Fluxonium and fluxonium-like qubit control via baseband voltage pulses

Abstract

Systems and methods for controlling a fluxonium-like qubit are provided. In one example, a quantum computing system may include a qubit. The qubit can include an inductance, a capacitance, and a Josephson junction connected in parallel. The quantum computing system may include a control system configured to implement a control signal to implement a quantum gate on the qubit. The control signal may include a voltage-bias pulse.

Description

FLUXONIUM AND FLUXONIUM-LIKE QUBIT CONTROL VIA BASEBAND VOLTAGE PULSES FIELD

[0001] The present disclosure relates generally to systems and methods for quantum computing. CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] The present application is based upon and claims the right of priority to each of U.S. Provisional Patent Application No.63 / 581,747, filed on September 11, 2023; U.S. Provisional Patent Application No.63 / 561,022, filed on March 4, 2024; and U.S. Provisional Patent Application No.63 / 562,390, filed on March 7, 2024; the disclosure of each of which is hereby incorporated by reference herein in its entirety for all purposes. BACKGROUND

[0003] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that enables the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0〉 + b |1〉 The “0” and “1” states of a digital computer are analogous to the |0〉 and |1〉 basis states, respectively of a qubit. SUMMARY

[0004] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or can be learned from the description, or can be learned through practice of the embodiments.

[0005] Example aspects of the present disclosure provide an example quantum computing system. In some implementations, the quantum computing system can include a qubit. In some implementations, the qubit can include an inductance, a capacitance, and a Josephson junction connected in parallel. In some implementations, the quantum computing system can include a control system configured to implement a control signal to implement aquantum gate on the qubit. In some implementations, the control signal can include a voltage- bias pulse.

[0006] Example aspects of the present disclosure provide an example method. In some implementations, the method can include obtaining a target rotation for a qubit having a fluxonium mode. In some implementations, the method can include determining, based at least in part on the target rotation, a target Z rotation about a Z axis and a target Y rotation about a Y axis. In some implementations, the method can include determining, based at least in part on the target Z rotation and the target Y rotation, a Z rotation time and a Y rotation time. In some implementations, the method can include determining, based at least in part on the Z rotation time and the Y rotation time, a target pulse shape for a time-dependent charge- bias pulse. In some implementations, the method can include providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse. In the example method, the target pulse shape can be configured to cause the qubit to rotate about the Z axis for the Z rotation time; and rotate about the Y axis for the Y rotation time.

[0007] Example aspects of the present disclosure provide an example method. The example method can include obtaining a target rotation for a qubit having an effective Hamiltonian comprising an inductive energy term and a phase-slip energy term. In some implementations, the method can include determining, based at least in part on the target rotation, a target Z rotation about a Z axis and a target Y rotation about a Y axis. In some implementations, the method can include determining, based at least in part on the target Z rotation and the target Y rotation, a Z rotation time and a Y rotation time. In some implementations, the method can include determining, based at least in part on the Z rotation time and the Y rotation time, a target pulse shape for a time-dependent charge-bias pulse. In some implementations, the method can include providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse. In the example method, the target pulse shape can be configured to cause the qubit to rotate about the Z axis for the Z rotation time; and rotate about the Y axis for the Y rotation time.

[0008] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the related principles.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0010] FIG.1 depicts an example quantum computing system according to example aspects of the present disclosure;

[0011] FIG.2A depicts an example baseband voltage drive according to example aspects of the present disclosure;

[0012] FIG.2B depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0013] FIG.3A depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0014] FIG.3B depicts example quantization axes according to example aspects of the present disclosure;

[0015] FIG.4 depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0016] FIG.5 depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0017] FIG.6 depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0018] FIG.7 depicts an example charge-bias pulse according to example aspects of the present disclosure;

[0019] FIG.8 depicts an example of a quantum computing system according to example aspects of the present disclosure;

[0020] FIG.9 depicts a flowchart diagram of an example quantum computing method according to example aspects of the present disclosure;

[0021] FIG.10 depicts a flowchart diagram of an example quantum computing method according to example aspects of the present disclosure;

[0022] FIG.11 depicts a flowchart diagram of an example quantum computing method according to example aspects of the present disclosure; and

[0023] FIG.12 depicts a block diagram of an example computing system according to example aspects of the present disclosure.DETAILED DESCRIPTION

[0024] Example embodiments according to some aspects of the present disclosure are directed to systems and methods for single-qubit control of fluxonium qubits, fluxonium-like qubits, and related qubits using baseband voltage pulses. In some embodiments, systems and methods of the present disclosure can perform any arbitrary single-qubit operation on a fluxonium, fluxonium-like, or related qubit, including a Pauli Z rotation of any arbitrary angle; a Pauli Y rotation of any arbitrary angle; a Pauli X rotation of any arbitrary angle; an identity operation over any arbitrary time period; and any combination thereof.

[0025] Although aspects of the present disclosure are discussed with reference to fluxonium qubits for purposes of illustration and discussion, the control techniques provided herein are applicable to fluxonium and fluxonium-like qubits. For instance, aspects of the present disclosure may be applicable to: a fluxonium qubit; a variant of fluxonium (e.g., fluxonium-like qubit having two or more Josephson junctions in parallel or Josephson junction replaced with a superconducting quantum interference device (“SQUID”), etc.); a qubit with a fluxonium mode within its circuit (e.g., zero-pi qubit, etc.); or a qubit that reduces in some limit to a phase-slip-junction device (e.g., having an effective Hamiltonian similar to equation 18 below); or other suitable qubit or device.

[0026] Performing a Z or Y rotation can include determining, based on a desired angle of rotation and a qubit frequency associated with the fluxonium-like qubit, a target amount of time to rotate the qubit around the Z or Y axis. Performing a Z or Y rotation can include providing, for the target amount of time, a charge bias configured to rotate the qubit around the Z or Y axis.

[0027] Performing an X rotation or identity operation can include, for example, combining two or more Z rotations or Y rotations. For example, performing an X rotation can include combining one or more (e.g., two) Z rotations with one or more (e.g., one) Y rotations.

[0028] Performing an identity operation can include, for example, performing a plurality of rotations configured to cause no net change to the fluxonium-like qubit’s state over a target time period. For example, an identity operation can include two or more rotations about a same axis of rotation (e.g., two Z rotations, four Z rotations, etc.). In some instances, an identity operation can include at least one rotation in a first direction of rotation and at least one rotation in a second direction of rotation opposite the first direction. For example, in some instances, performing an identity operation can include determining, based on a target identity operation time, a first rotation time and a second rotation time (e.g., half the targetidentity operation time, etc.). In some instances, performing an identity operation can include rotating the fluxonium-like qubit around the Z axis in a first direction for the first rotation time and in a second direction opposite the first direction for the second rotation time. In some instances, an identity operation can use more than two (e.g., four) Z rotations in combination to prevent or reduce unintended Y rotations that may occur during transitions between charge biases.

[0029] In some instances, a charge-bias pulse can be provided by providing a voltage- bias line coupled to a fluxonium-like circuit by a capacitor. For example, a time-dependentcharge-bias pulse ^^^^^^^^(^^^^) ∝ ^^^^(^^^^) can be implemented by a baseband-voltage drive V(t).

[0030] In some instances, a charge-bias pulse can be shaped to suppress a leakage. For example, a smooth pulse shape (e.g., a shape that is not perfectly square) can be configured to cause reduced leakage relative to a square pulse shape. In some instances, a charge-bias pulse can be characterized by a first transition time, a hold time, and a second transition time. As an illustrative example, performing a Y rotation can include transitioning, over a first transition time, to a charge bias configured to cause rotation about a Y axis (e.g., from a zero charge bias, etc.); holding the charge bias constant for a period of time (“hold time”); and transitioning, over a second transition time, from the charge bias configured to cause rotation about the Y axis (e.g., to a zero charge bias, etc.). In some instances, the first and second transition times can be numerically optimized to suppress leakage.

[0031] Example embodiments according to some aspects of the present disclosure can provide for a number of technical effects and benefits, such as improvements to computing technology (e.g., quantum computing technology). Example technical effects and benefits can include increased coherence times, reduced gating times, increased gate fidelity, and reduced hardware requirements compared to alternate systems and methods. In some instances, increased coherence times and reduced gate times can provide for increased complexity of quantum computations by allowing more gating operations to be performed before qubit decoherence. In some instances, increased gate fidelities can increase quantum computation accuracy and decrease a computational cost (e.g., electricity cost, hardware usage, etc.) of a quantum computation.

[0032] For example, systems and methods according to examples of the present disclosure can provide faster fluxonium gate times relative to microwave control, without sacrificing the high gate fidelities and long coherence times associated with some alternate fluxonium-based systems and methods. In some instances, systems and methods according toexamples of the present disclosure can control a fluxonium-like qubit (e.g., fluxonium qubit) without using a rotating-wave approximation, which can in some instances lead to faster and higher-fidelity gates. For example, in some instances, a gating speed can be equal to a qubit frequency of the fluxonium qubit, operating at the speed limit set by the qubit parameters. In some example simulations according to the present disclosure, provided example systems and methods achieved coherence times above one millisecond, single-qubit gate fidelities greater than 99.999 percent (accounting for unitary over rotations and leakage), and single-qubit gate times as low as 2.1 nanoseconds.

[0033] As another example, systems and methods according to examples of the present disclosure can achieve higher gate fidelities and longer coherence times relative to baseband- flux control, which requires flux-biasing the qubit away from its flux sweet spot. In contrast, systems and methods according to examples of the present disclosure can provide any arbitrary single-qubit operation without changing the qubit energy during the gate, enabling a fluxonium-like qubit (e.g., fluxonium qubit) to remain at its flux sweet spot throughout operation. Enabling a fluxonium qubit to remain at its flux sweet spot can, for example, increase fidelity and coherence times relative to flux biasing. Additionally, improved gate fidelities associated with examples of the present disclosure can in some instances be robust against errors in control amplitude and timing.

[0034] As another example, systems and methods according to examples of the present disclosure can operate without fast-flux lines, thereby reducing an amount of hardware required to control a fluxonium qubit compared to alternate methods. This reduced hardware requirement can, for example, improve the extensibility and scalability of fluxonium architecture.

