Non-adiabatic embodiments of iSWAP quantum logic gates

Abstract

Methods, systems, and apparatus for generating a dip process for implementing an iSWAP quantum logic gate between a first qubit and a second qubit. In one aspect, a dip process defining a trajectory of a detuning between a frequency of a first qubit and a frequency of a second qubit includes: during a first phase, non-adiabatically driving the detuning between the frequency of the first qubit and the frequency of the second qubit to pass through a first avoidance crossover in a leakage channel; during a second phase, driving a detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossover by a second avoidance in the exchange channel; during a third phase, allowing the first qubit and the second qubit to freely evolve and interact; during the fourth phase, performing the second phase in the opposite order; and during the fifth phase, performing the first phase in the opposite order.

Description

[0001] This application is a divisional application of PCT patent application No. 201980093611.6, filed on March 5, 2019, entitled "Non-Adiabatic Implementation of iSWAP Quantum Logic Gate". Technical Field

[0002] This manual relates to quantum information processing. Background Technology

[0003] This specification describes the techniques used to implement iSWAP logic gates in quantum computers. Summary of the Invention

[0004] In general, an innovative aspect of the subject matter described in this specification can be implemented in a method for implementing an iSWAP quantum logic gate between a first qubit and a second qubit, the method comprising: implementing a plugging schedule that defines a detuning trajectory between the frequencies of the first qubit and the second qubit, including: during a first phase, non-adiabatically driving the detuning between the frequencies of the first qubit and the second qubit to avoid crossover via a first in a leakage channel; during a second phase, driving the detuning between the frequencies of the first qubit and the second qubit to avoid crossover via a second in a swapping channel; during a third phase, allowing the first qubit and the second qubit to freely evolve and interact; during a fourth phase, implementing the second phase in reverse order; and during a fifth phase, implementing the first phase in reverse order.

[0005] Other embodiments of this aspect include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or combinations thereof on the system, which, in operation, cause the system to perform these actions. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing apparatus, cause the apparatus to perform these actions.

[0006] The foregoing and other embodiments may each optionally include one or more of the following features, individually or in combination. In some embodiments, the descent process is based on a trapezoidal ramp function, characterized by the ramp rise time, hold time, and variance of a Gaussian filter function.

[0007] In some implementations, the method further includes generating a descent process, including determining a pulse for: the detuning between the frequencies of the first qubit and the second qubit, the frequency of the first qubit, and the frequency of the second qubit.

[0008] In some implementations, the generation of the sag process also includes synchronizing errors in the switching channel and the leaking channel.

[0009] In some implementations, the pulse determined for the frequency of the first qubit is equal to the sum of the pulse determined for the second qubit and the determined detuning pulse.

[0010] In some implementations, the frequency of the first qubit depends on an asymmetric parameter equal to the difference between the interaction frequency and the initial frequency of the second qubit, divided by the idle detuning.

[0011] In some implementations, the pulse that determines the frequency of the second qubit is equal to ,in This indicates the initial frequency of the second qubit. Indicates the frequency of interaction. Indicates the overshoot frequency, and This represents the trapezoidal ramp function.

[0012] In some implementations, the pulse determined for detuning is equal to ,in Indicates initial detuning. Indicates the overshoot frequency, and This represents the trapezoidal ramp function.

[0013] In some implementations, synchronizing errors in the exchange channel and the leakage channel includes: for a full group exchange, determining a detuning trajectory between the frequencies of the first qubit and the second qubit, and minimizing the leakage channel error of that trajectory via time-dependent inter-qubit interaction strength.

[0014] In some implementations, the strength of the interaction between qubits is proportional to the square root of the product of the frequency of the first qubit and the frequency of the second qubit.

[0015] In some implementations, synchronizing errors in the switching channel and the leakage channel includes adjusting a cost function that minimizes the probability of a leakage error plus the probability of a switching error: an interaction frequency representing the frequency at which the frequency trajectories of the first qubit and the second qubit meet, a hold time, and an overshoot frequency equal to the difference between the frequencies of the first qubit and the second qubit during the hold time.

[0016] In some implementations, adjusting the interaction frequency, hold time, and overshoot frequency to minimize the cost function involves repeatedly performing the following steps until it is determined that the value of the cost function is converging to a minimum: scanning the interaction frequency-hold time with an overshoot frequency constant; scanning the interaction frequency-overshoot frequency; and scanning the hold time-overshoot frequency.

[0017] In some implementations, the method also includes using a randomized benchmark to adjust the generation process to increase the fidelity of the iSWAP gate.

[0018] In some implementations, allowing the first and second qubits to evolve and interact freely includes allowing the first and second qubits to evolve and interact freely within a predetermined distance from the 10-01 resonance to achieve group exchange.

[0019] In some implementations, detuning between the frequency driving the first qubit and the frequency driving the second qubit to avoid crossover via a second crossover avoidance in the exchange channel includes thermally driving the detuning between the frequency driving the first qubit and the frequency driving the second qubit to avoid crossover via a second crossover avoidance in the exchange channel.

[0020] In some implementations, the first and second qubits include capacitively coupled Xmon qubits.

[0021] In some implementations, the leakage channel includes a manifold spanned by computational state 11 and two non-computational states 02 and 20, and wherein detuning between the frequency driving the first qubit and the frequency driving the second qubit to avoid crossover in the leakage channel includes detuning between the frequency driving the first qubit and the frequency driving the second qubit to resonate through states 11–20.

[0022] In some implementations, the exchange channel includes a manifold spanned by computed states 10 and 01, and wherein detuning between the frequency driving the first qubit and the frequency driving the second qubit to avoid crossover in the exchange channel includes detuning between the frequency driving the first qubit and the frequency driving the second qubit to resonate through states 10–01.

[0023] In some implementations, the second phase is performed in reverse order, including driving the detuning between the frequencies of the first and second qubits to achieve a complete group exchange between qubit states 10 and 01.

[0024] The subject matter described in this specification can be implemented in a particular manner to achieve one or more of the following advantages.

[0025] Existing implementations of iSWAP gates are relatively slow and prone to leakage. In particular, it is not clear a priori how to mitigate this leakage without compromising the fidelity of the SWAP operation performed by the iSWAP gate. For example, Rigetti Computing Inc. recently reported a 150 ns gate time and 94% gate fidelity for a parametrically modulated iSWAP gate implemented in a system consisting of fixed-frequency transmon qubits coupled to another frequency-tunable transmon qubit (see, for example, N. Didier, EA Sete, MP da Silva, CT Rigetti, arXivpreprint arXiv:1706.06566 (2017)).

[0026] The technique described so far uses a system consisting of two frequency-tunable Xmon qubits. This special arrangement allows for the synchronization of exchange and leakage errors by assigning different trajectories to the two qubits based on their frequency parking positions (asymmetric synchronization), and then tuning the interaction frequency to a specific value where both errors are strongly suppressed.

[0027] As a result, the currently described technique for executing the iSWAP gate provides a fast and robust ramp-down process in a non-adiabatic state, allowing for full SWAP operation with suppression of leakage errors and over 99% fidelity. For a given interqubit interaction strength, the gate execution time is greatly reduced to ~23 ÷ 25 ns, approaching the physical limit. =π / 2g, for example, for the typical interaction strength available in the current hardware implementation of the Xmon qubit. , Furthermore, the techniques described in this specification provide an effective protocol for automatic door calibration.

