Non-adiabatic implementation of iSWAP quantum logic gate

Abstract

Methods, systems, and apparatuses for generating a quench process for implementing an iSWAP quantum logic gate between a first qubit and a second qubit. In one aspect, a quench process that defines a trajectory of a detuning between a frequency of the first qubit and a frequency of the second qubit includes: during a first phase, driving the detuning between the frequency of the first qubit and the frequency of the second qubit non-adiabatically to pass through a first avoided crossing in a leakage channel; during a second phase, driving the detuning between the frequency of the first qubit and the frequency of the second qubit to pass through a second avoided crossing in an exchange channel; during a third phase, allowing the first qubit and the second qubit to evolve and interact freely; during a fourth phase, implementing the second phase in reverse order; and during a fifth phase, implementing the first phase in reverse order.

Description

Technical Field

[0001] This manual relates to quantum information processing. Background Technology

[0002] This specification describes the techniques used to implement iSWAP logic gates in quantum computers. Summary of the Invention

[0003] 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.

[0004] 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.

[0005] 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.

[0006] 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.

[0007] In some implementations, the generation of the sag process also includes synchronizing errors in the switching channel and the leaking channel.

[0008] 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.

[0009] 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.

[0010] In some implementations, the pulse that determines the frequency of the second qubit is equal to Where ω q ω represents the initial frequency of the second qubit. i Let f represent the interaction frequency, μ represent the overshoot frequency, and f ramp (t) represents the trapezoidal ramp function.

[0011] In some implementations, the pulse determined for detuning is equal to ∈(t)=∈0[1-f ramp (t)]-μf ramp (t), where ∈0 represents the initial detuning, μ represents the overshoot frequency, and f ramp (t) represents the trapezoidal ramp function.

[0012] 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.

[0013] 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.

[0014] 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.

[0015] 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.

[0016] In some implementations, the method also includes using a randomized benchmark to adjust the generation process to increase the fidelity of the iSWAP gate.

[0017] 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.

[0018] 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.

[0019] In some implementations, the first and second qubits include capacitively coupled Xmon qubits.

[0020] 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 and 20.

[0021] 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 state 10 01.

[0022] 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.

[0023] The subject matter described in this specification can be implemented in a particular manner to achieve one or more of the following advantages.

[0024] 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, E.A. Sete, MP. Da Silva, C.R. Rigetti, arXiv preprint arXiv:1706.06566 (2017)).

[0025] 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.

[0026] As a result, the currently described technique for executing the iSWAP gate provides a fast and robust sag 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 and approaches the physical limit t. min =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.

[0027] 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.

[0028] 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.

[0029] 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

[0030] Figure 1 An example system for implementing iSWAP quantum logic gates is described.

[0031] Figure 2 A graph comparing an example adiabatic protocol and an example non-adiabatic protocol used to implement an iSWAP gate is shown.

[0032] Figure 3 Two graphs of the energy levels of two coupled Xmon qubits constrained by a non-adiabatic protocol are shown.

[0033] 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.

[0034] Figure 5 Two example sudden drop processes are shown.

[0035] Figure 6 The transition probabilities for two example process shapes that allow full SWAP operation are shown.

[0036] Figure 7 An example of synchronization with leakage error and exchange error is shown.

[0037] Figure 8 An example detuning trajectory is shown during the implementation of the iSWAP gate.

[0038] Figure 9 Three example processes with different interaction frequencies are shown.

[0039] Figure 10 Three example values ​​for g are shown for different interaction frequencies.

[0040] Figure 11 Three equivalent processes with different q values ​​but the same initial detuning and interaction frequency are shown.

[0041] Figure 12 An example spectrum of the interaction frequency of a two-qubit iSWAP gate implemented by rectangular pulses is shown.

[0042] Figure 13 An example of a rabbi descent protocol is shown.

[0043] Figure 14 Example SWAP error and leakage error are shown.

[0044] Figure 15 The exchange error and leakage error as functions of gate time are shown.

[0045] Figure 16 This is a flowchart of an example process used to generate a sudden drop process.

[0046] Figure 17 A series of scans implementing the automatic calibration protocol are shown. Detailed Implementation

[0047] Overview

[0048] A complete swap operation is the trajectory maintenance, i.e., zero leakage, of a two-qubit system's swap channel state. and (This is written as states 10 and 01) A complete population transfer is achieved. Most generally, this operation can be described in terms of (|00>|01>|10>|11>) using a unitary matrix as follows:

[0049]

[0050] Where φ represents the phase shift, and angles θ1 and θ2 represent additional phases 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:

[0051]

[0052] 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.

[0053] Furthermore, performing a SWAP or iSWAP gate between the first and second qubits by driving the detuning ε(t) between the 10 and 01 states to zero (10–01 resonance) 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.

[0054] 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 population exchange in the |01)-|10) channels, accompanied by a reduced probability of exciting a single qubit to a higher-frequency anharmonic state.

