Stark-shift corrected iswap quantum gate

Abstract

Methods, systems and apparatus for implementing an iSWAP quantum gate between a first qubit and a second qubit. In one aspect, a method includes setting the first qubit and the second qubit on-resonance; and maintaining the first qubit and the second qubit on-resonance for a predetermined hold time to enable a population swap, wherein during the hold time the method further comprises: applying a first microwave drive to the first qubit and a second microwave drive to the second qubit, wherein the first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit.

Description

[0001]Attorney Docket: 56113-0565WO1 STARK-SHIFT CORRECTED ISWAP QUANTUM GATE BACKGROUND This specification relates to quantum computing. In quantum computing, a quantum gate is a rudimentary quantum circuit that operates on one or more qubits. Quantum gates are the analogues for quantum computing to classical logic gates for conventional digital computers. A universal quantum gate set is any set of gates to which any operation possible on a quantum computer can be reduced. Universal quantum gate sets require an entangling gate. Example entangling gates include controlled- NOT, controlled-Z, and iSWAP quantum gates. SUMMARY This specification describes technologies for implementing iSWAP quantum gates in quantum computers. In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for implementing an iSWAP quantum gate between a first qubit and a second qubit, the method comprising: setting the first qubit and the second qubit on-resonance; and maintaining the first qubit and the second qubit on-resonance for a predetermined hold time to enable a population swap; and applying, during the predetermined hold time, a first microwave drive to the first qubit and a second microwave drive to the second qubit, wherein the first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit. Another innovative aspect of the subject matter described in this specification can be implemented in a method that includes implementing an iSWAP quantum gate between a first qubit and a second qubit, comprising adding, during implementation of the iSWAP quantum gate, microwave tones to the first qubit and the second qubit to apply a Stark shift that adjusts a stray phase of an energy level for a 11 state in which the first qubit is in a 1- state and the second qubit is in a 1-state. Another innovative aspect of the subject matter described in this specification can be implemented in a method for implementing a fermionic simulation quantum gate between a first qubit and a second qubit, the method comprising: setting the first qubit and the second qubit on-resonance; and maintaining the first qubit and the second qubit on-resonance for a predetermined hold time to enable a population swap; and applying, during the predetermined hold time, a first microwave drive to the first qubit and a second microwave drive to the Attorney Docket: 56113-0565WO1 second qubit, wherein the first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit. Another innovative aspect of the subject matter described in this specification can be implemented in a method for calibrating an iSWAP quantum gate, the method comprising: calibrating an iSWAP-like quantum gate, comprising calibrating an interaction time and strength in a resonant condition and in the absence of any stark shifting; measuring a stray ZZ interaction; and using a resonant analysis model to determine drives that remove the stray ZZ interaction. Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations maintaining thefirst qubit and the second qubit on-resonance for the predetermined hold time causes a stray^^^^ interaction between the first qubit and the second qubit; and application of the respectiveStark shifts to the first qubit and to the second qubit cancels the stray ^^^^ interaction. For example, the method may comprise applying the respective Stark shifts to the first qubit and to the second qubit to cancel or inhibit a stray ^^^^ interaction between the first qubit and the second qubit. In some implementations setting the first qubit and the second qubit on-resonance comprises adjusting an operating frequency of the first qubit and an operating frequency of the second qubit such that energy levels for two-qubit states 01 and 10 have a same energy level, wherein the two-qubit state 01 represents a state in which the first qubit is in a 0-state and the second qubit is in a 1-state and the two-qubit state 10 represents a state in which the first qubit is in a 1-state and the second qubit is in a 0-state; maintaining the first qubit and the second qubit on-resonance for the predetermined hold time causes a stray phase of an energy level for the two-qubit state 11, wherein the two-qubit state 11 represents a state in which the first qubit is in a 1-state and the second qubit is in a 1-state; and application of the Attorney Docket: 56113-0565WO1 Stark shift to the first qubit and to the second qubit corrects the stray phase. That is, in some examples, the method comprising applying the Stark shift to the first qubit and to the second qubit to correct a stray phase that would otherwise have arisen from maintaining the first qubit and the second qubit on-resonance for the predetermined hold time. In some implementations the stray phase comprises an increase of the energy level for the