference will now be made in detail to the exemplary embodiments discussed in relation to the accompanying drawings. In some instances, like reference numbers can be used throughout the drawings and the following description to refer to like or similar parts. Unless defined otherwise, technical or scientific terms used in the disclosed embodiments have the same meaning as commonly understood by one of ordinary skill in the art. The disclosed embodiments are described in sufficient detail to enable those skilled in the art to practice the disclosed embodiments. It is to be understood that other embodiments can be utilized and that changes can be made without departing from the scope of the disclosed embodiments. Accordingly, the materials, methods, and examples are illustrative only and not intended to be limiting.

[0024] Quantum computers offer the ability to perform certain tasks (equivalently, solve certain problems) that are believed to be intractable for classical computers, including any possible future classical computers. To understand the advantages of quantum computers, it is useful to understand the contrast between quantum computers and classical computers. Classical computers operate according to digital logic. Digital logic refers to a system of logic that operates on units of information called bits. A bit can have two values, typically denoted 0 and 1, and is the smallest unit of information in digital logic. Operations are performed on bits using logic gates, which take one or more bits as input and give one or more bits as output. Typically, a logic gate has only one bit as output (although this single bit can be sent as input to multiple other logic gates), and the value of this bit typically depends on the values of at least some of the input bits. In modern computers, logic gates are typically composed of transistors, and bits are typically represented by voltage levels on wires connected to the transistors. A simple example of a logic gate is an AND gate, which (in its simplest form) takes two bits as input and gives one bit as output. The output of the AND gate is 1 if the values of both inputs are 1, and 0 otherwise. By connecting the inputs and outputs of various logic gates together in a particular way, a classical computer can implement arbitrarily complex algorithms to accomplish a wide variety of tasks.

[0025] At first glance, quantum computers operate in a similar fashion to classical computers. Quantum computers operate according to a system of logic that operates on units of information called qubits (a combination of "quantum" and "bit"). A qubit is the smallest unit of information in a quantum computer, and a qubit can have an arbitrary linear combination of two values, typically denoted |0> and |1>. In other words, for any combination of a and β, the value of a qubit, denoted |ψ>, can equal a|0> + β|1>, where a and β are complex numbers, and |a|2 + |β|2 = 1. Operations are performed on qubits using quantum logic gates, which take one or more qubits as input and give one or more qubits as output. Given the low-level nature of current quantum systems, quantum algorithms are typically represented in terms of their underlying quantum circuits. In turn, quantum circuits are composed of quantum gates, which are the basic components that directly manipulate qubits. 2 +|β| 2 = 1. Operations are performed on qubits using quantum logic gates, which take one or more qubits as input and give one or more qubits as output. Given the low-level nature of current quantum systems, quantum algorithms are typically represented in terms of their underlying quantum circuits. In turn, quantum circuits are composed of quantum gates, which are the basic components that directly manipulate qubits.

[0026] A controlled-Z gate is a quantum gate that acts on two qubits. When the first qubit is in the |0> state, the controlled-Z gate has no effect on the state of the second qubit. When the first qubit is in the |1> state and the second qubit is in the |0> state, the controlled-Z gate has no effect on the state of the second qubit. When both the first and second qubits are in the |1> state, then the controlled-Z gate inverts the phase of the second qubit.

[0027] A controlled Z gate can be constructed from two-qubit gates that are locally equivalent to a controlled Z gate. Such gates can differ from a controlled Z gate only in local operations. For example, a DDCZ gate can be locally equivalent to a controlled Z gate when executed with the following single-qubit operations:

[0028] Given an M gate with unitary where M + is the conjugate transpose of the M gate, and is the Kronecker product of two M gates (representing the parallel application of two M gates to two independent qubits).

[0029] Given an N gate with unitary where is the Kronecker product of two N gates (representing the parallel application of two N gates to two independent qubits).

[0030] A controlled Z gate (CZ) can then be constructed by applying two M + gates in parallel, applying a DDCZ gate, applying two M gates in parallel, and then applying two N gates in parallel (for a global factor e iπ / 4 that lacks physical relevance): DDCZ

[0031] Superconducting quantum circuits can be used to implement qubits and quantum gates. Such qubits can be based on current (e.g., flux qubits) or charge (e.g., charge qubits) or energy (e.g., phase qubits). Different implementations can have different characteristics, such as sensitivity to external noise, coherence times, or anharmonicity. For example, a transmon qubit, a type of charge qubit that includes a capacitively shunted Josephson junction, can exhibit reduced sensitivity to charge noise. As an additional example, a fluxonium qubit, a type of flux qubit that includes a Josephson junction shunted by a capacitor and an inductor (the latter of which can be implemented using an array of additional Josephson junctions), can exhibit long coherence times and large anharmonicity.

[0032] Providing a dynamic decoupling (DD) drive to one or more qubits can improve qubit performance. The DD drive can be or include a microwave drive that resonates with one or more qubits. Such a DD drive can refocus the time evolution of a qubit, resulting in noise located around a single qubit being effectively averaged out. In this way, the DD drive can at least partially isolate the qubit from dephasing noise with a correlation time longer than the period of the Rabi oscillations. Thus, the DD drive can extend the dephasing time of the qubit, allowing for more (or more complex) quantum computation.