[0055] 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.

[0056] Example Architecture

[0057] 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.

[0058] 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 1As 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.

[0059] 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.

[0060] The Hamiltonian describing two qubits can be given by the following equation (3).

[0061]

[0062] In equation (3), ω i (t) represents the time-dependent natural frequency of a single qubit, η i g(t) represents the anharmonicity / deharmonicity of a qubit, and g(t) represents the interaction strength between qubits. a i This indicates the creation operator and the annihilation operator, and The scalar operators are represented. Typical values ​​for η and g include η≈2π×200÷250MHz and g≈2π×15÷20MHz. Without loss of generality, ω1(t)=ω2(t)+ε(t), where ε(t) represents the controlled detuning ε(-t) with initial and final values. p / 2)=ε(t p / 2)=2π×1GHz, and t p This represents the pulse duration (gate time) applied to one or more qubits to implement a quantum logic gate.

[0063] The Hamiltonian described in equation (3) can be transformed into a second-qubit rotating frame and eliminated, for example, g(t)a i a i and Simplifying with the inverse rotation term, the Hamiltonian obtained in the rotating wave approximation (RWA) preserves the total number of excitations M, and thus the 9×9 Hilbert space is split into 5 subspaces corresponding to M = 0, 1, ..., 4. The three subspaces M = 0, 1, 2 are relevant to the qubit operations driven by ε(t). These are the ground state 00 (also denoted as |00>), the SWAP manifold spanned by the computed states 10 and 01, and the leak manifold spanned by the computed state 11 and the two uncomputed states 02 and 02.

[0064] Therefore, the sub-Hamiltonian matrix H describes the SWAP channel and the leakage channel in RWA. s (t) and H l (t) can be represented as

[0065]

[0066]

[0067] The process used to implement the iSWAP gate is implemented through the parameterization of detuning ε(t). Detuning can take the following form

[0068]

[0069] Where g0 represents the initial value of g(t), controlling the angle. Depending on the set of variable 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).

[0070] The parameters v and λ define two limiting cases known in the art: a non-adiabatic protocol (ν = 0, λ = 1) and an adiabatic protocol (ν = η1, λ = 1). The differences between these two protocols are as follows: Figure 2 and Figure 3 As shown.

[0071] 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.

[0072] 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.

[0073] 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.

[0074] Figure 2 A graph 200 is shown comparing an example adiabatic protocol 202 used to implement an iSWAP gate and an example non-diabatic protocol 204 used to implement an iSWAP gate. The graph includes a dimensionless time representation 2t / t. p (where t) p The horizontal axis 206 represents gate time, and the vertical axis 208 represents detuning ∈ (t) / 2π measured in GHz. 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.

[0075] The difference between adiabatic protocol 202 and non-adiabatic protocol 204 becomes clear when comparing the slope of ε(t) in leakage channel 212 near the 11–20 resonance (avoided level crossing) occurring at ε = η1. Non-adiabatic protocol 204 passes through crossing 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 ε = 0, and the behavior is reversed; non-adiabatic protocol 204 has an inflection point, while adiabatic protocol 202 drops almost vertically.

[0076] 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 ​​the local adiabatic evolution of 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."

[0077] 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.

[0078] Figure 3 Two plots, 300 and 350, are shown for the energy levels of two coupled Xmon qubits (e.g., qubits 106 and 108) subject to 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 a dimensionless time interval 2t / t. p (where t) p The horizontal axis represents gate time, and the vertical axis represents energy levels measured in GHz.

[0079] The proposed non-adiabatic iSWAP gate process

[0080] 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.

[0081] 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.

[0082] 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 t. min =π / 2g, for example, for an interaction intensity of g / 2π = 15MHz, t min =16.

[0083] Here, a definition of a nonadiabatic path based on the Landau-Ziner criterion is adopted. Specifically, for avoiding crossovers in the |11>-|20> range, according to classical Landau-Ziner theory, the probability of a nonadiabatic transition (i.e., the system remaining in state |11> after passing the crossover point) is given by P. D =1-P LZ =exp(-Γ) is given, where Γ = 2πJ 2 / v, J represent the matrix elements of the Hamiltonian between nonadiabatic states, and v represents the relative "velocity" of the nonadiabatic energy levels at the crossover point. Therefore, if The pathway is then considered non-adiabatic, i.e., Γ << 1. The non-adiabatic threshold Γ th From the typical value Γ th The convergence of the "instantaneous" time-dependent perturbation series of ~0.1÷0.2 is determined. (This is in the context of the descent protocol.) and In the case of, It is the maximum (also known as idle) mistuning, and This is a typical ascending (also known as a ramp) time. Therefore, It satisfies the non-adiabatic criterion very well.

[0084] 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.

[0085] 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.

[0086] 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.