[0035] In some instances, increased gate fidelity can increase an accuracy of a fluxonium-based quantum computation by reducing one or more error types associated with microwave control or baseband flux control. Additionally, increased gate fidelity can reduce a computational cost (e.g., electricity cost, hardware usage, etc.) associated with a quantum computation. For example, in some instances, determining a quantum computation result can include running a quantum computation circuit a plurality of times. In some instances, a number of times required to accurately determine a quantum computation result (e.g., within a target precision) can depend, at least in part, on an error rate (e.g., fidelity, coherence, etc.) associated with the quantum computation circuit. In such instances, improved accuracy according to examples of the present disclosure can enable a quantum computation result to be determined, within a target precision, using fewer quantum computations compared toalternate systems and methods. In this manner, for instance, a quantum computation result can be determined at a reduced computational cost compared to alternate systems and methods.

[0036] As used herein, the use of the terms “about,” “approximately,” or “near” in conjunction with a numerical value refers to within 10 percent of the stated numerical value. “Near zero” means within 0.025 of zero. “Approximately constant” means deviating by less than 10 percent from a mean value greater than or equal to 0.25 or deviating by less than 0.025 from a mean value smaller than 0.25.

[0037] With reference now to the Figures, example embodiments of the present disclosure will be discussed in further detail. Example Systems

[0038] FIG.1 depicts an example system for controlling a fluxonium-like qubit according to example embodiments of the present disclosure. A voltage bias source 102 can be coupled to a fluxonium-like qubit 106 by a first capacitor 104. The fluxonium-like qubit 106 can comprise, for example, one or more Josephson junction(s) 108, one or more inductor(s) 110, and one or more second capacitor(s) 112. In some instances, components of the fluxonium-like qubit can be connected in parallel (e.g., as depicted in FIG.1), or in another configuration (e.g.. in a zero-pi configuration or other suitable configuration, etc.).

[0039] The voltage bias source 102 can be, for example, any system (e.g., device, apparatus, etc.) configured to provide a voltage bias. In some instances, a voltage bias source 102 can be configured to provide a time-dependent voltage-bias pulse. In some instances, a voltage bias source 102 can comprise a controller configured to control a shape of a time- dependent voltage-bias pulse. In some instances, a voltage-bias source 102 can be configured to control one or more transition times in which a voltage-bias pulse transitions from a first voltage to a second voltage. In some instances, a voltage-bias source 102 can be configured to provide a voltage-bias pulse of a particular shape or combination of shapes (e.g., flat section with voltage bias equal to zero, non-zero voltage sections with approximately quadratic shape, etc.). In some instances, a shape of the voltage-bias pulse can be based at least in part on a mathematical function (e.g., quadratic function, cosine function, sine function, harmonic function, etc.) In some instances, a shape of the voltage-bias pulse can be continuous (i.e., without discontinuities). In some instances, a voltage-bias pulse can be shaped to generate a charge-bias pulse of a particular shape, as discussed below with respect to FIG.2.

[0040] In some instances, one or more of the first capacitor 104, fluxonium-like qubit 106, Josephson junction(s) 108, inductor(s) 110, and second capacitor(s) 112 can be standard quantum computing system components constructed according to known methods. In some instances, one or more of the first capacitor 104, fluxonium-like qubit 106, Josephson junction(s) 108, inductor(s) 110, and second capacitor(s) 112 can be configured to operate in a superconducting quantum computing environment. In some instances, one or more of the first capacitor 104, fluxonium-like qubit 106, Josephson junction(s) 108, inductor(s) 110, and second capacitor(s) 112 can be configured to operate at very low temperatures (e.g., at 3 Kelvin, etc.) and to dissipate very little heat during operation.

[0041] In some instances, the fluxonium-like qubit 106 can be a standard fluxonium qubit constructed according to known methods. In some instances, the fluxonium-like qubit 106 can be a variant of fluxonium (e.g., fluxonium-like qubit having two or more Josephson junctions in parallel or Josephson junction replaced with a superconducting quantum interference device (“SQUID”), etc.); a qubit with a fluxonium mode within its circuit (e.g., zero-pi qubit, etc.); a qubit that reduces in some limit to a phase-slip-junction device (e.g., having an effective Hamiltonian similar to equation 18 below); or other suitable qubit or device.

[0042] In some instances, a fluxonium-like qubit 106 can include a circuit having a fluxonium mode. In some instances, a fluxonium mode can include a circuit mode having a Hamiltonian comprising a charging-energy term, a Josephson-energy term, and an inductive energy term. In some instances, a fluxonium mode can include a circuit mode corresponding to a parallel combination of a capacitance, an inductance, and a Josephson junction. In some instances, a fluxonium-like qubit 106 can include a circuit having a fluxonium mode and having or not having one or more additional circuit modes (sometimes referred to as degrees of freedom) in addition to a fluxonium mode. As a non-limiting illustrative example, a fluxonium-like qubit 106 can include a zero-pi qubit having a fluxonium mode and one or more additional modes (e.g., transmon mode, etc.). In some instances, a circuit having a fluxonium mode can include any circuit having a linear combination of circuit modes (e.g., circuit modes identifiable via standard circuit analysis methods) wherein the linear combination comprises a fluxonium mode. For example, in some instances, the linear combination can lead to a degree of freedom that, when disregarding interactions with any additional modes of the fluxonium-like qubit 106, has a Hamiltonian comprising a charging- energy term, a Josephson-energy term, and an inductive energy term.

[0043] In some instances, a fluxonium mode can include a mode that corresponds, in a lumped-element circuit model, to a parallel combination of a capacitance, an inductance, and a Josephson junction. For example, in some instances, a fluxonium-like qubit 106 can include a circuit comprising an inductance, a capacitance, and a Josephson junction (e.g., with or without one or more additional circuit elements). As a non-limiting illustrative example, a fluxonium-like qubit 106 can in some instances comprise a zero-pi qubit having a plurality (e.g., two, etc.) of inductances, a plurality (e.g., two, etc.) of capacitances, and a plurality (e.g., two, etc.) of Josephson junctions. Continuing the non-limiting illustrative example, the zero-pi qubit can include one or more parallel triplets, each parallel triplet comprising an inductance, a capacitance, and a Josephson junction in parallel.

[0044] In some instances, a fluxonium-like qubit 106 can include a qubit that reduces in a limit to a phase-slip-junction device. A qubit that reduces in a limit to a phase-slip-junction device can include, for example, a qubit having an effective Hamiltonian having an inductive term and a phase-slip energy term. In some instances, an inductive term can quantify an energy of m flux quanta in a superconducting loop defined by a device’s geometry. In some instances, a phase-slip energy term can quantify an energy cost or gain from tunneling of a flux quanta in or out of a superconducting loop. Further details of one or more example effective Hamiltonians are provided below with respect to equations (1) through (21) and accompanying text.

[0045] In some instances, a qubit that reduces in a limit to a quantum phase-slip device can include a qubit having a qubit frequency dispersion, in the presence of an external flux bias, that approximates or otherwise corresponds to an energy spectrum of a phase-slip- junction device. For example, in some instances, a fluxonium-like qubit 106 can include a device having a qubit frequency dependence that is approximately linear or hyperbolic between a first qubit frequency (e.g., maximum frequency) and a second qubit frequency (e.g., a lower frequency). For example, in some instances, a fluxonium-like qubit 106 can include a device having a qubit frequency dependence wherein a second qubit frequency (e.g., lower frequency) can be associated with the value of the phase-slip energy of the fluxonium-like qubit 106, and a slope of the qubit transition between the first and second qubit frequencies can be associated with the inductive energy of the fluxonium-like qubit 106.

[0046] In some instances, the fluxonium-like qubit 106 can be characterized by one or more fluxonium parameters (e.g., charging energy, Josephson energy, inductive energy, qubit frequency, anharmonicity, etc.). In some instances, the parameters of the fluxonium-like qubit106 can correspond or not correspond to one or more known or published parameter sets associated with one or more types of fluxonium qubit (e.g., MIT fluxonium, Princeton heavy fluxonium, etc.). However, this is not required, and systems and methods according to examples of the present disclosure can be used in combination with other fluxonium-like qubits (e.g., fluxonium-like qubits characterized by unpublished parameters; improved fluxonium-like qubits manufactured according to novel methods; etc.) without going outside the scope of the present disclosure. Example Baseband-Voltage Drives

[0047] FIGS.2A and 2B depict an example baseband-voltage drive 204 for generating an example charge bias pulse 202. An increasing baseband-voltage drive 204 can be provided during a first transition time 208 to cause a charge bias 202 to increase. A near-constant (e.g., constant) baseband-voltage drive 204 can be provided during a hold time 210 to cause a charge bias 202 to remain unchanged or approximately unchanged. And a baseband-voltage drive 204 having a decreasing voltage can be provided during a second transition time 212 to cause a charge bias 202 to decrease.

[0048] In some instances, a baseband-voltage drive 204 can comprise a time-dependent voltage-bias pulse. In some instances, a shape of the baseband-voltage drive 204 can be controlled by a voltage-bias source 102. In some instances, a baseband-voltage drive 204 can be characterized by one or more pulse shapes or combinations of pulse shapes (e.g., flat section with voltage bias approximately constant, increasing-voltage and decreasing-voltage sections with approximately sigmoidal shape or “S” shape, etc.). In some instances, a pulse shape of the baseband-voltage drive 204 can be based at least in part on a mathematical function (e.g., sigmoidal function such as logistic function, hyperbolic tangent, arctangent, etc.; non-sigmoidal function such as cosine function, sine function, harmonic function, quadratic function, etc.) In some instances, a pulse shape of the baseband-voltage drive 204 can be continuous (i.e., without discontinuities). In some instances, a pulse shape of the baseband-voltage drive 204 can be smooth, wherein a first derivative with respect to time of the baseband-voltage drive 204 can be continuous (i.e., without discontinuities).

[0049] In some instances, a time-dependent charge bias pulse 202 can be generated by obtaining a target charge-bias pulse shape; determining, based on a mathematical relationship (e.g., proportionality constant, etc.) between voltage bias and charge bias, a target voltage- bias pulse shape for the baseband voltage drive 204, wherein the target voltage-bias pulse shape is configured to create a charge-bias pulse 202 corresponding to the target charge-bias pulse shape; and providing, by a voltage-bias source 102, a voltage-bias pulse based on thetarget voltage-bias pulse shape. In some instances, a mathematical relationship between atime-dependent voltage bias and a time-dependent charge bias can be expressed as ^^^^^^^^(^^^^) ∝^^^^(^^^^), where V(t) is a baseband voltage at a particular time t, and ^^^^^^^^(^^^^) is a charge bias at a particular time t. In some instances, a relationship between a time-dependent charge bias and time-dependent voltage bias can be determined based at least in part on one or more properties (e.g., resistance, capacitance, time constant, etc.) of a capacitor 104, fluxonium- like qubit 106, or other components of a circuit comprising a voltage bias source 102, first capacitor 104, and fluxonium-like qubit 106.