[0028] The techniques described so far for implementing the iSWAP gate utilize only low-frequency control of detuning between qubit frequencies and are suitable for recent quantum computing architectures. Furthermore, since the iSWAP gate is computationally difficult to simulate using classical computers, the techniques described in this specification can facilitate recent experiments on quantum advantage and have a direct impact on the field of quantum computing.

[0029] The implementation of the techniques described above can reduce circuit design complexity and provide a path to scalable quantum computing architectures with high-fidelity multi-qubit gates. This, in turn, is an important step toward the long-term goal of developing error-correcting quantum computers.

[0030] Details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the specification, drawings, and claims. Attached Figure Description

[0031] Figure 1 An example system for implementing iSWAP quantum logic gates is described.

[0032] Figure 2 A graph comparing an example adiabatic protocol and an example non-adiabatic protocol used to implement an iSWAP gate is shown.

[0033] Figure 3 Two graphs of the energy levels of two coupled Xmon qubits constrained by a non-adiabatic protocol are shown.

[0034] Figure 4 This is a flowchart of an example process for implementing the iSWAP quantum logic gate between the first and second qubits according to the sag process.

[0035] Figure 5 Two example sudden drop processes are shown.

[0036] Figure 6 The transition probabilities for two example process shapes that allow full SWAP operation are shown.

[0037] Figure 7 An example of synchronization with leakage error and exchange error is shown.

[0038] Figure 8 An example detuning trajectory is shown during the implementation of the iSWAP gate.

[0039] Figure 9 Three example processes with different interaction frequencies are shown.

[0040] Figure 10 Three example values ​​for g are shown for different interaction frequencies.

[0041] Figure 11 Three equivalent processes with different q values ​​but the same initial detuning and interaction frequency are shown.

[0042] Figure 12 An example spectrum of the interaction frequency of a two-qubit iSWAP gate implemented by rectangular pulses is shown.

[0043] Figure 13 An example of a rabbi descent protocol is shown.

[0044] Figure 14 Example SWAP error and leakage error are shown.

[0045] Figure 15 The exchange error and leakage error as functions of gate time are shown.

[0046] Figure 16 This is a flowchart of an example process used to generate a sudden drop process.

[0047] Figure 17 A series of scans implementing the automatic calibration protocol are shown. Detailed Implementation

[0048] Overview

[0049] A complete swap operation is the trajectory maintenance, i.e., zero leakage, of a two-qubit system's swap channel state. and A complete population transfer can be achieved between states 10 and 01 (referred to as states 10 and 01 in this text). Most generally, this operation can be performed on... In the basics, this is described using the following unitary matrix:

[0050]

[0051] in Indicates phase shift and angle and This represents the additional phase that can be corrected using a single qubit z-rotation. Since these additional phases can be ignored, the SWAP gate and the iSWAP gate can be described by the unitary matrix given in equation (2) below:

[0052]

[0053] A challenging aspect of implementing SWAP or iSWAP gates is that the implementation must be consistent with general quantum computing architectures, such as capacitively coupled qubits. In such architectures, qubit frequencies are arranged in a so-called zigzag order, with the frequencies of their nearest-neighbor qubits separated by ~1 GHz. After applying a gate operation to the qubits, they return to their original placement. This makes some protocols, such as Landau-Zener type direct passage driving schedules, difficult or impossible to apply.

[0054] Furthermore, by detuning the energy levels between the 10 and 01 states... Driving to zero (10–01 resonance) to execute a SWAP or iSWAP gate between the first and second qubits inevitably requires first passing through an 11–20 leakage resonance. As a result, simple trapezoidal ramp functions are insufficient to reliably avoid leakage to the non-computational parts of the Hilbert space. Existing hardware implementations of SWAP or iSWAP gates also suffer from slow execution.

[0055] The techniques described in this specification address these challenges. Specifically, this specification describes a non-adiabatic protocol that can be applied to qubits to implement two-qubit gates, such as SWAP gates or iSWAP gates, with improved fidelity. The described non-adiabatic protocol utilizes low-frequency control of detuning between qubit frequencies. More specifically, the described technique employs a multi-parameter set of bias-controlled timing protocols configured to suppress errors in both the SWAP and leaky quantum channels at the end of gate execution. The defined pulse shape and duration are obtained using motion synchronization in both channels, resulting in high-fidelity SWAP operation exceeding 99%, which is a complete group exchange in the |01)-|10) channels, accompanied by a reduced probability of exciting a single qubit to a higher-frequency anharmonic state.

[0056] For convenience, the techniques described in this specification are presented with reference to implementing the iSWAP gate. However, these techniques can also be applied to implementing the SWAP gate, as well as other gates based on SWAP operations, such as... Door.

[0057] Example Architecture

[0058] Figure 1 An example system 100 for implementing iSWAP quantum logic gates on a two-qubit subsystem is depicted. Example system 100 is an example of a system implemented as part of a quantum computing device in which the systems, components, and techniques described in this specification can be implemented.

[0059] System 100 includes a two-qubit subsystem 102 that communicates with control electronics 104. The two-qubit subsystem 102 includes a first qubit 106 and a second qubit 108. In some cases, such as... Figure 1 As shown, the first qubit 106 and the second qubit 108 can be capacitively coupled Xmon qubits. For example, the first qubit 106 and the second qubit 108 can be part of a linear chain of Xmon qubits included in a quantum computing device. However, in other cases, qubits can include flux qubits, phase qubits, or qubits with frequency interaction.

[0060] The first qubit 106 and the second qubit 108 can be operated by adjusting the qubit frequency, for example, applying pulses generated by the control electronics 104 to the qubit. In the case that the first qubit 106 and the second qubit 108 are Xmon qubits, the qubit frequencies can be arranged at predetermined distances from each other and in a zigzag pattern relative to other qubits that may be included in the quantum computing device.

[0061] The Hamiltonian describing two qubits can be given by the following equation (3).

[0062]

[0063] In equation (3), The time-dependent natural frequency of a single qubit. This indicates the anharmonicity or deharmonicity of a qubit. Indicates the strength of interaction between qubits. This indicates the creation operator and the annihilation operator, and Represents a quantity operator. Typical values ​​include MHz and MHz. Without sacrificing versatility, ,in This represents controlled detuning with initial and final values. GHz, and This represents the pulse duration (gate time) applied to one or more qubits to implement a quantum logic gate.

[0064] The Hamiltonian described in equation (3) can be transformed into a rotating frame of the second qubit and eliminated, for example, and Simplify by using the inverse rotation term. The Hamiltonian obtained in the rotating wave approximation (RWA) preserves the total number of excitations M, and therefore, the 9 × 9 Hilbert space is split into 5 subspaces corresponding to M = 0, 1 . . , 4. The three subspaces for M = 0, 1, 2 are... This is relevant to the qubit operations being driven. These are the ground state 00 (also represented as...). ), the SWAP manifold spanned by computed states 10 and 01, and the leak manifold spanned by computed state 11 and two non-compute states 02 and 02.