[0087] See below for reference. Figure 5 In some embodiments, the proposed ramp-down process implemented using Example Process 400 is generated based on a trapezoidal waveform of the control angle θ(t) of the frequency trajectory of the first qubit defined during protocol execution. During the proposed protocol, the motion of the control vector corresponding to the control angle accelerates in the middle of the ramp-down process (end of the first phase) and then decelerates (end of the second phase). As described below, the acceleration near the avoidance of crossover of the leakage channel is controlled by a non-adiabatic Rabi process, which results in the leakage error relative to t. p 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.

[0088] Implementing iSWAP gates using the Rabbi protocol.

[0089] 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.

[0090] A known local adiabatic condition can be given by the following equation (7).

[0091]

[0092] In equation (7), Ψ0 and Ψ1 represent the Hamiltonians H describing the switching channel. s The instantaneous adiabatic eigenstate of (t), and Indicates for H s The time-dependent gap of (t) is where ∈ denotes the detuning between energy levels, and g denotes the interaction strength between qubits. (It is convenient to initially assume g(t) = g0 to introduce this protocol; however, this assumption will be relaxed thereafter.)

[0093] The process used to implement 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 ∈ i >0 to ∈ fThe two-qubit 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.

[0094] The Hamiltonian H in equation (4) describes the switching channel. s (t) can also be expressed as

[0095]

[0096] Among them, control angle

[0097]

[0098] Let represent the angle between the control vector (effective magnetic field b = (2g, ∈ (t))) 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 local adiabatic condition given by equation (7) implies

[0099]

[0100] Based on this meaning, the problem can be solved precisely by transforming the time-dependent Schrödinger equation to the natural time scale:

[0101]

[0102] t(τ)=∫0 τ sin[θ(τ′)]dτ′ (12)

[0103] These equations define a 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 w in the natural time frame. min =2g, and the motion of the system is entirely determined by the correlation between the control angle and natural time θ(τ) = θ[t(τ)].

[0104] Based on the expression for τ(t) in equation (12), we obtain Furthermore, based on the local adiabatic conditions of equations (7) and (10), the following is derived: In other words

[0105]

[0106] Where θ0 and θ f Let represent the initial angle and the final angle, respectively, and τ pLet 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.

[0107] In equation (8), the control vector rotates around the y-axis from the North Pole to the South Pole in the x, z plane. To make the similarity to the Rabi problem more apparent, the current coordinate system is rotated by π / 2 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:

[0108]

[0109] Where ψ(τ)=χ + (τ)|α>+χ - (τ)|β> denotes the two-component spinor, and |α>=(1,0) and |β>=(0,1) denote the eigenvectors 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.

[0110] function χ -,+ (τ) satisfies two separate Schrödinger-like equations:

[0111] -4χ″ -,+ (x)-(γ 2 ±2iθ″(x)+θ′(x) 2 )χ ± (x)=0 (15) Its initial condition is chosen to be the Hamiltonian given by equation (6) at t=-t p One of the intrinsic states (labeled as 0 or 1) at / 2. χ ± The second boundary condition for the derivative can be obtained directly from equation (8). Here, x = 2τ / t p Let γ represent the dimensionless natural time during the gate operation, and γ = g·t p This represents the dimensionless total duration of the gate, i.e., the gate time.

[0112] The following sections use time-independent (or “non-adiabatic”) bases 0 and 1 associated with the eigenstates of the Hamiltonian at the initial time as the levels move further apart. The primary quantity of interest is at the end of the gate, t = t p The probability of transitioning from initial state 0 to final state 1 at point / 2 From the perspective of SWAP operations, P 01 The probability of success is always P.s =1-P 01 It is the probability of the SWAP error. Using equation (15) and boundary conditions, the probability P of the SWAP error satisfying the Schrödinger-like equation is... s Given by the following formula

[0113]

[0114] 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.

[0115]

[0116] For convenience, P s It is represented as the dimensionless gate time γ = g·t p The function. The detuning process generated by the Rabi protocol can be represented by the relation θ(t) = arccot[∈(t) / 2g] and To determine, that is

[0117]

[0118] 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 the pulse time and has a set of times, where P... s =0. If the pulse time is tuned to one of these intervals, the SWAP gate can be executed with very high fidelity.

[0119] The method for generating the proposed descent process

[0120] In the following sections, time-independent nonadiabatic bases 0 and 1 are used, which are associated with the eigenstates of the Hamiltonian at the initial time as the levels are far apart, because this is more convenient for numerical implementations. For convenience, a return process compatible with the Xmon architecture is also considered, and the time interval (0, t) is... p The gate simulation is performed within the gate. Similarly, the main quantity of interest in the SWAP channel is the probability P that transitions from the initial state 0 to the final state 1 at the end of the gate. 01 (t p In the leakage channel, the main quantity of interest is the probability of transitioning from the computed state |11> to the non-computed state |20>, and P1 = P 01 (t p ) indicates leakage error.