[0050] In some instances, a charge bias 202 can be characterized by a non-square pulse shape. A non-square pulse shape can include any shape that is not perfectly square, such as an approximately square shape (e.g., flat-top-cosine with small transition times 208, 212) or a shape that is not approximately square (e.g., approximately parabolic shape characterized by a small or zero hold time 210, etc.). In some instances, a charge bias 202 can be characterized by a continuous pulse shape having no discontinuities. In some instances, a charge bias 202 can be characterized by a smooth pulse shape, wherein a derivative of the charge bias 202 with respect to time can be continuous (i.e., without discontinuities).

[0051] In some instances, a transition time 208, 212 can be a time during which a charge bias 202 is changing (e.g., increasing, decreasing, etc.). In some instances, a transition time 208, 212 can be a time during which a voltage bias of the baseband voltage drive 204 is increasing or decreasing. In some instances, a transition time 208, 212 can be a time between a first hold time 210 and a second hold time 210. In some instances, a transition time 208, 212 can be a time between a first time in which a charge bias 202 is equal to a first charge bias 202 value of interest, and a second time in which a charge bias 202 is equal to a second charge bias 202 value of interest. In some instances, a voltage bias of the baseband voltage drive 204 can be constant or approximately constant throughout a transition time 208, 212.

[0052] In some instances, a hold time 210 can be a time during which a charge bias 202 is constant or approximately constant (e.g., constant plus or minus a small variance ^^^^, actual charge bias 202 generated based on a target charge bias that is constant, etc.). In some instances, a hold time 210 can be a time during which a voltage bias associated with the baseband voltage drive 204 is constant or approximately constant. In some instances, a hold time 210 can be a time between a first transition 208 and second transition time 212. In some instances, a hold time 210 can be a time during which a charge bias 202 is equal or approximately equal to a charge bias 202 value of interest. In some instances, a charge bias202 value of interest can be a charge bias 202 value configured to cause the fluxonium-like qubit 106 to rotate about a particular axis of rotation, e.g., as described below with respect to FIGS.3-7. Example Discussion of Fluxonium Hamiltonians and Generalized Theories of Fluxonium

[0053] FIGS.3-7 depict example charge-bias pulses for single-qubit control of a fluxonium-like qubit (e.g., fluxonium qubit). In some instances, example charge-bias pulses for controlling a fluxonium-like qubit according to example aspects of the present disclosure can be understood or described in the context of one or more example two-level Hamiltonians or other properties of fluxonium-like qubits (e.g., fluxonium qubits), such as an example two- level Hamiltonian for a fluxonium qubit that preserves a parametric charge-bias dependence. To aid understanding of various systems and methods according to examples of the present disclosure, example discussions of fluxonium Hamiltonians and generalized theories of fluxonium are provided below.

[0054] Although some aspects of the discussion below may include specific reference to one or more properties of fluxonium qubits, systems and methods according to the present disclosure can be applied to other fluxonium-like qubits (e.g., qubits having a fluxonium mode; qubits that reduce in a limit to a phase-slip-junction device; qubits having one or more inductances, capacitances, and Josephson junctions in parallel; zero-pi qubits; etc.) without deviating from the scope of the present disclosure.

[0055] Example discussions of the present disclosure can include discussion of a fluxonium model incorporating an offset charge parameter, which can in some instances tracked throughout several theoretical steps which may not be present in prior disclosures. In some instances, discussion of such a model can lead to discussion of a simplified, two-level Hamiltonian with a parametric ngdependence. In some instances, example methods for universal single-qubit control of a fluxonium qubit can be understood or described in the context of such a Hamiltonian.

[0056] In a single-mode approximation, a fluxonium Hamiltonian can in some instances be written aswhere ^^^^C, ^^^^Jand ^^^^Lare the qubit's charging, Josephson, and inductive energies, respectively,^^�^^ and ^�^^^ are the qubit's phase and reduced charge operators satisfying�^^�^^, ^�^^^� = ^^^^, and ^^^^^^^^ and^^^^^^^^^^^^^^^^are the externally controlled charge and phase biases, which may be time-dependent.

[0057] In some instances, a fluxonium qubit can operate in the approximate parameter regime ^^^^L< ^^^^C< ^^^^Jand can be characterized by a low frequency (e.g., in the range 1-800 MHz) and an anharmonicity of several GHz. A large anharmonicity can help to suppress leakage during single-qubit operations, but a low qubit frequency can make single-qubit operations slow when implemented via a microwave drive. However, the lower the qubit frequency is, the better the qubit T1will be (dielectric loss is proportional to the qubit frequency), leading to a design tradeoff necessary to balance coherent errors and high qubit coherence.

[0058] Example aspects of the present discussion can include isolating the 2^^^^-periodic part of a fluxonium Hamiltonian written as ^�^^^p = 4^^^^C(^�^^^ − ^^^^^^^^)2 − ^^^^J^^^^^^^^^^^^^^�^^, (2)where, as a difference of the transmon (or charge) qubit, ^^^^ ∈ (−∞,∞) is a noncompactdegree of freedom. (In other words, ^^^^ ± 2^^^^^^^^ ^^^^^^^^^^^^ℎ ^^^^ ∈ ℤ are distinguishable potentialminima.) According to Bloch's theorem, the eigenfunctions of ^�^^^pare of the form ^^^^^^^^ ^^^^^^^^^^^^ ^^^^^^^(^^^^) = ^^^^ ^^^^ ^^^^^(^^^^), (3)where ^^^^ ∈ [−12, 12) is a continuous quantum number which plays the role of “crystal momentum," n is a band index, and ^^^^^^^^^^^^(^^^^)is a 2^^^^-periodic function of the phase. These Blochstates satisfy the orthogonality relation ^^^^^^^^^|^^^^^^^^′^^^^′^ =the inneris extended to the entire real axis.

[0059] In some instances, to find the statesone can specifyeigenvalue equationand simplify it, arrivingwhere 2^^^^-periodic boundary conditions are imposed to the phase. In some instances, equation (4) can be equivalent to the eigenvalue problem defined by a transmon-qubit Hamiltonianwith identical ^^^^C and ^^^^J parameters and offset-charge bias ^^^^^^^^ − ^^^^.

[0060] In some instances, equation (4) can be solved, and the resulting basis can be used to represent equation (1). In some instances, a phase operator can be written as ^^�^^ ≡ ^^^^^^^^^^^^ + Ω� , (5)where Ω�is an interband coupling potential that can be computed to beFor example, this expression can be derived assuming a gauge where ^^^^^^^^^^^^^|^^^^ ^^^^^^^^^^^^^^^^^ = 0 for all n∈ �− 12, 12�, which can be ensured numerically. Similarly, a fluxonium charge operator can take the formI some instances, the charge matrix elementscan be calculated usingtransmon-like eigenstates defined in equation (4).

[0061] With the above definitions, the fluxonium Hamiltonian of equation (1) can be written as

[0062] The first term represents the `transmon part' within the fluxonium Hamiltonian, which is however defined for a noncompact phase and therefore diagonalized by Bloch (instead of charge) states. The second term represents the inductive shunt which leads to a displacement along the Bloch bands (term proportional to ^^^^^^^^^^^^) and a coupling between the bands. Note that because the inductive potential breaks translational symmetry, k is no longer a good quantum number and should instead be thought as a new, compact degree of freedom in terms of which the fluxonium Hamiltonian is specified.

[0063] Although the Hamiltonian in Eq. (8) is useful, example aspects of the present discussion include additional discussion of example Hamiltonians that may further facilitate understanding and discussion of example systems and methods for quantum control based on the offset-charge dependence. For example, in some instances, a change of basis can be implemented where the conjugate-charge operator associated with k, which can be denoted^�^^^ , is diagonal. In particular, the derivative operator can take the form^^^^^^^^^^^^ ≡ −2^^^^^�^^^ , (9)where ^�^^^ = ∑^^^^∈ℤ ^^^^|^^^^〉^^^^^| . (The factor of 2^^^^ appears because of the 1-periodicity of k.)Due to the relation in Eq. (5), ^�^^^ can be regarded as a measure of the phase across theJosephson junction ^^�^^ in units of 2^^^^. For reasons that may become clear below, this basis can be referred to as a phase-slip representation. This change of frame can correspond to aFourier transform implemented by a unitary ^�^^^†^^^^. When applied to the full Hamiltonian in Eq. (8), this transformation can lead to the form(10) where ^̂^^^^^^^is a potential-energy operator associated with the nth Bloch band. More precisely, ^̂^^^^^^^can correspond to the Fourier transform of the nth eigenvalue of Eq. (4),as a function of the `coordinate' k. Such an eigenvalue is 1-periodic in k - ng. This can be seen, forexample, by noting that the shift k - ng → ^^^^ − ^^^^^^^^ ± 1 can be compensated by a unitarytransformation acting on Eq. (4) and displacing the charge operator by one unit ^�^^^ → ^�^^^ ∓ 1.This unitary leaves invariant the ^^^^^^^^^^^^^^�^^ operator, returning the Hamiltonian to its originalform. Hence, one can write the general Fourier decomposition of the eigenvaluewhere ^�^^^^^^^represents the average frequency of the nth band, and ^^^^^^^^^^^^^^^^is a coefficient withenergy units. Introducing the phase-slip operatorwehave:for l > 0, which can allow equation (11) to be rewritten as

[0064] Because (^�^^^+)^^^^can represent a `phase slip' of 2^^^^l units (e.g., ignoring the interband coupling term), and ℏ−1^^^^^^^^^^^^^^^^can be a rate associated with this process (including l >0 and l < 0), ^^^^^1^^^0can be referred to as the phase-slip energy of fluxonium. In some instances,^^^^1^^^^0 can also define the qubit energy at its flux-frustration point ^^^^^^^^^^^^^^^^ = ^^^^.

[0065] In some instances, the formulation described above can lead to a two-level fluxonium Hamiltonian describing the low-energy physics of the circuit and incorporating a parametric charge-bias dependence. For example, one can first drop the interband couplingterm in Eq. (6) by assuming a large frequency gap�^^^^^^^^ ^^^^ ^^^^0 − ^^^^^^^^� ≫�^^^^0� between the ground-and nth-excited bands, such that second-order corrections to^^^^due to Ω�can be This can be reasonable because these frequency transitions are of GHz order, as they are defined by the `transmon parameters' EC, EJ. Furthermore, one can verify by direct numerical implementation and diagonalization of the full model in Eq. (8) (including the interband coupling term) that this is an excellent approximation for EC, EJand ELparameters in thefluxonium regime. Within this approximation, the fluxonium Hamiltonian emerges from the lowest band n = 0: ^�^^^^ ^^^ L0 =2(−2^^^^^�^^^ + ^^^^^^^^^^^^^^^^)2 + ^̂^^^0,after dropping the constant energy shift ℏω�0. This equation also considers the limit of singlephase slips, where ^^^^1^^^^0 ≫ ^^^^^^^^>1^^^^0 . This assumption is justified below and constitutes a verygood approximation for all EC, EJparameters in the fluxonium regime.