two-qubit state 11 by more than double an interaction strength between the first qubit and the second qubit. In some implementations the Stark shift applied by the first microwave drive or thesecond microwave drive has a frequency in a range ±[20 ^^^^^^, 200^^^^^^].In some implementations the drive detuning applied by the first microwave drive or the second microwave drive is approximately 150MHz. In some implementations the Stark shifts applied by the first microwave drive and the second microwave drive are approximately the same. In some implementations maintaining the first qubit and the second qubit on- resonance for the predetermined hold time to enable a population swap comprises allowing the first qubit and the second qubit to directly interact or interact via a coupler that couples the first qubit to the second qubit. In some implementations setting the first qubit and the second qubit on-resonance comprises driving a detuning between energy levels of 10 and 01 states to zero. In some implementations the first qubit and the second qubit comprise frequency- controllable qubits. In some implementations the first qubit and the second qubit comprise superconducting qubits. In some implementations the first qubit and second qubit comprise capacitively coupled Xmon qubits. The subject matter described in this specification can be implemented in particular ways so as to realize one or more of the following advantages. Unlike conventional techniques for executing SWAP or iSWAP quantum gates that implement interactions that are close to a true SWAP or iSWAP operation (referred to in the art as “SWAP-like” operations), the presently described techniques can be used to implement interactions that effect true SWAP or iSWAP operations (or, more generally, can be used to implement any true fermionic simulation gate operations). Accordingly, the presently described techniques are applicable to a wide range of settings including quantum error correction. Attorney Docket: 56113-0565WO1 Further, the presently described techniques utilize only low-frequency control of detuning between qubit frequencies and are applicable to near term quantum computing architectures. Further, since quantum circuits that are constructed using iSWAP gates are computationally hard to simulate using classical computers (e.g., harder to simulate than same size circuits that are constructed using CZ gates or CNOT gates), the techniques described in this specification can facilitate state of the art experiments on beyond classical computation and an immediate impact on the field of quantum computing. In addition, the presently described techniques preserve the leakage-free nature of the resonant iSWAP interactions, which is highly impactful on both quantum error correction and noisy intermediate scale quantum applications. Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS FIG.1 depicts an example system for implementing iSWAP quantum gates. FIG.2 illustrates an example implementation of an iSWAP-like gate between a first qubit and a second qubit. FIG.3 illustrates an example implementation of a stark-shift corrected iSWAP quantum gate between a first qubit and a second qubit. FIG.4 is a flow diagram of an example process for implementing a stark-shift corrected iSWAP quantum gate between a first qubit and a second qubit. FIG.5 shows two plots that illustrate the error in an experimental demonstration of a stark-shift corrected iSWAP quantum gate. FIG.6 plots the empirical cumulative distribution function for different distance from the target angle for an iSWAP-like quantum gate and the presently described Stark-shifted iSWAP quantum gate. FIG.7 shows an example quantum computing device. DETAILED DESCRIPTION A complete SWAP operation is a trace preserving, i.e., zero-leakage, transformation of a two-qubit system that enables complete population transfer between the states |1^^|0^ Attorney Docket: 56113-0565WO1 and |0^^|1^ (herein written as states 10 and 01) of the SWAP channel. Most generally, such an operation is a fermionic simulation gate which can be described by the unitary matrix 10 0 0^^ = ^0 ^^ ^^ 0^^∗ ^^ (1)where ^^ = cos ^^ , ^^ = −^^ , angle on the |01^|10^ subspacein radians (e.g., determined by a strength and duration of a ^^^^ + ^^^^ interaction) and ^^represents a controlled phase angle in radians, which determines how much the |11^ state is phased. SWAP and iSWAP gates can be described by the unitary matrices given in equation (2) below 10 0 0 1 0 0 0= ^0 0 ^^ 0 ^ = 0 0 ^^ 0 where ^^ = ^^,^^ = ଶ for a SWAP and ^^ = 0,^^ = ^^ / 2 for an iSWAP.A challenging aspect of implementing SWAP or iSWAP quantum gates is that some quantum computing architectures, e.g., superconducting architectures, produce qubit interactions that are close to a SWAP or iSWAP gate but do not implement a true SWAP or iSWAP. Such gates are referred to as “SWAP-like” or “iSWAP-like” quantum gates. Whilst these gates can be implemented with high fidelity, they cannot be used in settings such as quantum error correction, e.g., because of gate fidelity. A good iSWAP-like gate can be comparable to a true iSWAP with a very bad error rate. In some implementations a gate fidelity of 99% is considered the bare minimum for a useful two qubit gate, and in the long term for quantum error correction 99.9% is targeted. However, trying to use a iSWAP-like as a true iSWAP has lower