FLUXONIUM AND FLUXONIUM-like qubit control via baseband voltage pulses

Abstract

The invention provides a system and a method for controlling a fluxonium-like quantum bit. In one example, a quantum computing system may include qubits. The qubit may include an inductor, a capacitor, 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 a qubit. The control signal may include a voltage bias pulse.

Description

Technical Field

[0001] This disclosure generally relates to systems and methods for quantum computing.

[0002] Cross-reference to related applications

[0003] This application is based on and claims priority to U.S. Provisional Patent Application No. 63 / 581,747, filed September 11, 2023; U.S. Provisional Patent Application No. 63 / 561,022, filed March 4, 2024; and U.S. Provisional Patent Application No. 63 / 562,390, filed March 7, 2024; the disclosure of each of these U.S. Provisional Patent Applications is incorporated herein by reference in its entirety for all purposes. Background Technology

[0004] Quantum computing is a computational method that utilizes quantum effects, such as superposition of ground states and entanglement, to perform specific calculations more efficiently than classical digital computers. Compared to digital computers that store and process information in bits (e.g., "1" or "0"), quantum computing systems can process information using qubits. A qubit can refer to a quantum device capable of superimposing multiple states, such as data in both "0" and "1" states, and / or to the superposition of data itself in multiple states. In conventional terminology, the superposition of "0" and "1" states in a quantum system can be represented, for example, as a... +b The "0" and "1" states of a digital computer are similar to those of a qubit. and Ground state. Summary of the Invention

[0005] Aspects and advantages of embodiments of this disclosure will be set forth in part in the description which follows, or may be learned from the description or by practice of the embodiments.

[0006] An exemplary aspect of this disclosure provides an example quantum computing system. In some implementations, the quantum computing system may include qubits. In some implementations, qubits may include inductors, capacitors, and Josephson junctions connected in parallel. In some implementations, the quantum computing system may include a control system configured to implement control signals to implement quantum gates on the qubits. In some implementations, the control signals may include voltage bias pulses.

[0007] An exemplary aspect of this disclosure provides an example method. In some implementations, the method may include obtaining a target rotation for a qubit having a fluxonium mode. In some implementations, the method may include determining a target Z-rotation about a Z-axis and a target Y-rotation about a Y-axis based at least in part on the target rotation. In some implementations, the method may include determining a Z-rotation time and a Y-rotation time based at least in part on the target Z-rotation and the target Y-rotation. In some implementations, the method may include determining a target pulse shape for a time-dependent charge bias pulse based at least in part on the Z-rotation time and the Y-rotation time. In some implementations, the method may include providing a baseband voltage drive configured to generate a time-dependent charge bias pulse. In the example method, the target pulse shape may be configured to cause the qubit to rotate about the Z-axis for a Z-rotation time and rotate about the Y-axis for a Y-rotation time.

[0008] An exemplary aspect of this disclosure provides an exemplary method. This exemplary method may include obtaining a target rotation for a qubit having an effective Hamiltonian, the effective Hamiltonian including an induced energy term and a phase-slip energy term. In some implementations, the method may include determining a target Z-rotation about a Z-axis and a target Y-rotation about a Y-axis based at least in part on the target rotation. In some implementations, the method may include determining a Z-rotation time and a Y-rotation time based at least in part on the target Z-rotation and the target Y-rotation. In some implementations, the method may include determining a target pulse shape for a time-dependent charge bias pulse based at least in part on the Z-rotation time and the Y-rotation time. In some implementations, the method may include providing a baseband voltage drive configured to generate a time-dependent charge bias pulse. In the exemplary method, the target pulse shape may be configured to cause the qubit to rotate about the Z-axis for a Z-rotation time and rotate about the Y-axis for a Y-rotation time.

[0009] These and other features, aspects, and advantages of the various embodiments of this disclosure will be better understood with reference to the following description and the appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate exemplary embodiments of the disclosure and, together with the description, explain the relevant principles. Attached Figure Description

[0010] Referring to the accompanying drawings, a detailed discussion of embodiments is set forth in this specification for those skilled in the art, in which:

[0011] Figure 1 An example quantum computing system according to an example aspect of this disclosure is described;

[0012] Figure 2AAn example baseband voltage drive according to an example aspect of this disclosure is depicted;

[0013] Figure 2B An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0014] Figure 3A An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0015] Figure 3B An example quantized axis is depicted according to an example aspect of this disclosure;

[0016] Figure 4 An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0017] Figure 5 An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0018] Figure 6 An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0019] Figure 7 An example charge bias pulse according to an example aspect of this disclosure is depicted;

[0020] Figure 8 An example of a quantum computing system according to an exemplary aspect of this disclosure is depicted;

[0021] Figure 9 A flowchart is depicted for an example quantum computing method according to an example aspect of this disclosure;

[0022] Figure 10 A flowchart is depicted for an example quantum computing method according to an example aspect of this disclosure;

[0023] Figure 11 A flowchart depicts an example quantum computing method according to an example aspect of this disclosure; and

[0024] Figure 12 A block diagram of an example computing system according to an example aspect of this disclosure is depicted. Detailed Implementation

[0025] Example embodiments of some aspects of this disclosure relate to systems and methods for single-qubit control of fluxonium qubits, fluxonium-like qubits, and associated qubits using baseband voltage pulses. In some embodiments, the systems and methods of this disclosure can perform arbitrary single-qubit operations on fluxonium, fluxonium-like, or associated qubits, including arbitrary Pauli Z-rotations, arbitrary Pauli Y-rotations, arbitrary Pauli X-rotations, identity operations over any time period, and any combination thereof.

[0026] Although aspects of this disclosure are discussed with reference to fluxonium qubits for purposes of illustration and discussion, the control techniques provided herein are applicable to fluxonium qubits and fluxonium-like qubits. For example, aspects of this disclosure may be applied to: fluxonium qubits; variations of fluxonium qubits (e.g., fluxonium-like qubits having two or more Josephson junctions in parallel or Josephson junctions replaced by superconducting quantum interference devices (“SQUIDs”); qubits having fluxonium modes within their circuitry (e.g., zero-pi qubits); or qubits that reduce certain constraints on phase-slip junction devices (e.g., having an effective Hamiltonian similar to Equation 18 below); or other suitable qubits or devices.

[0027] Performing a Z or Y rotation may include determining a target amount of time to rotate the qubit around the Z or Y axis based on the desired rotation angle and the qubit frequency associated with the fluxonium-like qubit. Performing a Z or Y rotation may include providing a charge bias configured to rotate the qubit around the Z or Y axis for the target amount of time.

[0028] Performing an X rotation or identity operation may include, for example, combining two or more Z rotations or Y rotations. For instance, performing an X rotation may include combining one or more (e.g., two) Z rotations with one or more (e.g., one) Y rotations.

[0029] Performing an identity operation may include, for example, performing multiple rotations configured not to cause a net change in the state of a fluxonium-like qubit within a target time period. For example, an identity operation may include two or more rotations around the same rotation axis (e.g., two Z rotations, four Z rotations, etc.). In some cases, an identity operation may include at least one rotation in a first rotation direction and at least one rotation in a second rotation direction opposite to the first rotation direction. For example, in some cases, performing an identity operation may include determining a first rotation time and a second rotation time based on a target identity operation time (e.g., half of the target identity operation time, etc.). In some cases, performing an identity operation may include rotating the fluxonium-like qubit around the Z-axis in a first direction for a first rotation time and rotating the fluxonium-like qubit in a second direction opposite to the first direction for a second rotation time. In some cases, an identity operation may use two or more (e.g., four) Z rotations in a combined manner to prevent or reduce unintended Y rotations that may occur during transitions between charge biases.

[0030] In some cases, charge bias pulses can be provided by supplying a voltage bias line coupled to a fluxonium-like circuit via a capacitor. For example, time-dependent charge bias pulses. This can be achieved by driving V(t) with the baseband voltage.

[0031] In some cases, the charge bias pulse can be shaped to suppress leakage. For example, a smooth pulse shape (e.g., not a perfectly square shape) can be configured to cause a reduction in leakage relative to a square pulse shape. In some cases, the 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 to a charge bias configured to cause rotation about the Y-axis (e.g., from a zero charge bias, etc.) within a first transition time; holding the charge bias constant for a certain period of time (“hold time”); and transitioning from the charge bias configured to cause rotation about the Y-axis (e.g., to a zero charge bias, etc.) within a second transition time. In some cases, the first transition time and the second transition time can be numerically optimized to suppress leakage.

[0032] Example embodiments based on some aspects of this disclosure can provide numerous technical effects and benefits, such as improvements to computing techniques (e.g., quantum computing techniques). Compared to alternative systems and methods, example technical effects and benefits may include increased coherence time, reduced gating time, increased gate fidelity, and reduced hardware requirements. In some cases, increased coherence time and reduced gating time can provide increased quantum computing complexity by allowing more gating operations to be performed before the qubits become decoherent. In some cases, increased gate fidelity can increase quantum computing accuracy and reduce the computational cost of quantum computing (e.g., power costs, hardware usage, etc.).

[0033] For example, the systems and methods of the examples according to this disclosure can provide faster fluxonium gate times relative to microwave control without sacrificing the high gate fidelity and long coherence time associated with some alternative fluxonium-based systems and methods. In some cases, the systems and methods of the examples according to this disclosure can control fluxonium-like qubits (e.g., fluxonium qubits) without using a rotating wave approximation, which in some cases can lead to faster and higher fidelity gates. For example, in some cases, the gate speed can be equal to the qubit frequency of the fluxonium qubit operating under speed limits set by the qubit parameters. In some example simulations according to this disclosure, the provided example systems and methods achieve coherence times of more than one millisecond, single-qubit gate fidelity greater than 99.999% (considering unitary over-rotation and leakage), and single-qubit gate times as low as 2.1 nanoseconds.

[0034] As another example, the systems and methods of the examples according to this disclosure can achieve higher gate fidelity and longer coherence time relative to baseband flux control, which requires biasing the qubit flux away from its flux optimum. In contrast, the systems and methods of the examples according to this disclosure can provide any arbitrary single-qubit operation without altering the qubit energy during gate operation, thereby enabling fluxonium-like qubits (e.g., fluxonium qubits) to remain at their flux optimum throughout the operation. For example, enabling fluxonium qubits to remain at their flux optimum can increase fidelity and coherence time relative to the flux bias. Additionally, in some cases, the improved gate fidelity associated with the examples of this disclosure is robust against errors in control amplitude and timing.

[0035] As another example, the systems and methods according to this disclosure can operate without a fast-throughput line, thereby reducing the amount of hardware required to control fluxonium qubits compared to alternative methods. This reduced hardware requirement can, for example, improve the scalability and extensibility of the fluxonium architecture.