[0065] Therefore, the sub-Hamiltonian matrix describing the SWAP and leakage channels in RWA and It can be represented as

[0066]

[0067]

[0068] The process used to implement the iSWAP gate is through detuning. It is implemented through parameterization. Mistuning can take the following forms.

[0069] (6)

[0070] in express The initial value, control angle Depending on the set of variational parameters {c} containing M ≥ 2 elements, and two additional parameters ν (shift) and λ (scaling) can be used in extended tuning / optimization processes (e.g., error synchronization processes described below).

[0071] parameter λ and λ define two known limiting cases in the art, namely, non-adiabatic protocols ( = 0, λ = 1) and adiabatic protocol ( The differences between these two protocols are as follows: Figure 2 and Figure 3 As shown.

[0072] The control electronics 104 includes control devices, such as an arbitrary waveform generator, capable of operating the first qubit 106 and the second qubit 108. For example, the control electronics 104 may include control devices that tune the frequencies of the first qubit 106 and the second qubit 108 by applying control signals (e.g., voltage pulses) to the qubits via corresponding control lines.

[0073] Furthermore, the control device may include a measurement device, such as a readout resonator, capable of performing measurements of the first qubit 106 and the second qubit 108 via corresponding qubit control lines. The control electronics 104 may be configured to store, display, and / or further process the measurement results of the first qubit 106 and the second qubit 108.

[0074] In some embodiments, control electronics 104 may include a data processing device and an associated memory. The memory may include a computer program with instructions that, when executed by the data processing device, cause the data processing device to perform one or more functions described herein, such as applying control signals to quantum bits.

[0075] Figure 2A graph 200 is shown comparing an example adiabatic protocol 202 used to perform iSWAP gates and an example non-diabatic protocol 204 used to perform iSWAP gates. The graph includes dimensions-free time representations. (in The horizontal axis 206 represents gate time, and the detuning is measured in GHz. The vertical axis is 208. The first horizontal line 210 defines the level crossing points in the SWAP channel. The second horizontal line 212 defines the level crossing points in the leakage channel.

[0076] When comparison occurs In the leakage channel 212 near the 11–20 resonance (avoided level crossing) at the location. When considering the slope, the difference between adiabatic protocol 202 and non-diabatic protocol 204 is clear. Non-diabatic protocol 204 passes through cross 212 at a very high velocity, while adiabatic protocol 202 has an inflection point corresponding to the minimum relative velocity of the energy levels. For SWAP channel 210, the 10–01 resonance occurs... At the same point, the behavior is the opposite, that is, non-insulating protocol 204 has an inflection point, while insulating protocol 202 drops almost vertically.

[0077] like Figure 2 As shown, adiabatic protocol 202 takes the shape of a "cascade waterfall," which rapidly descends and decelerates near level 212, forming a "ledge," before rapidly descending again and forming a ledge once more. This behavior reflects the idea of ​​local adiabatic evolution in a system with several energy level crossings. In other words, the process behaves as a cascade, decelerating near each avoided crossing and accelerating again afterward. Non-adiabatic protocol 204 (the subject of this specification) descends directly, exhibiting a "plunge waterfall."

[0078] exist Figure 3 Furthermore, for non-insulated protocols, the energy eigenvalues ​​of the Hamiltonian given by equation (1) are shown in the leakage channel and the SWAP channel.

[0079] Figure 3 Two plots, 300 and 350, illustrate the energy levels of two coupled Xmon qubits (e.g., qubits 106 and 108) constrained by a non-adiabatic protocol. Plot 300 shows the energy levels in a leaky manifold. Plot 350 shows the energy levels in a swapped manifold. Both plots include dimensionless time representations. (in The horizontal axis represents gate time, and the vertical axis represents energy levels measured in GHz.

[0080] The proposed non-adiabatic iSWAP gate process

[0081] The proposed protocol for implementing the iSWAP gate between the first and second qubits includes a plugging schedule that defines the detuning trajectory between the frequencies of the first and second qubits. The plugging schedule comprises multiple phases: a two-stage ramp-down path, a stationary phase, and a two-stage ramp-up path in reverse order, which maintains the overall time-reversal symmetry of the protocol. (See below for reference.) Figure 4 Describe each stage in detail.

[0082] Figure 4 This is a flowchart of an example process 400 for implementing the proposed sag process, which defines the detuning trajectory between the frequencies of the first and second qubits. For convenience, process 400 will be described as being executed by quantum hardware communicating with control electronics located at one or more locations. For example, quantum hardware appropriately programmed according to this specification... Figure 1 System 100 can execute process 400.

[0083] During the first phase, the system non-adiabatically drives the detuning between the frequencies of the first and second qubits to avoid crossovers in the leakage channel, thereby avoiding leakage errors (step 402). That is, the detuning changes rapidly enough that the curve representing the detuning descends almost vertically through the crossover. This provides a reduction in ramp time overhead, and for a given inter-qubit interaction strength g, the estimated gate time can be approximately equal to the physical limit. For example, regarding interaction strength In other words, .

[0084] This section uses a definition of a non-adiabatic path based on the Landau-Ziner criterion. Specifically, for... To avoid crossings, according to the classic Landau-Ziner theory, non-adiabatic transitions (i.e., the system will remain in the same state after passing through a crossing point) are avoided. The probability of ) is given by Given, among which , The matrix elements represent the Hamiltonians between non-adiabatic states, and This represents the relative "velocity" of the non-adiabatic energy levels at the intersection. Therefore, if Then the pathway is considered non-adiabatic, that is... Non-adiabatic threshold From typical values The convergence of the "instantaneous" time-dependent perturbation series is determined by this. In the descent protocol... and = In the case of, 1 GHz is the maximum (also known as idle) detuning, and ns is the typical rise (also known as ramp) time. Therefore, It satisfies the non-adiabatic criterion very well.

[0085] During the second phase, the system drives a detuning between the frequencies of the first and second qubits to avoid crossover in the SWAP channel, thereby achieving full group exchange (step 404). In some embodiments, the driving of the detuning frequency between the frequencies of the first and second qubits during the second phase is adiabatic.

[0086] During the third phase (stationary phase), the system allows the first and second qubits to evolve freely (step 406). More specifically, during the third phase, the qubit frequencies are very close to the resonant 10⁻⁶ (e.g., within a predetermined distance) to enable group exchange. The first and second qubits are allowed to interact and exchange groups while the entire two-qubit system remains in hold time. It evolves freely during this period.

[0087] During the fourth phase, the system performs the second phase in reverse order to achieve full group switching in the switching channel (step 408). During the fifth phase, the system performs the first phase in reverse order to avoid leakage errors (step 410). The fourth and fifth phases maintain time-reversal symmetry. The example sag process implemented using steps 402-410 of example process 400... Figure 2 , Figure 3 and Figure 5 As shown in the image.

[0088] See below for reference. Figure 5 In some embodiments, the proposed use example process 400 implements a descent process based on a control angle defined in the frequency trajectory of the first qubit during protocol execution. The trapezoidal waveform is generated. During the proposed protocol, the motion of the control vector corresponding to the control angle accelerates in the middle of the ramp descent process (end of the first stage) and then decelerates (end of the second stage). As described below, the acceleration near the avoidance of the leakage channel crossing is controlled by a non-adiabatic Rabi process, which leads to the leakage error affecting... The Rabi oscillation, the amplitude of which is Proportional. In other embodiments, see below. Figures 8-17 The proposed descent process can be generated by defining the frequency trajectories of the first and second qubits during protocol execution.

