Pulse shaping for fast, low-leakage parametrically modulated quantum computing gates

Abstract

Systems and methods are provided for shaping a control pulse for controlling a quantum computing system. In some instances, a directly controllable physical parameter can be used to indirectly control a derived quantum parameter. In one example, a method may include obtaining a target pulse shape associated with a derived quantum parameter of interest. The target pulse shape may correspond to parametric modulation. The method may include determining, based on the target pulse shape, a second pulse shape associated with the controllable physical parameter. The method may include generating, based on the second pulse shape, a control pulse.

Description

PULSE SHAPING FOR FAST, LOW-LEAKAGE PARAMETRICALLY MODULATED QUANTUM COMPUTING GATES CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application is based upon and claims the right of priority to U.S. Provisional Patent Application No.63 / 611,520, filed on December 18, 2023, the disclosure of which (including any appendices) is hereby incorporated by reference herein in its entirety for all purposes. FIELD

[0002] The present disclosure relates generally to systems and methods for quantum computing. BACKGROUND

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

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

[0005] Example aspects of the present disclosure provide an example method. In some implementations, the example method can include obtaining a target pulse shape associated with a quantum parameter of interest. In the example method, the target pulse shape can correspond to parametric modulation. The example method can include determining, based on the target pulse shape, a second pulse shape associated with aphysically controllable parameter. The example method can include generating, based on the second pulse shape, a control pulse.

[0006] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the related principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0007] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0008] FIG.1 depicts an example of a quantum computing system according to example aspects of the present disclosure;

[0009] FIG.2 is a flowchart diagram of an example method according to example aspects of the present disclosure.

[0010] FIG.3 is a flowchart diagram of an example method according to example aspects of the present disclosure.

[0011] FIG.4 is a flowchart diagram of an example method according to example aspects of the present disclosure.

[0012] FIG.5 depicts a block diagram of an example computing system according to example aspects of the present disclosure. DETAILED DESCRIPTION

[0013] Example embodiments according to some aspects of the present disclosure are directed to systems and methods for controlling a quantum computing system, wherein a directly controllable physical parameter λ (e.g., external offset-charge or phase bias, etc.) can be used to indirectly control a derived quantum parameter (e.g., a λ-dependent qubit frequency, effective two-qubit coupling, etc.). More particularly, the present disclosure is directed to systems and methods for controlling a pulse shape associated with a controllable physical parameter λ over time t, written as λ(t), for a parametrically modulated quantum computing system.

[0014] A parametrically modulated quantum computing system can be a quantum computing system characterized by time-periodic modulation of one or more derivedquantum parameters. For example, in some instances, a derived parameter f can be proportional or approximately proportional to a sine wave (e.g., f ∝ sin(xt), wherein x can be a modulation frequency of interest). Parametric modulation can have a variety of technical effects and benefits, such as providing resonant behavior between frequency-detuned quantum systems; enabling two-qubit couplings with large on-off ratios; etc.

[0015] Prior systems and methods have disclosed parametric modulation using, for example, sinusoidal modulation of a directly controllable physical parameter λ. However, quantum gates based on such methods can in some instances be slow (e.g., at small drive amplitudes) or be associated with high leakage (e.g., at large drive amplitudes). Thus, it is desirable to develop systems and methods for generating faster, lower-leakage parametrically modulated quantum gates.

[0016] In an example aspect of the present disclosure, pulse-shaping systems and methods are provided to remedy various causes of leakage and poor gate fidelities, thereby enabling faster, lower-leakage parametrically modulated quantum gates compared to prior disclosures.

[0017] An example method of the present disclosure can include obtaining a target derived waveform associated with a derived quantum parameter. The example method can include determining, based on the target derived waveform, a control waveform associated with a controllable physical parameter and configured to generate or approximate the target derived waveform.

[0018] In some instances, determining a control waveform can include determining an adiabatic component (e.g., using spectroscopy measurements, etc.) of a relationship between the controllable physical parameter and the derived quantum parameter. In some instances, determining a control waveform can include determining an inverse of the adiabatic component.

[0019] In some instances, a relationship between the derived quantum parameter and the controllable physical parameter can be known to have no non-adiabatic component (i.e., non-adiabatic component exactly equal to zero). In such instances, a control waveform can be generated based on the target derived waveform and the inverse of the adiabatic component.

[0020] In other instances, a relationship between the derived quantum parameter and the controllable physical parameter can have one or more non-zero non-adiabatic components, whose magnitude in some instances cannot be determined directly. In such instances, determining a control waveform can further comprise approximating the non-adiabatic components of the relationship between the derived quantum parameter and the controllable physical parameter. In such instances, a control waveform can be determined based on the target derived waveform, the inverse of the adiabatic component, and one or more approximations of the non-adiabatic component(s). However, in instances where a nonadiabatic component can be derived exactly (e.g., nonadiabatic component known to be zero), a control waveform can be determined without using approximations of any nonadiabatic component.

[0021] In some instances, determining a control waveform can include determining a harmonic decomposition (e.g., Fourier series, etc.) of a second waveform configured to generate the target derived waveform. In some instances, determining a control waveform can include truncating the harmonic decomposition. For example, a first n (e.g., 5, 10, 15, 20, etc.) harmonics of a Fourier series associated with the second waveform can be determined, and a control waveform can be generated using a multi-harmonic pump.