[0033] ​Certain DD driving for controlled Z gates can use continuous waveforms to isolate the qubits from dephasing noise during gate operation. However, such DD driving includes a π phase shift in the middle of applying the continuous waveform. This phase shift acts to reduce the phase accumulation due to the qubit frequency since the phase accumulation during the second part of the DD driving cancels out the phase accumulation during the first part of the DD driving. Moreover, this approach requires the magnitude of the difference in Rabi frequencies between the DD driving applied to each qubit to greatly exceed the magnitude of the interaction term in the two-qubit Hamiltonian:

[0034] |Ω A -Ω B |>>|λ| (1)

[0035] where Ω A and Ω B are the Rabi frequencies of each DD driving, and λ is the interaction term in the two-qubit Hamiltonian. However, the time required to complete the gate operation is inversely proportional to the magnitude of the interaction term:

[0036]

[0037] Thus, shorter gate durations (which facilitate performing more quantum operations) require larger interaction terms. However, larger interaction terms require larger magnitude DD driving, which can result in significant leakage errors (e.g., increasing the probability of transitioning to a quantum state other than the one used for computation). Moreover, hardware limitations can limit the achievable DD magnitude. Such hardware limitations can include power limitations on the waveform generator that creates the DD driving, limitations on the heat generated by the DD driving (e.g., resulting from limitations of the cryogenic environment in which the qubits are operated), effects of filters used to reduce noise in the cryogenic environment, or other hardware limitations.

[0038] The disclosed embodiments include improved DD drive systems and methods that are not constrained by Equation 1. In some embodiments, the contemplated DD drive includes continuous phase drive (e.g., drive with constant phase; drive lacking phase shift, such as a phase inversion midway through a gate duration; or the like). In various embodiments, the contemplated DD drive can synchronize an integer number of Rabi periods with a gate duration. Specifically, the disclosed embodiments contemplate fine-tuning DD amplitudes and removing phase flips in the dynamic decoupling drive. As described herein, gate fidelity can oscillate with DD amplitude. Thus, rather than relying on large amplitudes, DD amplitudes can be selected to match peaks in the fidelity-amplitude relationship, thereby avoiding the large amplitude requirement of conventional DD drive methods. Moreover, by removing phase flips, the minimum DD amplitude required for high-fidelity gates can be significantly reduced. By operating at lower DD amplitudes, leakage errors can be significantly reduced. Moreover, the DD amplitudes described herein can be implemented using existing experimental equipment. Thus, the disclosed embodiments are capable of significantly reducing DD amplitudes and supporting high-fidelity quantum gates with short gate durations. Although described with respect to controlled Z gates for convenience, the disclosed embodiments are not limited thereto.

[0039] Figure 1A A schematic diagram of an exemplary circuit 110 suitable for implementing a controlled Z gate in accordance with the disclosed embodiments is depicted. Circuit 110 can include two fluxonium qubits (qubit 111 and qubit 115). Each fluxonium qubit can be implemented using a Josephson junction shunted by a capacitor and an inductor. Each of these inductors can be realized by an array of Josephson junctions. Each qubit can be configured to operate at a local minimum frequency with respect to a bias magnetic flux. In this non-limiting example, the local minimum of qubit 115 can be lower than the local minimum of qubit 111. The qubits can be coupled by coupler 113. Coupler 113 can be implemented using a capacitor. In some cases, circuit 110 can be set up to realize lateral resonant coupling between the qubits (e.g., charge coupling, or the like). However, such coupling can require alignment of the qubit frequencies. When the gate is not in operation, the qubits can remain at different frequencies.

[0040] In some embodiments, circuit 110 can be implemented using a chip containing the qubits and the coupling between the qubits. In some embodiments, the chip can include coupling to bias drive source 120, DD drive source 130, and readout device 140.

[0041] Bias drive source 120 can be coupled to circuit 110. In some embodiments, bias drive source 120 can be configured to provide a bias flux to at least one of qubit 111 or qubit 115. Consistent with the disclosed embodiments, qubit 111 and qubit 115 can be configured to have different frequencies. Bias drive source 120 can be configured to drive the qubits into resonance, thereby enabling gate operations. In some cases, bias drive source 120 can cause a magnetic bias flux to be provided to a qubit (e.g., qubit 115 in this non-limiting example) at a lower frequency. In some embodiments, the bias flux can be provided by causing a current to flow through a coil external to circuit 110. In various embodiments, the bias flux can be provided by causing a current to flow through a coil on the chip. The disclosed embodiments are not limited to a particular method of biasing the qubits.

[0042] DD drive source 130 can be coupled to circuit 110. In some embodiments, DD drive source 130 can be configured to provide continuous microwave drive to the qubits. The microwave drive can be alternating (e.g., sinusoidal, square wave, biphasic pulse train, etc.). In this example, the disclosed embodiments are described with reference to DD drive provided to a single qubit. The DD drive can be applied to the same qubit as the bias flux. The bias flux can increase the qubit's sensitivity to flux noise by tuning the qubit out of a local frequency minimum with respect to the flux where it is less sensitive to flux noise. The DD drive can at least partially address this increased noise sensitivity by mitigating the effects of certain noise (e.g., noise with a correlation time longer than the period of the Rabi oscillation). In some embodiments, the bias bend and DD drive can be applied to a lower frequency qubit (e.g., qubit 115). Additionally or alternatively, applying DD drive to only one qubit can reduce calibration and control requirements for circuit 110, as well as reduce amplitude tuning requirements. Furthermore, leakage errors due to drive applied to qubit 111 can be avoided. However, the disclosed embodiments are not limited to applying DD drive to only one qubit. In some embodiments, two different microwave drives can be provided to two different qubits in circuit 110.

[0043] The readout device 140 can be coupled to the circuit 110. The readout device 140 can enable a computing device to perform a measurement on one or more qubits of the circuit. In some cases, a quantum computing device including the circuit 110 can use the readout device 140 to measure the state of one or more qubits of the circuit 110 upon completion of a sequence of quantum operations. The sequence of quantum operations can include performing a controlled-Z gate (e.g., by performing a DDCZ gate and additional single-qubit gates, etc.). In some embodiments, the readout device 140 can include an arbitrary waveform generator configured to provide a probe signal (e.g., a microwave probe tone) to a coupled resonator. In various embodiments, the readout device 140 can include a detector configured to determine an amplitude and a phase of an output signal received from the coupled resonator in response to the provision of the microwave probe tone.

[0044] In some embodiments, a single device can perform the functions of two or more of the bias drive source 120, the DD drive source 130, and the readout device 140. For example, the DD drive source 130 can provide both the DD drive and the probe signal. As another example, the current that generates the bias flux can be combined with the DD drive.