[0089] Implementing iSWAP gates using the Rabbi protocol.

[0090] Alternative approaches to implementing the iSWAP gate include protocols that satisfy local adiabatic evolution conditions. Local adiabatic evolution conditions imply that detuning can change rapidly far from where crossovers are avoided, and must be slowed down near the minimum gap between energy levels. Therefore, the conditions for local adiabatic evolution can be defined more broadly.

[0091] A known local adiabatic condition can be given by the following equation (7).

[0092]

[0093] In equation (7), and This represents the Hamiltonian describing the switching channel. The instantaneous adiabatic intrinsic state, and Indicates for The time-related gap, its represents the detuning between energy levels, and g represents the interaction strength between qubits. (To introduce this protocol, it was initially assumed that...) It is convenient, however, this assumption will subsequently be relaxed.

[0094] The process for implementing the iSWAP gate, which satisfies the local adiabatic condition given by equation (5), includes a forward single-path process. In the forward single-path process, the detuned scan starts from... > 0 to The system exhibits high energies (<0) and undergoes cross-avoidance only once. Since qubits do not return to their initial placement state, this process cannot be directly applied to the Xmon architecture. However, analysis of the protocol provides insights for designing the proposed non-adiabatic descent protocol.

[0095] The Hamiltonian of the switching channel is described in equation (4). It can also be expressed as

[0096]

[0097] Among them, control angle

[0098]

[0099] Indicates the control vector (effective magnetic field) The angle between the qubit and the z-axis on the Bloch sphere of one of the two qubits. The control angle defines the motion of the qubit whose frequency changes during the implementation of the iSWAP gate. The locally adiabatic condition given by equation (7) implies

[0100] .

[0101] Based on this meaning, the problem can be solved precisely by transforming the time-dependent Schrödinger equation to the natural time scale:

[0102]

[0103]

[0104] These equations define the bijection describing the accelerating reference frame, where the magnitude of the effective magnetic field (control vector) is a time-independent constant. In other words, the Hamiltonian of the bilevel system has a constant gap in the natural time frame. Furthermore, the system's motion is entirely controlled by the angle relative to natural time. The correlation was determined.

[0105] According to equation (12) The expression yields Furthermore, based on the local adiabatic conditions of equations (7) and (10), the following is derived: In other words

[0106]

[0107] in and Let these represent the initial angle and the final angle, respectively. Let represent the pulse time (gate time) in the natural time scale. Equation (9) shows that, by applying the local adiabatic conditions given in equation (7), a known Rabi problem concerning the motion of the magnetic moment in a uniform rotating magnetic field of fixed amplitude is obtained.

[0108] In equation (8), the control vector rotates around the y-axis in the x, z plane from the North Pole to the South Pole. To make the similarity to the Rabi problem more apparent, the current coordinate system is rotated around the x-axis. This places the control vector in the x, y plane (instead of the x, z plane in standard qubit terminology). The transformed Schrödinger equation then takes the following form:

[0109]

[0110] in This represents a two-component spinor, and and Let denote the eigenvector of the z-Pauli operator in the rotated coordinate system (i.e., the z-Pauli operator in the original frame). Equations (13) and (14) describe a specific case of the Rabi problem with zero magnetic field in the z direction.

[0111] function It satisfies two separate Schrödinger-like equations:

[0112] (15)

[0113] Its initial condition is chosen to be the Hamiltonian given by equation (6) in One of the intrinsic states (marked as 0 or 1) at the location. The second boundary condition for the derivative can be obtained directly from equation (8). Here, This represents the dimensionless natural time during the gate operation, and This represents the dimensionless total duration of the gate, i.e., the gate time.

[0114] The following content uses time-independent (or “non-adiabatic”) bases 0 and 1 associated with the eigenstates of the Hamiltonian at the initial time as the levels move away from each other. The primary quantity of interest is at the end of the gate. The probability of transitioning from initial state 0 to final state 1 From the perspective of swap operations, The probability of always succeeding, and This is the probability of the SWAP error. Using equation (15) and boundary conditions, the probability of the SWAP error satisfying the Schrödinger-like equation is... Given by the following formula

[0115]

[0116] Equations (14) and (15) can be solved directly, reflecting that the Hamiltonian in equation (8) becomes time-independent in the rotating frame associated with the uniform rotating control vector. The probability of the SWAP error is given by the following equation.

[0117]

[0118] For the sake of convenience, Represented as dimensionless gate time The function. The detuning process generated by the Rabi protocol can be used using relations. and To determine, that is

[0119]

[0120] This detuning process, which produces the SWAP error given by equation (17), has several interesting properties. Importantly, for the process described so far, the error oscillates as a function of pulse time and has a set of times, where If the pulse time is tuned to one of these intervals, the SWAP gate can be executed with very high fidelity.

[0121] The method for generating the proposed steep drop process

[0122] In the following sections, time-independent, non-adiabatic basis 0s and 1s are used, which are associated with the eigenstates of the Hamiltonian at the initial time as the levels become increasingly distant from each other, because this is more convenient for numerical implementations. For convenience, a return process compatible with the Xmon architecture is also considered, and time intervals are also included. Internal execution gate simulation. Similarly, the primary quantity of interest in the SWAP channel is the probability of transitioning from initial state 0 to final state 1 at the end of the gate. In the leakage path, the main quantity of interest is derived from the computational state. Transition to non-computational state The probability, and This indicates leakage error.

[0123] Processes are based on angles in the natural time scale The trapezoidal waveform. The trapezoidal waveform can be given by the following formula.

[0124]

[0125] in The time interval representing the length of a slope ascent (or descent) The angular velocity during the period. This is determined by defining the relative ramp time. The sudden drop process is composed of two parameters s and To define, where , It is the maximum angular distance traveled by the control vector. Therefore, And maintain time .

[0126] The meaning of the total gate time in the natural scale can be derived from equation (12) and related to the laboratory gate time. Related:

[0127]

[0128] Figure 5 The curve 500 shows the angle in natural time. Two example trapezoidal waveforms, 502 and 504. Figure 5The curve 550 shows two corresponding detuning processes for each trapezoidal waveform 503 and 504. The detuning process 552 corresponds to the trapezoidal waveform 502 and includes parameter values: The detuning process 554 corresponds to the trapezoidal waveform 504 and includes parameter values: For convenience, graph 550 also shows avoidance crosses 556 and 558, respectively, representing avoidance crosses in the leakage channel and SWAP channel.