gate fidelity. The techniques described in this specification address these challenges. In particular, this specification describes an implementation of a SWAP or iSWAP quantum gate that introduces two microwave drives on the target qubits. The drives are applied in such a way as to correct stray phases incurred during the swap, e.g., stray ZZ interactions, to achieve a Attorney Docket: 56113-0565WO1 true SWAP or iSWAP quantum gate. The SWAP or iSWAP gate is therefore referred to herein as a Stark-shift corrected SWAP or iSWAP quantum gate. For convenience, the techniques described in this specification are presented with reference to implementing iSWAP gates. However, the techniques can be equally applied toimplementing SWAP gates, as well as other gates that are based on SWAP operations, e.g. a√^^^^^^^^ gate.FIG.1 depicts an example system 100 for implementing an iSWAP quantum gate on a two-qubit subsystem. The 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. The system 100 includes a two-qubit subsystem 102 in communication with control electronics 104. The two-qubit subsystem 102 includes a first qubit “Q0” 106 and a second qubit “Q1” 108. In some implementations, the first qubit 106 and second qubit 108 can directly interact (without a coupler). In other implementations, as shown in FIG.1, the first qubit 106 and second qubit 108 can interact through a controllable coupler “C” 110. For example, the first qubit 106 and second qubit 108 can be capacitively coupled Xmon qubits, e.g., included in an array of Xmon qubits in a quantum computing device. In other examples, the qubits can include flux qubits, phase qubits, or qubits with frequency interactions. The first qubit 106 and second qubit 108 can be operated by adjusting respective qubit frequencies, e.g., through application of pulses generated by the control electronics 104 to the qubits. In cases where the first qubit 106 and the second qubit 108 are Xmon qubits, the qubit frequencies can be parked at a predetermined distance from one another, and in a zig- zag position with respect to other qubits that may be included in the quantum computing device. A Hamiltonian ^^ describing the two-qubit subsystem 102 can be given by Equation (3) below: ଶ1^^ ^^^^ ற ற ^^ ^^ற − ^^றଶ൯ (3) In Equation (3), ^^ is an index that represents the first qubit 106 and second qubit 108, ^^^(^^)represents time-dependent natural frequencies of the individual qubits, ^^^(^^) Attorney Docket: 56113-0565WO1qubit’s anharmonic detunings, ^^(^^) represents the interqubit interaction strength,^^ற,^^ represent creation and annihilation oper ற^ ^ ators, and ^^^ ^^^ represents the number operator.Typical values of η, g include ^^ ≈ 2^^ × 200 ÷ 250 MHz and ^^ ≈ 2^^ × 15 ÷ 20 MHz.Without loss of generality ^^^(^^) = ^^ଶ(^^) + ^^(^^), where ^^(^^) represents the controlleddetuning with the initial and final values ε(−tp / 2) = ε(tp / 2)= 2π × 1 GHz, and tprepresents the duration of a pulse applied to the qubit or qubits to implement a quantum gate (the gate time). The Hamiltonian described in Equation (3) can be simplified by transforming it to arotating frame of the second qubit and eliminating the counter-rotating terms such as^^(^^)^^^^^^ and ^^(^^)^^ற^ ^^ற^ . The resulting Hamiltonian in the rotating wave approximation(RWA) conserves the total number of excitations ^^ and, therefore, the 9 × 9 Hilbert spacesplits into 5 subspaces corresponding to ^^ = 0, 1... , 4. Three of these subspaces with^^ = 0, 1, 2 are relevant for the qubit operations driven by ^^(^^). These are the ground state00 (also denoted as |00^), the SWAP manifold spanned by the computational states 10 and 01 and the leakage manifold spanned by the computational state 11 and two non-computational states 02 and 20. The sub-Hamiltonian matrices ^^^(^^) and ^^^(^^) that describe the SWAP channel and the leakage channel in the RWA can be expressed as ^^^(^^) = ൬^^(^^) ^^(^^)^^ 0 ^ (4) A schedule for implementing an iSWAP gate can then achieved through parametrization of the detuning ^^(^^). In some implementations the detuning can take the form ^^(^^) = ^^ + 2^^^ · ^^ ^^^^^^ [^^ (^^, {^^})] (6)where ^^^represents an initial value of the interqubit interaction strength ^^(^^), the control angle ϑ depends on a set of variational parameters {c} containing M ≥ 2 elements and the Attorney Docket: 56113-0565WO1 two additional parameters ν (shift) and λ (scaling) that define two limiting cases known in theart as non-adiabatic (^^ = 0, λ = 1) and adiabatic (^^ = ^^^, ^^ = ^2) protocols, respectively.The control electronics 104 include control devices, e.g., arbitrary waveform generators, that can operate the first qubit 106 and qubit 108. For example, the control electronics 104 includes control devices that tune the frequency of the first qubit 106 and second qubit 108 by applying control signals, e.g., voltage pulses, to the qubits through respective control lines. The control electronics 104 can further include control devices that apply other control signals, e.g., microwave tones that implement a stark shift, to the first qubit 106 and second qubit 108, as described in more detail below with reference to FIGS.3 and 4. In addition, the control devices may include measurement devices, e.g., readout resonators, that can perform measurements of the first qubit 106 and the second qubit 108 through respective qubit control lines. The