[0121] The process is based on a trapezoidal waveform of angle θ(τ) in the natural time scale. The trapezoidal waveform can be given by the following equation.

[0122]

[0123] Where ω r The time interval τ representing the length of the slope ascent (or descent) r The angular velocity during this period. This is determined by defining the relative ramp time s = τ. r / τ p The sudden drop process is determined by two parameters s and θ. max To define, where θ max It is the maximum angular distance traveled by the control vector. Therefore, ω r =θ max / (sτ p And hold for time τ h =(1-2s)τ p .

[0124] The meaning of the total gate time in the natural scale can be derived from equation (12) and related to the laboratory gate time t. p Related:

[0125]

[0126] Figure 5 The graph 500 shows two example trapezoidal waveforms 502 and 504 of angle θ / π over natural time. Figure 5 Graph 550 shows two corresponding detuning processes ∈(t) / 2π for each trapezoidal waveform 503 and 504. Detuning process 552 corresponds to trapezoidal waveform 502 and includes parameter values: s = 0.39, θ max =0.59π. The detuning process 554 corresponds to the trapezoidal waveform 504 and includes the parameter values: s = 0.25, θ max =0.7π. For convenience, graph 550 also shows avoidance crossovers 556 and 558, respectively, representing avoidance crossovers in the leakage channel and SWAP channel.

[0127] 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, we obtain the evolution matrix U describing the SWAP unitary as a product of the transition matrices M corresponding to each time interval:

[0128]

[0129] Where τ p =2τ r +τ h Furthermore, the expression for the universal transfer matrix depends on three variables: the time interval τ, the angular velocity κ, and the initial phase θ0 of the control vector. The explicit form of the transfer matrix is ​​given by the following equation:

[0130]

[0131] in As shown in equation (21), the variables κ, θ0, and τ are assumed to 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:

[0132]

[0133] Using equations (21)-(23), the transition probability P is obtained. 01 The final expression:

[0134]

[0135] in Equation (20) can be used to determine the explicit correlation P. 01 (γ), and the transition probability is expressed in the same form as the aforementioned forward process.

[0136] Figure 6 The transition probabilities P of two example processes that allow full swap operations are shown. 01 =1. Figure 6 This includes four graphs (a), (b), (c), and (d). Graphs (a) and (b) show the transition probabilities P of two example processes as a function of pulse times in nanoseconds. 01 As shown in graph (a), for the pulse time corresponding to the first maximum value of graph (a), which is approximately 12 ns, a full SWAP operation P can be performed. 01 =1. Similarly, as shown in graph (b), for the pulse time corresponding to the third maximum value in graph (b), which is approximately 20 ns, a full SWAP operation P can be performed. 01 =1. Graph (c) shows θ as... max The phase diagram of the function with parameter s. Each line in graph (c) represents a pair (s, θ) max ), (s,θ max The process with complete group exchange is defined. Graph (d) illustrates this as θ maxA phase diagram of the pulse time in nanoseconds as a function of θ. Each line in graph (c) represents a pair (gate time, θ) max (gate time, θ) max It defines a process with complete group exchange.

[0137] As mentioned above, P 01 (γ) describes the success probability of the SWAP operation. Therefore, condition P 01 =1 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 a pulse time t... p =γ0 / 2g makes P 01 (γ0) = 1, which can identify the relationship between control parameters s and θ. max The relationship between them, such as Figure 6 As shown.

[0138] To determine the pulse time t p =γ0 / 2g, such that, by introducing the dimensionless parameter α = 2g / ω r Furthermore, the success probability of a swap operation can be expressed as:

[0139]

[0140] Where ξ(α)=gτ h =αθ max (1 / (2s-1)), and the relationship between α and γ is given by the following equation:

[0141]

[0142] By determining the function P in equation (25) 01 The maximum value of (α) can analytically determine the line of complete exchange, that is, for the curve P(γ) reaching the point P(γ0) = 1, s and θ max The relationship between them. This relationship can be expressed in parametric form as follows:

[0143]

[0144]

[0145] Where 0 ≤ α ≤ 1, and equations (27) and (28) describe Figure 6 The upper curve in graph (c). Figure 6 It is also shown that s(θ) max) 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 α = 0, s = 0, θ max =π / 2 (rectangle) from α=1, s=0.5, Different shapes of trapezoid θ(τ) ending in a triangle. Although the shapes are different, they are all along the line of complete swap ( Figure 6 The pulse time of the bottom line of (d) is almost exactly the same as θ. max Irrelevant. 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 t. min =π / 2g 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]

[0153]

[0154] Where H s (t), H l (t) represents 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 H represents 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. s and H l The eigenvalues. Here, we assume the gate starts at... And it ends at t p =2. Therefore, the leakage error can be expressed as:

[0158] P s =|<Ψ s (r p / 2)|χ s 10 >| 2 (34)

[0159] P l =1-|<Ψ l (r p / 2)|χ l 11 >| 2 (35)

[0160] In t p Determine P within the range of other parameters s and P l Afterwards, the error can be synchronized. Error synchronization utilizes the leakage error t. p The fact that t is an oscillating function p Having a set of points such that P l (t p ) = 0. Figure 7 This is illustrated by using η as a fitting parameter. However, in practice, since η is uncontrollable during gate execution, error synchronization must be performed in reverse. This is a challenging task, as will be elaborated in the following sections. However, the techniques described in this specification provide an efficient solution that facilitates the implementation of a fast and robust automated protocol for qubit calibration.