[0129] Now we compute a portion of the unitary evolution matrix corresponding to the SWAP channel. Since the process defined by equation (19) is piecewise, equation (15) is solved in each time interval, and boundary conditions are used to match the solution at each time "wall". Eliminating the state vector corresponding to the intermediate time will propagate the solution from the beginning to the end. As a result, the transition matrix corresponding to each time interval is obtained. The product of the product, describing the evolution matrix U of SWAP unitary:

[0130]

[0131] in Furthermore, the expression for the general transition matrix depends on three variables: time interval. angular velocity and the initial phase of the control vector The explicit form of the transition matrix is ​​given by the following equation:

[0132] in As shown in equation (21), the variables , , Assume they have specific values ​​for each time interval in equation (22). Finally, matrix u rotates the coordinate system back to its initial orientation by rotating the control vector in the xz plane:

[0133]

[0134] Using equations (21)-(23), the transition probabilities are obtained. The final expression:

[0135]

[0136] in Equation (20) can be used to determine explicit correlations. And the transition probability is expressed in the same form as the aforementioned forward process.

[0137] Figure 6The transition probabilities of two example processes that allow full swap operations are shown. . Figure 6 Includes four graphs (a), (b), (c), and (d). Graphs (a) and (b) show the transition probabilities of two example processes as a function of pulse time in nanoseconds. As shown in graph (a), a full swap operation can be performed for the pulse time corresponding to the first maximum value of graph (a), which is approximately 12 ns. Similarly, as shown in graph (b), a full swap operation can be performed for the pulse time corresponding to the third maximum value in graph (b), which is approximately 20 ns. The graph (c) shows the curve as... The phase diagram of the function parameter s. Each line in graph (c) represents a pair , A process with complete group exchange is defined. Graph (d) illustrates this as... A phase diagram of the pulse time in nanoseconds as a function of the signal. Each line in graph (c) represents a pair of... , A process with full group exchange is defined.

[0138] As mentioned above, This describes the success probability of the SWAP operation. Therefore, the condition... This ensures a complete group exchange of qubits. Unlike the forward process discussed above, this condition is not guaranteed for any particular return process. However, fortunately, such a process can be found in the case of the protocol presented in this specification, and for the existence of pulse times... Make It can identify the control parameter s and The relationship between them, such as Figure 6 As shown.

[0139] To determine the pulse time This allows for the introduction of dimensionless parameters. Furthermore, the success probability of a swap operation can be expressed as:

[0140]

[0141] in ,and and The relationship between them is given by the following formula:

[0142]

[0143] By determining the function in equation (25) The maximum value of can analytically determine the line of complete exchange, that is, for the curve Arrival point Speaking of and The relationship between them. This relationship can be expressed in parametric form as follows:

[0144]

[0145] in And equations (27) and (28) describe Figure 6 The upper curve in graph (c). Figure 6 It was also shown that, It is a multivalued function, with multiple branches originating from the arctangent terms in equations (27) and (28). Other curves describe the process with multiple Rabi oscillations during the hold time (i.e., longer gates).

[0146] Figure 6 Each point on the upper curve in (c) corresponds to a point from... (Rectangle) starting to (Triangle) Ending Trapezoid Different shapes. Although the shapes are different, along the line of complete swap ( Figure 6 The pulse time of the bottom line of (d) is almost exactly the same as that of the pulse time of the bottom line of (d). More specifically, the pulse time is limited to the following intervals.

[0147]

[0148] Therefore, the slowest door in this family is only slightly slower than the physical limit. Length 2.2%.

[0149] Error synchronization

[0150] The first step in the error synchronization process is to determine the process corresponding to a complete SWAP operation in the target region of the gate time. This determination can be performed analytically or numerically by solving Schrödinger for the SWAP channel and the leakage channel. The next step is to simultaneously fine-tune and solve for both channels to synchronize the minimum error and find the parameter set corresponding to the maximum gate fidelity.

[0151] The Schrödinger equation can be expressed as:

[0152] (30)

[0153] (31)

[0154] in Let represent the 2 × 2 and 3 × 3 matrices given by equations (2) and (3), respectively. Equations (30) and (31) can be solved using the following boundary conditions:

[0155]

[0156]

[0157] in and The time-independent Hamiltonian, representing the computational states 10 and 11 (i.e., the idle qubits residing in their positions) when the detuning between qubit frequencies is at its maximum, is... and Eigenvalues. Here, we assume the gate starts at... And ended at Therefore, leakage error can be expressed as:

[0158]

[0159]

[0160] exist and Afterwards, the errors can be synchronized. Error synchronization utilizes leakage error. The fact of the oscillation function, Having a set of points such that . Figure 7 By using This is illustrated by the fitting parameters. However, in practice, due to... Since gate execution is uncontrollable, error synchronization must be performed in reverse. This is a challenging task, as will be illustrated in the following sections. However, the techniques described in this specification provide an efficient solution that facilitates the implementation of fast and robust automated protocols for qubit calibration.

[0161] Figure 7 An example of synchronization of leakage error and SWAP error is shown. Dashed line 702 represents the gate time corresponding to zero leakage. Horizontal line 704 represents... Door Time .like Figure 7 As shown, the gate time can be selected to satisfy the third zero point at the leakage error of 706. .

[0162] Asymmetric synchronization of errors

[0163] As mentioned above, Figure 2 , Figure 3 and Figure 5 An example qubit trajectory is shown in a stationary reference frame of an idle second qubit. And the energy levels of a two-qubit system. That is, the detuning trajectory between the frequencies of the first and second qubits is defined by adjusting the frequency of the first qubit while maintaining the frequency of the second qubit at a constant value. However, in some implementations, it may be more practical to adjust the frequencies of both qubits and allow them to move symmetrically. In these implementations, the typical trajectory of the qubits is implemented as two symmetrically shaped pulses resembling rounded trapezoids, such as... Figure 8 As shown.

[0164] Figure 8 The graph (a) shows the curve covering the entire detuning. Example detuning trajectories during iSWAP gate implementation under normal scaling. In graph (a), the x-axis represents time (ns) and the y-axis represents qubit frequency (GHz). Line 802a represents the frequency of the first qubit implemented using circular pulses. Line 802b represents the frequency of the first qubit implemented using bare pulses. Line 804a represents the frequency of the second qubit implemented using circular pulses. Line 804b represents the frequency of the second qubit implemented using bare pulses.

[0165] The trajectories of the first and second qubits are at the midpoint. Meeting This is called the interaction frequency. The midpoint is represented by line 806. At the midpoint, the qubit frequency is very close to resonance, and the two-qubit system evolves freely while the qubits interact strongly with each other. The time interval of this interaction... This is called the hold time. As long as the coupling constant g is time-independent, all hold-off detuning is possible. The trajectory selections are all equivalent to global phase and have the same probability results.

[0166] Figure 8 The graph (b) shows an example detuning trajectory of the graph (a) magnified near the interaction frequency.

[0167] Assumption The results can be illustrated using an example of the instantaneous sag protocol. Consider the instantaneous sag of the first qubit as follows: Resonance, and assuming the second quantum bit is at rest. It is time-independent. The probability of group exchange is given by the formula for Rabi oscillations:

[0168]

[0169] Due to resonance From equation (5), it can be seen that the state is... and The non-adiabatic energy level degrades, which allows for the introduction of so-called "bright" and "dark" states:

[0170]

[0171]

[0172] In state and Based on this, the Hamiltonian assumption of equation (5) is as follows:

[0173]

[0174] Therefore, state and Formation with detuning η and coupling A two-level system, while the dark state Completely decoupled from other states. Due to the state Initially unfilled, leakage into the non-computational subspace occurs solely due to the state. and The leakage error is caused by the Rabi oscillation between them, and can be obtained by the following formula.