control electronics 104 can be configured to store, display, and / or further process the results of measurements of the first qubit 106 and the second qubit 108. In some implementations, the control electronics 104 may include a data processing apparatus and associated memory. The memory may include a computer program having instructions that, when executed by the data processing apparatus, cause the data processing apparatus to perform one or more functions described herein, such as applying a control signal to a qubit. FIG.2 illustrates an example implementation of an iSWAP-like quantum gate between a first qubit and a second qubit, e.g., the first qubit Q0 and second qubit Q1 described above with reference to FIG.1. In this example, the first qubit Q0 and second qubit Q1 are assumed to be coupled via a tunable coupler C. The example implementation of the iSWAP-like quantum gate is illustrated using pulse diagrams and energy diagrams. In each pulse diagram, the horizontal axis represents time. In each energy diagram, the vertical axis represents energy and the labelled bars represent energy levels (where “01” indicates that qubit Q0 is in a 0-state and qubit Q1 is in a 1-state, etc.). During a first stage of the implementation of the iSWAP-like quantum gate, the operating frequencies of the first qubit Q0 and / or the second qubit Q1 are adjusted to bring the first qubit Q0 and second qubit Q1 on-resonance. For example, as shown in plots (a) and (b) of FIG.2, the operating frequency 202 of the first qubit Q0 and the operating frequency 204 of the second qubit Q1 can be adjusted so that each qubit frequency is close to, e.g., Attorney Docket: 56113-0565WO1 within a predetermined distance from, a resonance 10-01 to enable a population swap. The adjustment of the operating frequencies of the first qubit Q1 and / or the second qubit Q1 causes the energy levels 01 and 10 to have a same energy 206. During a second stage of the implementation of the iSWAP-like quantum gate, the coupler C is turned on (e.g., the operating frequency 208 of the coupler C is adjusted, as shown in plot (c)) to allow the first qubit Q0 and second qubit Q1 to interact at an interaction strength ^^. As shown in plot (d) of FIG.2, the interaction between the first qubit Q0 and second qubit Q1 causes the qubits to hybridize (where the excitation “1” moves between the first qubit and the second qubit). After the coupler C is turned off (e.g., by adjusting the operating frequency 208 of the coupler C), the energy levels 01 and 10 are swapped, producing the target iSWAP behavior. However, during the second stage, the interaction between the first qubit Q0 and second qubit Q1 also causes the higher energy levels 02 and 20 to interact 210, as shown in plot (e). This interaction affects the 11 energy level, i.e., the interaction “pushes” on the 11 energy level so that the energy level increases and is larger than intended, e.g., larger than approximately double the interaction strength ^^. This in turn creates a stray (e.g., unintended) ZZ interaction between the first qubit Q0 and second qubit Q1 (that is, the increase in energy causes a Z excitation on both qubits, phasing the 11 energy level). Accordingly, the iSWAP quantum gate is not a true iSWAP quantum gate, and is an iSWAP-like quantum gate. FIG.3 illustrates an example implementation of a stark-shift corrected iSWAP quantum gate between a first qubit and a second qubit, e.g., the first qubit Q0 and second qubit Q1 described above with reference to FIG.1. Again, in this example, the first qubit Q0 and second qubit Q1 are assumed to be coupled via a tunable coupler C, however the first qubit Q0 and second qubit Q1 could directly interact without a coupler (which would be equivalent to a setting where the qubits are coupled by a coupler that is always turned on). The example implementation of the stark-shifted iSWAP quantum gate is illustrated using pulse diagrams and energy diagrams, as described above with reference to FIG.2. During a first stage of the implementation of the stark-shift corrected iSWAP quantum gate, the operating frequencies of the first qubit Q0 and / or the second qubit Q1 are adjusted to bring the first qubit Q0 and second qubit Q1 on-resonance. The first stage of the implementation of the stark-shift corrected iSWAP quantum gate is similar to the first stage of the implementation of the iSWAP-like quantum gate described in FIG.2, where the operating frequency 202 of the first qubit Q0 and the operating frequency 204 of the second Attorney Docket: 56113-0565WO1 qubit Q1 are be adjusted so that each qubit frequency is close to, e.g., within a predetermined distance from, the resonance 10-01 to enable a population swap. During a second stage of the implementation of the stark-shift corrected iSWAP quantum gate, the coupler C is turned on to allow the first qubit Q0 and second qubit Q1 to interact at an interaction strength parameterized by coupling constant ^^. As described above with reference to FIG.2, the interaction between the first qubit Q0 and second qubit Q1 causes a stray ZZ interaction between the first qubit Q0 and second qubit Q1. To correct this, a first microwave tone 302 is applied to the first qubit Q0 and a second microwave tone 304 is applied to the second qubit Q1. Application of the microwave tones apply a Stark shift that counteracts the push on the 11 energy