[0045] A Hamiltonian can be constructed for the circuit 110, assuming that the DD drive is applied only to the qubit 115 and in a frame co-rotating with the qubit and applying the rotating wave approximation:

[0046]

[0047] where λ is a coefficient of the interaction term, σ + ≡ |1><0| (e.g., σ + is a matrix that, when given the quantum state |0>, returns the quantum state |1>) and σ - ≡ |0><1| (e.g., σ - is a matrix that, when given the quantum state |1>, returns the quantum state |0>) and Ω B is the amplitude of the DD drive. The first term of Equation 3 is referred to herein as the “static Hamiltonian,” and the second term is referred to herein as the “DD Hamiltonian.” The phases on the |0> and |1> states are chosen so that λ is positive. The duration of the DD drive can be determined using Equation 2 above.

[0048] Based on the elements depicted in FIG. 1, a circuit Hamiltonian can be constructed for the circuit 110 as follows:

[0049]

[0050] In this Hamiltonian, n A is the charge on the junction capacitance of the qubit 111, expressed as a number of Cooper pairs; n Bis the charge on the junction capacitance of qubit 115, denoted as the number of Cooper pairs; Φ A is the loop flux through the inductor of qubit 111, denoted in units consistent with the units of the phase difference across the Josephson junction of qubit 111; Φ B is the loop flux through the inductor of qubit 115, denoted in units consistent with the units of the phase difference across the Josephson junction of qubit 115; E CA is the energy scaling factor of the junction capacitance of qubit 111; E CB is the energy scaling factor of the junction capacitance of qubit 115; E JA is the energy scaling factor of the Josephson junction in qubit 111; E JB is the energy scaling factor of the Josephson junction in qubit 115. J C is the energy scaling factor of the coupling capacitance. Φ ext,A may be the bias flux of qubit 111, and can be set to π, putting the qubit at the local minimum frequency with respect to the magnetic flux. Φ ext,B may be the bias flux of qubit 115, and can be set to a value that matches the frequency of qubit 115 to the frequency of qubit 111.

[0051] Figure 1B depicts a table of values for an exemplary instance of the controlled Z gate depicted in Figure 1A Utilizing these values, Φ ext,B may be set to 1.97 to match the frequencies between the qubits in circuit 110.

[0052] By sandwiching H C between the first two states of the circuit, the circuit Hamiltonian can be projected onto the span of the two lowest states of the circuit (the states used for computation). The result is Figure 1C the Hamiltonian coefficient matrix depicted in B The effect of the DD drive can be incorporated into equation 4 by adding a term proportional to n Figure 1D Once the resulting circuit Hamiltonian is projected onto the span of the two lowest states of the circuit, the DD drive term takes on the matrix form depicted in By expressing the resulting Hamiltonian in the two-qubit rotating reference frame, and applying the rotating wave approximation, the Hamiltonian depicted in equation 1 can be obtained, where the value of the interaction term λ = 0.0634 GHz-h.

[0053] For different values of the DD amplitude, the time evolution operator can be determined for the gate length given in equation 2. The time evolution operator can be used to compute the unitary matrix corresponding to the DDCZ gate Figure 1Eaverage fidelity of the DDCZ gate. In some embodiments, the fidelity of the DDCZ gate can bound the fidelity of a CZ gate constructed using the DDCZ gate. In a non-limiting example, the average fidelity is computed as a Haar average over the states. Figure 2A The gate fidelity and state evolution depicted in FIG. 4C are generated using the gate durations given in Equation 2.

[0054] Figure 2A The DDCZ gate fidelity as a function of the DD drive amplitude for a regular DD drive is depicted, excluding leakage effects. The DD drive provided in this example is continuous and sinusoidal. As described above, the phase of the DD drive is inverted at the midpoint of the gate duration. From the plot, it can be observed that the fidelity trajectory resembles an underdamped step response, with the first peak at about 0.25 GHz h (or about 4 times the amplitude of the interaction term in the Hamiltonian). The quasi-periodic oscillations decrease as the drive amplitude increases.

[0055] Figure 2B The DDCZ gate fidelity as a function of the DD drive amplitude, according to disclosed embodiments, is depicted, excluding leakage effects. In this example, a sinusoidal dynamic drive is provided, without phase inversion. As shown, the fidelity in this example is a quasi-periodic function of the amplitude (as the period of oscillation between peaks is not constant). Figure 2B The approximate period of oscillation in FIG. 4B is about Figure 2A Half the approximate period of oscillation in FIG. 4B. Thus, the first maximum in the fidelity trajectory is reached at a DD amplitude of about 0.125 GHz h (or about 2 times the amplitude of the interaction term in the Hamiltonian). A DD drive consistent with disclosed embodiments can have an amplitude corresponding to a peak in the fidelity trajectory. However, unlike Figure 2A Unlike the relationship depicted in FIG. 4B, the fidelity varies greatly with the DD amplitude. Thus, a DD drive according to disclosed embodiments can require additional calibration to ensure the fidelity of the DDCZ gate.

[0056] Leakage of qubits in circuit 110 to states outside the intended computational subspace (e.g., the space of |00>, |01>, |10>, and |11>) can reduce the fidelity of a DDCZ gate implemented using circuit 110. In the presence of such leakage, the upper bound on the fidelity of the DDCZ gate can be established via the Cauchy-Schwartz formula as:

[0057]

[0058] In this bound, Π S is the projector onto the computational subspace:

[0059] ΠS = |00><00| + |01><01| + |10><10| + |11><11| (6)

[0060] and V is the implemented unitary (possibly with leakage error), and denotes the squared Frobenius norm. In other words, L is the average over the ensemble outside the computational subspace of the state resulting from the implemented unitary acting on the state in the computational subspace.