[0161] Figure 7 An example synchronization of leakage error and SWAP error is shown. Dashed line 702 represents the gate time corresponding to zero leakage. Horizontal line 704 represents the time that makes P... s (t p Gate time t = 0 p .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 ωi(t) and the energy levels of a two-qubit system are shown in a static reference frame with an idle second qubit. That is, the detuned 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 Graph (a) shows an example detuning trajectory during the implementation of the iSWAP gate on a normal scale covering the entire detuning range of ~1 GHz. The x-axis of graph (a) represents time (ns), and the y-axis represents the qubit frequency (GHz). Line 802a represents the frequency of the first qubit implemented using a circular pulse. Line 802b represents the frequency of the first qubit implemented using a bare pulse. Line 804a represents the frequency of the second qubit implemented using a circular pulse. Line 804b represents the frequency of the second qubit implemented using a bare pulse.

[0165] The trajectories of the first and second qubits meet at the midpoint ωi, which is called the interaction frequency. This midpoint is represented by line 806. At the midpoint, the qubit frequencies are very close to resonance, and the two-qubit system evolves freely while the qubits interact strongly with each other. The time interval th of this interaction is called the hold time. As long as the coupling constant g is time-independent, all trajectory choices for maintaining detuning ∈(t) = ω1(t) - ω2(t) are equivalent to global phase and have the same probabilistic outcome.

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

[0167] The result of assuming g = const can be illustrated using the example of an instantaneous sag protocol. Consider the instantaneous sag of the first qubit as |10>-|01> resonance, and assume the second qubit is at rest, with g being time-independent. The probability of a group exchange is given by the formula for Rabi oscillations:

[0168] P 10→01 =sin 2 (gtp) (36)

[0169] Since at the resonance ε(t) = 0, it can be seen from equation (5) that the nonadiabatic energy levels of states |20> and |02> become degenerate, which allows for the introduction of so-called “bright” and “dark” states:

[0170]

[0171]

[0172] Based on the states |11>, |ψb>, and |ψd>, the Hamiltonian assumption of equation (5) is as follows:

[0173]

[0174] Therefore, states |11> and |ψb> form a two-level system with detuned η and coupled 2g, while the dark state |ψd> is completely decoupled from the other states. Since state |ψd> is initially unfilled, leakage into the non-computational subspace is caused only by Rabi oscillations between states |11> and |ψb>, and the leakage error can be obtained via the following equation.

[0175]

[0176] According to the complete commutation condition P 10→01 =1 and equation (36), thus obtaining

[0177]

[0178] Substituting tp from equation (39) into equation (38) and applying the zero-leakage condition Pl = 0, the condition for error synchronization is generated:

[0179]

[0180] Where n≥2 are integers.

[0181] Therefore, if η has a fixed value and g = const, the error cannot be synchronized at any g. Consequently, g 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 if g(t) deviates from its initial value relatively moderately (~10% or less) during gate execution, it is sufficient to achieve the desired synchronization and develop fast and robust qubit calibration protocols.

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

[0183]

[0184] Here, ni 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. E Ci =e 2 / (2C ∑i C∑ is the charging energy, C∑i is the total equivalent capacitance of each qubit, Cqq is the coupling capacitance, and EJi is the (tunable) Josephson energy controlled by the external flux bias applied to qubit 1. The qubit operates in a coupled anharmonic oscillator state, defined by the inequality EJi / ECi >> 1. 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] Where A is a dimensionless constant proportional to the coupling capacitance:

[0191]

[0192] The value of A is based on the typical capacitance of the current Xmon qubit.

[0193] Next, assume that the frequencies of both qubits are shifting and instantly drop to the point ωq + q ∈ 0, where ωq is the frequency of the second qubit at its current position, 0 ≤ q ≤ 1, and ∈ 0 = 2π × 1 GHz is the idle detuning. Equation (35) must be satisfied at the interaction frequency (meeting point), which produces

[0194]

[0195] In a typical contemporary Xmon qubit, the placement frequency is between 4.5 and 6.5 GHz. Combining this with equation (46) for inter-qubit coupling, it implies that the minimum value of n that can satisfy equation (47) is n = 4. For example, if ∈0 = 2π × 1 GHz, ωq = 2π × 5.11 GHz, and η = 2π × 240 MHz, then from equation (47) we can find q ≈ 0.6, and if n = 4, then ω i The value ≈2π×5.7GHz falls within the interval between the two placement frequencies, meaning that error synchronization is indeed possible.