level. That is, application of the microwave tones moves the 11 energy level to its correct position (i.e., lowers the energy) and cancels the stray ZZ interaction. The frequency and strength of the microwave tones applied to the first qubit Q0 and to the second qubit Q1 is dependent on the specific quantum hardware implementation. Generally, the frequency of the microwave tone is far enough from the qubit operating frequency, e.g., larger than or smaller than, so that the application of the microwave tone does not excite the respective qubit, e.g., at least ±20MHz from the qubit frequency, and closerthan a frequency that ceases to effectively push on the 11 energy level, e.g., closer than±200MHz. For example, the frequency of the microwave tones applied to the first qubit Q0and to the second qubit Q1 can be approximately ±150MHz (depending on whether the microwave is driven above or below the qubit frequency. After the coupler C is then turned off, the energy levels 01 and 10 are swapped and appropriately phased / entangled (for the case of implementing an iSWAP not a SWAP gate), producing the target iSWAP behavior. Since the stray ZZ interaction has been corrected, the stark-shift corrected iSWAP quantum gate is a true iSWAP quantum gate. FIG.4 is a flow diagram of an example process 400 for implementing an iSWAP quantum gate between a first qubit and a second qubit. The first qubit and second qubit can be any physical qubit device that is frequency tunable. For example, in some implementations the first qubit and second qubit can be superconducting qubits, e.g., capacitively coupled Xmon qubits. For convenience, the process 400 will be described as being performed by quantum hardware in communication with control electronics located in Attorney Docket: 56113-0565WO1 one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 400. The system sets the first qubit and the second qubit on-resonance by adjusting an operating frequency of the first qubit and / or an operating frequency of the second qubit such that energy levels for two-qubit states 01 and 10 have a same energy (step 402). For example, the system can drive the detuning between energy levels for the two-qubit states 10 and 01 to zero, e.g., by adjusting the operating frequency of the first qubit whilst maintaining the operating frequency of the second qubit frequency at a constant value or by adjusting both operating frequencies symmetrically. The system maintains the first qubit and the second qubit on-resonance and allows the first qubit and the second qubit to freely evolve and interact (step 404). In implementations where the first qubit and second qubit interact via a coupler, the system allows the first qubit and second qubit to interact by turning the coupler on, i.e., by adjusting an appropriate operating frequency of the coupler. The first qubit and the second qubit are allowed to interact and swap populations while the whole two-qubit system freely evolves during a predetermined hold time, e.g., a hold time between approximately 10 and 20ns such as 18ns. Maintaining the first qubit and the second on-resonance for the hold time causes the energy level for the two-qubit state 11 to increase, e.g., by more than double an interaction strength between the first qubit and the second qubit. This increase in energy phases the energy level for the 11 state and causes a stray ^^^^ interaction between the first qubit and the second qubit, which a true iSWAP quantum gate cannot have. Therefore, to correct the stray ZZ interaction, the system applies a first microwave drive to the first qubit and a second microwave drive to the second qubit during the hold time (step 406). The first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit. Application of the Stark shifts correct the phase of the energy level for the two-qubit state 11 and cancels the stray ^^^^ interaction. The Stark shift applied by the first microwave drive or the second microwave drive can have a strength in a range ±[20 MHz,200MHz]. For example, in some implementations the drive detuning (or drive frequency) applied by the first microwave drive or the second microwave drive is approximately 150MHz (where the drive detuning is the frequency the off-res drive is applied at in order to induce a stark shift). The drive detuning applied by the first microwave drive and the second microwave drive can be approximately the same. Attorney Docket: 56113-0565WO1 The system terminates the interaction between the first qubit and the second qubit, e.g., by turning the coupler off, and sets the first qubit and second qubit off-resonance, e.g., by returning the operating frequencies of the first qubit and the second qubit to their initial values (step 408). In some implementations, example process 400 can be performed after the iSWAP gate has been calibrated. The presently described techniques operate in the resonant regime, which require a different analytical approach to the perturbation theory (degenerate instead of non-degenerate) compared to, e.g., techniques that operate in the off-resonant regime. A primary difference between operating in the off-resonant and resonant regime is that the relationship between drive detuning, drive strength, and stark shift are qualitatively different, which determines how the calibration of the stark shift should be performed. For example, to calibrate the presently