[0061] Figure 3A DDCZ gate fidelities as a function of DD drive amplitude for conventional DD driving are depicted, including leakage effects. As Figure 2A shown in FIG. 2A, such conventional driving is sinusoidal and includes a phase inversion midway through the gate duration. However, Figure 3A differs from Figure 2A in that higher energy levels of each qubit in the circuit 110 are modeled in the simulation. As Figure 3A depicted in FIG. 2B, the upper bound on gate fidelity starts at approximately 1 and decreases as the DD amplitude increases (as transitions to states outside the computational subspace become more likely). Also depicted is the simulated gate fidelity under this model, increasing from 0.6 to over 0.95 as the DD amplitude increases toward 0.20 GHz h (or approximately 3 times the amplitude of the interaction term in the Hamiltonian). In contrast to Figure 2A , the location of the peak in the fidelity trajectory shifts toward higher DD amplitudes. This figure further demonstrates that the conventional DD driving results require drive amplitudes that can lead to reduced gate fidelity.

[0062] Figure 3B DDCZ gate fidelities as a function of DD drive amplitude for conventional DD driving are depicted, including leakage effects. As Figure 3A shown in FIG. 2A, higher energy levels of each qubit in the circuit 110 are modeled in the simulation. However, the driving does not include a phase inversion midway through the gate duration. As Figure 3B depicted in FIG. 2B, the upper bound on gate fidelity starts at approximately 1 and decreases as the DD amplitude increases (as transitions to states outside the computational subspace become more likely). However, this decrease is not as fast as observed in Figure 3A . The simulated gate fidelity increases to a peak of approximately 0.125 GHz h (or approximately 2 times the amplitude of the interaction term in the Hamiltonian). As Figure 3A shown in FIG. 2C, Figure 3B the fidelity trajectory in

[0063] In general, the upper bound on leakage decreases as the DD amplitude decreases, worsening the leakage error. This decrease does not appear to be strictly monotonic, as the overall population that leaks out of the computational subspace can return after some time. Furthermore, the peak in the simulated fidelity trajectory as a function of DD amplitude is very close to the upper bound. This result suggests that the gate fidelity can be limited by the states that leak out of the computational subspace. In turn, this suggests that implementations using lower DD amplitudes, such as the disclosed embodiments, can support higher fidelities than conventional approaches, thereby improving the performance of quantum computers. As shown, by tuning the DD amplitude to the peak in the fidelity-amplitude relationship, one can achieve improved fidelity. By removing the phase flips, one can even achieve lower DD amplitudes and higher fidelities: Figure 3B The first peak in Figure 3A has a higher fidelity at a lower DD amplitude than the first peak in

[0064] Consistent with the disclosed embodiments, the DD drive can be tuned so that the Rabi oscillations caused by the DD drive are synchronized with the gate duration. Figure 4A FIGS. 4A and 4B depict the state populations of the circuit 110 using a DD drive that includes phase flips Figure 4B and a DD drive that does not include phase flips Figure 4A In each figure, the independent variable is time, and the dependent variable is the population of each of the four states in the computational subspace, which in these two examples is initially entirely in the |00> state. The desired output at the end of the operation is to have the population evenly distributed between the |00> state and the |11> state.

[0065] Figure 4A FIG. 4C depicts the state evolution of a DDCZ gate according to the disclosed embodiments. Based on the value of the first peak in Figure 2B the amplitude of the DD drive is chosen to be 0.125 GHz h. The evolution of the populations can be explained via a Suzuki-Trotter expansion:

[0066]

[0067] This equation states that, as n approaches infinity, the limit of the matrix power of the Hamiltonian of the circuit 110 (equation 3) is equal to the product of two matrix powers: the matrix power corresponding to the static Hamiltonian and the matrix power corresponding to the DD drive Hamiltonian. As an intuitive explanation, consider applying the DD drive Hamiltonian and the static Hamiltonian alternately and infinitesimally.

[0068] The evolution of the populations between the states can then be considered in terms of the dynamics of the three components:

[0069] 1. Rabi oscillations between the |00> and |01> states caused by the DD drive Hamiltonian.

[0070] 2. Oscillation between |01> and |10> states caused by the static Hamiltonian.

[0071] 3. A Rabi oscillation between |10> and |11> states also caused by the DD- driven Hamiltonian, (because the DD drive can cause the state of qubit 115 to oscillate between |0> and |1>, independent of the state of qubit 111).

[0072] It can be seen that the Rabi oscillation in the second qubit between |00> and |01> states causes a decrease in population of the |00> state, and an increase in population of the |01> state. The transfer term in the static Hamiltonian begins to transfer population of the |01> state to the |10> state. Finally, another Rabi oscillation in the second qubit causes a decrease in population of the |10> state, and an increase in population of the |11> state.

[0073] The correct unitary (as given in Equation 2) can be obtained by synchronizing these first and second dynamic Rabi oscillation periods with the gate duration. Thus, by setting: Figure 1E

[0074] π / Ω B ≈π / 2λ (8)

[0075] or

[0076] Ω B ≈2λ (9)

[0077] A high-fidelity gate can be obtained. The above Equations 8 and 9 can be approximate, because the DD-driven Hamiltonian and the static Hamiltonian do not commute, so these matrix powers are not independent. In the example explored herein, 2λ = 2*0.0634 GHz-h = 0.127 GHz-h, while Figure 2A Ω B at the peak in Equation 6 is 0.122 GHz-h. However, in some embodiments, a drive value within some percentage of 2λ will suitably appropriate fidelity DDCZ gates. In some cases, this percentage can be between 5% and 10%.

[0078] Furthermore, Figure 2B the other peaks shown in Equation 6 characterize integer numbers of Rabi oscillations approximately synchronized with the gate duration. Thus, by fine-tuning the DD amplitude and without using phase flips, a high-fidelity DDCZ gate operation can be achieved.