[0036] In some cases, increased gate fidelity can increase the accuracy of fluxonium-based quantum computing by reducing one or more error types associated with microwave control or baseband flux control. Additionally, increased gate fidelity can reduce the computational costs associated with quantum computing (e.g., power costs, hardware usage, etc.). For example, in some cases, determining a quantum computing result may involve running the quantum computing circuit multiple times. In some cases, the number of times required to accurately determine the quantum computing result (e.g., within a target accuracy) may depend at least in part on the error rate (e.g., fidelity, coherence, etc.) associated with the quantum computing circuit. In such cases, the improved accuracy according to the examples of this disclosure can make it possible to determine the quantum computing result within a target accuracy using fewer quantum computations compared to alternative systems and methods. In this way, for example, quantum computing results can be determined at a reduced computational cost compared to alternative systems and methods.

[0037] As used herein, the terms “approximately,” “around,” or “close to” when used with numerical values ​​mean within 10% of the stated value. “Close to zero” means within 0.025 of zero. “Approximately constant” means less than 10% of the average value that deviates from 0.25 or greater, or less than 0.025 of the average value that deviates from 0.25 or less.

[0038] Exemplary embodiments of this disclosure will now be discussed in more detail with reference to the accompanying drawings.

[0039] Example System

[0040] Figure 1 An example system for controlling a fluxonium-like qubit according to an exemplary embodiment of this disclosure is depicted. A voltage bias source 102 can be coupled to a fluxonium-like qubit 106 via a first capacitor 104. The fluxonium-like qubit 106 may include, for example, one or more Josephson junctions 108, one or more inductors 110, and one or more second capacitors 112. In some cases, the components of the fluxonium-like qubit may be connected in parallel (e.g., as shown in the diagram). Figure 1 (as depicted), or connected in another configuration (e.g., in a zero-pi configuration or other suitable configuration, etc.).

[0041] For example, voltage bias source 102 can be any system (e.g., apparatus, device, etc.) configured to provide voltage bias. In some cases, voltage bias source 102 can be configured to provide time-dependent voltage bias pulses. In some cases, voltage bias source 102 can include a controller configured to control the shape of the time-dependent voltage bias pulses. In some cases, voltage bias source 102 can be configured to control one or more transition times in which the voltage bias pulse transitions from a first voltage to a second voltage. In some cases, voltage bias source 102 can be configured to provide voltage bias pulses of a specific shape or combination of shapes (e.g., a flat segment with an equal-zero voltage bias, a non-zero voltage segment with an approximately square shape, etc.). In some cases, the shape of the voltage bias pulse can be at least partially based on a mathematical function (e.g., a quadratic function, a cosine function, a sine function, a harmonic function, etc.). In some cases, the shape of the voltage bias pulse can be continuous (i.e., without discontinuities). In some cases, the voltage bias pulse can be shaped to generate charge bias pulses of a specific shape, as discussed below with reference to FIG2.

[0042] In some cases, one or more of the first capacitor 104, the fluxonium-like qubit 106, the Josephson junction 108, the inductor 110, and the second capacitor 112 may be standard quantum computing system components constructed according to known methods. In some cases, one or more of the first capacitor 104, the fluxonium-like qubit 106, the Josephson junction 108, the inductor 110, and the second capacitor 112 may be configured to operate in a superconducting quantum computing environment. In some cases, one or more of the first capacitor 104, the fluxonium-like qubit 106, the Josephson junction 108, the inductor 110, and the second capacitor 112 may be configured to operate at very low temperatures (e.g., at 3 Kelvin) and dissipate very little heat during operation.

[0043] In some cases, the fluxonium-like qubit 106 can be a standard fluxonium qubit constructed according to known methods. In some cases, the fluxonium-like qubit 106 can be a variant of fluxonium (e.g., a fluxonium-like qubit with two or more Josephson junctions in parallel or a Josephson junction replaced by a superconducting quantum interference device (“SQUID”); a qubit having a fluxonium mode within its circuit (e.g., a zero-pi qubit); a qubit that reduces certain constraints on the phase-slip junction device (e.g., having an effective Hamiltonian similar to Equation 18 below); or other suitable qubits or devices.

[0044] In some cases, the fluxonium-like qubit 106 may include a circuit having a fluxonium mode. In some cases, the fluxonium mode may include a circuit mode having a Hamiltonian, which includes a charging energy term, a Josephson energy term, and an induction energy term. In some cases, the fluxonium mode may include a circuit mode corresponding to a parallel combination of a capacitor, an inductor, and a Josephson junction. In some cases, the fluxonium-like qubit 106 may 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 the fluxonium mode. As a non-limiting illustrative example, the fluxonium-like qubit 106 may include a zero-pi qubit having a fluxonium mode and one or more additional modes (e.g., a transmon mode, etc.). In some cases, a circuit having a fluxonium mode may include any circuit having a linear combination of circuit modes (e.g., circuit modes identifiable via standard circuit analysis methods), wherein the linear combination includes the fluxonium mode. For example, in some cases, linear combinations can result in degrees of freedom for Hamiltonians including charging energy, Josephson energy, and induced energy terms when interactions with any additional modes of fluxonium qubit 106 are neglected.

[0045] In some cases, the fluxonium mode can include a mode corresponding to a parallel combination of capacitors, inductors, and Josephson junctions in a lumped-element circuit model. For example, in some cases, the fluxonium-like qubit 106 can include a circuit comprising inductors, capacitors, and Josephson junctions (e.g., with or without one or more additional circuit elements). As a non-limiting illustrative example, in some cases, the fluxonium-like qubit 106 can include a zero-pi qubit having multiple (e.g., two equal) inductors, multiple (e.g., two equal) capacitors, and multiple (e.g., two equal) Josephson junctions. Continuing with the non-limiting illustrative example, the zero-pi qubit can include one or more parallel triplets, each parallel triplet comprising an inductor, a capacitor, and a Josephson junction connected in parallel.

[0046] In some cases, a fluxonium qubit 106 may include a qubit with reduced confinement in relation to a phase-slip junction device. A qubit with reduced confinement in relation to a phase-slip junction device may include, for example, a qubit with an effective Hamiltonian having an induction term and a phase-slip energy term. In some cases, the induction term may quantize the energy of m flux quanta in a superconducting loop defined by the geometry of the device. In some cases, the phase-slip energy term may quantize the energy cost or gain from tunneling flux quanta into and out of the superconducting loop. Further details of one or more example effective Hamiltonians are provided below with respect to equations (1) through (21) and the accompanying text.

[0047] In some cases, qubits that reduce the confinement of quantum phase-slip devices can include qubits with an energy spectrum approximating that of a phase-slip junction device in the presence of an external flux bias, or qubits with frequency dispersions corresponding to the energy spectrum of a phase-slip junction device. For example, in some cases, the 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., the maximum frequency) and a second qubit frequency (e.g., a lower frequency). For example, in some cases, the fluxonium-like qubit 106 can include a device with qubit frequency dependence, wherein the second qubit frequency (e.g., a lower frequency) can be associated with the value of the phase-slip energy of the fluxonium-like qubit 106, and the slope of the qubit transition between the first and second qubit frequencies can be associated with the induced energy of the fluxonium-like qubit 106.

[0048] In some cases, the fluxonium-like qubit 106 can be characterized by one or more fluxonium parameters (e.g., charging energy, Josephson energy, induced energy, qubit frequency, anharmonicity, etc.). In some cases, the parameters of the fluxonium-like qubit 106 can correspond to one or more known or disclosed sets of parameters associated with one or more types of fluxonium qubits (e.g., MIT fluxonium qubit, Princeton heavy fluxonium, etc.). However, this is not required, and the systems and methods of the examples according to this disclosure can be used in conjunction with other fluxonium-like qubits (e.g., fluxonium-like qubits characterized by undisclosed parameters; improved fluxonium-like qubits fabricated according to novel methods, etc.) without departing from the scope of this disclosure.

[0049] Example baseband voltage drive

[0050] Figure 2A and Figure 2B An example baseband voltage drive 204 for generating an example charge bias pulse 202 is depicted. An increased baseband voltage drive 204 may be provided during a first transition time 208 to increase the charge bias 202. A near-constant (e.g., constant) baseband voltage drive 204 may be provided during a hold time 210 to keep the charge bias 202 constant or approximately constant. A baseband voltage drive 204 with a decreased voltage may be provided during a second transition time 212 to decrease the charge bias 202.

[0051] In some cases, the baseband voltage driver 204 may include time-dependent voltage bias pulses. In some cases, the shape of the baseband voltage driver 204 may be controlled by the voltage bias source 102. In some cases, the baseband voltage driver 204 may be characterized by one or more pulse shapes or combinations of pulse shapes (e.g., flat segments with approximately constant voltage bias, voltage increase and decrease segments with approximately inverted or "S"-shaped voltages, etc.). In some cases, the pulse shape of the baseband voltage driver 204 may be at least partially based on mathematical functions (e.g., inverted functions such as logistic functions, hyperbolic tangents, arctangents, etc.; non-inverted functions such as cosine functions, sine functions, harmonic functions, quadratic functions, etc.). In some cases, the pulse shape of the baseband voltage driver 204 may be continuous (i.e., without discontinuities). In some cases, the pulse shape of the baseband voltage driver 204 may be smooth, wherein the first derivative of the baseband voltage driver 204 with respect to time may be continuous (i.e., without discontinuities).

[0052] In some cases, a time-dependent charge bias pulse 202 can be generated by obtaining a target charge bias pulse shape, determining a target voltage bias pulse shape for baseband voltage drive 204 based on a mathematical relationship between voltage bias and charge bias (e.g., a proportionality constant), and providing a voltage bias pulse from voltage bias source 102 based on the target voltage bias pulse shape. The target voltage bias pulse shape is configured to create a charge bias pulse 202 corresponding to the target charge bias pulse shape. In some cases, the mathematical relationship between time-dependent voltage bias and time-dependent charge bias can be expressed as follows: Where V(t) is the baseband voltage at a specific time t, and It is the charge bias at a specific time t. In some cases, the relationship between time-dependent charge bias and time-dependent voltage bias can be determined at least in part based on one or more properties (e.g., resistance, capacitance, time constant, etc.) of capacitor 104, fluxonium-like qubit 106, or other components of the circuit including voltage bias source 102, first capacitor 104, and fluxonium-like qubit 106.

[0053] In some cases, the charge bias 202 may 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., a flat-topped cosine with small transition times 208, 212) or a shape that is not approximately square (e.g., an approximately parabolic shape characterized by a small hold time or zero hold time 210, etc.). In some cases, the charge bias 202 may be characterized by a continuous pulse shape without discontinuities. In some cases, the charge bias 202 may be characterized by a smooth pulse shape, wherein the derivative of the charge bias 202 with respect to time may be continuous (i.e., without discontinuities).

[0054] In some cases, transition times 208, 212 may be the time during which the charge bias 202 is changing (e.g., increasing, decreasing, etc.). In some cases, transition times 208, 212 may be the time during which the voltage bias of the baseband voltage drive 204 is increasing or decreasing. In some cases, transition times 208, 212 may be the time between a first hold time 210 and a second hold time 210. In some cases, transition times 208, 212 may be the time between a first time when the charge bias 202 equals a first charge bias 202 value of interest and a second time when the charge bias 202 equals a second charge bias 202 value of interest. In some cases, the voltage bias of the baseband voltage drive 204 may be constant or approximately constant throughout the transition times 208, 212.