[0066] In general, the phase-slip energies can be determined by diagonalizing the transmon-like Hamiltonian within the fluxonium model, and implementing a numerical Fourier transform of the resulting energy dispersion as a function of k. However, in the transmon regime where EJ≫ EC, the ground-state energy of the transmon qubit is approximated by the asymptotic expression− ^^^^^^^^,withIt can be observed that Eq. (16) is a very good approximation for most fluxonium parameters, and that the limit of single phase slips can be well justified. Indeed, an amplitude of Fourier harmonics with l > 1 can be several orders of magnitude smaller than that of l = 1 for typical fluxonium parameters.

[0067] Based on the above equations, a Hamiltonian can be approximately expressed by:where ^^^^0 may be calculated from equation (16) in the ^^^^J ≫ ^^^^C regime, or numerically in anyother case.

[0068] In the fluxonium regime, one has ^^^^0≲ ^^^^L, leading to a weak coupling of the fluxon (also known as persistent-current) states {|^^^^〉}. Thus, near the flux-frustrationcondition ^^^^^^^^^^^^^^^^ ≈ ^^^^, the eigenstates of Eq. (17) are spanned by a coherent superposition of|^^^^ = 0〉 and |^^^^ = 1〉, which are exactly degenerate for ^^^^^^^^^^^^^^^^ = ^^^^. Assuming sweet-spotoperation, and truncating the Hamiltonian to the two-fluxon subspace, Eq. (17) simplifies towhere the phase slip operators have been replaced by their two-level forms^�^^^+ ^ ^^�^^+ = |1〉 ^0|

[0069] Without loss of generality, one can in some instances assume that the reference charge-bias that defines the qubit quantization axis at time t = 0 is ng= 0, such that the ground and excited states of equation (18) are, respectively:

[0070] In some instances, the qubit Pauli operators can be reintroduced in the basis defined in equation (20) above, the Hamiltonian can be rotated according to the unitary: ^^^^† = |+〉^0| + |−〉^1|and equation 18 can map to:with Z = |+〉^+| − |−〉^−| and Y = -i(|+〉^−| − |−〉^+|).

[0071] In some instances, equation (21) can provide a form of the fluxonium Hamiltonian that can be convenient for understanding or describing single-qubit control based on baseband voltage-bias pulses. For example, in some instances equation (21) can be interpreted as the Hamiltonian of a spin-12 particle in a “magnetic field” whose direction (in the Y-Z plane) is set by the offset charge. In some instances, for example, single-qubit operations can be implemented by introducing nonadiabatic changes of the quantization axis of the spin. In some instances, nonadiabatic changes of the quantization axis of the spin can be introduced by controlling the direction of the magnetic field, i.e., the value of ng. Note that, in contrast to baseband-flux control, baseband-voltage control can in some instances be performed without changing the qubit energy (which according to Eq. (21) can be constant and equal to ^^^^0). This can in some instances lead to preserved qubit coherence times during baseband-voltage pulses. Moreover, because the rate at which the equivalent spin precesses can be the qubit frequency ^^^^0 / ℏ, systems and methods according to example aspects of the present disclosure can operate at the speed limit set by the qubit frequency. Example Charge Bias Pulses for Single-Qubit Control

[0072] FIGS.3A and 3B depict two related views associated with an example rotation of a fluxonium qubit about a Y axis. FIG.3A depicts an example charge-bias pulse for causingthe fluxonium qubit to rotate about the Y axis, while FIG.3B depicts an example relationship between charge bias and an axis of rotation of the fluxonium qubit.

[0073] FIG.3A depicts an example charge bias pulse for causing a fluxonium qubit to rotate about the Y axis in a positive direction. Such a rotation can be used, for example, to perform a Pauli Y-gate, and can be used in combination with other rotations to implement any arbitrary single-qubit operation. The example charge bias pulse can transition, during a first transition time 208, from a pre-transition charge bias (e.g., at or near zero in the example depiction) to a Y rotation charge bias 302 configured to cause the fluxonium qubit to rotate about the Y axis in a positive direction. The example charge bias pulse can remain, during a hold time 210, at or near the Y rotation charge bias 302. The example charge bias pulse can transition, during a second transition time 212, from the Y rotation charge bias 302 to another charge bias (e.g., at or near zero in the example depiction).

[0074] In some instances, a Y rotation charge bias 302 can be about 0.25 or -0.25. For example, in instances when a computational quantization axis is the Z axis and is associated with a charge bias of 0.0, a charge bias of about 0.25 or -0.25 can cause the fluxonium qubit to rotate about the Y axis (e.g., in opposite directions). In some instances, a Y rotation charge bias 302 can be a charge bias configured to cause an axis of rotation 308 of the fluxonium qubit (e.g., an instantaneous quantization axis) to be parallel with a Pauli Y axis.

[0075] In some instances, a Y rotation charge bias 302 can be determined based on anapproximate fluxonium Hamiltonian written as ^�^^^0where ^^^^^^^^is an externally controlled charge bias and ^^^^0is a value discussed above with respect to equation (16). For example, in some instances, such an approximate fluxonium Hamiltonian can be interpreted as the Hamiltonian of a spin-12 particle in a “magnetic field” whose direction (in the Y-Z plane) is set by the offset charge. In some instances, for example, single-qubit operations can be implemented by introducing nonadiabatic changes of the quantization axis of the spin. In some instances, nonadiabatic changes of the quantization axis of the spin can be introduced by controlling the direction of the magnetic field, i.e., the value of ng. In such instances, the equivalent spin can precess about the quantization axis of the spin at a rate equal to the qubit frequency ^^^^0 / ℏ, where ^^^^0and ℏ are values discussed above with respect to equations (15) and (16). In some instances, a state of the fluxonium qubit can be represented by a spin precessing around an instantaneous quantization axis. Example relationships between charge bias and rotation axis are further discussed below with respect to FIG.3B.

[0076] In some instances, a first transition time 208, hold time 210, and second transition time 212 can be determined based at least in part on a desired angle of rotation and a qubit frequency. For example, in some instances, a fluxonium qubit can precess around an instantaneous quantization axis at a rate equal to a qubit frequency ^^^^0 / ℏ. In such instances, a total time required to complete a desired angle of rotation ^^^^ can be approximately equal towith small adjustments to account for a smooth and not perfectly square shape of the charge- bias pulse. For example, in the case of a perfectly square pulse shape with transition times 208, 212 of exactly zero, an example hold time 210 can be exactly equal to|^^^^|(^^^^0 / ℏ). In example instances where a transition time 208, 212 is larger than zero, a total time associated with the charge-bias pulse (e.g., equal to a sum of transition and hold times 208, 210, 212) can be approximately equal to (e.g., slightly larger than)and the hold time 210 alone can besmaller (e.g., slightly smaller) than.

[0077] In some instances, a charge-bias pulse can have a pulse shape configured to reduce leakage. In some instances, reducing leakage can include selecting a pulse shape that suppresses high-frequency components that are resonant with fluxonium transitions outside the computational subspace. For example, in some instances, a charge-bias pulse can have a smooth (e.g., having one or more continuous derivatives) pulse shape configured to reduce leakage. For instance, a non-limiting example of a simple smooth pulse shape can be a flat- top-cosine pulse shape. In some instances, transition times 208, 212 can be numerically optimized to minimize gate infidelity. In some instances, numerically optimizing transition times 208, 212 to minimize gate infidelity can include selecting transition times 208, 212 that minimize leakage.

[0078] In some instances, numerically optimizing transition times 208, 212 can include simulating an effect of a candidate charge-bias pulse on a fluxonium qubit, and comparing the simulated effect to a desired effect (e.g., Y(−^^^^2) rotation, etc.). In some instances, simulating an effect of a charge-bias pulse on a fluxonium qubit can include solving a time-dependent Schrodinger equation for equation (1) above where ^^^^^^^^ → ^^^^^^^^(^^^^). In someinstances, a simulation can include simulating the time-evolving the Schrodinger equation in the eigenbasis of the fluxonium Hamiltonian. For example, in some instances, charge and phase operators in equation (1) above can be written in terms of bosonic ladder operators. Insome instances, the Hamiltonian can be diagonalized, and a plurality (e.g., 100, etc.) of eigenstates of the Hamiltonian can be used to propagate the time-dependent Schrodinger equation.

[0079] In some example experiments according to the present disclosure, example charge-bias pulses for performing a Y(−^^^^2) rotation were tested for multiple transition times 208, 212 and multiple fluxonium parameter sets (e.g., charging, Josephson, and inductive energies; qubit frequencies; etc.). In the experiments, average gate fidelities of example gates according to example aspects of the present disclosure were extremely high (e.g., 99.999 to 99.99999 percent, etc.) and robust to errors in drive amplitude ngor pulse timing. For example, in some example experiments, a range of Josephson energies (between 3 and 18 times a charging energy) was tested for each of a plurality of transition times 208, 212 and each of a plurality of inductive energies. In the example experiments, one or more transition times 208, 212 existed, for every combination of Josephson and inductive energies tested, for which unitary gate infidelities fell below 10-4, often by several orders of magnitude (e.g., about 10-12for some Josephson energies; about 10-8, about 10-6, etc.). In some example experiments, gate fidelities were tested for a range of drive amplitude ngvalues, and gate infidelities remained below 10-4for values within about 2-4 percent of an optimal drive amplitude ngvalue, indicating that gate fidelities can be robust to both drive amplitude deviations and timing errors. However, in some instances, best-case gate infidelities were even lower (e.g., below 10-7, etc.) for optimal drive amplitude ngvalues, indicating that fine- tuning a drive amplitude may be advantageous in some implementations. Additionally, example gating times were extremely fast in the example experiments. For example, in some example experiments (e.g., associated with a qubit frequency of 222 MHz, etc.), gating times for a Y(−^^^^2) rotation were as low as about 2-4 ns (e.g., 2.1 ns, etc.). Additional details regarding example experiments are provided in the Appendix to U.S. Provisional Patent App. No.63 / 561,022, which forms a part of this disclosure.