[0079] Figure 4B The state evolution process for a DDCZ gate with conventional DD driving is shown. For comparison purposes, the amplitude of the DD driving was set to approximately 0.125 GHz-h, similar to the drive amplitude used in generating Figure 4A the DDCZ gate shown in Figure 6. As can be seen, the DDCZ gate with conventional DD driving is not as accurate as the DDCZ gate with the DD driving described herein.​Figure 2A As shown, this amplitude will not produce a high-fidelity gate. Figure 4B The overall evolution shown can also be explained using the Suzuki-Trotter expansion. In this simulation, the first half of the evolution is identical to that with the improved DD drive. However, because the conventional DD drive uses phase flipping, the second half of the evolution is different. Figure 4B As shown, applying a DD drive with opposite phase can have the effect of reversing the Rabi oscillation. Specifically, when passing through the middle of the gate, the Rabi oscillation between |10> and |11> is reversed, thus significantly interfering with the synchronization between the static Hamiltonian evolution and the Rabi oscillation.

[0080] Figure 4C The state evolution of the DDCZ gate for a conventional DD drive is shown, where the DD magnitude is selected as matched. Figure 2A The peak value in the fidelity amplitude trajectory depicted. In this example, the DD drive amplitude is Ω. B =0.253 GHz·h, approximately 4λ = 4 * 0.0634 GHz·h = 0.254 GHz·h. For example... Figure 4C As shown, the Rabi oscillations from |00> to |01> and from |10> to |11> are all at critical points (e.g., local minimum or maximum). Therefore, the phase flip of the inverted Rabi oscillation does not affect the dynamics of these oscillations. In this way, the gate duration can be synchronized with the two Rabi oscillations. Thus, when using a conventional drive, the first peak in the fidelity occurs approximately at about twice the DD amplitude of the first peak in the fidelity when using a continuous phase drive (e.g., a constant phase drive or a drive without phase flips, etc.).

[0081] Figure 5 A flowchart is shown of an exemplary method 500 for operating a DDCZ gate according to a disclosed embodiment. Method 500 may include steps of providing a bias drive, providing a tuned or continuous phase dynamics decoupling drive, and reading out the result of the controlled Z-gate operation. A quantum computing system may be used to perform method 500. The quantum computing system includes superconducting circuitry suitable for implementing the controlled Z-gate, such as… Figure 1AThe circuit 110 depicted in the diagram includes at least two qubits. Consistent with the disclosed embodiments, the qubits may be flux qubits, charge qubits, phase qubits, etc. In various embodiments, the qubits may be transmon qubits or fluxonium qubits. The qubits may be coupled (e.g., laterally coupled). The quantum computing system may include a bias drive source (e.g., bias drive source 120, etc.), a DD drive source (e.g., DD drive source 130, etc.), and a readout device (e.g., readout device 140). Conventional computing systems may include a protective environment for isolating DDZ gates, such as a dilution refrigerator or other suitable cryogenic environment. Quantum computing systems may include conventional computing devices (e.g., digital computing devices including a processor and one or more memories or caches, etc.). Consistent with the disclosed embodiments, the conventional computing device may coordinate the operation of the bias drive source, the kinetic decoupling drive source, and the readout device.

[0082] Consistent with the disclosed embodiments, the qubits of the circuit may be in a specific state prior to performing method 500. This state may be pre-defined (e.g., it may be an initial state, such as the ground state). The state may be the result of one or more previously calculated operations. In some embodiments, for example, gate operations may be applied to one or more qubits of circuit 110 to configure the states of these gates (e.g., qubits M, M...). + N, X, Y, Z, phase, or CNOT gate operations can be applied to one or more qubits to configure the states of those qubits. According to method 500, a quantum computing system can perform a DDCZ gate operation on the initial state of a qubit to obtain a new state.

[0083] In step 501, a conventional computing device may provide instructions to a bias drive source. Instructions may include commands (e.g., data values ​​representing a "start" command), trigger signals (e.g., timing pulses), codes, etc. These instructions may cause the bias drive source to apply magnetic flux to a qubit in the gate (e.g., a lower-frequency qubit). Applying magnetic flux may include supplying current to a coil, inductor, electromagnet, etc., to apply magnetic flux to one of the qubits. The amplitude of the applied magnetic field may also be selected so that the frequency of one qubit resonates with the frequency of another qubit.

[0084] In some embodiments, a baseline magnetic flux may be applied to one or both qubits, and applying the magnetic flux may include increasing (or decreasing) the baseline magnetic flux (e.g., by increasing or decreasing the current supplied to a coil, inductor, electromagnet, etc.).

[0085] In step 503, a conventional computing device may provide instructions to the DD drive source. Instructions may include commands (e.g., data values ​​representing a "start" command), trigger signals (e.g., timing pulses), codes, etc. The instructions may cause the drive source to apply DD drive to at least one qubit (e.g., the qubit with the same magnetic flux, another qubit affected by a DDCZ gate operation, multiple qubits affected by a DDCZ gate operation, etc.). As described in this disclosure, DD drive may be alternating. As described in this disclosure, DD drive may be a continuous phase drive. As described in this disclosure, the magnitude of the DD drive may depend on the magnitude of the interaction term between qubits in the Hamiltonian of the controlled Z-gate, which is specified by a frame that rotates with the qubits.

[0086] In some embodiments, the duration and amplitude of the DD drive can be configured to synchronize the duration of the DD drive with an integer number of Rabi oscillations. For example, the amplitude of the DD drive can be selected such that n∈Z occurs during the provision of the DD drive. + Rabi oscillations (e.g., between the |10> and |11> states and between the |00> and |01> states). In some cases, the amplitude of the DD drive can be selected such that one, two, or three Rabi oscillations occur during the provision of the DD drive.