[0055] In some cases, the hold time 210 can be during which the charge bias 202 is constant or approximately constant (e.g., constant plus or minus a small variance). The holding time 210 can be the time during which the actual charge bias 202 generated based on a constant target charge bias is constant. In some cases, the holding time 210 can be the time during which the voltage bias associated with the baseband voltage drive 204 is constant or approximately constant. In some cases, the holding time 210 can be the time between the first transition 208 and the second transition time 212. In some cases, the holding time 210 can be the time during which the charge bias 202 is equal to or approximately equal to the charge bias 202 value of interest. In some cases, the charge bias 202 value of interest can be the charge bias 202 value configured to rotate the fluxonium-like qubit 106 about a specific rotation axis, for example, as described below with reference to Figures 3 to 7.

[0056] Example discussion of Fluxonium Hamiltonian and generalized theory of Fluxonium

[0057] Figure 3 to Figure 7 Example charge bias pulses for single-qubit control of fluxonium-like qubits (e.g., fluxonium qubits) are described. In some cases, example charge bias pulses for controlling fluxonium-like qubits according to example aspects of this disclosure can be understood or described in the context of one or more example second-order Hamiltonians or other properties of fluxonium-like qubits (e.g., fluxonium qubits) such as example second-order Hamiltonians for fluxonium qubits that preserve parameter charge bias dependence. To aid in understanding the various systems and methods according to examples of this disclosure, an example discussion of fluxonium Hamiltonians and the generalized theory of fluxonium is provided below.

[0058] While some aspects of the discussion below may include specific references to one or more properties of fluxonium qubits, the systems and methods according to this disclosure can be applied to other types of fluxonium qubits (e.g., qubits with fluxonium modes; qubits with reduced confinement to phase-slip junction devices; qubits with one or more inductors, capacitors, and Josephson junctions in parallel; zero-pi qubits, etc.) without departing from the scope of this disclosure.

[0059] The exemplary discussion in this disclosure may include incorporating the offset charge parameter n g The discussion of the fluxonium model, in some cases, allows for tracking the offset charge parameter throughout several theoretical steps, where it may not exist in the prior disclosure. In some cases, this model discussion can lead to having a parameter n... gA simplified discussion of the two-level Hamiltonian with dependencies. In some cases, example methods for general single-qubit control of fluxonium qubits can be understood or described within the context of this Hamiltonian.

[0060] In the single-mode approximation, the fluxonium Hamiltonian can be written as follows in some cases:

[0061]

[0062] in , and These are the charging energy of the qubit, the Josephson energy, and the sensing energy. It is to satisfy The phase and reduced charge operator of the qubit, and and It is an externally controlled charge and phase bias, which can be time-dependent.

[0063] In some cases, fluxonium qubits can be used in approximate parameter schemes. < < Operation can be performed at medium frequencies and can be characterized by anharmonicity at low frequencies (e.g., in the range of 1 to 800 MHz) and several GHz. Large anharmonicity may help suppress leakage during single-qubit operation, but when implemented via microwave drive, low qubit frequencies can slow down single-qubit operation. However, the lower the qubit frequency, the better the qubit T1 will be (dielectric loss is proportional to qubit frequency), leading to the design trade-offs necessary to balance coherence error and high qubit coherence.

[0064] Example aspects of this discussion may include isolating fluxonium Hamiltonians. For the periodic part, the fluxonium Hamiltonian is written as

[0065]

[0066] The difference, which is the transmon (or charge) qubit, ( ) is a non-tight degree of freedom. (In other words, It is a distinguishable potential minimum). According to Bloch's theorem, The characteristic function has the following form

[0067]

[0068] in It is a continuous quantum number that acts as "crystal momentum," and n is the band index. It is phase 2 - Periodic functions. These Bloch states satisfy orthogonality relations. The inner product extends across the entire real axis.

[0069] In some cases, in order to find the state You can specify the eigenvalue equation. And simplify it to obtain

[0070]

[0071] Among them 2 - Periodic boundary conditions are applied to the phase. In some cases, equation (4) can be equivalent to equation (4) with the same... Parameters and offset charge bias The eigenvalue problem of the definition of the transmon-qubit Hamiltonian.

[0072] In some cases, equation (4) can be solved, and the resulting basis can be used to represent equation (1). In some cases, the phase operator can be written as

[0073]

[0074] in It can be calculated as the following interband coupling potential

[0075]

[0076] For example, suppose that for all n and k have The specification allows us to derive this expression, which can be ensured numerically. Similarly, the fluxonium charge operator can be taken in the following form.

[0077]

[0078] In some cases, charge matrix elements The transmon eigenstates defined in equation (4) can be used to calculate the eigenstates.

[0079] According to the above definition, the fluxonium Hamiltonian of equation (1) can be written as:

[0080]

[0081] The first term represents the "transmon part" within the fluxonium Hamiltonian; however, this "transmon part" is defined for non-compact phases and is therefore diagonalized via Bloch (not charge) states. The second term represents an induced shunting, which results in a displacement along the Bloch band (and...). The coupling between the proportional terms and the band. Note that because the induced potential violates the translation symmetry, k is no longer a good quantum number and should instead be considered as a new tight degree of freedom according to its specified fluxonium Hamiltonian.

[0082] While the Hamiltonian in equation (8) is useful, the exemplary aspects of this discussion include additional discussions of exemplary Hamiltonians that can further facilitate the understanding and discussion of exemplary systems and methods for offset-charge-dependent quantum control. For example, in some cases, a change in the basis can be achieved where the conjugate charge operator associated with k is diagonal, and this conjugate charge operator can be represented as In particular, the derivative operator can take the following form:

[0083]

[0084] in (Due to the 1-periodicity of k, it appears as 2.) (factors). Due to the relationship in equation (5), It can be considered as 2 Josephson knots as units The measure of phase on the basis. For reasons that will become clear below, this basis can be called a phase-slip representation. This frame change can be compared with unitary... The corresponding Fourier transform is implemented. When applied to the complete Hamiltonian in equation (8), this transformation can lead to the following form.

[0085]

[0086] in It is the potential energy operator associated with the nth Bloch band. More precisely, The nth eigenvalue of equation (4) can be used as a function of 'coordinate' k. The corresponding Fourier transform. Such eigenvalues ​​in k - n g The middle is 1-period. For example, this can be observed by noting the shift k - n. g This can be achieved by applying it to equation (4) and shifting the charge operator by one unit. We can see this by compensating with a unitary transform. This unitary transform makes... The operator remains unchanged, thus restoring the Hamiltonian to its original form. Therefore, the general Fourier decomposition of the eigenvalues ​​can be written as...

[0087]

[0088] in The average frequency of the nth band is represented by and It is a coefficient with units of energy. The phase-slip operator is introduced. We obtain:

[0089]

[0090] For l > 0, this allows equation (11) to be rewritten as

[0091]

[0092] because It can represent 2 l units of "phase slip" (e.g., ignoring interband coupling terms), and This can be the rate associated with the process (including l > 0 and l < 0), therefore This can be referred to as the phase-slip energy of fluxonium. In some cases, It can also be defined at its flux frustration point. The energy of qubits.

[0093] In some cases, the formulas described above can lead to a second-order fluxonium Hamiltonian describing the low-energy physics of the circuit and incorporating the parameter charge bias dependence. For example, the interband coupling term in equation (6) can be discarded first by assuming a large frequency gap between the baseband and the nth excitation band. This makes it possible to ignore the fact that The resulting The second-order correction. This is likely reasonable because these frequency shifts are on the order of GHz, as these frequency shifts are determined by the "transmon parameter" E. C E J Definition. Furthermore, this can be verified by direct numerical implementation and diagonalization of the full model in equation (8) (including the interband coupling term) to be E in the fluxonium scheme. C E J and E L A superior approximation of the parameters. Within this approximation, the constant energy shift is discarded. Then, the fluxonium Hamiltonian appears from the lowest band n = 0:

[0094]

[0095] This equation also considers the limit of single-phase slip, where This hypothesis is confirmed below and constitutes the complete E in the fluxonium scheme. C E J A very good approximation of the parameters.

[0096] Generally, the phase-slip energy can be determined by diagonalizing the transmon-like Hamiltonian within the fluxonium model and performing a numerical Fourier transform of the resulting energy dispersion as a function of k. However, in this case, E J E C In the transmon scheme, the ground state energy of the transmon qubit is approximated by an asymptotic expression.

[0097]

[0098] in

[0099]

[0100] It can be observed that equation (16) is a very good approximation for most fluxonium parameters, and the limit of single-phase slip can be well verified. In fact, for typical fluxonium parameters, the amplitude of Fourier harmonics with l > 1 can be several orders of magnitude smaller than the amplitude with l = 1.

[0101] Based on the above equations, the Hamiltonian can be approximately expressed as follows:

[0102]

[0103] in From Equation (16) in the scheme is calculated, or in any other case, it is calculated numerically.

[0104] In the fluxonium scheme, it has This leads to the magnetic flux quantum (also known as the continuous current) state { The weak coupling of}. Therefore, near the flux frustration condition In the case of equation (17), the characteristic state is obtained through and These eigenstates are included by the coherent superposition of the features. It is completely degenerate. Assuming optimal point operations and truncating the Hamiltonian to the two-flux quantum space, equation (17) simplifies to

[0105]

[0106] The phase-slip operators have been replaced by their two-level forms.

[0107]

[0108] Without loss of generality, in some cases, it can be assumed that the reference charge bias of the quantization axis of the qubit at time t = 0 is n. g = 0, so that the ground state and excited state of equation (18) are respectively:

[0109]

[0110]

[0111] In some cases, the qubit Pauli operator can be reintroduced into the basis defined in equation (20) above, and the Hamiltonian can be rotated according to the unitary transformation:

[0112]

[0113] And equation 18 can be mapped to:

[0114]

[0115] Where Z = And Y = -i( .

[0116] In some cases, equation (21) can provide a form that can be readily understood or described for a fluxonium Hamiltonian controlled by a single qubit based on a baseband voltage bias pulse. For example, in some cases, equation (21) can be interpreted as a spin in a “magnetic field” whose direction (in the YZ plane) is set by a bias charge. The Hamiltonian of a particle. For example, in some cases, single-qubit operations can be achieved by introducing a non-adiabatic change in the quantization axis of the spin. In other cases, this can be achieved by controlling the direction of the magnetic field, i.e., n. g The value of is used to introduce a nonadiabatic change in the spin quantization axis. Note that, unlike baseband flux control, baseband voltage control can be performed in some cases without changing the qubit energy (which, according to equation (21), can be constant and equal to ). In some cases, this can result in the qubit coherence time maintained during the baseband voltage pulse. Furthermore, since the rate of equivalent spin precession can be the qubit frequency... Therefore, the systems and methods according to the exemplary aspects of this disclosure can operate within a speed limit set by the qubit frequency.