[0022] Example embodiments according to some aspects of the present disclosure can provide for a number of technical effects and benefits, such as improvements to computing technology (e.g., quantum computing technology). For example, systems and methods of the present disclosure can provide fast parametric two-qubit gates with highly suppressed leakage rates and suppressed ac-Stark shifts. For example, two significant causes of poor fidelities of some quantum operations (e.g., two-qubit gates) can include breaking of a rotating-wave approximation, and spurious nonadiabatic transitions of an effective Hamiltonian. In some example embodiments, systems and methods according to aspects of the present disclosure can operate without the need for a rotating wave approximation and can account for nonadiabatic components of an effective Hamiltonian. In this manner, for instance, infidelities (e.g., due to leakage) of quantum operations can be reduced, and faster parametric operations can be enabled without becoming limited by leakage at large drive amplitudes. Additionally, it will be appreciated that faster gating can increase a fidelity of a quantum circuit by reducing the occurrence of quantum decoherence.

[0023] It will be further appreciated that parametric modulation can have technical effects and benefits that have previously been limited by the effects of slow speeds or high leakage rates. For example, parametric modulation makes it possible to induce resonant behavior (i.e., the exchange of energy) between quantum systems that are frequency-detuned. In superconducting quantum computing systems, parametric modulation can facilitate two- qubit couplings with large on-off ratios, which can in some instances be a building block forhigh-fidelity two-qubit gates. However, it will be appreciated that slow speeds can in some instances limit the effectiveness of parametric gates due to quantum decoherence, and leakage can similarly limit the effectiveness of a quantum gate.

[0024] Additionally, systems and methods of the present disclosure can be combined with additional pulse-shaping methods to further reduce leakage. For example, systems and methods of the present disclosure can be combined with derivative removal by adiabatic gate (DRAG) pulse-shaping methods to avoid frequency collisions responsible for leakage or crosstalk.

[0025] With reference now to the Figures, example embodiments of the present disclosure will be discussed in further detail.

[0026] FIG.1 depicts an example quantum computing system 100. The example system 100 is an example of a system on one or more classical computers or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing structures or systems can be used without deviating from the scope of the present disclosure.

[0027] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The quantum hardware 102 includes components for performing quantum computation. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits. In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0028] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0029] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or morecontrol devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0030] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout devices(s) 114 may implement one or more quantum logic gates that operate on the quantum system 102 through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0031] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout devices(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0032] The readout device(s) 114 can take advantage of a difference in the impedance for the |0〉 and |1〉 states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state |0〉 or the state |1〉, due tothe nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0033] In some implementations, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the two- dimensional grid 122 depicted in FIG.1 includes 16 qubits arranged in a square formation, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength. In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit 128. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit. That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0034] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed by the calibration system. Some operating frequencies are better than other operating frequencies. One metric for assessing how good a particular operating frequency is for a particular qubit is energy relaxation time (T1) for the qubit at the frequency. Lower energy relaxation times can lead to larger quantum computational errors.

[0035] In various implementations, the example system 100 can be implemented as a client device, a server device, or both. The example system 100 can be implemented as part of a distributed computing system. The example system 100 can be implemented along with other example systems, which may be the same or different. The example system 100 can beimplemented in a server farm or other facility that operates multiple computing systems to provide computational services to or on behalf of a plurality of client systems. Advantageously, techniques according to example aspects of the present disclosure can provide for improved calibration and maintenance of computing facilities, increasing service uptime, decreasing failure rates, etc.

[0036] Figure 2 depicts a flowchart diagram of an example method for generating a parametric modulation control pulse according to example embodiments of the present disclosure. Although Figure 2 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of example method 200 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0037] At 202, example method 200 can include obtaining a target pulse shape associated with a quantum parameter of interest, wherein the target pulse shape corresponds to parametric modulation. In some instances, the quantum parameter of interest can be a derived parameter (e.g., tunable qubit frequency, two-qubit coupling strength, etc.). In some instances, parametric modulation can include time-periodic modulation of a derived parameter (e.g., f(t) = sin(kωt), where f is a derived parameter, k is an integer, and ω is a modulation frequency). In some embodiments, a derived parameter can be related to one or more physically controllable parameters (e.g., phase bias, flux, charge bias, etc.).

[0038] In some instances, a derived parameter can be defined based on a change of frame from a true Hamiltonian (e.g., Hamiltonian describing a single qubit, Hamiltonian describing a two-qubit coupling, etc.) to an effective Hamiltonian. In some instances, a change of frame from a true Hamiltonian to an effective Hamiltonian can be based at least in part on a unitary transformation, such aswhere U can be a unitary that depends on the problem, and H can be an effective Hamiltonian. In some instances, the effective Hamiltonian can be directly related to a derived parameter and indirectly related to a corresponding physically controllable parameter. In some instances, an effective Hamiltonian can be linear with respect to the derived parameterf, such that the derived parameter can follow= from a matrix element of ^�^^^^^^^^^^^^^^^(^^^^), or moregenerally a trace operation of the formwhere Π�can be a λ-independent operator (e.g., projector). In some instances, a linear dependence between the derived parameter f and the effective Hamiltonian can be independent of an amplitude associated with the physically controllable parameter (e.g., associated with a time-dependent oscillation of the physically controllable parameter; associated with a second pulse shape or control pulse shape; etc.). In some instances, ^^^^(^^^^) can have units of 1 / time, such as when f corresponds to a frequency or coupling strength. However, this is not required.