[0087] In various embodiments, the magnitude of the DD drive can correspond to a peak in the relationship between the DD magnitude and fidelity of a dynamically decoupled controlled Z-gate. This peak can be the first peak in the relationship or a subsequent peak. In the non-limiting example provided herein, the interaction term is 0.0634 GHz·h, the peak in the fidelity of a conventional DD drive is 0.253 GHz·h, and the peak in the fidelity of a continuous-phase DD drive is 0.128 GHz·h. A suitable DD magnitude can be determined by selecting an initial DD magnitude based on the formulas provided herein and then fine-tuning the DD drive magnitude to obtain the desired gate fidelity. In some embodiments, this fine-tuning can occur as part of the initial configuration of the quantum computing device before the sequence of quantum computing operations is performed. In some cases, the magnitude of the DD drive can be one to three times the magnitude of the interaction term.

[0088] In some embodiments, in addition to tuning the DD amplitude or other alternative schemes, the duration of the gate can be fine-tuned. In some cases, as described herein, the duration of the DD drive can be inversely proportional to the amplitude of the interaction term in the circuit Hamiltonian. In some embodiments, according to Equation 2, the DD drive can be applied for a duration approximately equal to (e.g., within 10%, etc.) the quotient of π divided by the amplitude of the interaction term in the circuit Hamiltonian. For example, when the quotient of π divided by the amplitude of the DD drive is 25 ns, the duration can be between 22.5 ns and 27.5 ns.

[0089] In some embodiments, the magnetic flux and the DD drive can coincide. In various embodiments, the magnetic flux can begin before the DD drive or end after the DD drive. In some embodiments, the cessation of the bias flux can cause the two qubits to de-resonate.

[0090] In step 505, the quantum computing device can obtain the result at least in part based on the operation of the DDCZ gate. In some embodiments, the controlled Z-gate may be the final gate computed before the result is read out. In various embodiments, one or more other gates or operations may be executed using the qubits in the controlled Z-gate before the state of the quantum computing system is read out. In some cases, the state of the quantum computing system may depend at least in part on the execution of the controlled Z-gate. For example, a conventional computing device can enable a quantum computer to execute a quantum computing algorithm. Executing the controlled Z-gate may be a step in the quantum computing algorithm. When the algorithm has been executed completely, the result of the quantum computing algorithm can be read out from the quantum computing system.

[0091] The results can be read using the readout device 140. In some embodiments, a conventional computing device can provide instructions to the readout device to read the results of the quantum computing. Results can also be read from one or more qubits of a controlled Z-gate or from one or more other qubits of the quantum computing device.

[0092] In some embodiments, the readout device 140 can be used for the distributed readout of qubit states, or another suitable readout method. The readout device 140 can be coupled to one or more readout resonators. Each of the one or more readout resonators can be coupled to a qubit of circuit 110, and the state of the qubit can be inferred from the state-related frequency shift of the coupled readout resonators. The readout device 140 can provide a probe signal to the coupled readout resonators and measure the output signal received from the coupled readout resonators. The state of the qubit can be determined based on the amplitude and phase of the output signal. According to embodiments of this disclosure, the output signal can be a reflected or transmitted signal.

[0093] In some embodiments, a non-volatile computer-readable storage medium including instructions is also provided, and these instructions can be executed by a device (e.g., the disclosed encoder and decoder) to perform the methods described above. Common forms of non-volatile media include, for example, floppy disks, hard disks, solid-state drives, magnetic tape or any other magnetic data storage media, optical disc read-only memory (CD-ROM), any other optical data storage media, any physical media with a perforated pattern, random access memory (RAM), programmable read-only memory (PROM) and erasable programmable read-only memory (EPROM), flash memory (FLASH-EPROM) or any other flash memory, volatile random access memory (NVRAM), cache, registers, any other memory chip or cassette memory and its network version. The device may include one or more processors (CPUs), input / output interfaces, network interfaces and / or memory.

[0094] The foregoing description is for illustrative purposes. This description is not exhaustive and is not limited to the precise forms or embodiments disclosed. Modifications and adaptations to the embodiments will be apparent from the detailed description and practice of the disclosed embodiments. For example, the described implementations include hardware, but systems and methods conforming to this disclosure can be implemented in both hardware and software. Furthermore, while some components have been described as coupled to each other, these components may be integrated with each other or distributed in any suitable manner.

[0095] Furthermore, although illustrative embodiments have been described herein, the scope includes any and all embodiments based on this disclosure that have equivalent elements, modifications, omissions, combinations, adjustments, or variations (e.g., aspects spanning various embodiments). Elements in the claims will be interpreted broadly based on the language used in the claims and are not limited to the examples described in this specification or in the course of the application, which will be interpreted as non-exclusive. Moreover, the steps of the disclosed method can be modified in any way, including reordering steps or inserting or deleting steps.

[0096] It should be noted that the relational terms used in this document (e.g., “first” and “second”) to distinguish one entity or operation from another do not require or imply any actual relationship or order between these entities or operations. Furthermore, the words “including,” “having,” “containing,” and “comprising,” as well as other similar forms, are intended to be semantically equivalent and open-ended, as one or more items following any of these terms do not imply an exhaustive list of those items, nor do they imply limitation to the listed items.

[0097] The features and advantages of this disclosure are apparent from the detailed description, and therefore the appended claims are intended to cover all systems and methods falling within the true spirit and scope of this disclosure. As used herein, the indefinite articles “a” and “an” mean “one or more.” Similarly, the use of plural terms does not necessarily indicate multiple unless it is explicit in the given context. Furthermore, since many modifications and variations will readily arise from a study of this disclosure, it is not intended to limit this disclosure to the exact constructions and operations shown and described; therefore, all suitable modifications and equivalents may be considered to fall within the scope of this disclosure.

[0098] As used herein, unless otherwise expressly stated, the term "or" includes all possible combinations unless impractical. For example, if it is specified that a database may include A or B, then unless otherwise specified or impractical, the database may include A, B, or A and B. As a second example, if it is specified that a database may include A, B, or C, then unless otherwise specified or impractical, the database may include A, or B, or C, or A and B, or A and C, or B and C, or A and B and C.