described stark-shift corrected iSWAP quantum gate, the system can calibrate the iSWAP-like gate, where the interaction time and strength in the resonant condition in the absence of any stark shifting is calibrated. Then, the system can measure the stray ZZ and apply drives to remove it (using, e.g., a suitable resonant analysis model). This differs to other approaches where the shift itself constituted the gate, so the calibration process begins by calibrating the stark shift drives (using the off-resonant model), not with calibrating a resonant interaction (which is absent from such approaches). FIG.5 shows two plots that illustrate the error in an experimental demonstration of astark-shift corrected iSWAP quantum gate. The first plot 502 shows error in the swap angle^^. The second plot 504 shows error in the phase angle ^^. In each plot, the horizontal axisrepresents qubit coupling strength g (in MHz) and the vertical axis represents microwave amplitude. The first plot 502 shows a low error curve 506. The second plot 504 shows a low error curve 508. Optimal gate parameters for the stark-shift corrected iSWAP quantum gate can be found where the low error curves 506 and 508 coincide - in this example atapproximately ^^ = −16^^^^^^ and ^^ = 0.425.FIG.6 plots the empirical cumulative distribution function (CDF) for different distance from the target angle for an iSWAP-like quantum gate and the presently described Stark-shifted iSWAP quantum gate. The x-axis represents the distance from the target angle and is a measure of the correctness of the gate, expressed as a rotation away from the ideal phase angle for the gate, where 0 is perfect. The y axis represents the CDF. The plotted curves shows how many gates are better than a given correctness as a percentage. Where a Attorney Docket: 56113-0565WO1 curve crosses 0.5 is the average accuracy of the implementations of the gate, which is marked by a dashed line, e.g., line 602. The second dashed line 604 is the median gate performance. The blue curve 606 represents the uncorrected iSWAP-like implementation, which is solidly far from 0. The orange curve 608 represents the corrected Stark iSWAP gates, which are much closer to 0 and therefore much better gates. FIG.7 depicts an example quantum computer 700 for performing the quantum operations described in this specification. The example quantum computer 700 includes an example quantum computing device 702. The quantum computing device 702 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions, are exemplary only, and do not limit implementations of the inventions described and / or claimed in this document. The example quantum computing device 702 includes a qubit assembly 752 and a control and measurement system 704. The qubit assembly includes multiple qubits, e.g., qubit 706, that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG.7 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubit assembly 752 also includes adjustable coupling elements, e.g., coupler 708, that allow for interactions between coupled qubits. In the schematic depiction of FIG.7, each qubit is adjustably coupled to each of its four adjacent qubits by means of respective coupling elements. However, this is an example arrangement of qubits and couplers and other arrangements are possible, including arrangements that are non-rectangular, arrangements that allow for coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits. Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realization of the multiple qubits and how they interact with one another is dependent on a variety of factors including the type of the quantum computing device 702 included in the example computer 700 or the type of quantum computations that the quantum computing device is performing. For example, in an atomic quantum computer the qubits may be realized via atomic, molecular or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer the qubits may be realized via superconducting qubits or semi-conducting qubits, e.g., superconducting transmon states. As another example, in a NMR quantum computer the qubits may be realized via nuclear spin states. Attorney Docket: 56113-0565WO1 In some implementations a quantum computation can proceed by loading qubits, e.g., from a quantum memory, and applying a sequence of unitary operators to the qubits. Applying a unitary operator to the qubits can include applying a corresponding sequence of quantum logic gates to the qubits, e.g., to implement the surface code circuits described in this specification. Example quantum logic gates include single-qubit gates, e.g., Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z), Hadamard gates, S gates, rotations, two-qubit gates, e.g., controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT) controlled swap gates (also referred to as CSWAP), iSWAP gates, and gates involving three or more qubits, e.g., Toffoli gates. The quantum logic gates can be implemented by applying control signals 710 generated by the control and measurement system 704 to the qubits and to the couplers. For example, in some implementations the qubits in the qubit assembly 752 can be frequency tunable. In these examples, each qubit can have associated operating frequencies that can be adjusted through application of voltage pulses via one or more drive-lines coupled to the qubit. Example operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits, and