[0039] In some instances, determining a target pulse shape can include configuring a target pulse shape to reduce one or more sources of error (e.g., leakage, crosstalk, etc.). In some instances, determining a target pulse shape can include derivative removal by adiabatic gate (DRAG) pulse shaping. For example, in some instances, spectral components of a target pulse shape associated with a derived quantum parameter can be engineered to avoid frequency collisions with spurious transitions of an effective Hamiltonian (e.g., frequency collisions responsible for leakage or cross-talk).

[0040] At 204, example method 200 can include determining, based on the target pulse shape, a second pulse shape associated with a physically controllable parameter.

[0041] In some example implementations, the second pulse shape can be a pulse shape configured to cause a derived quantity to oscillate with single-harmonic (e.g., single- frequency) oscillation. In some instances, the second pulse shape can include an amplitude- scalable shape, wherein the derived quantity experiences single-harmonic oscillation for any amplitude of the second pulse shape. In some instances, the derived quantity can experience single-harmonic oscillation without the need of a rotating wave approximation. In some instances, such a second pulse shape can advantageously improve fidelity of a quantum computing operation (e.g., quantum gate). For example, breaking of a rotating-wave approximation can in some instances be a significant cause of poor two-qubit gate fidelities of alternative parametric modulation methods, which can be mitigated by eliminating a need for a rotating wave approximation.

[0042] In some instances, determining a second pulse shape for generating amplitude- independent single-harmonic oscillation can be based on a differential equation. In some instances, the differential equation can be a differential equation derived from a frame inwhich f(t) is defined (e.g., based on a time-dependent version of equation (1) above). In some instances, a differential equation for determining a second pulse shape can be written asIn some instances, the derived quantity of interest can be redefined aswhere ^^^^^^^^^^^^^^^^^^^^(^^^^)can be a desired parametric modulation (e.g., target pulse shape, etc.). In equation (4), the first term ^^^^(^^^^) can represent an adiabatic contribution to the effectivederived parameter, and the second term ^^^^ℏ^^^^^^^^�Π�^�^^^†(^^^^)^^^^^^^^^�^^^(^^^^)�^̇^^^(^^^^) of equation (4) canrepresent a nonadiabatic contribution. In some instances, a mapping between a derived parameter f and a physically controllable parameter λ can be one-to-one, and equation (4) can be rewritten asIn some instances, equation (5) can be solved to determine an ideal physical-parameter modulation waveform, which can correspond to ^^^^(^^^^) = ^^^^−1[^^^^(^^^^)]. (6)

[0043] In some instances, a second pulse shape can correspond to the ideal physical- parameter modulation waveform of equation (6). For example, in instances, where a nonadiabatic response of a system is identically zero, an adiabatic portion[^^^^(^^^^)]can be extracted from steady-state measurements (e.g., spectroscopy, etc.) coupled with time-domain experiments (e.g., Ramsey sequence, etc.), and equation (6) can be solved based on the adiabatic portion (e.g., based on an inverseof the adiabatic portion f(λ)).

[0044] In some instances, a second pulse shape can correspond to an approximation of an ideal physical-parameter modulation waveform of equation (6). For example, in some instances, a nonadiabatic portion of equation (4) can be unknown in practice, and an approximate solution to equation (5) can be determined based solely on the adiabatic contribution. In some instances, determining an approximate solution based on the adiabatic contribution can include using equation (6) to derive λ(t), and then approximating the nonadiabatic term of equation (5) in a second step, leading to a second pulse shape corresponding to an approximate form for based on the proposed ^^^^(^^^^). In someinstances, approximating the nonadiabatic term of equation (5) can include performing one or more numerical approximations (e.g., Schrieffer-Wolff transformations, etc.).

[0045] At 206, example method 200 can include generating, based on the second pulse shape, a control pulse. In some instances, the control pulse can be a time-dependent pulse of the physically controllable parameter. In some instances, the second pulse shape can include two or more (e.g., 5, 10, 15, etc.) harmonics. In some instances, the control pulse can have a pulse shape including two or more harmonics. In some instances, determining the second pulse shape or determining a control pulse shape can include performing a harmonic decomposition (e.g., Fourier decomposition, etc.) of a pulse shape determined based at least in part on an adiabatic portion of a relationship between the quantum parameter of interest and the physically controllable parameter. In some instances, determining the second pulse shape or determining a control pulse shape can include truncating the harmonic decomposition. In some instances, generating the control pulse can include using a multi- harmonic pump to generate a multi-harmonic waveform associated with the physically controllable parameters.

[0046] Example details associated with two example applications of example aspects of method 200 are further described below with respect to FIG.3 (tunable qubit frequency) and FIG.4 (two-qubit coupling). However, method 200 is not limited to the example applications of FIGS.3-4, and other applications are possible without going outside the scope of the present disclosure.

[0047] Figure 3 depicts a flowchart diagram of an example method 300 for generating a parametric modulation control pulse according to example embodiments of the present disclosure. Although Figure 3 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of example method 300 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0048] At 302, example method 300 can include obtaining a target pulse shape associated with a quantum parameter of interest, wherein the target pulse shape corresponds to parametric modulation, and wherein the quantum parameter of interest is a tunable qubit frequency. Although example method 300 describes pulse shaping methods associated with a tunable qubit frequency, the various steps of method 300 are not limited to tunable qubitfrequencies and can be adapted to other quantum parameters of interest without deviating from the scope of the present disclosure.