[0117] Example charge bias pulse for single-qubit control

[0118] Figure 3A and Figure 3B Two related views are depicted in association with an example rotation of the fluxonium qubit around the Y-axis. Figure 3A An example charge bias pulse for rotating fluxonium qubits around the Y-axis is depicted, while Figure 3B An example relationship between the charge bias and rotation axis of a fluxonium qubit is depicted.

[0119] Figure 3A An example charge bias pulse is depicted for rotating a fluxonium qubit in the positive direction about the Y-axis. For example, this rotation can be used to perform a Pauli Y gate and can be combined with other rotations to achieve any arbitrary single-qubit operation. During a first transition time 208, the example charge bias pulse can transition from a pre-transition charge bias (e.g., at or near zero in the example description) to a Y-rotation charge bias 302 configured to rotate the fluxonium qubit in the positive direction about the Y-axis. During a hold time 210, the example charge bias pulse can remain at or near the Y-rotation charge bias 302. The example charge bias pulse can transition from the Y-rotation charge bias 302 to another charge bias (e.g., at or near zero in the example description) during a second transition time 212.

[0120] In some cases, the Y-rotation charge bias 302 can be approximately 0.25 or -0.25. For example, in cases where the computational quantization axis is the Z-axis and associated with a charge bias of 0.0, a charge bias of approximately 0.25 or -0.25 can cause the fluxonium qubit to rotate around the Y-axis (e.g., in the opposite direction). In some cases, the Y-rotation charge bias 302 can be a charge bias configured to make the rotation axis 308 of the fluxonium qubit (e.g., the instantaneous quantization axis) parallel to the Pauli Y-axis.

[0121] In some cases, the Y-rotation charge bias 302 can be determined based on an approximate fluxonium Hamiltonian, which is written as... ,in It is an externally controlled charge bias, and It is the value discussed above regarding equation (16). For example, in some cases, this approximate fluxonium Hamiltonian can be interpreted as a spin in the "magnetic field" whose direction (in the YZ plane) is set by the offset charge. The Hamiltonian of a particle. For example, in some cases, single-qubit operations can be achieved by introducing a non-adiabatic change in the quantization axis of the spin. In other cases, this can be achieved by controlling the direction of the magnetic field, i.e., n. g The value of is used to introduce a nonadiabatic change in the quantization axis of the spin. In this case, the equivalent spin can be equal to the qubit frequency. The rate of precession around the quantization axis of the spin, where These are the values ​​discussed above regarding equations (15) and (16). In some cases, the state of a fluxonium qubit can be represented by spin precession around the instantaneous quantization axis. An example relationship between charge bias and the spin axis will be referenced below. Figure 3B Further discussion.

[0122] In some cases, the first transition time 208, the hold time 210, and the second transition time 212 can be determined at least in part based on the desired rotation angle and the qubit frequency. For example, in some cases, the fluxonium qubit can be at a frequency equal to the qubit frequency. The rate of precession around the instantaneous quantization axis. In this case, the desired rotation angle is achieved. The total time required can be approximated as equal to

[0123]

[0124] It incorporates minor adjustments to account for the smooth, imperfect square shape of the charge bias pulse. For example, in the case of a perfect square pulse shape with transition times 208, 212 that are exactly zero, the example hold time 210 can be exactly equal to... In the example where transition times 208 and 212 are greater than zero, the total time associated with the charge bias pulse (e.g., equal to the sum of transition times 208 and 212 and hold time 210) can be approximately equal to (e.g., slightly greater than) [the sum of these two times]. Furthermore, the retention time of 210 can be less than (e.g., slightly less) on its own. .

[0125] In some cases, the charge bias pulse may have a pulse shape configured to reduce leakage. In some cases, reducing leakage may include selecting a pulse shape that suppresses high-frequency components of fluxonium transition resonances outside the computational subspace. For example, in some cases, the charge bias pulse may have a smooth pulse shape (e.g., having one or more continuous derivatives) configured to reduce leakage. For example, a non-limiting example of a simple smooth pulse shape could be a flat-topped cosine pulse shape. In some cases, transition times 208, 212 may be numerically optimized to minimize gate distortion. In some cases, numerically optimizing transition times 208, 212 to minimize gate distortion may include selecting transition times 208, 212 that minimize leakage.

[0126] In some cases, numerically optimizing the transition times 208, 212 may include simulating the effect of candidate charge bias pulses on the fluxonium qubit and comparing the simulated effect with the desired effect (e.g., Y( (Rotation, etc.) are compared. In some cases, simulating the effect of charge bias pulses on fluxonium qubits may involve solving the time-dependent Schrödinger equation for equation (1) above, where In some cases, simulations can include simulating the time-evolving Schrödinger equation under the characteristic basis of the fluxonium Hamiltonian. For example, in some cases, the charge and phase operators in equation (1) above can be written according to the boson ladder operator. In some cases, the Hamiltonian can be diagonalized, and multiple (e.g., 100, etc.) characteristic states of the Hamiltonian can be used to propagate the time-dependent Schrödinger equation.

[0127] In some example experiments according to this disclosure, tests were conducted for multiple transition times 208, 212 and multiple fluxonium parameter sets (e.g., charging energy, Josephson energy, and induced energy; qubit frequency, etc.) to perform Y( A rotating example charge bias pulse. In experiments, the example gate according to an example aspect of this disclosure exhibits extremely high average gate fidelity (e.g., 99.999% to 99.99999%, etc.) and high driving amplitude n. g The timing error is robust. For example, in some example experiments, the range of Josephson energies (between 3 and 18 times the charging energy) was tested for each of multiple transition times 208, 212 and each of multiple induced energies. In the example experiments, for each combination of Josephson and induced energies tested, there were one or more transition times 208, 212, for which the gate distortion decreased to 10. -4Below this, the decrease typically drops by several orders of magnitude (e.g., for some Josephson energies, approximately 10). -12 Approximately 10 -8 Approximately 10 -6 (etc.). In some example experiments, for the driving amplitude n g The range of values ​​tested the gate fidelity, and for the optimal drive amplitude n g The value is approximately 2 to 4% of the value, and the gate fidelity remains below 10. -4 This indicates that the gate fidelity is robust to drive amplitude deviation and timing error. However, in some cases, for the optimal drive amplitude n... g The value, in the best case, has even lower gate distortion (e.g., below 10). -7 (etc.), which indicates that fine-tuning the drive amplitude can be advantageous in some implementations. Additionally, in the example experiments, the example gating time is very fast. For example, in some example experiments (e.g., associated with a qubit frequency of 222 MHz, etc.), the gating time for Y( The rotational gating time is as low as approximately 2 to 4 ns (e.g., 2.1 ns, etc.). Additional details regarding example experiments are provided in the appendix to U.S. Provisional Patent Application No. 63 / 561,022, which forms part of this disclosure.

[0128] Although Figure 3A Example positive charge biases are depicted, but negative charge biases can be used to rotate fluxonium qubits about the Y-axis without exceeding the scope of this disclosure. For example, in some cases, a charge bias at or near approximately -0.25 can also cause the fluxonium qubit to rotate about the Pauli Y-axis (e.g., in the opposite direction to the rotation caused by a charge bias of approximately 0.25). For example, in cases where the quantization axis is calculated to be the Z-axis and associated with a charge bias of 0.0, a charge bias of approximately -0.25 can cause the fluxonium qubit to rotate about the Y-axis (e.g., in the opposite direction to the charge bias of approximately 0.25).

[0129] Figure 3B An example relationship between the charge bias of a fluxonium qubit (e.g., a Y-rotation charge bias 302) and a rotation axis 308 is depicted. The charge bias allows the fluxonium qubit to rotate about the rotation axis 308, which can form an angle 304 relative to a computational quantization axis 306. In some cases, the computational quantization axis 306 can be the Z-axis 310. When the charge bias is a Y-rotation charge bias 302, the rotation axis 308 can be the Y-axis 312.

[0130] In some cases, computational quantization axis 306 can be used to measure or define the state of fluxonium qubits (e.g., such as...). or The axis, direction, or dimension of the ground state, including, or otherwise corresponding to the axis, direction, or dimension. In some cases, the computational quantization axis 306 may be an axis associated with a computational state space that is associated with a quantum computation being performed using fluxonium qubits.

[0131] Rotation axis 308 may include the axis around which the state of the fluxonium qubit rotates (e.g., relative to a Bloch sphere). In some cases, rotation axis 308 may be the same as or different from computational quantization axis 306. In some cases, rotation axis 308 may be the instantaneous quantization axis of the fluxonium qubit. In some cases, the instantaneous quantization axis at a given time t may depend on the charge bias at time t. (For example, as described below).

[0132] In some cases, the angle 304 between the rotation axis 308 and the computational quantization axis 306 can be written as Approximate fluxonium Hamiltonian and charge bias Related, among which It is an externally controlled charge bias. For example, in some cases, the fluxonium Hamiltonian can be interpreted as a spin in a "magnetic field" whose direction (in the YZ plane) is set by the offset charge. The Hamiltonian of a particle, and the quantization axis of its spin can be related to n. g The value is proportional. In some cases, the angle 304 between the rotation axis 308 and the computational quantization axis 306 can be equal to... ,in This is a charge bias (e.g., Y-rotation charge bias 302). For example, in calculating the quantized axis 306 in relation to a charge bias of 0.0, angle 304 could be equal to... In some cases, the rotation axis 308 associated with the Y-rotation charge bias 302 can be angled 304 with a π / 2 radian angle to the calculated quantized axis 306.

[0133] In some cases, the Z-axis 310 and the Y-axis 312 can be the Pauli Z-axis and the Pauli Y-axis, respectively.

[0134] Figure 4An example charge bias pulse is depicted for rotating a fluxonium qubit in the positive direction around the Z-axis 310. For example, this rotation can be used to perform a Pauli Z-gate and can be combined with other rotations to achieve any arbitrary single-qubit operation. During a first transition time 208, the example charge bias pulse can transition from a pre-transition charge bias (e.g., at or near zero in the example description) to a positive Z-rotation charge bias 402, which is configured to rotate the fluxonium qubit in the positive direction around the Z-axis. During a hold time 210, the example charge bias pulse can remain at or near the positive Z-rotation charge bias 402. During a second transition time 212, the example charge bias pulse can transition from the positive Z-rotation charge bias 402 to another charge bias (e.g., at or near zero in the example description), which is configured to rotate the fluxonium qubit in the positive direction around the Z-axis.

[0135] In some cases, the positive Z-rotation charge bias 402 can be approximately 0.5 or -0.5. For example, in cases where the calculated quantization axis 306 is the Z-axis and associated with a charge bias of 0.0, a charge bias of approximately 0.5 or -0.5 can cause the fluxonium qubit to rotate around the Z-axis (e.g., in the positive direction in both cases). In some cases, the positive Z-rotation charge bias 402 can be a charge bias configured to make the rotation axis 308 of the fluxonium qubit (e.g., the instantaneous quantization axis) parallel to the Pauli Z-axis. In some cases, the rotation axis 308 associated with the positive Z-rotation charge bias 402 can be an angle 304 having π radians relative to the calculated quantization axis 306. In some cases, according to... Figures 3A to 3B The example system and method described herein, wherein the positive Z-rotation charge bias 402 can be determined by determining the charge bias associated with the appropriate angle 304 between the Z-axis 310 and the calculated quantization axis 306.