[0080] Although FIG.3A depicts an example positive charge bias, a negative charge bias can be used to rotate a fluxonium qubit about a Y axis without going outside the scope of the present disclosure. For example, in some instances, a charge bias at or near about -0.25 can also cause a fluxonium qubit to rotate about a Pauli Y axis (e.g., in a direction opposite a rotation caused by a charge bias of about 0.25). For example, in instances when a computational quantization axis is the Z axis and is associated with a charge bias of 0.0, acharge bias of about -0.25 can cause the fluxonium qubit to rotate about the Y axis (e.g., in a direction opposite a charge bias of about 0.25).

[0081] FIG.3B depicts an example relationship between a charge bias (e.g., Y rotation charge bias 302) and an axis of rotation 308 of the fluxonium qubit. A charge bias can cause a fluxonium qubit to rotate about an axis of rotation 308, which can form an angle 304 with respect to a computational quantization axis 306. In some instances, a computational quantization axis 306 can be a Z axis 310. In instances where a charge bias is a Y rotation charge bias 302, an axis of rotation 308 can be a Y axis 312.

[0082] In some instances, a computational quantization axis 306 can be, comprise, or otherwise correspond to an axis, direction, or dimension on which a state of the fluxonium qubit (e.g., basis state such as |0〉 or |1〉) is measured or defined. In some instances, a computational quantization axis 306 can be an axis associated with a computational state space associated with a quantum computation being performed using the fluxonium qubit.

[0083] The axis of rotation 308 can comprise an axis about which a state of a fluxonium qubit is rotating (e.g., in relation to a Bloch sphere). In some instances, the axis of rotation 308 can be the same as or different from the computational quantization axis 306. In some instances, the axis of rotation 308 can be an instantaneous quantization axis of the fluxonium qubit. In some instances, an instantaneous quantization axis at a given time t can depend on a charge bias ^^^^^^^^(^^^^) at the time t (e.g., as described below).

[0084] In some instances, an angle 304 between the axis of rotation 308 and the computational quantization axis 306 can be related to a charge bias ^^^^^^^^according to anapproximate fluxonium Hamiltonian written as ^�^^^0where ^^^^^^^^is an externally controlled charge bias. For example, in some instances, a fluxonium Hamiltonian can be interpreted as the Hamiltonian of a spin-12 particle in a “magnetic field” whose direction (in the Y-Z plane) is set by the offset charge, and a quantization axis of the spin can be proportional to the value of ng. In some instances, an angle 304 between the axis of rotation 308 and the computational quantization axis 306 can be equal to 2^^^^^^^^^^^^, where ^^^^^^^^is a charge bias (e.g., Y rotation charge bias 302). For example, in instances where a computational quantization axis 306 is associated with a charge bias of 0.0, an angle 304 can be equal to 2^^^^^^^^^^^^. In some instances, an axis of rotation 308 associated with a Y rotation charge bias 302 can have an angle 304 with a computational quantization axis 306 of π / 2 radians.

[0085] In some instances, a Z axis 310 and Y axis 312 can be a Pauli Z axis and Pauli Y axis, respectively.

[0086] FIG.4 depicts an example charge bias pulse for causing a fluxonium qubit to rotate about a Z axis 310 in a positive direction. Such a rotation can be used, for example, to perform a Pauli Z-gate, and can be used in combination with other rotations to implement any arbitrary single-qubit operation. The example charge bias pulse can transition, during a first transition time 208, from a pre-transition charge bias (e.g., at or near zero in the example depiction) to a positive-Z-rotation charge bias 402 configured to cause the fluxonium qubit to rotate about the Z axis in a positive direction. The example charge bias pulse can remain, during a hold time 210, at or near the positive-Z-rotation charge bias 402. The example charge bias pulse can transition, during a second transition time 212, from the positive-Z- rotation charge bias 402 configured to cause the fluxonium qubit to rotate about the Z axis in a positive direction to another charge bias (e.g., at or near zero in the example depiction).

[0087] In some instances, a positive-Z-rotation charge bias 402 can be about 0.5 or -0.5. For example, in instances when a computational quantization axis 306 is the Z axis and is associated with a charge bias of 0.0, a charge bias of about 0.5 or -0.5 can cause the fluxonium qubit to rotate about the Z axis (e.g., in a positive direction in both cases). In some instances, a positive-Z-rotation charge bias 402 can be a charge bias configured to cause an axis of rotation 308 of the fluxonium qubit (e.g., an instantaneous quantization axis) to be parallel with a Pauli Z axis. In some instances, an axis of rotation 308 associated with a positive-Z-rotation charge bias 402 can have an angle 304 with a computational quantization axis 306 of π radians. In some instances, a positive-Z-rotation charge bias 402 can be determined by determining a charge bias associated with an appropriate angle 304 between the Z axis 310 and a computational quantization axis 306 according to example systems and methods described with respect to FIGS.3A-3B.

[0088] In some instances, a first transition time 208, hold time 210, and second transition time 212 can be determined based at least in part on a desired angle of rotation and a qubit frequency. For example, in some instances, a fluxonium qubit can precess around an instantaneous quantization axis at a rate equal to a qubit frequency ^^^^0 / ℏ. In such instances, a total time required to complete a desired angle of rotation ^^^^ can be approximately equal towith small adjustments to account for a smooth and not perfectly square shape of the charge- bias pulse. For example, in the case of a perfectly square pulse shape with transition times208, 212 of exactly zero, an example hold time 210 can be exactly equal to|^^^^|. In exampleinstances where a transition time 208, 212 is larger than zero, a total time associated with the charge-bias pulse (e.g., equal to a sum of transition and hold times 208, 210, 212) can be approximately equal to (e.g., slightly larger than)(^^^^0 / ℏ), and the hold time 210 alone can be smaller (e.g., slightly smaller) than(^^^^0 / ℏ).

[0089] FIG.5 depicts an example charge bias pulse for causing a fluxonium qubit to rotate about the Z axis in a negative direction. Such a rotation can be used, for example, to perform a Pauli Z-gate, and can be used in combination with other rotations to implement any arbitrary single-qubit operation. In the example charge bias pulse, a charge bias can remain, throughout a hold time 210, at a negative-Z-rotation charge bias value 502 configured to cause the fluxonium qubit to rotate about the Z axis in a negative direction (e.g., at or near zero).

[0090] In some instances, a negative-Z-rotation charge bias 502 can be 0.0. For example, in instances when a computational quantization axis 306 is the Z axis and is associated with a charge bias of 0.0, a charge bias of 0.0 can cause the fluxonium qubit to rotate about the Z axis in a negative direction. In some instances, a negative-Z-rotation charge bias 502 can be a charge bias configured to cause an axis of rotation 308 of the fluxonium qubit (e.g., an instantaneous quantization axis) to be parallel with a Pauli Z axis. In some instances, an axis of rotation 308 associated with a negative-Z-rotation charge bias 502 can have an angle 304 with a computational quantization axis 306 of zero radians. In some instances, a negative-Z- rotation charge bias 502 can be determined by determining a charge bias associated with an appropriate angle 304 between the Z axis and a computational quantization axis 306 according to example systems and methods described with respect to FIGS.3A-3B.

[0091] In some instances, a hold time 210 can be determined based at least in part on a desired angle of rotation and a qubit frequency. For example, in some instances, a fluxonium qubit can precess around an instantaneous quantization axis at a rate equal to a qubit frequency ^^^^0 / ℏ. In such instances, a total time required to complete a desired angle of rotation ^^^^ can be approximately equal towith possible adjustments to account for effects of one or more transition times 208, 212 (e.g., as depicted in other figures). For example, in the case of a Z rotation performed withoutany transition times 208, 212, an example hold time 210 can in some instances be exactly equal to (^^^^0 / ℏ). For example, in instances where a computational quantization axis 306 is the Z axis and is associated with a charge bias of 0.0, an example Z rotation of angle ^^^^ can in some instances be performed using a charge bias pulse that remains at 0.0 for an example hold time 210 exactly equal

[0092] Although FIG.5 does not expressly depict a transition to or from a charge bias value other than the negative-Z-rotation charge bias 502, a charge-bias pulse for negative Z rotation can include one or more transitions without going outside the scope of the present disclosure. For example, in some instances, an identity operation may include one or more transitions between a negative-Z-rotation charge bias 502 and a positive-Z-rotation charge bias 402. As another example, an X rotation may include one or more transitions between a negative-rotation Z rotation charge bias 502 and a charge bias configured to cause a rotation about the Pauli Y axis. Other transitions are possible.

[0093] FIG.6 depicts an example charge bias pulse for causing a fluxonium qubit to rotate about the Pauli X axis. Such a rotation can be used, for example, to perform a Pauli X- gate, and can be used in combination with other rotations to implement any arbitrary single- qubit operation. The example charge bias can remain, throughout a first hold time 610, at a negative-Z-rotation charge bias 502. The example charge bias can transition to a Y rotation charge bias 302 and remain at the Y rotation charge bias 302 for a second hold time 612. The example charge bias can transition to a negative-Z-rotation charge bias 502 and remain there for a third hold time 614.

[0094] In some instances, a first hold time 610, second hold time 612, and third hold time 614 can be, comprise, or be comprised by a hold time 210. In addition to the depicted and labelled hold times 610, 612, 614, an example charge bias pulse of FIG.6 can in some instances be characterized by one or more transition times 208, 212 not labelled in FIG.6 (e.g., between first hold time 610 and second hold time 612; between second hold time 612 and third hold time 614; etc.).

[0095] In some instances, a first hold time 610, second hold time 612, and third hold time 614 can be determined based at least in part on a target X rotation angle ^^^^^^^^. For example, in some instances an X rotation having a magnitude of ^^^^^^^^can be decomposed intoone or more (e.g., two) Z rotations having magnitudes of ^^^^^^^^1,^^^^^^^^2, etc., and one or more (e.g.,one) Y rotations ^^^^^^^^1, etc. For example, in some instances, an X rotation (i.e., a rotation aboutan X axis) can be decomposed into Z and Y rotations according to standard geometrymethods, wherein an X, Y, and Z axis can be treated as orthogonal axes in a three- dimensional space. In such instances, a target rotation time associated with each Z and Y rotation can be determined based at least in part on a target rotation angle (e.g.,^^^^^^^^1,^^^^^^^^2,^^^^^^^^1, etc.) associated with that rotation. In such instances, a first hold time 610,second hold time 612, and third hold time 614 can be determined according to methods described with respect to FIGS.3-5 for determining hold times 210.