[0049] In some example applications of aspects of the present disclosure, a parametrically modulated quantum computing system can include a tunable qubit with a periodically modulated frequency. In some instances, a periodically modulated frequency can be a derived parameter that is indirectly controlled via one or more physically controllable parameters λ. In some instances, a tunable qubit can include a superconducting qubit (e.g., comprising one or more Josephson junctions). In some instances, a tunable qubit can include a transmon qubit.

[0050] At 304, example method 300 can include determining an adiabatic component of a relationship between the quantum parameter of interest and a physically controllable parameter.

[0051] In some instances, a physically controllable parameter for controlling a periodically modulated frequency of a tunable qubit can include a time-dependent phase bias. In some example embodiments, a tunable qubit can be used as a component of another quantum computing system or method (e.g., single-qubit gate, two-qubit gate, qubit reset, leakage removal, etc.).

[0052] In some instances, a change of frame can be configured such that a nonadiabatic component of a relationship between a derived parameter and a physically controllable parameter can exactly vanish (e.g., be exactly equal to zero). For example, in some applications of aspects of the present disclosure, a change of frame associated with a tunable transmon qubit can be defined by equation 1 above, wherein the unitary ^�^^^(^^^^) can bewhere|^^^^^^^^(^^^^^^^^^^^^^^^^)^are the parametric eigenstates of a qubit Hamiltonian (e.g., transmon Hamiltonian) with associated eigenenergies ħωi(^^^^^^^^^^^^^^^^), |^^^^^denotes a canonical-basis vector (e.g., |1^= (0, 1, 0, ... , 0)T, etc.), and D is a number of qubit eigenstates (e.g., transmoneigenstates) kept. In some instances, a projector Π�associated with equations (2) and (7) canbe |1^^1| − |0^^0|. In such instances, a nonadiabatic Hamiltonian of equation (4) can exactlyvanish. For example, in instances when one or more qubit eigenstates (e.g., transmoneigenstates) obey the parallel transportHellman-Feynmantheorem can be used to arrive at the expression^(9) with zero diagonal components in the canonical basis.

[0053] At 306, example method 300 can include determining an inverse of the adiabatic component. In some instances, determining a second pulse shape can include solving an algebraic equation associated with an adiabatic component of a relationship between a derived parameter and a physically controllable parameter. For example, in instances where a nonadiabatic component is equal to zero, equation (4) can reduce to an algebraic equation for ^^^^^^^^^^^^^^^^(^^^^) without approximations. For example, if a target pulse shape associated with adesired derived parameter is a qubit-frequency modulation ^^^^ (^^^^) =− ^^^^0(^^^^), thencorresponding second pulse shape (e.g., phase-bias modulation pulse shape) can follow fromthe inverse

[0054] At 308, example method 300 can include determining that a non-adiabatic component of the relationship between the quantum parameter of interest and the physically controllable parameter is equal to zero. For example, as explained above, a non-adiabatic component of a relationship between a tunable qubit frequency and a phase bias can exactly vanish when an effective Hamiltonian is defined according to equations (1) and (7) above.

[0055] At 310, example method 300 can include determining, based on the target pulse shape and the inverse of the adiabatic component, a second pulse shape associated with the physically controllable parameter.

[0056] In some instances, a second pulse shape can include a plurality of harmonics. For example, for some target pulse shapes ^^^^01(^^^^) associated with a tunable qubit frequency, determining a second pulse shape can include determining a plurality of harmonics associated with a fundamental frequency ωfundassociated with the second pulse shape. In some instances, a Fourier decomposition of a second pulse shape can include a plurality of kthharmonics having Fourier-series amplitudes calculated aswhere T =2^^^^^^^^^^^^^^^^^^^^^^^^, ^^^^^^^^can be amplitudes associated with odd values of k (e.g., first harmonic, third harmonic, fifth harmonic, etc.), and ^^^^^^^^can be amplitudes associated with even values ofk (e.g. second harmonic, fourth harmonic, tenth harmonic, etc.). In some instances, determining a second pulse shape can include truncating a harmonic decomposition (e.g., Fourier series), wherein only a first n (e.g., 5, 10, 20, etc.) harmonics (e.g., having amplitudes ^^^^^^^^and ^^^^^^^^) are included in a second pulse shape. In some instances, a subset (e.g., only odd- numbered harmonics, only even, etc.) of the first n harmonics can be included in a second pulse shape, and a subset of the first n harmonics can be omitted from a second pulse shape.

[0057] At 312, example method 300 can include generating, based on the second pulse shape, a control pulse. In some instances, generating a control pulse can include one or more actions described above with respect to method 200 at 206.

[0058] Figure 4 depicts a flowchart diagram of an example method for generating a parametric modulation control pulse according to example embodiments of the present disclosure. Although Figure 4 depicts steps performed in a particular order for purposes of illustration and discussion, the methods of the present disclosure are not limited to the particularly illustrated order or arrangement. The various steps of example method 400 can be omitted, rearranged, combined, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0059] At 402, example method 400 can include obtaining a target pulse shape associated with a quantum parameter of interest, wherein the target pulse shape corresponds to parametric modulation, and wherein the quantum parameter of interest is a two-qubit coupling strength. Although example method 400 describes pulse shaping methods associated with a two-qubit coupling strength, the various steps of method 400 are not limited to two- qubit coupling strengths, and can be adapted to other quantum parameters of interest without deviating from the scope of the present disclosure.