[0136] In some cases, the first transition time 208, the hold time 210, and the second transition time 212 can be determined at least in part based on the desired rotation angle and the qubit frequency. For example, in some cases, the fluxonium qubit can be at a frequency equal to the qubit frequency. The rate of precession around the instantaneous quantization axis. In this case, the desired rotation angle is achieved. The total time required can be approximated as equal to

[0137]

[0138] It incorporates minor adjustments to account for the smooth, imperfect square shape of the charge bias pulse. For example, in the case of a perfect square pulse shape with transition times 208, 212 that are exactly zero, the example hold time 210 can be exactly equal to... In the example where transition times 208 and 212 are greater than zero, the total time associated with the charge bias pulse (e.g., equal to the sum of transition times 208 and 212 and hold time 210) can be approximately equal to (e.g., slightly greater than) [the sum of these two times]. Furthermore, the retention time of 210 can be less than (e.g., slightly less) on its own. .

[0139] Figure 5 An example charge bias pulse is depicted for rotating a fluxonium qubit in the negative direction around the Z-axis. This rotation can be used, for example, to perform a Pauli Z-gate and can be combined with other rotations to achieve any arbitrary single-qubit operation. In the example charge bias pulse, the charge bias can be maintained at a negative Z-rotation charge bias value 502 throughout the hold time 210, which is configured to rotate the fluxonium qubit in the negative direction around the Z-axis (e.g., at or near zero).

[0140] In some cases, the negative Z-rotation charge bias 502 can be 0.0. For example, in the case where the calculated quantization axis 306 is the Z-axis and is associated with a charge bias of 0.0, the charge bias of 0.0 can cause the fluxonium qubit to rotate in the negative direction around the Z-axis. In some cases, the negative Z-rotation charge bias 502 can be a charge bias configured to make the rotation axis 308 of the fluxonium qubit (e.g., the instantaneous quantization axis) parallel to the Pauli Z-axis. In some cases, the rotation axis 308 associated with the negative Z-rotation charge bias 502 can be an angle 304 with zero radians relative to the calculated quantization axis 306. In some cases, according to... Figures 3A to 3B The example system and method described herein, wherein the 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 the computational quantization axis 306.

[0141] In some cases, the retention time 210 can be determined at least in part based on the desired rotation angle and the qubit frequency. For example, in some cases, the fluxonium qubit can be held at a frequency equal to the qubit frequency. The rate of precession around the instantaneous quantization axis. In this case, the desired rotation angle is achieved. The total time required can be approximated as equal to

[0142]

[0143] It is possible to adjust for the effects of one or more transition times 208, 212 (e.g., as depicted in other figures). For example, in the case of performing a Z-rotation without any transition times 208, 212, the example hold time 210 may in some cases be exactly equal to For example, in instances where the quantization axis 306 is the Z-axis and associated with a charge bias of 0.0, in some cases, it is possible to maintain a charge bias of 0.0 for an example holding time 210 that is exactly equal to the value of 0.0. To execute the angle using charge bias pulses Example Z-rotation.

[0144] Although Figure 5 Transitions to or from charge bias values ​​other than the negative Z-rotation charge bias 502 are not explicitly described, but charge bias pulses for negative Z-rotation may include one or more transitions without exceeding the scope of this disclosure. For example, in some cases, an identity operation may include one or more transitions between the negative Z-rotation charge bias 502 and the positive Z-rotation charge bias 402. As another example, X-rotation may include one or more transitions between the negative Z-rotation charge bias 502 and a charge bias configured to cause rotation about the Pauli Y-axis. Other transitions are also possible.

[0145] Figure 6 An example charge bias pulse is depicted for rotating a fluxonium qubit around the Pauli X-axis. This rotation, for example, can be used to perform a Pauli X-gate and can be combined with other rotations to achieve any arbitrary single-qubit operation. The example charge bias can be maintained at a negative Z-rotation charge bias 502 for the entire first hold time 610. The example charge bias can transition to a Y-rotation charge bias 302 and remain at 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 at negative Z-rotation charge bias for a third hold time 614.

[0146] In some cases, the first holding time 610, the second holding time 612, and the third holding time 614 may be holding time 210, include holding time 210, or be included by holding time 210. In addition to the holding times 610, 612, and 614 depicted and marked, Figure 6 Example charge bias pulses can be generated in some cases by Figure 6 Characterized by one or more unmarked transition times 208, 212 (e.g., between the first hold time 610 and the second hold time 612; between the second hold time 612 and the third hold time 614, etc.).

[0147] In some cases, the first holding time 610, the second holding time 612, and the third holding time 614 can be at least partially based on the target X rotation angle. To determine. For example, in some cases, having The X-rotation of the amplitude can be decomposed into having One or more (e.g., two) Z-rotations and one or more (e.g., one) Y-rotations of amplitude. For example, in some cases, an X rotation (i.e., rotation about the X-axis) can be decomposed into Z and Y rotations according to standard geometric methods, where the X, Y, and Z axes can be considered as orthogonal axes in three-dimensional space. In this case, the target rotation time associated with each Z and Y rotation can be based at least in part on the target rotation angle associated with that rotation (e.g., ...). The first holding time 610, the second holding time 612, and the third holding time 614 can be determined according to the figures 3 to 4. Figure 5 The method described is used to determine the holding time 210.

[0148] although Figure 6 Two Z-rotations performed using a negative Z-rotation charge bias 502 are depicted, but an X-rotation can also be performed using one or more Z-rotations using a positive Z-rotation charge bias 402 without departing from the scope of this disclosure. For example, in some cases, a negative rotation of N radians (D degrees) is equivalent to 2 A positive rotation in radians (360–D degrees) is possible, and in some cases, equivalent rotations can be used interchangeably. However, in some cases, two rotationally equivalent operations can have different properties (e.g., different gating times, fidelity, etc.), which, for a specific use case, can make one operation superior to the other. For example, when the negative Z-rotation charge bias 502 is equal to 0.0 or equal to the charge bias associated with the computation of the quantized axis 306, the negative Z-rotation charge bias 502 can be associated with increased fidelity compared to the positive Z-rotation charge bias 402. As another example, when the desired Z-rotation is a small positive rotation (e.g., less than...),... In cases involving radians, etc., a positive Z-rotation charge bias 402 can be associated with a faster gating time than a negative Z-rotation charge bias 502. In some cases, the 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 Z-rotation and one negative Z-rotation). Similarly, in some cases, the 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.

[0149] Figure 7Example charge bias pulses for enabling fluxonium qubits to perform identity operations are depicted. For example, this rotation can be used alone or in combination with other rotations to achieve any arbitrary single-qubit operation. The example charge bias can transition to a positive Z-rotation charge bias 402 and remain at positive Z-rotation charge bias 402 throughout the first hold time 710. The example charge bias can transition to a negative Z-rotation charge bias 502 and remain at 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 at this second positive Z-rotation charge bias for a third hold time 714. The example charge bias can transition to a negative Z-rotation charge bias 502 and remain at the negative Z-rotation charge bias for a fourth hold time 716.

[0150] In some cases, the second positive Z-rotation charge bias 702 can be approximately 0.5 or -0.5. In some cases, the second positive Z-rotation charge bias 702 can differ from the positive Z-rotation charge bias 402. For example, in some cases, the second positive Z-rotation charge bias 702 can have a similar (e.g., identical) magnitude and an opposite sign compared to the positive Z-rotation charge bias 402. For example, in cases where the quantization axis 306 is calculated to be the Z-axis and associated with a charge bias of 0.0, a charge bias of approximately 0.5 or -0.5 can each cause the fluxonium qubit to rotate around the Z-axis (e.g., in the positive direction in both cases). In some cases, the second positive Z-rotation charge bias 702 can be a charge bias configured to make the rotation axis 308 of the fluxonium qubit (e.g., the instantaneous quantization axis) parallel to the Pauli Z-axis. In some cases, the rotation axis 308 associated with the second positive Z-rotation charge bias 702 can be an angle 304 having π or -π radians relative to the calculated quantized axis 306. In some cases, according to... Figures 3A to 3B In the example systems and methods described, the 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 the calculated quantization axis 306.

[0151] In some cases, the identity operation can be configured to avoid any unwanted Y rotation. For example, in some cases, an identity operation that transitions beyond the Y rotation charge bias 302 can be configured to cancel any unwanted Y rotation caused by this transition. For example, in some cases, a positive Z rotation charge bias 402 and a second positive Z rotation charge bias 702 can be configured such that a transition beyond the Y rotation charge bias 302 (e.g., a charge bias of about 0.25) can be paired with a transition beyond the second charge bias (e.g., about -0.25), resulting in a Y rotation in the opposite direction compared to the Y rotation charge bias 302. Pairing transitions beyond the Y rotation charge bias can include, for example, selecting positive Z rotation charge bias 402 and a second positive Z rotation charge bias 702 with opposite signs (e.g., 0.5 and -0.5). Pairing transitions beyond the Y-rotation charge bias can include, for example, shaping the charge bias pulses such that transitions associated with opposite signs are associated with similar (e.g., identical) transition times 208, 212 or similar pulse shapes (e.g., slopes with similar or identical amplitudes and opposite signs when exceeding the Y-rotation charge bias). Identity operations involving two or more positive Z-rotation charge biases 402, 702 with opposite signs can be viewed as a cascade of, for example, two smaller identity operations with a buffer time (e.g., a second hold time 612) (e.g., approximately 0). 0.5 The charge bias pulse, with approximately 0 0.5 (charge bias pulse cascade).

[0152] In some cases, the first holding time 710, the second holding time 712, the third holding time 714, and the fourth holding time 716 may be one or more holding times 210, include one or more holding times, or be included by one or more holding times. In some cases, Figure 7 An example charge bias pulse can be characterized by one or more transition times 208, 212 (e.g., between one or more hold times 710, 712, 714, 716, etc.).

[0153] In some cases, the first hold time 710, the second hold time 712, the third hold time 714, and the fourth hold time 716 can be selected at least in part based on the target total time associated with the identity operation. In other words, the hold times 710, 712, 714, and 716 can be selected at least in part based on the target duration of the identity operation. For example, in some cases, the sum of the first hold time 710, the second hold time 712, the third hold time 714, the fourth hold time 716, and any transition times 208 and 212 associated with the example charge bias pulse can be equal to the target total time or the target duration of the identity operation.

[0154] In some cases, the target total time associated with an identity operation can be determined based on the target total time associated with multiple operations (e.g., Z, Y, and X rotations and identity operations). For example, the target total time associated with multiple operations can be determined based on a comparison (e.g., mathematical difference) between the current time and the target time at which the fluxonium qubit must be in the target state. For example, the target qubit state and the target final time can be obtained. The target total operation time can be determined based on the target final time and the start time of the start of the fluxonium qubit control operation. Based on the target qubit state, multiple transition times 208, 212 and hold times 210 can be determined for rotating the qubit to the target qubit state (e.g., via multiple Z and Y rotations). Based on the target total operation time and the target transition times and hold times 208, 210, and 212, the target identity operation time (e.g., the total operation time minus the sum of all relevant transition times and hold times) can be obtained.