[0099] It should be understood that the above embodiments can be implemented by hardware, software (program code), or a combination of hardware and software. If implemented by software, it can be stored in the above-described computer-readable medium. When executed by a processor, the software can perform the disclosed methods. The computing units and other functional units described in this disclosure can be implemented by hardware, software, or a combination of hardware and software. Those skilled in the art will also understand that multiple of the above modules / units can be combined into one module / unit, and each of the above modules / units can be further divided into multiple sub-modules / sub-units.

[0100] In the foregoing description, numerous specific details have been described with reference to embodiments, which may vary depending on the implementation. Certain adjustments and modifications may be made to the described embodiments. Other embodiments will be apparent to those skilled in the art in light of the detailed description and practice of the invention disclosed herein. The description and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the appended claims. The sequence of steps shown in the figures is also intended for illustrative purposes only and is not intended to limit one to any particular order of steps. Therefore, those skilled in the art will understand that these steps may be performed in a different order when implementing the same method.

[0101] The embodiments may be further described using the following terms:

[0102] 1. A quantum computing system for performing dynamically decoupled controlled Z-gate operations, comprising: a superconducting circuit, including:

[0103] First quantum bit;

[0104] The second qubit is laterally coupled to the first qubit; and at least one computing device is configured to: provide a first instruction to a first driving source, causing the first driving source to apply an external magnetic flux to the second qubit so that the frequency of the second qubit resonates with the frequency of the first qubit; and provide a second instruction to the second driving source, causing the second driving source to apply a continuous alternating drive with a continuous phase to the second qubit.

[0105] 2. The quantum computing system according to Clause 1, wherein: the duration and amplitude of the continuously alternating drive are set to synchronize the gate duration of the dynamically decoupled controlled Z-gate operation with an integer number of Rabi oscillation periods.

[0106] 3. The quantum computing system according to Clause 1, wherein: the amplitude of the continuously alternating drive is selected based on the amplitude of the interaction term between the first and second qubits in the Hamiltonian of the superconducting circuit, the Hamiltonian being specified for a frame that rotates together with the first and second qubits.

[0107] 4. The quantum computing system according to Clause 3, wherein: the amplitude of the continuously alternating drive is one to three times the amplitude of the interaction term.

[0108] 5. A quantum computing system according to any one of Clauses 1 to 4, wherein: the duration of the continuous alternating drive is inversely proportional to the amplitude of the continuous alternating drive.

[0109] 6. The quantum computing system according to any one of clauses 1 to 5, wherein: the duration of the continuous alternating drive is selected to be within 10% of the quotient of π divided by the magnitude of the continuous alternating drive.

[0110] 7. A quantum computing system according to any one of Clauses 1 to 5, wherein: the amplitude of the continuously alternating drive is selected as the peak value corresponding to the relationship between the amplitude and fidelity of the dynamically decoupled controlled Z-gate operation.

[0111] 8. The quantum computing system according to Clause 7, wherein: the peak includes the first peak in the relation.

[0112] 9. A quantum computing system according to any one of Clauses 1 to 8, wherein: the first qubit and the second qubit are both fluxonium qubits.

[0113] 10. The quantum computing system according to any one of clauses 1 to 9, wherein: at least one computing device is further configured to provide a third instruction to read out the state of the quantum computing system after providing a second instruction to a second driving source.

[0114] 11. A method for performing a dynamically decoupled controlled Z-gate operation, comprising: providing an external magnetic flux to a second qubit of a superconducting circuit of a quantum computing system via a first driving source to tune the frequency of the second qubit to the frequency of a first qubit of the superconducting circuit, the first qubit being laterally coupled to the second qubit; providing a continuous alternating drive to the second qubit via a second driving source, the amplitude of the continuous alternating drive corresponding to a peak in the relationship between the amplitude and fidelity of the dynamically decoupled controlled Z-gate operation; and reading out the state of the quantum computing system after providing the second driving source.

[0115] 12. The method according to Clause 11, wherein: the amplitude of the continuously alternating drive is selected based on the amplitude of the interaction term between the first and second qubits in the Hamiltonian of the superconducting circuit, the Hamiltonian being specified for the frame rotating with the first and second qubits.

[0116] 13. The method according to Clause 12, wherein: the magnitude of the continuous alternating drive is one to three times the magnitude of the interaction term.

[0117] 14. The method according to any one of clauses 11 to 13, wherein: the duration of the continuous alternating drive is inversely proportional to the amplitude of the continuous alternating drive.

[0118] 15. The method according to any one of clauses 11 to 14, wherein: the duration of the continuous alternating drive is selected to be within 10% of the quotient of π divided by the amplitude of the continuous alternating drive.

[0119] 16. The method according to any one of Clauses 11 to 15, wherein: the peak includes the first peak in the relation.

[0120] 17. The method according to any one of clauses 11 to 16, wherein: the duration and amplitude of the continuous alternating drive are selected such that the gate duration of the dynamically decoupled controlled Z-gate operation is synchronized with an integer number of Rabi oscillation periods.

[0121] 18. The method according to any one of clauses 11 to 17, wherein: the first qubit and the second qubit are both fluxonium qubits.

[0122] 19. A non-volatile computer-readable medium comprising instructions that, when processed by a quantum computing system, cause the quantum computing system to perform a first operation for implementing a dynamically decoupled controlled Z-gate operation, the first operation comprising: providing an external magnetic flux to a second qubit of a superconducting circuit via a first driving source to tune the frequency of the second qubit to the frequency of a first qubit of the superconducting circuit, the first qubit being laterally coupled to the second qubit; providing a continuous alternating drive to the second qubit via a second driving source, the duration of the continuous alternating drive being inversely proportional to the amplitude of the continuous alternating drive; and within 10% of the quotient of π divided by the amplitude of the continuous alternating drive; and reading out the state of the quantum computing system after providing the second driving source.

[0123] 20. The non-volatile computer-readable medium according to Clause 19, wherein: the amplitude of the continuously alternating drive is selected as a first peak value corresponding to the relationship between the amplitude and fidelity of the dynamically decoupled controlled Z-gate operation.