[0060] In some example applications of aspects of the present disclosure, a parametrically modulated quantum computing system can include two qubits coupled by a tunable coupler. In some instances, the coupler can mediate an off-resonant two-qubit coupling of exchange type, where the coupling strength, ^^^^, can be a function of the coupler frequency set by an external phase bias ^^^^^^^^^^^^^^^^. If the qubits are detuned by an amount|Δ|>>|^^^^|, a time-independent exchange interaction can lead to a weak (spurious) cross-Kerr(or 'ZZ') coupling between the qubits. In contrast, if the effective two-qubit coupling is modulated at frequency Δ via a time-dependent phase bias ^^^^^^^^^^^^^^^^(^^^^), the exchange interaction can become resonant in the rotating frame of the qubits, making it possible to engineer a two-qubit SWAP-like gate. In some instances, pulse shaping methods according to aspects ofthe present disclosure can strongly suppress leakage and ac-Stark shifts of such a two-qubit gate (e.g., as compared to alternative parametric two-qubit gates, which may be implemented by considering a perfectly sinusoidal modulation of the coupler phase bias).

[0061] At 404, example method 400 can include determining an adiabatic component of a relationship between the quantum parameter of interest and a physically controllable parameter.

[0062] In some instances, determining a second pulse shape ^^^^^^^^^^^^^^^^(^^^^) based on a target pulse shape ^^^^(^^^^) can include identifying a frame in which a derived parameter can emerge as a linear parameter in an effective Hamiltonian. For example, for two-qubit coupler applications, a frame can be identified in which a two-qubit interaction strength mediated by a coupler mode emerges as a linear parameter in the effective two-qubit Hamiltonian. Such a frame can be defined, for example, by a composite unitary. The composite unitary can include, for example, a first unitary in which a non-interacting coupler Hamiltonian can be diagonal for all values of a phase bias ^^^^^^^^^^^^^^^^(^^^^). In some instances, a first unitary can comprise the single-body unitary of equation (7), wherein {|^^^^^^^^(^^^^^^^^^^^^^^^^)^} can correspond to the parametric eigenstates of a coupler Hamiltonian. Such a unitary can be referred to herein as^�^^^^^^^(^^^^^^^^^^^^^^^^). In a frame defined by ^�^^^^^^^(^^^^^^^^^^^^^^^^), a qubit-coupler interaction ^�^^^1, which can in someinstances be provided by a capacitive coupling between such modes, can become block-offdiagonal with respect to the coupler basis,partial trace that removes the qubits. In this frame, a second unitary can block-diagonalize the two-qubit+coupler Hamiltonian with respect to the coupler mode. In some instances, block-diagonalization can amount to removing offdiagonal couplings betweeen the qubits’ eigenstates, such that the two-qubit+coupler Hamiltonian in the new frame can become a direct sum of the form(11) is a projector onto the subspace spanned by the coupler’s ith eigenstate. In some instances (e.g., when a coupler remains in its parametric ground state at all times), equation(11) can be approximated based on a zeroth eigenstate, ^�^^^^^^^^^^^^^^^[^^^^^^^^^^^^^^^^] ≈ ^�^^^(0)^^^^^^^^^^^^[^^^^^^^^^^^^^^^^].

[0063] In some instances, identifying a frame in which a derived parameter can emerge as a linear parameter in an effective Hamiltonian can include numerically approximating one or more unitaries (e.g., second unitary of a composite unitary, etc.). For example, in someinstances, determining a unitary associated with the effective Hamiltonian of equation (11) can include a recursive algorithm for numerically approximating the unitary. In some instances, numerically approximating a unitary can include performing one or more Schrieffer-Wolff transformations. In some instances (e.g., when there are no qubit-coupler resonances), a numerical approximation can converge as a function of the order of a plurality of perturbations. In some instances, a unitary associated with the effective Hamiltonian of equation (11) can include a Hermitian operator (e.g., Schrieffer-Wolff generator, etc.). For example, a unitary can be of the formwhere ^̂^^^ is a Schrieffer-Wolff generator. A composite frame unitary can thus beleading to the effective Hamiltonian(14) where an explicit ^^^^^^^^^^^^^^^^-dependence has been omitted to simplify notation. The first term in this equation can represent an adiabatic contribution to the effective Hamiltonian. The second term can represent a nonadiabatic contribution to the effective Hamiltonian, due to the phase- bias-dependent coupler eigenstates. The third term can represent a nonadiabatic contribution to the effective Hamiltonian, due to the phase-bias-dependent coupler eigenfrequencies. The expression for the effective two-qubit coupling can follow from Eq. (14) and Eq. (4), such as whenΠ�= |100^^010|,where the first and second positions in the ket / bra can represent the first and second qubits respectively, and the last position can correspond to the coupler state.

[0064] Momentarily disregarding the nonadiabatic contributions in Eq. (14), the effective two-qubit Hamiltonian (assuming the coupler remains in its parametric ground state) may be approximated aswhere ^�^^^ and ^�^^^ correspond to lowering operators for the first and second qubit, respectively.