[0175]

[0176] According to the complete exchange condition And equation (36), we get

[0177]

[0178] The from equation (39) Substitute into equation (38) and apply the zero-leakage condition. The conditions for error synchronization are:

[0179]

[0180] in It is an integer.

[0181] Therefore, if It has a fixed value and Therefore, the error cannot be arbitrary. Synchronized. Therefore, It must be tunable, i.e., time-dependent. This time dependence is inherent in some qubit implementations, such as the Xmon qubit, and can be used to synchronize errors, even during gate execution. A relatively moderate deviation from its initial value (~10% or less) is sufficient to achieve the desired synchronization and develop a fast and robust qubit calibration protocol.

[0182] For simplicity, the following analysis considers a system of two capacitively coupled Xmon qubits. The functional correlation can be derived from the primitive Hamiltonian describing a system of two capacitively coupled Xmon qubits, expressed in terms of charge and phase operators:

[0183]

[0184] here, and Let represent the canonical conjugate operator corresponding to the number of Cooper pairs and the superconducting phase difference across the i-th qubit junction, respectively. , It is charging energy. i is the total equivalent capacitance of each qubit. It is a coupling capacitor, and The Josephson energy is controlled by an external flux bias applied to qubit 1 and is a (tunable) energy. The qubit operates in a coupled anharmonic oscillator state, which is determined by the inequality... Definition. In this state, the boson creation and annihilation operators can be introduced via the following equation.

[0185]

[0186] By using this transformation, applying RWA, and keeping only the leading anharmonic terms, the Hamiltonian in equation (41) can be simplified to the form given in equation (3), where the parameters are identified as:

[0187]

[0188]

[0189]

[0190] in It is a dimensionless constant proportional to the coupling capacitance:

[0191]

[0192] The values ​​are based on the typical capacitance of current Xmon qubits.

[0193] Next, assume that the frequencies of both qubits are shifting and instantly drop to a point. Where ωq is the frequency of the second qubit at its placement position. ,and This is idle detuning. Equation (35) must be satisfied at the interaction frequency (meeting point), which produces

[0194]

[0195] In typical contemporary Xmon qubits, the placement frequency is between 4.5 and 6.5 GHz. Combined with equation (46) for inter-qubit coupling, it implies that equation (47) can be satisfied. The minimum value is For example, if , GHz, and MHz, then we can find from equation (47) ,if ,but It is indeed possible for the error to fall within the interval between the two placement frequencies.

[0196] parameter This characterizes the asymmetry of the trajectory of a qubit relative to its placement frequency, i.e. Corresponding to a symmetrical trajectory, while These correspond to fixed frequencies for qubits 1 and 2, respectively. This is why the described process is called "asymmetric synchronization". Typical processes with different q values ​​and the time dependence of the corresponding g(t) given by equation (45) are shown respectively in Figure 9 and Figure 10 middle.

[0197] Note the interaction frequency. (instead of asymmetric parameters) ) plays an important role, such as Figure 11 As shown. Figure 11 Three different values ​​with the same initial detuning are shown. and interaction frequency The equivalent process. The interaction frequencies show that, regardless of their placement, the frequency of the qubits must be tuned to one of the fixed frequencies of equation (40). Thus, for a given pair of qubits, these interaction frequencies can be considered as the “spectrum” of the iSWAP gate, see [link to iSWAP gate]. Figure 12 , Figure 12 An example spectrum of the interaction frequency of a two-qubit iSWAP gate implemented by rectangular pulses is shown.

[0198] For any qubit with tunable coupling, equation (40) must be satisfied. For example, in order to perform a complete exchange and for To suppress leakage at MHz, it is necessary to... Operate near one of these amplitudes:

[0199]

[0200] here,

[0201] for .

[0202] Asymmetric synchronization using the Rabi descent protocol

[0203] The time dependence can be illustrated in the rabbi-type protocol described in this paper. Since in this approach, processes are based on a control perspective... The correlation with natural time, therefore the frequency of the qubit and the final It needs to be done by angle This can be represented as follows. As shown below, this can be achieved by solving... This can be accomplished using a quadratic equation. By taking advantage of the fact that the exact solution to this problem is known, the perfect swap path in the parameter space is also known. Therefore, the remaining task is to minimize the leakage error of the perfect swap path.

[0204] A trapezoidal ramp function generated in natural time. The trapezoidal ramp function can be defined as...

[0205]

[0206] in ,Right now ,and It is an auxiliary function:

[0207]

[0208] The following equations all use the symbol Equations (49) and (50) describe the equations with A rounded trapezoid, in which These are the parameters responsible for smearing and rounding. The control angle can then be expressed as:

[0209]

[0210] in Time-related mistuning is as defined above:

[0211]

[0212] This is equation (9) in terms of time-related factors. The generalization of the case. Similarly, the generalization of the natural time transformation given by equations (11) and (12) can be given by the following equation:

[0213]

[0214]

[0215] Through this transformation, the Schrödinger equation in the SWAP channel will assume the form of equation (14). Therefore, the solution in natural time is entirely derived from... Regardless of What is the explicit form of this? However, the latter requires generating a mapping between natural time and physical time, and ultimately generating the process. And find the leakage error. This can be achieved as follows.

[0216] The frequency of a quantum bit is defined as:

[0217]

[0218]

[0219] Among them, as mentioned above, asymmetric parameters By interaction frequency The initial frequency of the second qubit and idle detuning To define:

[0220]

[0221] The quadratic equation:

[0222]

[0223] The relevant solutions to this equation are as follows:

[0224]

[0225] The mapping can be generated using equations (51), (59), and (54):

[0226]

[0227] And inverse mapping (or ) can be obtained by mapping the equation (60) This is obtained by performing a numerical inversion. Ultimately, and Equations (59) and (52) can be used to calculate this.

[0228] As mentioned above, for an ideal trapezoid (Right now For example, the exact solution given by equation (25) and The explicit form is irrelevant. This means that equations (27) and (28) define the plane. The path to a perfect swap in the equations (49) through (51). This is true even for rounded trapezoids, as demonstrated by the finite... Numerical calculations have been confirmed. The SWAP error and leakage error calculated based on equations (36) and (38) are... Figure 14 The figure shows a function of the asymmetric parameter q of the rabbinic process, where GHz, GHz, (See) Figure 13 , Figure 13 An example Rabi sag protocol is shown, and the device anharmonicity parameters for qubits 1 and 2 are respectively MHz and MHz.

[0229] Using the equations (27) and (28) respectively and For intervals The differences in The error was calculated. SWAP error. For all All below And it is actually independent of q, that is, regardless of finiteness Indeed, this is a near-perfect swap line. As expected of an asymmetric synchronization process, all leakage curves show... The nearest minimum value. It has... The curve in In the case of The most obvious minimum value is found at that point. Therefore, the parameter... and Defined and All below The optimization process. Figure 15 The exchange error and leakage error as a function of gate time are shown. As expected, synchronization occurs at the fourth zero point of the leakage error, and the gate time is very short. In summary, note that the Rabi sag protocol implements a very fast, high-fidelity iSWAP gate that can accommodate asymmetric synchronization of switching and leakage errors; however, its practical implementation may be challenging due to the limited bandwidth of the control electronics.