[0124] 21. The non-volatile computer-readable medium according to any one of Clauses 19 to 20, wherein: the duration and amplitude of the continuously alternating drive are selected such that the gate duration of the dynamically decoupled controlled Z-gate operation is synchronized with an integer number of Rabi oscillation periods.

[0125] 22. The non-volatile computer-readable medium according to any one of Clauses 19 to 21, wherein: the first qubit and the second qubit are both fluxonium qubits.

[0126] Exemplary embodiments have been disclosed in the accompanying drawings and description. However, many variations and modifications can be made to these embodiments. Therefore, although specific terminology has been used, it is used in a general and descriptive sense and not to limit or constrain the scope of the embodiments as defined by the appended claims.

Claims

1. A quantum computing system for performing a dynamically decoupled controlled-Z gate operation, comprising: a superconducting circuit, wherein the superconducting circuit comprises: a first qubit; a second qubit laterally coupled to the first qubit; and at least one computing device configured to: provide first instructions to a first drive source such that the first drive source applies an external magnetic flux to the second qubit to cause a frequency of the second qubit to resonate with a frequency of the first qubit; and provide second instructions to a second drive source such that the second drive source applies a continuous alternating drive to the second qubit with a continuous phase, a duration of the continuous alternating drive and an amplitude of the continuous alternating drive configured to synchronize a gate time of the dynamically decoupled controlled-Z gate operation to an integer number of Rabi oscillation periods.

2. The quantum computing system of claim 1, wherein: the amplitude of the continuous alternating drive is selected based on an amplitude of an interaction term between the first qubit and the second qubit in a Hamiltonian of the superconducting circuit, the Hamiltonian specified in a frame co-rotating with the first and second qubits.

3. The quantum computing system of claim 2, wherein: the amplitude of the continuous alternating drive is one to three times the amplitude of the interaction term.

4. The quantum computing system of claim 1, wherein: the duration of the continuous alternating drive is inversely proportional to the amplitude of the continuous alternating drive.

5. The quantum computing system of claim 1, wherein: the duration of the continuous alternating drive is selected to be within 10% of a quotient of π divided by the amplitude of the continuous alternating drive.

6. The quantum computing system of claim 1, wherein: the amplitude of the continuous alternating drive is selected to correspond to a peak in a relationship between an amplitude and a fidelity of the dynamically decoupled controlled-Z gate operation.

7. The quantum computing system of claim 6, wherein: the peak comprises a first peak in the relationship.

8. The quantum computing system of claim 1, wherein: the first qubit and the second qubit are each a fluxonium qubit.

9. The quantum computing system of claim 1, wherein: the at least one computing device is further configured to provide third instructions to read out a state of the quantum computing system after providing the second instructions to the second drive source.

10. A method for performing a dynamically decoupled controlled-Z gate operation, comprising: providing, by a first drive source, an external magnetic flux to a second qubit of a superconducting circuit of a quantum computing system to tune a frequency of the second qubit to a frequency of a first qubit of the superconducting circuit, the first qubit laterally coupled to the second qubit; providing, by a second drive source, a continuous alternating drive to the second qubit, an amplitude of the continuous alternating drive corresponding to a peak in a relationship between an amplitude and a fidelity of the dynamically decoupled controlled-Z gate operation, a duration of the continuous alternating drive and the amplitude selected to synchronize a gate time of the dynamically decoupled controlled-Z gate operation to an integer number of Rabi oscillation periods; and providing, by a second drive source, a continuous alternating drive to the second qubit, an amplitude of the continuous alternating drive corresponding to a peak in a relationship between an amplitude and a fidelity of the dynamically decoupled controlled-Z gate operation, a duration of the continuous alternating drive and the amplitude selected to synchronize a gate time of the dynamically decoupled controlled-Z gate operation to an integer number of Rabi oscillation periods; and after providing the second drive source, reading out a state of the quantum computing system.

11. The method of claim 10, wherein: the amplitude of the continuous alternating drive is selected based on an amplitude of an interaction term between the first qubit and the second qubit in a Hamiltonian of the superconducting circuit, the Hamiltonian being specified in a frame co-rotating with the first and second qubits.

12. The method of claim 11, wherein: the amplitude of the continuous alternating drive is one to three times the amplitude of the interaction term.

13. The method of claim 10, wherein: the duration of the continuous alternating drive is inversely proportional to the amplitude of the continuous alternating drive.

14. The method of claim 10, wherein: the duration of the continuous alternating drive is selected to be within 10% of a quotient of π divided by the amplitude of the continuous alternating drive.

15. The method of claim 10, wherein: the peak includes a first peak in the relationship.

16. The method of claim 10, wherein: the first qubit and the second qubit are each fluxonium qubits.

17. A non-transitory computer readable medium comprising instructions that, when processed by a quantum computing system, cause the quantum computing system to perform first operations for implementing a dynamically decoupled controlled-Z gate operation, the first operations comprising: providing, by a first drive source, an external magnetic flux to a second qubit of a superconducting circuit to tune a frequency of the second qubit to a frequency of a first qubit of the superconducting circuit, the first qubit being transversely coupled to the second qubit; providing, by a second drive source, a continuous alternating drive to the second qubit, a duration of the continuous alternating drive being inversely proportional to an amplitude of the continuous alternating drive, the duration and the amplitude of the continuous alternating drive being selected to synchronize a gate duration of the dynamically decoupled controlled-Z gate operation to an integer number of Rabi oscillation periods; and within 10% of a quotient of π divided by the amplitude of the continuous alternating drive; and after providing the second drive source, reading out a state of the quantum computing system.

18. The non-transitory computer readable medium of claim 17, wherein: the amplitude of the continuous alternating drive is selected to correspond to a first peak in a relationship between an amplitude and a fidelity of the dynamically decoupled controlled-Z gate operation.

19. The non-transitory computer readable medium of claim 17, wherein: the first qubit and the second qubit are each fluxonium qubits.

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