[0155] In some cases, the first hold time 710, the second hold time 712, the third hold time 714, and the fourth hold time 716 can be configured not to cause net rotation (e.g., a net Z-rotation of zero radians, etc.). For example, the target rotation angle... It can be selected such that the sum of the target rotation angles is zero, and can be determined according to Figure 3 to... Figure 5 The method 210 described herein for determining the hold time determines multiple hold times 710, 712, 714, and 716 based at least in part on the target rotation angle. In some cases, the buffer time (e.g., a second hold time 612) can be selected such that the total accumulated phase disappears while suppressing clutter rotation and leakage. Due to the self-echoing nature of the example pulse scheme, this condition can be satisfied in most cases using a charge bias pulse according to an example aspect of this disclosure.

[0156] In some cases, the example control pulses of this disclosure can be advantageously combined with additional qubit control methods. For example, in some cases, a phase gate (e.g., a unity-fidelity phase gate) can be appended to the end of the identity operation sequence. In some cases, such a phase gate can, for example, prevent or compensate for unwanted Z-rotations.

[0157] Although Figure 7 A four-segment identity operation comprising four hold times is described, but the identity operation may include different numbers of segments (e.g., two, three, eight, etc.) without exceeding the scope of this disclosure.

[0158] Example quantum computing system

[0159] Figure 8 An example quantum computing system 800 is depicted. Example system 800 is an example of a system on one or more classical computers or quantum computing devices at one or more locations in which the systems, components, and techniques described below can be implemented. Those skilled in the art will understand using the disclosure provided herein that other quantum computing structures or systems can be used without departing from the scope of this disclosure.

[0160] System 800 includes quantum hardware 802 that communicates data with one or more classical processors 804. Quantum hardware 802 includes components for performing quantum computing. For example, quantum hardware 802 includes a quantum system 810, a control device 812, and a readout device 814 (e.g., a readout resonator). Quantum system 810 may include one or more multilevel quantum subsystems, such as registers of qubits. In some implementations, the multilevel quantum subsystem may include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0161] The type of multilevel quantum system used in system 800 may vary. For example, in some cases, it may be convenient to include one or more readout devices 814 attached to one or more superconducting qubits (e.g., transmon qubits, flux qubits, gmon qubits, xmon qubits, or other qubits). In other cases, ion traps, photonic devices, or superconducting cavities may be used (e.g., which allow for state preparation without the need for qubits). Further examples of implementations of multilevel quantum systems include fluxmon qubits, silicon quantum dots, or phosphorus-impurity qubits.

[0162] Quantum circuits can be constructed and applied to registers of qubits included in a quantum system 810 via multiple control lines coupled to one or more control devices 812. Example control devices 812 operating on qubit registers can be used to implement quantum gates or quantum circuits with multiple quantum gates, such as Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T-gates, multi-qubit quantum gates, coupler quantum gates, etc. One or more control devices 812 can be configured to operate the quantum system 810 via one or more corresponding control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystem can be superconducting qubits, and the control devices 812 can be configured to provide control pulses to the control lines to generate magnetic fields to adjust the frequency of the qubits.

[0163] The quantum hardware 802 may further include a readout device 814 (e.g., a readout resonator). Measurement results 808 obtained via the measurement device can be provided to a classical processor 804 for processing and analysis. In some implementations, the quantum hardware 802 may include quantum circuitry, and the control device 812 and the readout device 814 may implement one or more quantum logic gates that operate the quantum system 802 via physical control parameters (e.g., microwave pulses) transmitted through wires included in the quantum hardware 802. Further examples of the control device include an arbitrary waveform generator, where a DAC (digital-to-analog converter) creates the signal.

[0164] The readout device 814 can be configured to perform a quantum measurement on the quantum system 810 and send the measurement result 808 to the classical processor 804. Additionally, the quantum hardware 802 can be configured to receive data from the classical processor 804 specifying a physical control qubit parameter value 806. The quantum hardware 802 can use the received physical control qubit parameter value 806 to update the actions of the control device 812 and the readout device 814 on the quantum system 810. For example, the quantum hardware 802 can receive data specifying a new value representing the voltage intensity of one or more DACs included in the control device 812, and the quantum hardware can update the actions of the DACs on the quantum system 810 accordingly. The classical processor 804 can be configured, for example, to initialize the quantum system 810 in an initial quantum state by sending data specifying an initial parameter set 806 to the quantum hardware 802.

[0165] The readout device 814 can utilize elements of a quantum system (such as qubits). and The impedance difference between states is used to measure the state of an element (e.g., a qubit). For example, due to the nonlinearity of the qubit, when the qubit is in a state... or state The resonant frequency of the readout resonator can be different. Therefore, the microwave pulse reflected from the readout device 814 has 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 814 to block microwave propagation at the qubit frequency.

[0166] In some implementations, the quantum system 810 may include, for example, a plurality of qubits 820 arranged in a two-dimensional grid 822. For clarity, Figure 1 The two-dimensional grid 822 depicted includes 16 qubits arranged in a square pattern; however, in some implementations, system 810 may include fewer or more qubits. In some embodiments, multiple qubits 820 may interact with each other through multiple qubit couplers (e.g., qubit coupler 824). A qubit coupler can define the nearest-neighbor interaction between multiple qubits 820. In some implementations, the strength of the multiple qubit couplers is an adjustable parameter. 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). Data qubits are qubits that participate in the computation performed by system 800. Measurement qubits are qubits that can be used to determine the result of the computation performed by the data qubits. That is, during computation, the unknown state of the data qubits is transferred to the measurement qubits using appropriate physical operations and measured via appropriate measurement operations performed on the measurement qubits.

[0167] In some implementations, each of the multiple 820 qubits can operate using a corresponding operating frequency, such as an idle frequency and / or an interaction frequency and / or a readout frequency and / or a reset frequency. Different qubits may operate at different frequencies. For example, each qubit may be idle at a different operating frequency. The operating frequencies of the 820 qubits can be selected before the calibration system performs computations. Some operating frequencies are better than others. One measure of the merits of a particular operating frequency for a particular qubit is the energy relaxation time (T1) of the qubit at that frequency. A lower energy relaxation time can lead to larger quantum computation errors.

[0168] In various implementations, example system 800 can be implemented as a client device, a server device, or both. Example system 800 can be implemented as part of a distributed computing system. Example system 800 can be implemented together with other example systems, which may be the same or different. Example system 800 can be implemented in a server farm or other facility that operates multiple computing systems to provide computing services to or on behalf of multiple client systems. Advantageously, the techniques according to the example aspects of this disclosure can provide improved calibration and maintenance of computing facilities, increased service uptime, reduced failure rates, etc.

[0169] Example Method

[0170] Figure 9 A flowchart depicts an example method for controlling fluxonium qubits according to an example embodiment of this disclosure. Although Figure 9 For illustrative and discussion purposes, the steps are described in a particular order, but the method of this disclosure is not limited to the specifically described order or arrangement. The steps of the example method 900 may be omitted, rearranged, combined, and / or adapted in various ways without departing from the scope of this disclosure.

[0171] At 902, example method 900 may include obtaining a target rotation angle for rotating a qubit (e.g., a fluxonium-like qubit 106) about at least one rotation axis. In some cases, the at least one rotation axis may include a Z-axis 310. In some cases, the at least one rotation axis may include a Y-axis 312. In some cases, the at least one rotation axis may include an X-axis, and example method 900 at 902 may include determining a target rotation angle about the Z-axis and a target rotation angle about the Y-axis based on the target rotation angle about the X-axis. In some cases, the at least one rotation axis may be a rotation axis 308, include a rotation axis 308, or be included by a rotation axis 308. In some cases, the qubit may include a qubit having inductance, capacitance, and Josephson junctions in parallel. In some cases, the qubit may include a qubit having a fluxonium mode. In some cases, the qubit may include a qubit having an effective Hamiltonian, which includes an induced energy term and a phase-slip energy term. In some cases, example method 900 at 902 may include using one or more systems or performing actions related to Figures 3 to 4. Figure 7 One or more activities described.

[0172] At 904, example method 900 may include obtaining the qubit frequency associated with the qubit. In some cases, obtaining the qubit frequency may include retrieving the qubit frequency (e.g., from a non-transitory computer-readable medium, 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...). In some cases, example method 900 at 904 may include using one or more systems or performing actions related to Figure 3. Figure 7 One or more activities described.

[0173] At 906, the example method 900 may include determining the rotation time based at least in part on the target rotation angle and the qubit frequency. In some cases, at least one rotation axis may include an X-axis, and the example method 900 at 906 may include determining the Z-rotation time and Y-rotation time based on the target angles of the Z-rotation and Y-rotation. In some cases, the rotation time may be transition times 208, 212 or hold times 210, 610 to 614, 710 to 716, including or being included by transition times 208, 212 or hold times 210, 610 to 614, 710 to 716. In some cases, the rotation time may include the sum of multiple transition times 208, 212 and / or hold times 210, 610 to 614, 710 to 716. In some cases, the example method 900 at 906 may include using one or more systems or performing operations related to Figures 3 to 4. Figure 7 One or more activities described.

[0174] At 908, example method 900 may include determining a target pulse shape for a time-dependent charge bias pulse based at least in part on rotation time and at least one rotation axis. In some cases, the target pulse shape may include a non-square pulse shape configured to suppress leakage. In some cases, the non-square pulse shape may be configured to suppress high-frequency components resonating with fluxonium transitions outside the computational subspace of the qubit. In some cases, the non-square pulse shape may include a first transition portion in which the charge bias of the non-square pulse shape varies within a first time amount; a holding portion in which the charge bias of the non-square pulse shape is constant within a second time amount; and a second transition portion in which the charge bias of the non-square pulse shape varies in a direction opposite to the direction of change associated with the first transition portion within a third time amount. In some cases, determining the target pulse shape may include determining at least one charge bias based on at least one rotation axis and rotation direction, the at least one charge bias being configured to rotate the qubit about the rotation axis in the rotation direction. In some cases, example method 900 may include an identity operation, and the target pulse shape may include a first portion configured to perform a first rotation of the fluxonium qubit about at least one axis, and a second portion configured to perform a second rotation of the fluxonium qubit about at least one axis, wherein the second rotation is in a rotation direction opposite to that of the first rotation. In some cases, the target pulse shape may include a first portion having a positive charge bias and a second portion having a negative charge bias. In some cases, the target pulse shape may include a buffer portion between the first and second portions, wherein the buffer portion is configured to reduce the total accumulated phase of the fluxonium qubit. In some cases, the charge-biased pulse may include a charge bias 202. In some cases, example method 900 at 908 may include using one or more systems or performing operations related to Figures 3 to 4. Figure 7 One or more activities described.