where it may be used to perform single-qubit gates. As another example, in cases where qubits interact via couplers with fixed coupling, qubits can be configured to interact with one another by setting their respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. In other cases, e.g., when the qubits interact via tunable couplers, qubits can be configured to interact with one another by setting the parameters of their respective couplers to enable interactions between the qubits and then by setting the qubit’s respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed in order to perform multi-qubit gates. The type of control signals 710 used depends on the physical realizations of the qubits. For example, the control signals may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system. A quantum computation can be completed by measuring the states of the qubits, e.g., using a quantum observable such as X or Z, using respective control signals 710. The measurements cause readout signals 712 representing measurement results to be Attorney Docket: 56113-0565WO1 communicated back to the measurement and control system 704. The readout signals 712 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or the qubits. For convenience, the control signals 710 and readout signals 712 shown in FIG.7 are depicted as addressing only selected elements of the qubit assembly (i.e. the top and bottom rows), but during operation the control signals 710 and readout signals 712 can address each element in the qubit assembly 752. The control and measurement system 704 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 752, as described above, as well as other classical subroutines or computations. The control and measurement system 704 includes one or more classical processors, e.g., classical processor 714, one or more memories, e.g., memory 716, and one or more I / O units, e.g., I / O unit 718, connected by one or more data buses. The control and measurement system 704 can be programmed to send sequences of control signals 710 to the qubit assembly, e.g. to carry out a selected series of quantum gate operations, and to receive sequences of readout signals 712 from the qubit assembly, e.g. as part of performing measurement operations. The processor 714 is configured to process instructions for execution within the control and measurement system 704. In some implementations, the processor 714 is a single-threaded processor. In other implementations, the processor 714 is a multi-threaded processor. The processor 714 is capable of processing instructions stored in the memory 716. The memory 716 stores information within the control and measurement system 704. In some implementations, the memory 716 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 716 can include storage devices capable of providing mass storage for the system 704, e.g. a hard disk device, an optical disk device, a storage device that is shared over a network by multiple computing devices (e.g., a cloud storage device), and / or some other large capacity storage device. The input / output device 718 provides input / output operations for the control and measurement system 704. The input / output device 718 can include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, whereby to send control signals 710 to and receive readout signals 712 from the qubit assembly, as appropriate for the physical scheme for the quantum computer. In some implementations, the input / output device 718 can also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and / or a wireless interface device, e.g., an 802.11 card. In some implementations, the input / output device 718 Attorney Docket: 56113-0565WO1 can include driver devices configured to receive input data and send output data to other external devices, e.g., keyboard, printer and display devices. Although an example control and measurement system 704 has been depicted in FIG. 7, implementations of the subject matter and the functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the digital and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-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 combinations of one or more of them. The term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators. Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, 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, or to control the operation of, data processing apparatus. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, Attorney Docket: 56113-0565WO1 electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper. A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer Attorney Docket: 56113-0565WO1 program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data. The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers. For a system of one or more digital and / or quantum computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum processors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof. The essential elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum Attorney Docket: 56113-0565WO1 simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices. Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the Attorney Docket: 56113-0565WO1 combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products. Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. What is claimed is:

Claims

Attorney Docket: 56113-0565WO1 CLAIMS 1. A method for implementing an iSWAP quantum gate between a first qubit and a second qubit, the method comprising: setting the first qubit and the second qubit on-resonance; and maintaining the first qubit and the second qubit on-resonance for a predetermined hold time to enable a population swap; and applying, during the predetermined hold time, a first microwave drive to the first qubit and a second microwave drive to the second qubit, wherein the first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit.

2. The method of claim 1, wherein: maintaining the first qubit and the second qubit on-resonance for the predetermined hold time causes a stray ^^^^ interaction between the first qubit and the second qubit; and application of the respective Stark shifts to the first qubit and to the second qubit cancels the stray ^^^^ interaction.

3. The method of claim 1 or claim 2, wherein: setting the first qubit and the second qubit on-resonance comprises adjusting an operating frequency of the first qubit and an operating frequency of the second qubit such that energy levels for two-qubit states 01 and 10 have a same energy level, wherein the two- qubit state 01 represents a state in which the first qubit is in a 0-state and the second qubit is in a 1-state and the two-qubit state 10 represents a state in which the first qubit is in a 1-state and the second qubit is in a 0-state; maintaining the first qubit and the second qubit on-resonance for the predetermined hold time causes a stray phase of an energy level for the two-qubit state 11, wherein the two- qubit state 11 represents a state in which the first qubit is in a 1-state and the second qubit is in a 1-state; and application of the Stark shift to the first qubit and to the second qubit corrects the stray phase.

4. The method of claim 3, wherein the stray phase comprises an increase of the energy level for the two-qubit state 11 by more than double an interaction strength between the firstAttorney Docket: 56113-0565WO1 qubit and the second qubit.

5. The method of any one of the preceding claims, wherein the Stark shift applied by thefirst microwave drive or the second microwave drive has a frequency in a range±[20 ^^^^^^, 200^^^^^^].

6. The method of any one of the preceding claims, wherein drive detuning applied by the first microwave drive or the second microwave drive is approximately 150MHz.

7. The method of any one of the preceding claims, wherein the stark shifts applied by the first microwave drive and the second microwave drive are approximately the same.

8. The method of any one of the preceding claims, wherein maintaining the first qubit and the second qubit on-resonance for the predetermined hold time to enable a population swap comprises allowing the first qubit and the second qubit to directly interact or interact via a coupler that couples the first qubit to the second qubit.

9. The method of any one of the preceding claims, wherein setting the first qubit and the second qubit on-resonance comprises driving a detuning between energy levels of 10 and 01 states to zero.

10. The method of any one of the preceding claims, wherein the first qubit and the second qubit comprise frequency-controllable qubits.

11. The method of claim 11, wherein the first qubit and the second qubit comprise superconducting qubits.

12. The method of claim 12, wherein the first qubit and second qubit comprise capacitively coupled Xmon qubits.

13. A method comprising: implementing an iSWAP quantum gate between a first qubit and a second qubit, comprising adding, during implementation of the iSWAP quantum gate, microwave tones to the first qubit and the second qubit to apply a Stark shift that adjusts a stray phase of anAttorney Docket: 56113-0565WO1 energy level for a 11 state in which the first qubit is in a 1-state and the second qubit is in a 1- state.

14. A method for implementing a fermionic simulation quantum gate between a first qubit and a second qubit, the method comprising: setting the first qubit and the second qubit on-resonance; and maintaining the first qubit and the second qubit on-resonance for a predetermined hold time to enable a population swap; and applying, during the predetermined hold time, a first microwave drive to the first qubit and a second microwave drive to the second qubit, wherein the first microwave drive and the second microwave drive each apply a respective Stark shift to the first qubit and the second qubit.

15. A method for calibrating an iSWAP quantum gate, the method comprising: calibrating an iSWAP-like quantum gate, comprising calibrating an interaction time and strength in a resonant condition and in the absence of any stark shifting; measuring a stray ZZ interaction; and using a resonant analysis model to determine drives that remove the stray ZZ interaction.

16. An apparatus comprising: a first qubit; a second qubit coupled to the first qubit; control electronics comprising one or more control devices that tune the frequency of the first qubit and second qubit through application of respective control signals, wherein the control electronics are configured to perform operations according to the method of any one of claims 1 to 15.

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