[0230] Automatic calibration protocol

[0231] The asymmetric synchronization process described above can be applied to a wider range of processes suitable for immediate implementation with current quantum computing hardware. The iSWAP optimization algorithm presented in this specification relies on a detuning mechanism combined with the aforementioned synchronization process. The rounded trapezoidal pulse shape. The resulting automatic calibration protocol requires no human intervention and can be easily implemented using currently available quantum hardware, such as Xmon qubits and control electronics.

[0232] Figure 16 This is a flowchart of an example automatic calibration process 1600. For convenience, process 1600 will be described as being performed by quantum hardware communicating with control electronics located in one or more locations. For example, appropriately programmed according to this specification... Figure 1 System 100 can execute process 1600.

[0233] Generate the three-parameter trapezoidal ramp function (step 1602). For example, the three-parameter trapezoidal ramp function can be given by the following equation.

[0234]

[0235] in

[0236]

[0237] The pulse shape described by equations (61) and (62) is the convolution of an ideal trapezoid with a Gaussian filter function. The trapezoid is determined by the rise time (ramp rise time). and holding time To characterize, such as Figure 8 As shown in (a). Parameters The variance of a Gaussian filter in the time domain is uniquely related to its cutoff frequency (or bandwidth):

[0238]

[0239] in It is the cutoff frequency of a 3 dB Gaussian filter, for example, at At MHz, the 3 dB filter ns.

[0240] Pulses are generated for detuning and for the frequencies of qubits 1 and 2 (step 1604). For example, the pulses can be given by the following formula.

[0241]

[0242]

[0243]

[0244] in It is the overshoot frequency equal to the difference between the frequencies of qubits 2 and 1 during the holding time interval (see [reference]). Figure 8 ).

[0245] By adjusting the interaction frequency (where the trajectories of the first and second qubits meet). Duration and overshoot frequency To perform asymmetric synchronization of leakage error and SWAP error (step 1606). This adjustment process is central to the automatic protocol because it implements synchronization by minimizing the cost function.

[0246]

[0247] This minimization is achieved by performing a series of two-dimensional scans in the parameter space. More specifically, the following steps can be performed:

[0248] by const is used as an initial guess for scanning. (Interaction frequency - hold time). When performing a scan, the cost function in equation (67) is calculated at each point of the two-dimensional mesh using equations (30)-(33) (for Value and The range of values ​​represents Numerical simulation can be used to perform the scan, or it can be performed with different... Value and The corresponding measurement (with other parameters kept constant) determines the corresponding value of the cost function. After scanning, the minimum value of the cost function can be identified, and the parameters corresponding to the identified minimum value can be updated. At the same time, new ones can be used Values ​​to narrow down the next scan The range.

[0249] Considering this type of scanning with rectangular pulses is beneficial. Because... Therefore, the minimum value of the cost function will correspond to the intersection of the exchange line and the leakage stripe line (see...). Figure 17 The upper left and lower left panels, Figure 17 A series of scans implementing the automatic calibration protocol are shown. Bright lines correspond to the minimum of the cost function. For visualization purposes, the data in the density plot is plotted on a logarithmic scale to create a sharper image. The equations for these lines are as follows:

[0250]

[0251]

[0252] And finding the intersection point will lead to equation (47).

[0253] scanning (Interaction frequency - overshoot frequency). As mentioned above, scanning can be performed using numerical simulation, or measurements can be performed. The value is updated, and The new value was used to narrow down the next scan. The range.

[0254] scanning (Holding time - overshoot frequency). As mentioned above, scanning can be performed using numerical simulation, or measurements can be performed. The value is updated. This marks the end of the loop, and the values ​​of all three parameters have been updated.

[0255] Repeat the loop as needed until convergence is achieved. Convergence can be checked after each iteration, meaning it is not necessary to complete every loop. In all simulations of the iSWAP gate with typical Xmon parameters, the number of loops never exceeded 2.

[0256] Optionally, hardware testing and optimization using a randomized benchmark can also be performed (step 1608), for example, further adjustments to increase the fidelity of the iSWAP gate.

[0257] for GHz ns, GHz and ns ( MHz), in Figure 17 The image shows a visual example of a simulated aggregation protocol. The optimal values ​​for the returned parameters are: GHz MHz and ns. Gate time ns, and the SWAP error and leakage error are respectively and In some cases, two-dimensional scanning can be used. Figure 17 Instead, a one-dimensional curve of the type shown is used, which clearly illustrates the idea of ​​asymmetric synchronization, that is, when the qubit frequency is tuned to... The optimal value is the alignment of the minimum value.

[0258] The digital and / or quantum themes and implementations of digital functional operations and quantum operations described in this specification may be implemented in digital electronic circuits, suitable quantum circuits, or more generally in quantum computing systems, in physically embodied 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 a combination of one or more of them. The term "quantum computing system" may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0259] The implementations of the digital and / or quantum themes described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., 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 storage device, one or more qubits, or a combination of one or more of these. Alternatively or additionally, the program instructions can be encoded on artificially generated propagation signals capable of encoding digital and / or quantum information, such as machine-generated electrical, optical, or electromagnetic signals, which are generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing device.

[0260] The terms quantum information and quantum data refer to information or data carried, stored, or preserved by quantum systems, the smallest non-trivial system being the qubit, i.e., the 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 appropriate context. Such quantum systems can include, for example, multi-level systems with two or more energy levels. For example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the fundamental computational state is identified using the ground state and the first excited state; however, it should be understood that other arrangements are possible where the computational state is identified using higher-order excited states.

[0261] 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, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or include special-purpose logic circuitry, such as FPGAs (Field-Programmable Gate Arrays), ASICs (Application-Specific Integrated Circuits), or quantum emulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. In particular, a quantum emulator is a special-purpose quantum computer that does not have the ability 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.

[0262] A digital computer program, which can also be referred to or described as a program, software, software application, module, software module, script, 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 a standalone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which can also be referred to or described as a program, software, software application, module, software module, script, 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 a quantum programming language such as QCL or Quipper.

[0263] Digital and / or quantum computer programs may, but are 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 collaborative files (e.g., a file storing one or more modules, subroutines, or portions of code). 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 in one location or distributed across multiple locations 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). Typically, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum data and digital data.

[0264] The processes and logic flows described in this specification can be executed by one or more programmable digital and / or quantum computers, as appropriate, operating in conjunction with one or more digital and / or quantum processors that execute 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 dedicated logic circuitry (e.g., FPGAs or ASICs) or quantum simulators, and the apparatus can also be implemented as dedicated logic circuitry, or executed by a combination of dedicated logic circuitry or quantum simulators and one or more programmable digital and / or quantum computers.

[0265] For a system of one or more digital and / or quantum computers, being “configured” to perform a specific operation or action means that the system has software, firmware, hardware, or a combination thereof installed on it, which, in operation, causes the system to perform those operations or actions. For one or more digital and / or quantum computer programs, being configured to perform a specific operation or action means that one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause that device to perform an operation or action. A quantum computer can receive instructions from a digital computer, which, when executed by a quantum computing device, causes that device to perform an operation or action.