[0065] Provided a desired pulse shape for ^^^^(^^^^), a corresponding time-dependent phase bias can be found by inverting ℏ^^^^(^^^^) = tr�Π�^�^^^†[^^^^^^^^^^^^^^^^(^^^^)]^�^^^[^^^^^^^^^^^^^^^^(^^^^)]^�^^^[^^^^^^^^^^^^^^^^(^^^^)]�,which can be equivalent to Eq. (6) with the replacement (λ, f) → (^^^^^^^^^^^^^^^^,^^^^).

[0066] At 406, example method 400 can include determining an approximation of one or more non-adiabatic components of the relationship between the quantum parameter of interest and the physically controllable parameter.

[0067] In some instances, the nonadiabatic part of the effective Hamiltonian can modifythe coupling as ^^^^(^^^^) → ^^^^^^^^^^^^^^^^^^^^(^^^^), where(18)

[0068] The nonadiabatic contributions in the second and third line of Eq. (18) can be expressed in terms of commutators of the Schrieffer-Wolff generator, for which we arrive at

[0069] Complemented with a numerically exact way of calculating the operator gradients ^^^^^^^^^�^^^^^^^and ^^^^^^^^^̂^^^ [via an expression similar to Eq. (9)], Eq. (19) and Eq. (20) can enable evaluating the impact of nonadiabatic corrections on the effective two-qubit coupling Eq. (18).

[0070] In some instances, equation (18) can specify the instantaneous two-qubit coupling as a function of time for arbitrary time-evolution, including baseband and parametric coupling modulation. In the second case, ^^^^(^^^^)is a rapidly oscillatory function of time for which it can in some instances be useful to approximate Eq. (18) as ^^^^^^^^^^^^^^^^^^^^(^^^^) ≈ ^^^^(^^^^)where the overline can represent a time-average of the nonadiabatic amplitudes associated with ^^^^(^^^^). In some instances, a time-average can be calculated in a single period of the parametric drive; the result can be time-dependent but can in some instances be assumed to vary slowly in the time-scale of the drive.

[0071] An important special case of Eq. (21) can be realized for ^^^^(^^^^) of constant amplitude, where ^^^^ can become time-independent. In such a case, the expression for ^^^^^^^^^^^^^^^^^^^^(^^^^) can be further simplified, arriving at(22) where(23) groups the time-averaged non-adiabatic amplitudes in equation (21).

[0072] In some example experiments according to the present disclosure, the above- described methods were applied to a prototype two-qubit+coupler setup having a qubit-qubit detuning Δ / 2^^^^ of about 1 GHz. Other implementations are possible. The prototype coupler was debiased at its sweet spot ^^^^^^^^^^^^^^^^= 0 and flux-pumped such that the effective two-qubit coupling was modulated as(24) where ^^^^0= ^^^^ ( ^^^^^^^^^^^^^^^^→0) and ^^^^1is a time-independent coupling amplitude. A time-dependent coupling amplitude ^^^^1→ ^^^^1(^^^^) is also possible. In some instances, the waveform in Eq. (24) can be configured such that the two-qubit coupling varies between its minimum value ^^^^0and a maximum value ^^^^0+ ^^^^1, with ^^^^1> 0. In the example experiments, the derivative of the two-qubit coupling with respect to coupler phase-bias was determined, as were the trace amplitudes associated with each nonadiabatic process. Combined, these quantities were used to estimate the nonadiabatic coefficient ℰ in Eq. (23), which can in some instances be a function (e.g., approximately linear function, etc.) of a coupling modulation amplitude.

[0073] At 408, example method 400 can include determining, based on the target pulse shape, the adiabatic component, and the approximation of the one or more non-adiabatic components, a second pulse shape associated with the physically controllable parameter.

[0074] For example, given Eq. (24) (drive of constant amplitude), Eq. (22) can lead to the expression ^^^^^^^^^^^^^^^^^^^^(^^^^) ≈ ^̅^^^ −^^^^12cos(2^^^^^^^^) + ^^^^ℰ(^̅^^^)^^^^1^^^^ sin(2^^^^^^^^),(25) whereis the average two-qubit coupling during the pulse. Because ^̅^^^ is the time-independent part of Eq. (25), it can lead to drive-amplitude-dependent energy shifts of the qubit transitions, including both conditional (ZZ) and unconditional ac-Stark shifts. To make the coupling modulation resonant with the qubit-qubit detuning, a drive frequency can be chosen to match the ac-Stark-shifted two-qubit transition: 2^^^^ = |Δ�|where Δ�= ^�^^^^^^^ − ^�^^^^^^^ is the detuning between the dressed qubit modes obtained for the averagetwo-qubit coupling ^̅^^^. Using Eq. (22), and performing a rotating-wave approximation validfor |^^^^ / Δ�| ≪ 1, the effect1ive two-qubit Hamiltonian can take the approximate formℏ ≈is the effective parametric exchange coupling between the two qubits.

[0075] In some example experiments, the value of 2J (anticrossing amplitude) was plotted as a function of ^^^^1and compared to exact values determined according to Floquet theory. In some example experiments, the value of 2J showed remarkably strong agreement with Floquet theory, particularly given the strong phase-bias amplitudes associated with some example coupling modulations tested (e.g., ^^^^1 / 2π > 10 MHz, etc.). This agreement demonstrates that, at least in some instances, the Schrieffer-Wolff result, can be used toefficiently predict the parametric coupling and ac-Stark shift ^̅^^^ = Δ� − Δ from an estimation ofthe ‘dc’ two-qubit coupling and related quantities. Advantageously, the Schrieffer-Wolff result can in some instances be determined at reduced computational cost compared to alternative methods (e.g., exact Floquet theory computations).