[0175] At 910, example method 900 may include providing a baseband voltage drive configured to generate time-dependent charge bias pulses. In some cases, the baseband voltage driver may be, include, or be comprised of baseband voltage driver 204. In some cases, example method 900 at 910 may include using one or more systems or performing actions related to... Figure 1 To one or more activities described in Figure 2.

[0176] Figure 10 A flowchart depicts an example method for controlling fluxonium qubits according to an example embodiment of this disclosure. Although Figure 10 For illustrative and discussion purposes, the steps are described in a specific order, but the method of this disclosure is not limited to the specifically described order or arrangement. The steps of the example method 1000 may be omitted, rearranged, combined, and / or adapted in various ways without departing from the scope of this disclosure.

[0177] At 1002, example method 1000 may include obtaining a target rotation for a qubit (e.g., a fluxonium-like qubit 106). In some cases, the qubit may include a qubit with parallel inductance, capacitance, and Josephson junction. In some cases, the qubit may include a qubit with a fluxonium mode. In some cases, the qubit may include a qubit with an effective Hamiltonian, which includes an induced energy term and a phase-slip energy term. In some cases, the target rotation may include one or more rotation axes 308 and one or more corresponding rotation angles. In some cases, example method 1000 at 1002 may include using one or more systems or performing operations related to Figures 3 to 4. Figure 7 One or more activities described.

[0178] At 1004, example method 1000 may include determining the target Z-rotation about the Z-axis and the target Y-rotation about the Y-axis based at least in part on the target rotation. In some cases, the target Y-rotation or the target Z-rotation may be zero or equal to zero (e.g., 2). The Z-axis can be any non-zero value (e.g., positive, negative, etc.), such as radians. In some cases, the Z-axis can be Z-axis 310, include Z-axis 310, or be included by Z-axis 310. In some cases, the Y-axis can be Y-axis 312, include Y-axis 312, or be included by Y-axis 312. In some cases, example method 1000 at 1004 may include using one or more systems or performing operations related to Figure 3. Figure 7 One or more activities described.

[0179] At 1006, example method 1000 may include determining the Z-rotation time and Y-rotation time based at least in part on the target Z-rotation and the target Y-rotation. In some cases, the Z-rotation time or Y-rotation time may be zero or non-zero. In some cases, the Z-rotation time or Y-rotation time may be transition times 208, 212 or hold times 210, 610 to 614, 710 to 716, including transition times 208, 212 or hold times 210, 610 to 614, 710 to 716, or included by transition times 208, 212 or hold times 210, 610 to 614, 710 to 716. In some cases, example method 1000 at 1006 may include using one or more systems or performing operations related to Figures 3 to 4. Figure 7 One or more activities described.

[0180] At 1008, example method 1000 may include obtaining the target total rotation time for the fluxonium qubit. In some cases, the target total rotation time may include the sum of one or more transition times 208, 212 and hold times 210, 610 to 614, 710 to 716. In some cases, example method 1000 at 1008 may include using one or more systems or performing actions related to Figures 3 to 4. Figure 7 One or more activities described.

[0181] At 1010, example method 1000 may include determining the identity rotation time based on the Z rotation time, Y rotation time, and target total rotation time. In some cases, the identity rotation time may be zero or non-zero. In some cases, the identity rotation time may include the sum of one or more transition times 208, 212 and hold times 710 to 716. In some cases, example method 1000 at 1010 may include using one or more systems or performing operations related to Figures 3 to 4. Figure 7 One or more activities described.

[0182] At 1012, example method 1000 may include determining a target pulse shape for a time-dependent charge bias pulse based at least in part on the Z-rotation time, Y-rotation time, and isochronous rotation time. In some cases, the target pulse shape may be configured to cause the fluxonium qubit to rotate around the Z-axis for a Z-rotation time. In some cases, the target pulse shape may be configured to cause the fluxonium qubit to rotate around the Y-axis for a Y-rotation time. In some cases, the target pulse shape may include a Y-rotation portion including a charge bias configured to cause the fluxonium qubit to rotate around the Y-axis for a Y-rotation time. In some cases, the Y-rotation portion may include a first transition portion in which the charge bias of the Y-rotation portion changes within a first time amount; a holding portion in which the charge bias of the Y-rotation portion is constant within a second time amount; and a second transition portion in which the charge bias of the Y-rotation portion changes in a direction opposite to the direction of change associated with the first transition portion within a third time amount. In some cases, the charge bias pulse may include a charge bias 202. In some cases, example method 1000 at 1012 may include using one or more systems or performing operations related to Figure 3. Figure 7 One or more activities described.

[0183] At 1014, example method 1000 may include providing a baseband voltage drive configured to generate time-dependent charge bias pulses. In some cases, the baseband voltage driver may be, include, or be comprised of baseband voltage driver 204. In some cases, example method 1000 at 1014 may include using one or more systems or performing operations related to... Figure 1 To one or more activities described in Figure 2.

[0184] Figure 11 An example method 1100 for performing quantum computing using a quantum circuit, according to example aspects of this disclosure, is described. For example, in some cases, the quantum circuit may include fluxonium qubits 106. Although Figure 11 For illustrative and discussion purposes, the steps are depicted in a specific order; however, the method of this disclosure is not limited to the specific order or arrangement shown. The individual steps of method 1100 may be omitted, rearranged, combined, and / or adjusted in various ways without departing from the scope of this disclosure. Method 700 can be implemented by any suitable computing system, such as a quantum computing system including quantum hardware communicating with one or more quantum control devices, such as… Figure 8 The quantum computing system 800.

[0185] At 1102, example method 1100 may include obtaining data indicating a quantum circuit. Obtaining data may include, for example, receiving data from a computing device (e.g., a user device, a server device); receiving data from a user (e.g., via an input / output device); reading data from one or more non-transitory computer-readable media; generating data (e.g., using an algorithm), etc. Data indicating a quantum circuit may include, for example, circuit design, circuit diagram, one or more unitary matrices, software code (e.g., quantum software code in a quantum computing language), etc.

[0186] At 1104, example method 1100 may include preparing one or more qubits in a known quantum state. Preparing one or more qubits in a known quantum state may include, for example, preparing one or more qubits in a known ground state (e.g., by manipulating multiple qubits such that a qubit characterized by a particular ground state, for example...). or This can be separated from qubits that are not characterized by this ground state (e.g., physically separated, individually identifiable, etc.). Preparing one or more qubits in a known quantum state can, for example, involve performing quantum gating using a control device 812 to generate a known multi-qubit ground state. Preparing one or more qubits can include, with respect to... Figure 8 The method described uses control device 812.

[0187] At 1106, example method 1100 may include applying one or more quantum gates to one or more qubits to perform a quantum algorithm. For example, in some cases, control device 812 may be used to... Figure 8 The described methods implement quantum gates or quantum circuits with multiple quantum gates, such as Pauli gates, Adama gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc.

[0188] At 1108, example method 1100 may include measuring the state of at least one of one or more qubits using a readout device. The readout device may be, for example, readout device 814, and step 1106 may in some cases be related to... Figure 8 Execute in the manner described.

[0189] Example computing system

[0190] Figure 12 A block diagram of an example computing system 5, which can perform aspects of exemplary embodiments of the present disclosure, is depicted. System 5 includes a computing device 50, a server computing system 60, and a third-party system 70, which are communicatively coupled via a network 49. System 5 also includes a quantum computing system 80 communicatively coupled to the server computing system.

[0191] The computing device 50 can be any type of computing device (e.g., a classic computing device), such as, for example, a mobile computing device (e.g., a smartphone or tablet), a personal computing device (e.g., a laptop or desktop computer), a workstation, a cluster, a game 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 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 a single processor or multiple processors 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, 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 the operations described herein.

[0192] The computing device 50 may also include one or more input components for receiving user input. For example, the user input component may be a touch-sensitive component (e.g., a touch-sensitive display or touchpad) that is sensitive to the touch of a user input object (e.g., a finger or stylus). Touch-sensitive components can be used to implement a virtual keyboard. Other example user input components include a microphone, a traditional keyboard, or other components through which the user can provide input.

[0193] The quantum computing system 80 may include one or more processors 81 (e.g., a classical processor 804) and memory 82. The one or more processors 81 may be any suitable processing device (e.g., a processor core, microprocessor, ASIC, FPGA, controller, microcontroller, etc.) and may be a single processor or multiple processors operatively connected. The memory 82 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, disks, etc., and combinations thereof. The memory 82 may store data 83 and instructions 84, which are executed by the processor 81 to cause the quantum computing system 80 to perform the operations described herein.

[0194] The quantum computing system 80 may also include a quantum system 85 for performing quantum computing. In some cases, the quantum system 85 may be quantum hardware 802, including or consisting of such quantum hardware, as referenced above. Figure 8 As described.

[0195] In some implementations, the quantum computing system 80 may include one or more server computing systems 60 or otherwise be implemented by such one or more server computing systems. In the case where the quantum computing system 80 includes multiple server computing devices, such server computing devices may operate according to a sequential computing architecture, a parallel computing architecture, or some combination thereof.

[0196] The third-party system 70 may include one or more processors 71 and memory 72. The one or more processors 71 may be any suitable processing device (e.g., processor core, microprocessor, ASIC, FPGA, controller, microcontroller, etc.) and may be a single processor or multiple processors operatively connected. The memory 72 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, disks, etc., and combinations thereof. The memory 72 may 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 one or more server computing devices or is otherwise implemented by one or more server computing devices.

[0197] Server computing system 60 may include one or more processors 61 and memory 62. The one or more processors 61 may be any suitable processing device (e.g., processor core, microprocessor, ASIC, FPGA, controller, microcontroller, etc.) and may be a single processor or multiple processors operatively connected. Memory 62 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, disks, etc., and combinations thereof. Memory 62 may store data 63 and instructions 64, which are executed by processor 61 to cause server computing system 60 to perform operations. In some implementations, server computing system 60 includes one or more server computing devices or is otherwise implemented by one or more server computing devices.

[0198] Network 49 can be any type of communication network (e.g., classical or quantum), such as a local area network (e.g., intranet), a wide area network (e.g., the Internet), or some combination thereof, and can include any number of wired or wireless links. Generally, communication conducted through network 49 can be carried over any type of wired or wireless connection using various 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).

[0199] Figure 12 An example computing system that can be used to implement this disclosure is shown. Other computing systems may also be used. For example, in some implementations, quantum computing system 80 may include server computing system 60, or vice versa. In some implementations, quantum computing system 80 may be communicatively coupled to computing device 50, third-party system 70, or server computing system 60 via network 49.

[0200] The implementations of digital, classical, and / or quantum themes, as well as digital function operations and quantum operations, described in this specification may be implemented in digital electronic circuit systems, suitable quantum circuit systems, or more generally, quantum computing 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 system" may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0201] The implementation of the digital and / or quantum themes 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 a data processing device or for controlling the operation of a data processing device). 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 qubit / qubit structures, or a combination thereof. Alternatively or additionally, program instructions can be encoded on an artificially generated propagation signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, which is then transmitted to a suitable receiver device for execution by the data processing device.