[0096] Although FIG.6 depicts two Z rotations performed using a negative-Z-rotation charge bias 502, an X rotation can also be performed using one or more Z rotations using a positive-Z-rotation charge bias 402 without going outside the scope of the present disclosure. For example, a negative rotation of N radians (D degrees) can in some instances be equivalentto a positive rotation of 2^^^^ − ^^^^ radians (360 – D degrees), and equivalent rotations can insome instances be used interchangeably. However, in some instances, two rotationally equivalent operations can have differing properties (e.g., different gating times, fidelities, etc.) that may make one operation preferable to another for a particular use case. For example, in instances where a negative-Z-rotation charge bias 502 is equal to 0.0 or equal to a charge bias associated with a computational quantization axis 306, a negative-Z-rotation charge bias 502 may be associated with increased fidelity compared to a positive-Z-rotation charge bias 402. As another example, in instances where a desired Z rotation is a small positive rotation (e.g., less than ^^^^ radians, etc.), a positive-Z-rotation charge bias 402 may be associated with faster gating times than a negative-Z-rotation charge bias 502. In some instances, an X rotation can include one or more (e.g., two) negative Z rotations; one or more (e.g., two) positive Z rotations; or any combination thereof (e.g., one positive and one negative). Similarly, in some instances, an X rotation can include one or more (e.g., one) negative Y rotations; one or more (e.g., one) positive Y rotations; or any combination thereof.

[0097] FIG.7 depicts an example charge bias pulse for causing a fluxonium qubit to perform an identity operation. Such a rotation can be used, for example, alone or in combination with other rotations to implement any arbitrary single-qubit operation. The example charge bias can transition to a positive-Z-rotation charge bias 402 and remain, throughout a first hold time 710, at the positive-Z-rotation charge bias 402. The example charge bias can transition to a negative-Z-rotation charge bias 502 and remain at the negative- Z-rotation charge bias 502 for a second hold time 712. The example charge bias can transition to a second positive-Z-rotation charge bias 702 and remain there for a third holdtime 714. The example charge bias can transition to a negative-Z-rotation charge bias 502 and remain there for a fourth hold time 716.

[0098] In some instances, a second positive-Z-rotation charge bias 702 can be about 0.5 or -0.5. In some instances, the second positive-Z-rotation charge bias 702 can be different from a positive-Z-rotation charge bias 402. For example, in some instances, a second positive-Z-rotation charge bias 702 can have a similar (e.g., same) magnitude and opposite sign compared to a positive-Z-rotation charge bias 402. For example, in instances when a computational quantization axis 306 is the Z axis and is associated with a charge bias of 0.0, a charge bias of about 0.5 or -0.5 can each cause the fluxonium qubit to rotate about the Z axis (e.g., in a positive direction in both cases). In some instances, a second positive-Z- rotation charge bias 702 can be a charge bias configured to cause an axis of rotation 308 of the fluxonium qubit (e.g., an instantaneous quantization axis) to be parallel with a Pauli Z axis. In some instances, an axis of rotation 308 associated with a second positive-Z-rotation charge bias 702 can have an angle 304 with a computational quantization axis 306 of π or -π radians. In some instances, a second positive-Z-rotation charge bias 702 can be determined by determining a charge bias associated with an appropriate angle 304 between the Z axis and a computational quantization axis 306 according to example systems and methods described with respect to FIGS.3A-3B.

[0099] In some instances, an identity operation can be configured to avoid any unwanted Y rotations. For example, in some instances, an identity operation that transitions past a Y rotation charge bias 302 can be configured to counteract any unwanted Y rotations caused by that transition. For example, in some instances, a positive-Z-rotation charge bias 402 and second positive-Z-rotation charge bias 702 can be configured so that a transition past a Y rotation charge bias 302 (e.g., charge bias of about 0.25) can be paired with a transition past a second charge bias (e.g., about -0.25) causing a Y rotation in an opposite direction compared to the Y rotation charge bias 302. Pairing transitions past Y rotation charge biases can include, for example, selecting a positive-Z-rotation charge bias 402 and second positive-Z- rotation charge bias 702 with opposite signs (e.g., 0.5 and -0.5). Pairing transitions past Y rotation charge biases can include, for example, shaping a charge bias pulse so that transitions associated with opposite signs are associated with similar (e.g., same) transition times 208, 212 or similar pulse shapes (e.g., having slopes of similar or same magnitude and opposite sign when passing Y rotation charge biases, etc.). An identity operation comprising two or more positive-Z-rotation charge biases 402, 702 having opposite signs can be viewed, for example, as a concatenation of two smaller identity operations (e.g. charge bias pulse ofabout 0 ^ 0.5 ^ 0, concatenated with charge bias pulse of about 0 ^ −0.5 ^ 0) with abuffer time (e.g., second hold time 612).

[0100] In some instances, a first hold time 710, second hold time 712, third hold time 714, and fourth hold time 716 can be, comprise, or be comprised by one or more hold times 210. In some instances, an example charge-bias pulse of FIG.7 can be characterized by one or more transition times 208, 212 (e.g., between one or more hold times 710, 712, 714, 716, etc.).

[0101] In some instances, a first hold time 710, second hold time 712, third hold time 714, and fourth hold time 716 can be selected based at least in part on a target total time associated with the identity operation. In other words, the hold times 710, 712, 714, 716 can be selected based at least in part on a target duration of the identity operation. For example, in some instances a sum of the first hold time 710, second hold time 712, third hold time 714, fourth hold time 716, and any transition times 208, 212 associated with the example charge- bias pulse can be equal to the target total time or target duration of the identity operation.

[0102] In some instances, a target total time associated with the identity operation can be determined based on a target total time associated with a plurality of operations (e.g., Z, Y, and X rotations and identity operations). A target total time associated with a plurality of operations can be determined, for example, based on a comparison (e.g., mathematical difference) between a current time and a target time at which a fluxonium qubit must be in a target state. For example, a target qubit state and a target final time can be obtained. Based on the target final time and a starting time at which a fluxonium qubit control operation is started, a target total operation time can be determined. Based on the target qubit state, a plurality of transition times 208, 212 and hold times 210 can be determined for rotating the qubit into the target qubit state (e.g., via a plurality of Z rotations and Y rotations). Based on the target total operation time and the target transition and hold times 208, 210, and 212, a target identity operation time can be obtained (e.g., total operation time minus sum of all relevant transition and hold times).

[0103] In some instances, the first hold time 710, second hold time 712, third hold time 714, and fourth hold time 716 can be configured to cause no net rotation (e.g., net Z rotationof zero radians, etc.). For example, target rotation angles ^^^^^^^^1,^^^^^^^^2, etc. can be selected suchthat a sum of the target rotation angles is zero, and a plurality of hold times 710, 712, 714, 716 can be determined based at least in part on the target rotation angles according to methods for determining hold times 210 as described in FIGS.3-5. In some instances, a buffer time (e.g., second hold time 612) can be chosen so that a total accumulated phasevanishes, while spurious rotations and leakage are suppressed. This condition can be satisfied in most cases using charge-bias pulses according to example aspects of the present disclosure, thanks to self-echoing properties of example pulse schemes.

[0104] In some instances, example control pulses of the present disclosure can be advantageously combined with additional qubit control methods. For example, in some instances, a phase gate (e.g., unit-fidelity phase gate) can be appended to the end of an identity operation sequence. In some instances, such a phase gate can, for example, prevent or compensate for unwanted Z rotations.

[0105] Although FIG.7 depicts a four-segment identity operation comprising four hold times, an identity operation can include a different number of segments (e.g., two, three, eight, etc.) without going outside the scope of the present disclosure. Example Quantum Computing Systems

[0106] FIG.8 depicts an example quantum computing system 800. The example system 800 is an example of a system on one or more classical computers or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing structures or systems can be used without deviating from the scope of the present disclosure.

[0107] The system 800 includes quantum hardware 802 in data communication with one or more classical processors 804. The quantum hardware 802 includes components for performing quantum computation. For example, the quantum hardware 802 includes a quantum system 810, control device(s) 812, and readout device(s) 814 (e.g., readout resonator(s)). The quantum system 810 can include one or more multi-level quantum subsystems, such as a register of qubits. In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0108] The type of multi-level quantum subsystems that the system 800 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 814 attached to one or more superconducting qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0109] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 810 via multiple control lines that are coupled to one or more control devices 812. Example control devices 812 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 812 may be configured to operate on the quantum system 810 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 812 may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0110] The quantum hardware 802 may further include readout devices 814 (e.g., readout resonators). Measurement results 808 obtained via measurement devices may be provided to the classical processors 804 for processing and analyzing. In some implementations, the quantum hardware 802 may include a quantum circuit and the control device(s) 812 and readout devices(s) 814 may implement one or more quantum logic gates that operate on the quantum system 802 through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 802. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0111] The readout device(s) 814 may be configured to perform quantum measurements on the quantum system 810 and send measurement results 808 to the classical processors 804. In addition, the quantum hardware 802 may be configured to receive data specifying physical control qubit parameter values 806 from the classical processors 804. The quantum hardware 802 may use the received physical control qubit parameter values 806 to update the action of the control device(s) 812 and readout devices(s) 814 on the quantum system 810. For example, the quantum hardware 802 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 812 and may update the action of the DACs on the quantum system 810 accordingly. The classical processors 804 may be configured to initialize the quantum system 810 in an initial quantum state, e.g., by sending data to the quantum hardware 802 specifying an initial set of parameters 806.

[0112] The readout device(s) 814 can take advantage of a difference in the impedance for the |0〉 and |1〉 states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readoutresonator can take on different values when a qubit is in the state |0〉 or the state |1〉, due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 814 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 814 to impede microwave propagation at the qubit frequency.

[0113] In some implementations, the quantum system 810 can include a plurality of qubits 820 arranged, for instance, in a two-dimensional grid 822. For clarity, the two- dimensional grid 822 depicted in FIG.1 includes 16 qubits arranged in a square formation, however in some implementations the system 810 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 820 can interact with each other through multiple qubit couplers, e.g., qubit coupler 824. The qubit couplers can define nearest neighbor interactions between the multiple qubits 820. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 800 may be couplers with a fixed coupling strength. In some implementations, the multiple qubits 820 may include data qubits, such as qubit 826 and measurement qubits, such as qubit 828. A data qubit is a qubit that participates in a computation being performed by the system 800. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit. That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0114] In some implementations, each qubit in the multiple qubits 820 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 820 can be chosen before a computation is performed by the calibration system. Some operating frequencies are better than other operating frequencies. One metric for assessing how good a particular operating frequency is for a particular qubit is energy relaxation time (T1) for the qubit at the frequency. Lower energy relaxation times can lead to larger quantum computational errors.