[0076] Note that the perfectly sinusoidal waveform specified in Eq. (24) for ^^^^(t) leads to nonzero Fourier amplitudes of the phase-bias modulation ^^^^^^^^^^^^^^^^(^^^^) for harmonics beyond the fundamental frequency ω. For example, in some example experiments according to the present disclosure, a phase-bias oscillation appeared to be approximately sinusoidal for weak coupling-modulation amplitudes, and the ^^^^^^^^^^^^^^^^(^^^^) waveform became less sinusoidal for larger values of ^^^^1. Example figures showing numerical values of example experimental results (e.g. harmonic decompositions of example ^^^^^^^^^^^^^^^^(^^^^) waveforms, etc.) are included in Appendix A to U.S. Provisional Patent Application No.63 / 611,520, which is incorporated by reference herein.

[0077] At 410, example method 400 can include generating, based on the second pulse shape, a control pulse. In some instances, generating a control pulse can include one or more actions described above with respect to method 200 at 206.

[0078] FIG.5 depicts a block diagram of an example computing system 1 that can perform aspects of example embodiments of the present disclosure. The system 1 includes a computing device 50, a server computing system 60, and a third-party system 70 that are communicatively coupled over a network 49. The system 1 also includes a quantum computing system 80 that is communicatively coupled to the server computing system.

[0079] The computing device 50 can be any type of computing device (e.g., classical computing device), such as, for example, a mobile computing device (e.g., smartphone or tablet), a personal computing device (e.g., laptop or desktop), a workstation, a cluster, a gaming console or controller, a wearable computing device, an embedded computing device, or any other type of computing device. In some embodiments, the computing device 50 can be a client computing device. The computing device 50 can include one or more processors 51 and a memory 52. The one or more processors 51 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 52 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 52 can store data 53 and instructions 54 which are executed by the processor 51 to cause the computing device 50 to perform operations as described herein.

[0080] The computing device 50 can also include one or more input components that receive user input. For example, a user input component can be a touch-sensitive component(e.g., a touch-sensitive display screen or a touch pad) that is sensitive to the touch of a user input object (e.g., a finger or a stylus). The touch-sensitive component can serve to implement a virtual keyboard. Other example user input components include a microphone, a traditional keyboard, or other means by which a user can provide user input.

[0081] The quantum computing system 80 can include one or more processors 81 (e.g., classical processor(s) 104) and a memory 82. The one or more processors 81 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 82 can include one or more non-transitory computer- readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 82 can store data 83 and instructions 84 which are executed by the processor 81 to cause the quantum computing system 80 to perform operations as described herein.

[0082] The quantum computing system 80 can also include a quantum system 85 for performing quantum computations. In some instances, the quantum system 85 can be, comprise, or be comprised by quantum hardware 102, described above with reference to FIG. 1.

[0083] In some implementations, the quantum computing system can 80 include or be otherwise implemented by one or more server computing systems 60. In instances in which the quantum computing system 80 includes plural server computing devices, such server computing devices can operate according to sequential computing architectures, parallel computing architectures, or some combination thereof.

[0084] The third-party system 70 can include one or more processors 71 and a memory 72. The one or more processors 71 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 72 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 72 can store data 73 and instructions 74 which are executed by the processor 71 to cause the third-party system 70 to perform operations. In some implementations, the third- party system 70 includes or is otherwise implemented by one or more server computing devices.

[0085] The server computing system 60 can include one or more processors 61 and a memory 62. The one or more processors 61 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 62 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 62 can store data 63 and instructions 64 which are executed by the processor 61 to cause the server computing system 60 to perform operations. In some implementations, the server computing system 60 includes or is otherwise implemented by one or more server computing devices.

[0086] The network 49 can be any type of communications network (e.g., classical or quantum), such as a local area network (e.g., intranet), wide area network (e.g., Internet), or some combination thereof and can include any number of wired or wireless links. In general, communication over the network 49 can be carried via any type of wired or wireless connection, using a wide variety of communication protocols (e.g., TCP / IP, HTTP, SMTP, FTP), encodings or formats (e.g., HTML, XML), or protection schemes (e.g., VPN, secure HTTP, SSL).

[0087] FIG.5 illustrates one example computing system that can be used to implement aspects of the present disclosure. Other computing systems can be used as well. For example, in some implementations, the quantum computing system 80 can include the server computing system 60 or vice versa. In some implementations, the quantum computing system 80 may be communicatively coupled through the network 49 to the computing device 50, third-party system 70, or server computing system 60.

[0088] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0089] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs (e.g., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus). The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0090] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit (i.e., a system that defines the unit of quantum information). It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qubits) are possible.

[0091] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that createsan 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.

[0092] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0093] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0094] 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 anASIC, 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.

[0095] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0096] 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 microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0097] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0098] 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 flashmemory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; and CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0099] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0100] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0101] 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 andsystems can generally be integrated together in a single software product or packaged into multiple software products.

[0102] 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.