[0266] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs can be based on general-purpose or dedicated digital and / or quantum processors or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0267] The fundamental components of a digital and / or quantum computer are a central processing unit (CPU) for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented or integrated therein by dedicated logic circuitry or a quantum simulator. Typically, a digital and / or quantum computer will also include, or be operatively coupled to, one or more mass storage devices for storing, receiving, transferring, or both of digital and / or quantum data; these mass storage devices are, for example, disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, digital and / or quantum computers do not require such devices.

[0268] 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 disks; magneto-optical disks; CD-ROMs and DVD-ROMs; 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 long periods with high fidelity and efficiency, for example, where light is used for the light-matter interface for transmission and the material is used to store and preserve quantum characteristics (such as superposition or quantum coherence) of the quantum data.

[0269] Control of the various systems or portions thereof described in this specification may be implemented in a digital and / or quantum computer program product, including instructions stored on one or more 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 system, which may include one or more digital and / or quantum processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0270] Although this specification contains many specific implementation details, these should not be construed as limiting the scope of the claims, but rather as descriptions of features specific to particular embodiments. Certain features described in the context of separate embodiments may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented separately in multiple embodiments or in any suitable sub-combination. Furthermore, although features may be described above as functioning in certain combinations, and even initially claimed in this way, in some cases, one or more features in a claimed combination may be removed from that combination, and a claimed combination may refer to a sub-combination or a variation of a sub-combination.

[0271] Similarly, although operations are described in a specific order in the accompanying drawings, this should not be construed as requiring these operations to be performed in the specific order or sequence shown, or requiring all of the shown operations to obtain the desired result. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above embodiments should not be construed as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated into a single software product or packaged into multiple software products.

[0272] Specific implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions listed in the claims can be performed in a different order and still achieve the desired result. As an example, the processes described in the drawings do not necessarily require the specific order or sequence shown to obtain the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method implemented by a system including a quantum computer, the method comprising: A generation process, wherein the process defines the detuning trajectory between the frequencies of a first qubit and a second qubit included in the quantum computer, the generation comprising: A first control pulse is determined for the frequency of the first qubit, a second control pulse for the frequency of the second qubit, and a third control pulse for the detuning between the frequencies of the first and second qubits, wherein: Each of the first, second, and third control pulses depends on a parameterized trapezoidal ramp function characterized by the hold time; and Each of the second and third control pulses depends on an overshoot frequency, said overshoot frequency being equal to the difference between the frequency of the first qubit and the frequency of the second qubit during the hold time; and The holding time is adjusted to minimize the cost function, which includes the probability of leakage error plus the probability of exchange error.

2. The method according to claim 1, wherein, The parameterized trapezoidal ramp function is also characterized by the ramp rise time and the variance of the Gaussian filter function.

3. The method according to claim 1, wherein the interaction frequency, hold time, and overshoot frequency representing the frequency at which the frequency trajectories of the first qubit and the second qubit meet are adjusted to minimize a cost function including the probability of leakage error plus the probability of exchange error to synchronize errors in the exchange channel and the leakage channel.

4. The method according to claim 3, wherein, The leakage path includes a manifold spanned by computed state 11 and two non-compute states 02 and 20.

5. The method according to claim 3, wherein, The exchange channel includes a manifold spanned by computational states 10 and 01.

6. The method according to claim 3, wherein, Synchronizing errors in the exchange channel and the leakage channel includes: for a complete group exchange, determining the detuning trajectory between the frequencies of the first qubit and the second qubit, and minimizing the leakage channel error of the trajectory via the time-dependent inter-qubit interaction strength.

7. The method according to claim 6, wherein, The strength of the interaction between qubits is proportional to the square root of the product of the frequency of the first qubit and the frequency of the second qubit.

8. The method according to claim 1, wherein, The first control pulse is equal to the sum of the second control pulse and the third control pulse.

9. The method according to claim 1, wherein, The second control pulse equals Where t represents time, This indicates the initial frequency of the second qubit. The interaction frequency represents the frequency at which the frequency trajectories of the first and second qubits meet. Indicates the overshoot frequency, and This represents the trapezoidal ramp function.

10. The method according to claim 1, wherein, The third control pulse equals ,in, Indicates idle disharmony. Indicates the overshoot frequency, and This represents the trapezoidal ramp function.

11. The method according to claim 1, wherein, Adjusting the hold time to minimize the cost function involves repeating the following steps until it is determined that the value of the cost function is converging to a minimum: Scan hold time - overshoot frequency.

12. The method of claim 1, further comprising adjusting the interaction frequency, representing the frequency at which the frequency trajectories of the first qubit and the second qubit meet, to minimize the cost function.

13. The method according to claim 12, wherein, Adjusting the interaction frequency involves repeating the following steps until it is determined that the value of the cost function is converging to a minimum: The interaction frequency-hold time is scanned using the overshoot frequency constant.

14. The method of claim 1, further comprising adjusting the overshoot frequency to minimize the cost function.

15. The method according to claim 14, wherein, Adjusting the overshoot function involves repeating the following steps until it is determined that the value of the cost function is converging to a minimum: Scan interaction frequency - overshoot frequency.

16. The method according to claim 11, wherein, The scan involves performing measurements at different values ​​of holding time and overshoot frequency using a quantum computer.

17. The method of claim 1, further comprising using a randomized benchmark to adjust the process to increase iSWAP gate fidelity.

18. The method according to claim 1, wherein, The process of defining the detuning trajectory between the frequencies of the first and second qubits in a quantum computer includes: During the first phase, the frequency of the first qubit is non-adiabatically driven to detune the frequency of the second qubit in order to avoid crossover through the first leakage channel; During the second phase, the frequency driving the first qubit is detuned to the frequency of the second qubit in order to avoid crossover through the second switching channel; During the third phase, the first and second qubits are allowed to evolve and interact freely within a predetermined distance from the 10-01 resonance to achieve group exchange; During the fourth phase, the second phase is implemented in reverse order; and During the fifth phase, the first phase is implemented in reverse order.

19. The method according to claim 1, wherein, The detuning corresponds to the difference between the frequency of the first qubit and the frequency of the second qubit.

20. An apparatus comprising: Classical computing systems; as well as A quantum computer that communicates data with a classical computing system, the quantum computer including a first qubit, a second qubit coupled to the first qubit, and control electronics including one or more control devices that tune the frequencies of the first qubit and the second qubit by applying corresponding control signals; wherein the devices are configured to perform operations including: A generation process, wherein the process defines the detuning trajectory between the frequencies of a first qubit and a second qubit included in the quantum computer, the generation comprising: A first control pulse is determined for the frequency of the first qubit, a second control pulse for the frequency of the second qubit, and a third control pulse for the detuning between the frequencies of the first and second qubits, wherein: Each of the first, second, and third control pulses depends on a parameterized trapezoidal ramp function characterized by the hold time; and Each of the second and third control pulses depends on an overshoot frequency, said overshoot frequency being equal to the difference between the frequency of the first qubit and the frequency of the second qubit during the hold time; and The holding time is adjusted to minimize the cost function, which includes the probability of leakage error plus the probability of exchange error.

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