[0115] In various implementations, the example system 800 can be implemented as a client device, a server device, or both. The example system 800 can be implemented as part of a distributed computing system. The example system 800 can be implemented along with other example systems, which may be the same or different. The example system 800 can beimplemented in a server farm or other facility that operates multiple computing systems to provide computational services to or on behalf of a plurality of client systems. Advantageously, techniques according to example aspects of the present disclosure can provide for improved calibration and maintenance of computing facilities, increasing service uptime, decreasing failure rates, etc. Example Methods

[0116] Figure 9 depicts a flowchart diagram of an example method for controlling a fluxonium qubit according to example embodiments of the present disclosure. Although Figure 9 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of example method 900 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0117] At 902, example method 900 can include obtaining a target rotation angle to rotate a qubit (e.g., fluxonium-like qubit 106) around at least one axis of rotation. In some instances, the at least one axis of rotation can include a Z axis 310. In some instances, the at least one axis of rotation can include a Y axis 312. In some instances, the at least one axis of rotation can include an X axis, and example method 900 at 902 can include determining, based on a target angle of rotation around the X axis, a target angle of rotation around the Z axis and a target angle of rotation around the Y axis. In some instances, the at least one axis of rotation can be, comprise, or be comprised by an axis of rotation 308. In some instances, the qubit can include a qubit having an inductance, a capacitance, and a Josephson junction in parallel. In some instances, the qubit can include a qubit having a fluxonium mode. In some instances, the qubit can include a qubit having an effective Hamiltonian comprising an inductive energy term and a phase slip energy term. In some instances, example method 900 at 902 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0118] At 904, example method 900 can include obtaining a qubit frequency associated with the qubit. In some instances, obtaining the qubit frequency can comprise retrieving the qubit frequency (e.g., from non-transitory computer-readable media, etc.); receiving the qubit frequency (e.g., from a user, computing device, etc.); or determining the qubit frequency (e.g., based on other parameters associated with the fluxonium qubit, such as ^^^^0and ℏ. In some instances, example method 900 at 904 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0119] At 906, example method 900 can include determining, based at least in part on the target rotation angle and the qubit frequency, a rotation time. In some instances, the at least one axis of rotation can include an X axis, and example method 900 at 906 can include determining, based on a target angle of Z rotation and a target angle of Y rotation, a Z rotation time and a Y rotation time. In some instances, a rotation time can be, comprise, or be comprised by a transition time 208, 212 or hold time 210, 610-614, 710-716. In some instances, a rotation time can comprise a sum of a plurality of transition times 208, 212 and / or hold times 210, 610-614, 710-716. In some instances, example method 900 at 906 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0120] At 908, example method 900 can include determining, based at least in part on the rotation time and the at least one axis of rotation, a target pulse shape for a time-dependent charge-bias pulse. In some instances, the target pulse shape can include a non-square pulse shape configured to suppress a leakage. In some instances, a non-square pulse shape can be configured to suppress high-frequency components that are resonant with fluxonium transitions outside a computational subspace of the qubit. In some instances, a non-square pulse shape can include a first transition portion in which a charge bias of the non-square pulse shape is changing over a first amount of time; a hold portion in which the charge bias of the non-square pulse shape is constant over a second amount of time; and a second transition portion in which the charge bias of the non-square pulse shape is changing over a third amount of time, in a direction opposite a direction of change associated with the first transition portion. In some instances, determining the target pulse shape can include determining, based on the at least one axis of rotation and a direction of rotation, at least one charge bias configured to cause the qubit to rotate around the axis of rotation in the direction of rotation. In some instances, example method 900 can include an identity operation, and a target pulse shape can comprise a first portion configured to cause the fluxonium qubit to perform a first rotation around the at least one axis and a second portion configured to cause the fluxonium qubit to perform a second rotation around the at least one axis, wherein the second rotation is in a direction of rotation opposite a direction of rotation of the first rotation. In some instances, a target pulse shape can include a first portion having a positive charge bias and a second portion having a negative charge bias. In some instances, a target pulse shape can include a buffer portion between a first portion and a second portion, wherein the buffer portion is configured to reduce a total accumulated phase of the fluxonium qubit. In some instances, a charge-bias pulse can comprise a charge bias 202. In some instances,example method 900 at 908 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0121] At 910, example method 900 can include providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse. In some instances, a baseband- voltage drive can be, comprise, or be comprised by a baseband voltage drive 204. In some instances, example method 900 at 910 can include using one or more systems or performing one or more activities described with respect to FIGS.1-2.

[0122] Figure 10 depicts a flowchart diagram of an example method for controlling a fluxonium qubit according to example embodiments of the present disclosure. Although Figure 10 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of example method 1000 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0123] At 1002, example method 1000 can include obtaining a target rotation for a qubit (e.g., fluxonium-like qubit 106). In some instances, the qubit can include a qubit having an inductance, a capacitance, and a Josephson junction in parallel. In some instances, the qubit can include a qubit having a fluxonium mode. In some instances, the qubit can include a qubit having an effective Hamiltonian comprising an inductive energy term and a phase slip energy term. In some instances, a target rotation can include one or more axes of rotation 308 and one or more corresponding angles of rotation. In some instances, example method 1000 at 1002 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0124] At 1004, example method 1000 can include determining, based at least in part on the target rotation, a target Z rotation about a Z axis and a target Y rotation about a Y axis. In some instances, a target Y rotation or target Z rotation can be zero or equivalent to zero (e.g., 2^^^^ radians, etc.), or can be any non-zero (e.g., positive, negative, etc.) value. In some instances, a Z axis can be, comprise, or be comprised by a Z axis 310. In some instances, a Y axis can be, comprise, or be comprised by a Y axis 312. In some instances, example method 1000 at 1004 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0125] At 1006, example method 1000 can include determining, based at least in part on the target Z rotation and the target Y rotation, a Z rotation time and a Y rotation time. In some instances, a Z rotation time or Y rotation time can be zero or non-zero. In someinstances, a Z rotation time or Y rotation time can be, comprise, or be comprised by a transition time 208, 212 or hold time 210, 610-614, 710-716. In some instances, example method 1000 at 1006 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0126] At 1008, example method 1000 can include obtaining a target total rotation time for the fluxonium qubit. In some instances, a target total rotation time can comprise a sum of one or more transition times 208, 212 and hold times 210, 610-614, 710-716. In some instances, example method 1000 at 1008 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0127] At 1010, example method 1000 can include determining, based on the Z rotation time, the Y rotation time, and the target total rotation time, an identity rotation time. In some instances, an identity rotation time can be zero or non-zero. In some instances, an identity rotation time can comprise a sum of one or more transition times 208, 212 and hold times 710-716. In some instances, example method 1000 at 1010 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0128] At 1012, example method 1000 can include determining, based at least in part on the Z rotation time, the Y rotation time, and the identity rotation time, a target pulse shape for a time-dependent charge-bias pulse. In some instances, the target pulse shape can be configured to cause the fluxonium qubit to rotate about the Z axis for the Z rotation time. In some instances, the target pulse shape can be configured to cause the fluxonium qubit to rotate about the Y axis for the Y rotation time. In some instances, the target pulse shape can include a Y rotation portion comprising a charge bias configured to cause the fluxonium qubit to rotate about the Y axis for the Y rotation time. In some instances, the Y rotation portion can include a first transition portion in which a charge bias of the Y rotation portion is changing over a first amount of time; a hold portion in which the charge bias of the Y rotation portion is constant over a second amount of time; and a second transition portion in which the charge bias of the Y rotation portion is changing over a third amount of time, in a direction opposite a direction of change associated with the first transition portion. In some instances, a charge-bias pulse can comprise a charge bias 202. In some instances, example method 1000 at 1012 can include using one or more systems or performing one or more activities described with respect to FIGS.3-7.

[0129] At 1014, example method 1000 can include providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse. In some instances, a baseband- voltage drive can be, comprise, or be comprised by a baseband voltage drive 204. In someinstances, example method 1000 at 1014 can include using one or more systems or performing one or more activities described with respect to FIGS.1-2.

[0130] FIG.11 depicts an example method 1100 for performing a quantum computation using a quantum circuit according to example aspects of the present disclosure. For example, a quantum circuit can include a fluxonium qubit 106 in some instances. Although FIG.11 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of the method 1100 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure. The method 700 can be implemented by any suitable computing system, such as a quantum computing system including quantum hardware in communication with one or more quantum control devices, such as quantum computing system 800 of FIG.8.

[0131] At 1102, example method 1100 can include obtaining data indicative of a quantum circuit. Obtaining data can include, for example, receiving data from a computing device (e.g. user device, server device); receiving data from a user (e.g. via input / output device); reading data from one or more non-transitory computer-readable media; generating data (e.g. using an algorithm); etc. Data indicative of a quantum circuit can include, for example, a circuit design, circuit diagram, one or more unitary matrices, software code (e.g. quantum software code in a quantum computing language), etc.

[0132] At 1104, example method 1100 can include preparing one or more qubits in a known quantum state. Preparing one or more qubits in a known quantum state can include, for example, preparing one or more qubits in a known basis state (e.g. by manipulating a plurality of qubits such that qubits characterized by a particular basis state, e.g. |0〉 or |1〉, can be separated from qubits not characterized by that basis state (e.g. physically separated, separately identified, etc.). Preparing one or more qubits in a known quantum state can include, for example, using a control device 812 to perform quantum gating to generate a known multi-qubit basis state. Preparing one or more qubits can include using a control device 812 in a manner described with respect to FIG.8.

[0133] At 1106, example method 1100 can include applying one or more quantum gates to one or more qubits to execute a quantum algorithm. For example, in some instances control devices 812 can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc., in a manner described with respect to FIG.8

[0134] At 1108, example method 1100 can include measuring, using a readout apparatus, a state of at least one of the one or more qubits. The readout apparatus can be, for example, a readout device 814, and step 1106 can in some instances be performed in a manner described with respect to FIG.8. Example Computing Systems

[0135] FIG.12 depicts a block diagram of an example computing system 5 that can perform aspects of example embodiments of the present disclosure. The system 5 includes a computing device 50, a server computing system 60, and a third-party system 70 that are communicatively coupled over a network 49. The system 5 also includes a quantum computing system 80 that is communicatively coupled to the server computing system.

[0136] The computing device 50 can be any type of computing device (e.g., classical computing device), such as, for example, a mobile computing device (e.g., smartphone or tablet), a personal computing device (e.g., laptop or desktop), a workstation, a cluster, a gaming console or controller, a wearable computing device, an embedded computing device, or any other type of computing device. In some embodiments, the computing device 50 can be a client computing device or a server computing device. The computing device 50 can include one or more processors 51 and a memory 52. The one or more processors 51 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 52 can include one or more non-transitory computer- readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 52 can store data 53 and instructions 54 which are executed by the processor 51 to cause the user computing device 50 to perform operations as described herein.