[0103] Aspects of the disclosure have been described in terms of illustrative implementations thereof. Numerous other implementations, modifications, or variations within the scope and spirit of the appended claims can occur to persons of ordinary skill in the art from a review of this disclosure. Any and all features in the following claims can be combined or rearranged in any way possible. Accordingly, the scope of the present disclosure is by way of example rather than by way of limitation, and the subject disclosure does not preclude inclusion of such modifications, variations or additions to the present subject matter as would be readily apparent to one of ordinary skill in the art. Moreover, terms are described herein using lists of example elements joined by conjunctions such as “and,” “or,” “but,” etc. It should be understood that such conjunctions are provided for explanatory purposes only. Lists joined by a particular conjunction such as “or,” for example, can refer to “at least one of” or “any combination of” example elements listed therein, with “or” being understood as “and / or” unless otherwise indicated. Also, terms such as “based on” should be understood as “based at least in part on.”

[0104] Those of ordinary skill in the art, using the disclosures provided herein, will understand that the elements of any of the claims, operations, or processes discussed herein can be adapted, rearranged, expanded, omitted, combined, or modified in various ways without deviating from the scope of the present disclosure. Some of the claims are described with a letter reference to a claim element for exemplary illustrated purposes and is not meant to be limiting. The letter references do not imply a particular order of operations. For instance, letter identifiers such as (a), (b), (c),..., (i), (ii), (iii),..., etc. can be used to illustrate operations. Such identifiers are provided for the ease of the reader and do not denote a particular order of steps or operations. An operation illustrated by a list identifier of (a), (i), etc. can be performed before, after, or in parallel with another operation illustrated by a list identifier of (b), (ii), etc.

Claims

WHAT IS CLAIMED IS:

1. A method for shaping a control pulse of a quantum computing system, comprising: obtaining a target pulse shape associated with a quantum parameter of interest; determining, based on the target pulse shape, a second pulse shape associated with a physically controllable parameter; and generating, based on the second pulse shape, a control pulse; wherein the target pulse shape corresponds to parametric modulation.

2. The method of claim 1, wherein determining the second pulse shape comprises: determining an adiabatic component of a relationship between the quantum parameter of interest and the physically controllable parameter; and determining, based on the adiabatic component, the second pulse shape.

3. The method of claim 2, wherein determining the adiabatic component comprises: performing one or more steady-state measurements; and performing one or more time-domain measurements.

4. The method of claim 2, wherein determining, based on the adiabatic component, the second pulse shape comprises: determining an inverse of the adiabatic component; and determining, based on the target pulse shape and the inverse of the adiabatic component, a second pulse shape.

5. The method of claim 2, wherein: determining the second pulse shape further comprises determining an approximation of one or more non-adiabatic components of the relationship between the quantum parameter of interest and the physically controllable parameter; and the second pulse shape is based, at least in part, on the approximation of the one or more non-adiabatic components.

6. The method of claim 5, wherein determining the approximation of the one or more non-adiabatic components comprises performing one or more Schrieffer-Wolff transformations.

7. The method of claim 1, wherein: determining the second pulse shape comprises determining one or more harmonics of the second pulse shape.

8. The method of claim 7, wherein determining one or more harmonics comprises determining a harmonic decomposition associated with the second pulse shape.

9. The method of claim 8, wherein determining one or more harmonics further comprises truncating the harmonic decomposition.

10. The method of claim 1, wherein the physically controllable parameter comprises a phase bias.

11. The method of claim 1, wherein the quantum parameter of interest comprises a tunable qubit frequency.

12. The method of claim 1, wherein the quantum parameter of interest comprises a coupling strength.

13. The method of claim 1, wherein the target pulse shape comprises a single- frequency oscillation.

14. The method of claim 1, wherein determining the second pulse shape comprises derivative removal by adiabatic gating.

15. The method of claim 1, wherein the second pulse shape is determined based at least in part on an effective Hamiltonian, wherein the quantum parameter of interest is linear with respect to the effective Hamiltonian.

16. The method of claim 15, wherein a linear dependence between the quantum parameter of interest and the effective Hamiltonian is independent of an amplitude of the physically controllable parameter.

17. The method of claim 15, wherein the effective Hamiltonian is based at least in part on a unitary comprising one or more parametric eigenstates of at least one of: a qubit Hamiltonian; and a coupler Hamiltonian.

18. The method of claim 1, wherein the target pulse shape has a constant amplitude.

19. A quantum computing system comprising: a superconducting qubit having a tunable qubit frequency; and one or more control devices operable to provide one or more control pulses to control the tunable qubit frequency; wherein the quantum computing system is configured to perform operations, the operations comprising: generating a control pulse to control the tunable frequency according to a target pulse shape for the tunable qubit frequency, wherein the target pulse shape was determined based on a target pulse shape and an adiabatic component of a relationship between the tunable frequency and a physically controllable parameter, and wherein the target pulse shape corresponds to parametric modulation of the tunable qubit frequency.

20. A quantum computing system comprising: a first qubit; a second qubit; and one or more control devices operable to provide one or more control pulses to control a coupling strength between the first qubit and the second qubit; wherein the quantum computing system is configured to perform operations, the operations comprising: generating a control pulse to control the coupling strength, wherein the control pulse was determined based on a target pulse shape and an adiabatic component of arelationship between the tunable frequency and a physically controllable parameter, and wherein the target pulse shape corresponds to parametric modulation of the coupling strength.

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