[0196] The parameter q characterizes the asymmetry of the qubit's trajectory relative to its placement frequency; q = 0.5 corresponds to a symmetric trajectory, while q = 0, 1 correspond to the fixed frequencies of qubit 1 or 2, respectively. This is why the described process is called "asymmetric synchronization." Typical processes with different q values ​​and the corresponding time correlations of g(t) given by equation (45) are shown in... Figure 9 and Figure 10 middle.

[0197] Note that the interaction frequency ωi (not the asymmetric parameter q) plays an important role, such as... Figure 11 As shown. Figure 11 Three equivalent processes with different q but the same initial detuning ∈ 0 and interaction frequency ωi are shown. The interaction frequency shows 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 relevant documentation]. 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, to perform a complete exchange and suppress leakage for η = 2π × 240 MHz, operation needs to be performed near one of these amplitudes of g(t):

[0199]

[0200] Here, g h = g(t) for |t|≤th / 2.

[0201] Asymmetric synchronization using the Rabi descent protocol

[0202] The time dependence of g(t) can be illustrated in the Rabi-type protocol described in this paper. Since the process is based on the dependence of the control angle θ on natural time in this method, the qubit frequency and the final g need to be expressed in terms of angle θ. This can be accomplished by solving a quadratic equation for g(θ), as shown below. By utilizing 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.

[0203] A trapezoidal ramp function generated in natural time. The trapezoidal ramp function can be defined as...

[0204] f ramp (x,s)=p(x,s)-p(s+x-1,s) (49)

[0205] in That is, 0 ≤ x ≤ 1, and p(x,s) is an auxiliary function:

[0206]

[0207] The symbol x = τ / τ is used in all the following equations. p Equations (49) and (50) describe a rounded trapezoid with δ, where δ is the parameter responsible for smearing and rounding.

[0208] Therefore, the control angle can be expressed as:

[0209]

[0210] Where θ in =arccot(∈0 / 2g0). Time-dependent detuning is as defined above:

[0211] ∈(x)=2g(x)cot(θ(x)) (52)

[0212] This is a generalization of equation (9) to the case of time-dependent g. Similarly, equations (11) and (12)...

[0213] The generalization of the natural time transformation can be given by the following formula:

[0214]

[0215]

[0216] 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 defined by θ(τ), regardless of the explicit form of g(t). However, the latter requires generating a mapping between natural and physical time, and ultimately generating process ∈(t) and finding the leakage error. This can be achieved as follows.

[0217] The frequency of a quantum bit is defined as:

[0218] ω1(x)=ω q +q∈0+(1-q)∈(x) (55)

[0219] ω2(x)=ω1(x)-∈(x), (56)

[0220] As mentioned earlier, the asymmetric parameter q is determined by the interaction frequency ω. i The initial frequency ω of the second quantum bit q =ω2(0) and idle detuning ∈0 = ∈(0) are defined as follows:

[0221]

[0222] Based on equations (45) and (52)-(56), the quadratic equation of g(x) is derived:

[0223] g 2 (x)=A 2 [ω q +q∈0+2(1-q)g(x)cot(θ(x))][ω q +q∈0-2qg(x)cot(θ(x))] (58)

[0224] The relevant solutions to this equation are as follows:

[0225]

[0226] The t(τ) mapping can be generated using equations (51), (59) and (54):

[0227]

[0228] Furthermore, the inverse mapping τ(t) (or x(t)) can be obtained by numerically reversing the mapping t(τ) generated by equation (60). Finally, g(t) = g[θ(x(t))] and ∈(t) = ∈[θ(x(t))] can be calculated using equations (59) and (52).

[0229] As mentioned above, for an ideal trapezoid θ(x) (i.e., δ = 0), the exact solution P given by equation (25) is... 01(α) is independent of the explicit form of g(t). This means that equations (27) and (28) define the plane s, θ max 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 ω q =5.45GHz, ∈0=1GHz, δ=0.04 (see Figure 13 , Figure 13 An example Rabi sag protocol is shown, and the device anharmonicity parameters for qubits 1 and 2 are η1 = 246 MHz and η2 = 242 MHz, respectively.

[0230] Using θ as defined by equations (27) and (28) respectively max The errors s(α) and s(α) were calculated for different α values ​​in the interval 0 ≤ α ≤ 1. The SWAP error P s (α,q) is less than 10 for all α < 0.5. -5 And it is actually independent of q, that is, regardless of the finite δ, this is indeed a line close to perfect SWAP. As expected of asymmetric synchronization processes, all leakage curves show The minimum value near P. The curve with α = 0.35 is at P. l ~10 -6 In the case of , it has the most obvious minimum at q = 0.59. Therefore, the parameters q = 0.59 and α = 0.35 define P. s and P l All below 10 -5 The optimization process. Figure 15 The switching error and leakage error as a function of gate time are shown. As expected, synchronization occurs at the fourth zero of the leakage error, and the gate time is very short—~16 ns. In summary, note that the Rabi sag protocol achieves 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.