[0137] The computing device 50 can also include one or more input components that receive user input. For example, a user input component can be a touch-sensitive component (e.g., a touch-sensitive display screen or a touch pad) that is sensitive to the touch of a user input object (e.g., a finger or a stylus). The touch-sensitive component can serve to implement a virtual keyboard. Other example user input components include a microphone, a traditional keyboard, or other means by which a user can provide user input.

[0138] The quantum computing system 80 can include one or more processors 81 (e.g., classical processor(s) 804) and a memory 82. The one or more processors 81 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors thatare operatively connected. The memory 82 can include one or more non-transitory computer- readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 82 can store data 83 and instructions 84 which are executed by the processor 81 to cause the quantum computing system 80 to perform operations as described herein.

[0139] The quantum computing system 80 can also include a quantum system 85 for performing quantum computations. In some instances, the quantum system 85 can be, comprise, or be comprised by quantum hardware 802, described above with reference to FIG. 8.

[0140] In some implementations, the quantum computing system can 80 include or be otherwise implemented by one or more server computing systems 60. In instances in which the quantum computing system 80 includes plural server computing devices, such server computing devices can operate according to sequential computing architectures, parallel computing architectures, or some combination thereof.

[0141] The third-party system 70 can include one or more processors 71 and a memory 72. The one or more processors 71 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 72 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 72 can store data 73 and instructions 74 which are executed by the processor 71 to cause the third-party system 70 to perform operations. In some implementations, the third- party system 70 includes or is otherwise implemented by one or more server computing devices.

[0142] The server computing system 60 can include one or more processors 61 and a memory 62. The one or more processors 61 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 62 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 62 can store data 63 and instructions 64 which are executed by the processor 61 to cause the server computing system 60 to perform operations. In some implementations, the server computing system 60 includes or is otherwise implemented by one or more server computing devices.

[0143] The network 49 can be any type of communications network (e.g., classical or quantum), such as a local area network (e.g., intranet), wide area network (e.g., Internet), or some combination thereof and can include any number of wired or wireless links. In general, communication over the network 49 can be carried via any type of wired or wireless connection, using a wide variety of communication protocols (e.g., TCP / IP, HTTP, SMTP, FTP), encodings or formats (e.g., HTML, XML), or protection schemes (e.g., VPN, secure HTTP, SSL).

[0144] FIG.12 illustrates one example computing system that can be used to implement the present disclosure. Other computing systems can be used as well. For example, in some implementations, the quantum computing system 80 can include the server computing system 60 or vice versa. In some implementations, the quantum computing system 80 may be communicatively coupled through the network 49 to the computing device 50, third-party system 70, or server computing system 60.

[0145] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0146] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs (e.g., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus). The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0147] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit (i.e., a system that defines the unit of quantum information). It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qubits) are possible.

[0148] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0149] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0150] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0151] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0152] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0153] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / orquantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0154] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0155] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; and CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0156] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0157] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0158] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0159] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

[0160] Aspects of the disclosure have been described in terms of illustrative implementations thereof. Numerous other implementations, modifications, or variations within the scope and spirit of the appended claims can occur to persons of ordinary skill in the art from a review of this disclosure. Any and all features in the following claims can be combined or rearranged in any way possible. Accordingly, the scope of the present disclosure is by way of example rather than by way of limitation, and the subject disclosure does not preclude inclusion of such modifications, variations or additions to the present subject matter as would be readily apparent to one of ordinary skill in the art. Moreover, terms are describedherein using lists of example elements joined by conjunctions such as “and,” “or,” “but,” etc. It should be understood that such conjunctions are provided for explanatory purposes only. Lists joined by a particular conjunction such as “or,” for example, can refer to “at least one of” or “any combination of” example elements listed therein, with “or” being understood as “and / or” unless otherwise indicated. Also, terms such as “based on” should be understood as “based at least in part on.”

[0161] Those of ordinary skill in the art, using the disclosures provided herein, will understand that the elements of any of the claims, operations, or processes discussed herein can be adapted, rearranged, expanded, omitted, combined, or modified in various ways without deviating from the scope of the present disclosure. Some of the claims are described with a letter reference to a claim element for exemplary illustrated purposes and is not meant to be limiting. The letter references do not imply a particular order of operations. For instance, letter identifiers such as (a), (b), (c),..., (i), (ii), (iii),..., etc. can be used to illustrate operations. Such identifiers are provided for the ease of the reader and do not denote a particular order of steps or operations. An operation illustrated by a list identifier of (a), (i), etc. can be performed before, after, or in parallel with another operation illustrated by a list identifier of (b), (ii), etc.

Claims

WHAT IS CLAIMED IS:

1. A quantum computing system, comprising: a qubit comprising an inductance, a capacitance, and a Josephson junction connected in parallel; and a control system configured to implement a control signal to implement a quantum gate on the qubit, wherein the control signal comprises a voltage-bias pulse.

2. The quantum computing system of claim 1, wherein the quantum gate comprises an arbitrary single-qubit operation comprising one or more of: an identity operation; one or more rotations about a Z axis; one or more rotations about a Y axis; and one or more rotations about an X axis.

3. The quantum computing system of claim 1, wherein implementing the quantum gate comprises performing at least one rotation around at least one axis of rotation by: obtaining a target rotation angle; obtaining a qubit frequency associated with the qubit; determining, based at least in part on the target rotation angle and the qubit frequency, a rotation time; determining, based at least in part on the rotation time and the at least one axis of rotation, a target pulse shape for a time-dependent charge-bias pulse; and providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse.

4. The quantum computing system of claim 3, wherein the at least one axis of rotation comprises a Z axis.

5. The quantum computing system of claim 3, wherein the at least one axis of rotation comprises a Y axis.

6. The quantum computing system of claim 3, wherein the at least one axis of rotation comprises an X axis, and performing the at least one rotation comprises:obtaining a first target angle of rotation about the X axis; determining, based at least in part on the first target angle of rotation about the X axis, a second target angle of rotation about a Y axis and a third target angle of rotation about a Z axis; determining, based at least in part on the second target angle of rotation and the qubit frequency, a Y rotation time; and determining, based at least in part on the third target angle of rotation and the qubit frequency, a Z rotation time; wherein the target pulse shape for the time-dependent charge-bias pulse is based at least in part on the Y rotation time and the Z rotation time.

7. The quantum computing system of claim 3, wherein implementing the quantum gate comprises performing at least one identity operation, wherein the identity operation comprises: performing the at least one rotation around the at least one axis of rotation; and performing at least a second rotation around the at least one axis of rotation; wherein the second rotation is configured to rotate the qubit around the at least one axis, in a direction opposite a direction of the at least one rotation.

8. The quantum computing system of claim 7, wherein the target pulse shape comprises a first portion having a positive charge bias and a second portion having a negative charge bias.

9. The quantum computing system of claim 8, wherein the target pulse shape comprises a buffer portion between the first portion and the second portion, wherein the buffer portion is configured to reduce a total accumulated phase of the qubit.

10. The quantum computing system of claim 3, wherein determining the target pulse shape comprises determining, based on the at least one axis of rotation and a direction of rotation, at least one charge bias configured to cause the qubit to rotate around the axis of rotation in the direction of rotation.

11. The quantum computing system of claim 3, wherein the target pulse shape comprises a non-square pulse shape configured to suppress a leakage.

12. The quantum computing system of claim 11, wherein the non-square pulse shape is configured to suppress high-frequency components that are resonant with fluxonium transitions outside a computational subspace of the qubit.

13. The quantum computing system of claim 11, wherein the non-square pulse shape comprises a first transition portion in which a charge bias of the non-square pulse shape is changing over a first amount of time; a hold portion in which the charge bias of the non-square pulse shape is constant over a second amount of time; and a second transition portion in which the charge bias of the non-square pulse shape is changing over a third amount of time, in a direction opposite a direction of change associated with the first transition portion.

14. A method for providing a baseband-voltage drive to control a fluxonium-like qubit, comprising: obtaining a target rotation for a qubit comprising a circuit having a fluxonium mode; determining, based at least in part on the target rotation, a target Z rotation about a Z axis and a target Y rotation about a Y axis; determining, based at least in part on the target Z rotation and the target Y rotation, a Z rotation time and a Y rotation time; determining, based at least in part on the Z rotation time and the Y rotation time, a target pulse shape for a time-dependent charge-bias pulse; and providing a baseband-voltage drive configured to generate the time-dependent charge-bias pulse; wherein the target pulse shape is configured to cause the fluxonium qubit to: rotate about the Z axis for the Z rotation time; and rotate about the Y axis for the Y rotation time.

15. The method of claim 14, further comprising: obtaining a target total rotation time; and determining, based on the Z rotation time, the Y rotation time, and the target total rotation time, an identity rotation time; wherein the target pulse shape is further configured to cause the fluxonium qubitto: perform an identity operation, wherein a duration of the identity operation is equal to the identity rotation time.

16. The method of claim 15, wherein the target pulse shape comprises a first portion having a positive charge bias and a second portion having a negative charge bias.

17. The method of claim 16, wherein the target pulse shape comprises a buffer portion between the first portion and the second portion, wherein the buffer portion is configured to reduce a total accumulated phase of the fluxonium qubit.

18. The method of claim 14, wherein the target pulse shape comprises a Y rotation portion comprising a charge bias configured to cause the fluxonium qubit to rotate about the Y axis for the Y rotation time.

19. The method of claim 18, wherein the Y rotation portion comprises: a first transition portion in which a charge bias of the Y rotation portion is changing over a first amount of time; a hold portion in which the charge bias of the Y rotation portion is constant over a second amount of time; and a second transition portion in which the charge bias of the Y rotation portion is changing over a third amount of time, in a direction opposite a direction of change associated with the first transition portion.

20. A method for providing a baseband-voltage drive to control a fluxonium-like qubit, comprising: obtaining a target rotation for a qubit having an effective Hamiltonian comprising an inductive energy term and a phase-slip energy term; determining, based at least in part on the target rotation, a target Z rotation about a Z axis and a target Y rotation about a Y axis; determining, based at least in part on the target Z rotation and the target Y rotation, a Z rotation time and a Y rotation time; determining, based at least in part on the Z rotation time and the Y rotation time, a target pulse shape for a time-dependent charge-bias pulse; andproviding a baseband-voltage drive configured to generate the time-dependent charge-bias pulse; wherein the target pulse shape is configured to cause the qubit to: rotate about the Z axis for the Z rotation time; and rotate about the Y axis for the Y rotation time.

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