[0202] The terms quantum information and quantum data refer to information or data carried, stored, or contained in quantum systems, the smallest nontrivial system being a qubit (i.e., a system that defines a unit of quantum information). It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, for example, systems with two or more levels. Examples of such systems include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational base state is considered as the ground state and the first excited state; however, it should be understood that other settings where the computational state is considered as a higher-level excited state (e.g., a qubit) are also possible.

[0203] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, or multiple digital and quantum processors or computers, and combinations thereof. The device may also be or include dedicated logic circuit systems, such as FPGAs (Field-Programmable Gate Arrays) or ASICs (Application-Specific Integrated Circuits), or quantum simulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the capability to perform general-purpose quantum computing. In addition to hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stacks, database management systems, operating systems, or combinations thereof.

[0204] Digital or classical computer programs, which can also be referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and can be deployed in any form, including as standalone programs or as modules, components, subroutines, or other units suitable for use in a digital computing environment. Quantum computer programs, which can also be referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages) and translated into a suitable quantum programming language, or can be written in quantum programming languages ​​such as QCL, Quipper, Cirq, etc.

[0205] Digital and / or quantum computer programs may, but do not necessarily, correspond to files in a file system. Programs may be stored as a portion of a file containing 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 coordinating files (e.g., files storing one or more modules, subroutines, or code sections). Digital and / or quantum computer programs may be deployed to execute on a single digital or quantum computer, or on multiple digital and / or quantum computers located at a single site or distributed across multiple sites and interconnected via digital and / or quantum data communication networks. A quantum data communication network is understood as a network that can transmit quantum data using quantum systems (e.g., qubits). Generally, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum data and digital data.

[0206] The processes and logic flows described in this specification can be executed by one or more programmable digital and / or quantum computers (which may operate using one or more digital and / or quantum processors, as appropriate) executing one or more digital and / or quantum computer programs to perform functions by manipulating input digital and quantum data and generating outputs. The processes and logic flows can also be executed by a dedicated logic circuit system (e.g., an FPGA or ASIC or a quantum simulator) or by a combination of a dedicated logic circuit system or a quantum simulator and one or more programmable digital and / or quantum computers, and the device can also be implemented as said dedicated logic circuit system or said combination.

[0207] For a system of one or more digital and / or quantum computers or processors that is “configured” or “operable to” perform a specific operation or action, it means that the system has software, firmware, hardware, or a combination thereof installed thereon that causes the system to perform the operation or action in operation. For one or more digital and / or quantum computer programs configured to perform a specific operation or action, it means that the one or more programs include instructions that cause the device to perform the operation or action when executed by a digital and / or quantum data processing device. A quantum computer can receive instructions from a digital computer that cause the device to perform the operation or action when executed by a quantum computing device.

[0208] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs may be based on general-purpose or special-purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, or random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0209] Some example elements of a digital and / or quantum computer are a central processing unit (CPU) that makes or executes instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented by or incorporated into a dedicated logic circuit system or quantum simulator. Generally, a digital and / or quantum computer will also include one or more mass storage devices for storing digital and / or quantum data, such as magnetic disks, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information, or operatively coupled to receive digital and / or quantum data from or to said one or more mass storage devices, or both. However, a digital and / or quantum computer need not have such devices.

[0210] 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 memories, media, and memory devices, including, for example, semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable hard disks; magneto-optical disks; and CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data for a long period with high fidelity and high efficiency, for example, using light for transmission and using matter for storage and preservation of quantum characteristics (such as superposition or quantum coherence) of the quantum data at an optical-material interface.

[0211] Control of the various systems or portions thereof described in this specification may be implemented using digital and / or quantum computer program products, which include instructions stored on one or more tangible, non-transitory, machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may each be implemented as an apparatus, method, or electronic system, which may include one or more digital and / or quantum processing devices and memory for storing executable instructions to perform the operations described in this specification.

[0212] While this specification contains numerous details of specific implementations, these details should not be construed as limiting the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features described in this specification within the context of individual implementations may also be implemented in combination within a single implementation. Conversely, individual features described within the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple implementations. Furthermore, although features are described above as functioning in certain combinations, and even initially claimed to be so, one or more features from a claimed combination may, in some cases, be removed from said combination, and the claimed combination may be for a sub-combination or a variation thereof.

[0213] Similarly, although operations are depicted in a specific order in the accompanying drawings, this should not be construed as requiring such operations to be performed in the specific order shown or in sequential order, or requiring all shown operations to be performed to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous. Furthermore, the separation of the various system modules and components in the implementation described above should not be construed 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 encapsulated in multiple software products.

[0214] Specific implementations of this subject matter have been described. Other implementations are within the scope of the appended claims. For example, the actions described in the claims can be performed in different orders and still achieve the desired result. As an example, the processes depicted in the figures do not necessarily require a specific order or sequence shown to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous.

[0215] Various aspects of this disclosure have been described with respect to their illustrative implementations. Numerous other implementations, modifications, and alterations within the scope and spirit of the appended claims will arise in those skilled in the art upon careful reading of this disclosure. Any and all features of the following claims can be combined or rearranged in any possible manner. Therefore, the scope of this disclosure is illustrative rather than limiting, and this disclosure does not exclude such modifications, alterations, and / or additions to the subject matter that will be readily understood by those skilled in the art. Furthermore, terms are described herein using a list of example elements connected by conjunctions such as “and,” “or,” and “however.” It should be understood that such conjunctions are provided for illustrative purposes only. For example, a list connected by a specific conjunction such as “or” may refer to “at least one” or “any combination” of the example elements listed therein, where “or” should be understood as “and / or” unless otherwise indicated. Furthermore, terms such as “based on” should be understood as “at least partially based on.”

[0216] Those skilled in the art will understand using the disclosure provided herein that elements of any claim, operation, or process discussed herein can be adjusted, rearranged, expanded, omitted, combined, or modified in various ways without departing from the scope of this disclosure. Some claims are described with letter references to claim elements for illustrative purposes and are not intended to be limiting. Letter references do not imply a particular order of operations. For example, letter identifiers such as (a), (b), (c)..., (i), (ii), (iii)... may be used to describe operations. Such identifiers are provided for the convenience of the reader and do not indicate a particular order of steps or operations. Operations indicated by list identifiers (a), (i), etc., may be performed before, after, or in parallel with another operation indicated by list identifiers (b), (ii), etc.

Claims

1. A quantum computing system, comprising: A qubit, wherein the qubit comprises an inductor, a capacitor, and a Josephson junction connected in parallel; as well as A control system configured to implement control signals to implement quantum gates on the qubits, wherein the control signals include voltage bias pulses.

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

3. The quantum computing system of claim 1, wherein implementing the quantum gate comprises performing at least one rotation about at least one rotation axis: Obtain the target rotation angle; Obtain the qubit frequency associated with the qubit; The rotation time is determined at least in part based on the target rotation angle and the qubit frequency; The target pulse shape for the time-dependent charge bias pulse is determined at least in part based on the rotation time and the at least one rotation axis; as well as A baseband voltage drive is provided, which is configured to generate the time-related charge bias pulse.

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

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

6. The quantum computing system of claim 3, wherein the at least one rotation axis includes an X-axis, and performing the at least one rotation comprises: Obtain the first target rotation angle around the X-axis; The second target rotation angle around the Y axis and the third target rotation angle around the Z axis are determined at least in part based on the first target rotation angle around the X axis; The Y rotation time is determined at least in part based on the second target rotation angle and the qubit frequency; And the Z rotation time is determined at least in part based on the rotation angle of the third target and the frequency of the qubit; The target pulse shape for the time-related charge bias pulse is at least partially based on the Y rotation time and the Z rotation time.

7. The quantum computing system of claim 3, wherein implementing the quantum gate includes performing at least one identity operation, wherein the identity operation includes: Perform the at least one rotation about the at least one rotation axis; as well as Perform at least a second rotation about the at least one rotation axis; The second rotation is configured to rotate the qubit about the at least one axis in a direction opposite to the 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 includes a buffer portion between the first portion and the second portion, wherein the buffer portion is configured to reduce the total accumulated phase of the qubit.

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

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

12. The quantum computing system of claim 11, wherein the non-square pulse shape is configured to suppress high-frequency components of fluxonium transition resonances outside the computational subspace of the qubit.

13. The quantum computing system of claim 11, wherein the non-square pulse shape includes a first transition portion in which the charge bias of the non-square pulse shape varies within a first time period; a holding portion in which the charge bias of the non-square pulse shape is constant within a second time period; and a second transition portion in which the charge bias of the non-square pulse shape varies in a direction opposite to the direction of change associated with the first transition portion within a third time period.

14. A method for providing baseband voltage drive to control fluxonium-like qubits, comprising: To obtain a target rotation of qubits for use in circuits including those with fluxonium modes; The target Z-rotation around the Z-axis and the target Y-rotation around the Y-axis are determined at least in part based on the target rotation. The Z-rotation time and Y-rotation time are determined at least in part based on the target Z-rotation and the target Y-rotation; The target pulse shape for the time-dependent charge bias pulse is determined at least in part based on the Z rotation time and the Y rotation time. as well as A baseband voltage drive is provided, the baseband voltage drive being configured to generate the time-related charge bias pulse; The target pulse shape is configured to make the fluxonium qubit: Rotation around the Z-axis for the duration of the Z-rotation; as well as Rotation around the Y-axis for the duration of the Y-rotation.

15. The method of claim 14, further comprising: Obtain the total rotation time of the target; as well as Based on the Z rotation time, the Y rotation time, and the target total rotation time, determine the constant rotation time; The target pulse shape is further configured to make the fluxonium qubit: Perform an identity operation, wherein the 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 includes a buffer portion between the first portion and the second portion, wherein the buffer portion is configured to reduce the total accumulated phase of the fluxonium qubit.

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

19. The method of claim 18, wherein the Y-rotation portion comprises: In the first transition section, the charge bias of the Y-rotation section changes within a first time period. In the holding portion, the charge bias of the Y-rotation portion is constant over a second time period; as well as In the second transition portion, the charge bias of the Y-rotation portion changes in a direction opposite to the direction of change associated with the first transition portion within a third time period.

20. A method for providing baseband voltage drive to control fluxonium-like qubits, comprising: To obtain a target rotation for a qubit with an effective Hamiltonian, the effective Hamiltonian including an induced energy term and a phase-slip energy term; The target Z-rotation around the Z-axis and the target Y-rotation around the Y-axis are determined at least in part based on the target rotation. The Z-rotation time and Y-rotation time are determined at least in part based on the target Z-rotation and the target Y-rotation; The target pulse shape for the time-dependent charge bias pulse is determined at least in part based on the Z rotation time and the Y rotation time. as well as A baseband voltage drive is provided, the baseband voltage drive being configured to generate the time-related charge bias pulse; The target pulse shape is configured such that the qubit: Rotation around the Z-axis for the duration of the Z-rotation; as well as Rotation around the Y-axis for the duration of the Y-rotation.

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