[0231] Automatic calibration protocol

[0232] The asymmetric synchronization process described herein 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 rounded trapezoidal pulse shape for detuning ∈(t) in conjunction with the aforementioned synchronization process. The resulting automatic calibration protocol requires no human intervention and can be readily implemented with currently available quantum hardware, such as Xmon qubits and control electronics.

[0233] 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.

[0234] Generate the three-parameter trapezoidal ramp function (step 1602). For example, the three-parameter trapezoidal ramp function can be given by the following equation.

[0235] f ramp (t)=f(t)+f(-t), (61)

[0236] in

[0237]

[0238] 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) t. r and holding time t h To characterize, such as Figure 8 As shown in (a). The parameter σ is the variance of the Gaussian filter in the time domain, which is uniquely related to its cutoff frequency (or bandwidth):

[0239]

[0240] Where f c It is the cutoff frequency of a 3dB Gaussian filter, for example, at f c At 240MHz, the σ of the 3dB filter is 0.55ns.

[0241] 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.

[0242] ∈(t)=∈0[1-f ramp (t)]-μf ramp (t) (64)

[0243]

[0244] ω1(t)=ω2(t)+∈(t) (66)

[0245] Where μ is the overshoot frequency, which is equal to the difference between the frequencies of qubits 2 and 1 during the holding time interval (see [reference]). Figure 8 ).

[0246] By adjusting the interaction frequency (where the trajectories of the first and second qubits meet) ω i Duration t h The overshoot frequency μ is used to perform asymmetric synchronization of leakage error and SWAP error (step 1606). This adjustment process is the core of the automatic protocol because it implements synchronization by minimizing the cost function.

[0247] P(ω i ,t r ,t h ,μ)=P l (ω i ,t r ,t h ,μ)+P s (ω i ,t r ,t h ,μ) (67)

[0248] This minimization is achieved by performing a series of two-dimensional scans in the parameter space. More specifically, the following steps can be performed:

[0249] ■Scan ω using μ = const as the initial guess i -t h (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 ω). i value and t h The range of values ​​for ω i -t h Numerical simulation can be used to perform the scan, or it can be performed with different ω. i value and t h The corresponding measurement (with other parameters taking constant values) determines the corresponding value of the cost function. After scanning, the minimum value of the cost function can be identified, and the parameter t corresponding to the identified minimum value can be updated. h At the same time, a new ω can be used i The value is used to narrow down ω in the next scan. i The range.

[0250] Considering this type of scanning with rectangular pulses is beneficial. Since g = Aω iTherefore, 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:

[0251]

[0252]

[0253] And finding the intersection point will lead to equation (47).

[0254] ■Scan ω i -μ(interaction frequency - overshoot frequency). As mentioned above, the scan can be performed using numerical simulation, or a measurement can be performed. ω i The value of μ is updated, and the new value of μ is used to narrow the range of μ in the next scan.

[0255] ■Scan t h -μ (hold time - overshoot frequency). As mentioned above, a scan can be performed using numerical simulation, or a measurement can be performed. The value of μ is updated. This is the end of the loop, and the values ​​of all three parameters have been updated.

[0256] ■ 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.

[0257] 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.

[0258] For ω q =5.45GHz, t r =2.5ns, ∈0=1GHz and σ=0.55ns(f c =240MHz), in Figure 17 A visual example of a simulated aggregation protocol is shown. The optimal value for the returned parameter is: ω i =5.88GHz, μ=4.68MHz and t h = 14.9ns. Gate time t p =t h +2t r +6σ = 23.26 ns, and the SWAP error and leakage error are respectively P s =4.4×10-6 and P l =1.99×10 -5 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 ω. i The optimal value is the alignment of the minimum value.

[0259] The digital and / or quantum themes and implementations of digital functional operations and quantum operations described in this specification can 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.

[0260] 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.

[0261] 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 the 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 where the computational state is identified using higher-order excited states are also possible.

[0262] 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 the 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.

[0263] 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.

[0264] 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.

[0265] 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., FPGA or ASIC) 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.

[0266] 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.

[0267] 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.

[0268] 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.

[0269] 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.

[0270] 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.

[0271] 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.

[0272] 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.

[0273] 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 for implementing an iSWAP quantum logic gate between a first qubit and a second qubit, the method comprising: implementing a quenching procedure that defines a trajectory of a detuning between a frequency of the first qubit and a frequency of the second qubit, including: during a first phase, driving the detuning between the frequency of the first qubit and the frequency of the second qubit non-adiabatically to pass through a first avoided crossing in a leakage channel; during a second phase, driving the detuning between the frequency of the first qubit and the frequency of the second qubit to pass through a second avoided crossing in an exchange channel; during a third phase, allowing the first qubit and the second qubit to evolve and interact freely; during a fourth phase, implementing the second phase in reverse order; and during a fifth phase, implementing the first phase in reverse order.

2. The method of claim 1, wherein, the quenching procedure is based on a trapezoidal ramp function characterized by a ramp up time, a hold time, and a variance of a Gaussian filter function.

3. The method of claim 1 or claim 2, further comprising generating the quenching procedure, including: determining pulses for: the detuning between the frequency of the first qubit and the frequency of the second qubit, the frequency of the first qubit, and the frequency of the second qubit.

4. The method of claim 3, wherein, generating the quenching procedure further includes synchronizing errors in the exchange channel and the leakage channel.

5. The method of claim 4, wherein, the pulse determined for the frequency of the first qubit is equal to the pulse determined for the second qubit plus a sum of the determined detuning pulses.

6. The method of claim 5, wherein, the frequency of the first qubit depends on an asymmetry parameter equal to a difference between an interaction frequency and an initial frequency of the second qubit divided by an idle detuning.

7. The method of any one of claims 4 to 6, wherein, The pulse determined for the frequency of the second qubit equals where ω q denotes the initial frequency of the second qubit, ω i denotes the interaction frequency, μ denotes the overshoot frequency, and f ramp (t) denotes a trapezoidal ramp function.

8. The method of any one of claims 4 to 6, wherein, The pulse determined for detuning is equal to ∈(t) = ∈0[1 - f ramp (t)] - μf ramp (t), where ∈0denotes the initial detuning, μ denotes the overshoot frequency, and f ramp (t) denotes a trapezoidal ramp function.

9. The method of any one of claims 4 to 6, wherein, synchronizing errors in the exchange channel and the leakage channel includes, for a full group exchange, determining a trajectory of the detuning between the frequency of the first qubit and the frequency of the second qubit and minimizing a leakage channel error of the trajectory via a time-dependent inter-qubit interaction strength.

10. The method of claim 7, wherein, the inter-qubit interaction strength is proportional to a square root of a product of the frequency of the first qubit and the frequency of the second qubit.

11. The method of any one of claims 4-6 and 10, wherein, synchronizing errors in the exchange channel and the leakage channel includes adjusting the following to minimize a cost function that includes a probability of a leakage error plus a probability of an exchange error: an interaction frequency representing a frequency at which a trajectory of the frequency of the first qubit and a trajectory of the frequency of the second qubit meet, a hold time, and an overshoot frequency equal to a difference between the frequency of the first qubit and the frequency of the second qubit during the hold time.

12. The method of claim 11, wherein, adjusting the interaction frequency, the hold time, and the overshoot frequency to minimize the cost function includes repeating the following until a value of the cost function is determined to be converging to a minimum value: scanning the interaction frequency - hold time with an overshoot frequency constant; scanning the interaction frequency - overshoot frequency; and scanning the hold time - overshoot frequency.

13. The method of claim 12, further comprising adjusting the generated procedure using a randomization benchmark to increase iSWAP gate fidelity.

14. The method of claim 1, wherein, allowing the first qubit and the second qubit to evolve and interact freely includes allowing the first qubit and the second qubit to evolve and interact freely within a predetermined distance from a 10-01 resonance to achieve group exchange.

15. The method of any one of claims 1-2, 4-6, 10, and 12-14, wherein, Driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the exchange channel by a second comprises adiabatically driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the exchange channel by a second.

16. The method of any one of claims 1-2, 4-6, 10, and 12-14, wherein, The first qubit and the second qubit comprise capacitively coupled Xmon qubits.

17. The method of any one of claims 1-2, 4-6, 10, and 12-14, wherein, The leakage channel comprises a manifold spanned by the computational states 11 and the two non-computational states 02 and 20, and wherein driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the leakage channel by a first comprises driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the state 11 - 20 resonance.

18. The method of any one of claims 1-2, 4-6, 10, and 12-14, wherein, The exchange channel comprises a manifold spanned by the computational states 10 and 01, and wherein driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the exchange channel by a second comprises driving the detuning between the frequency of the first qubit and the frequency of the second qubit to avoid crossing in the state 10 - 01 resonance.

19. The method of any one of claims 1-2, 4-6, 10, and 12-14, wherein, Implementing the second phase in reverse order comprises driving the detuning between the frequency of the first qubit and the frequency of the second qubit to achieve full population exchange between the qubit states 10 and 01.

20. An apparatus for implementing an iSWAP quantum logic gate, comprising: a first qubit; a second qubit coupled to the first qubit; control electronics comprising one or more control devices to tune the frequency of the first qubit and the second qubit by applying respective control signals, wherein the control electronics are configured to implement the quenching procedure according to any one of claims 1 to 19.

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