High-accuracy calibration of an analog quantum simulator

Abstract

One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The QCS includes a quantum processor device that includes a set of qubits and a set of qubit couplers. The method includes determining a set of dressed qubit frequencies. Each dressed qubit frequency of the set of dressed qubit frequencies corresponds to a separate qubit pair of the set of qubits. A set of bare qubit frequencies is generated based on the set of dressed qubit frequencies. A spin-Hamiltonian for the set of qubits is approximated based on the set of bare qubit frequencies

Description

HIGH-ACCURACY CALIBRATION OF AN ANALOG QUANTUM SIMULATOR PRIORITY CLAIM

[0001] This application claims priority to United States Provisional Application Number 63 / 639,509 entitled HIGH-ACCURACY CALIBRATION OF AN ANALOG QUANTUM SIMULATOR, filed on April 26, 2024, the contents of which are herein incorporated in their entirety. FIELD

[0002] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to determining the fidelity of quantum error correction (QEC) circuits. 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] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The QCS includes a quantum processor device that includes a set of qubits and a set of qubit couplers. The method includes determining a set of dressed qubit frequencies. Each dressed qubit frequency of the set of dressed qubit frequencies corresponds to aseparate qubit pair of the set of qubits. A set of bare qubit frequencies is generated based on the set of dressed qubit frequencies. A spin-Hamiltonian for the set of qubits is approximated based on the set of bare qubit frequencies.

[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 quantum computing system according to example embodiments of the present disclosure.

[0009] FIG.2 shows a schematic view for a quantum simulation pipeline, according to various embodiments.

[0010] FIG.3 shows a device Hamiltonian employed in the device model, according to various embodiments.

[0011] FIG.4A shows coupling pathways modeled in the device Hamiltonian of FIG.3, according to various embodiments.

[0012] FIG.4B indicates the multiple types of coupling efficiencies that are employed for each of the coupling types in the device model, according to various embodiments.

[0013] FIG.5A shows an overview of the calibration steps, according to various embodiments.

[0014] FIG.5B shows details for performing the swap spectroscopy measurements of the various embodiments.

[0015] FIG.6A shows a spin-Hamiltonian, according to various embodiments.

[0016] FIG.6B shows the coefficients of the higher-order terms in the spin-Hamiltonian of FIG.6A, according to various embodiments.

[0017] FIG.7 shows a flowchart for a method for a method 700 for operating a quantum computing system, according to various embodiments.DETAILED DESCRIPTION

[0018] Example aspects of the present disclosure are directed to methods, architectures, and hardware configurations for implementing a scalable quantum computing system (QCS) calibration protocol. More specifically, the embodiments achieve low error in operating a QCS by explicitly calibrating the bare frequencies for qubits and qubit couplers. As used throughout, the term “quantum devices” may be used to refer to both qubits and qubit couplers. Although the embodiments are not limited to superconducting quantum devices, the following discussion assumes superconducting quantum devices, such as but not limited to transmon qubits and transmon qubit couplers. However, the embodiments are not limited to superconducting quantum devices and it is understood that the methods and protocols discussed herein may be generalized to other qubit architectures, such as but not limited to neutral atom qubits, trapped ion qubits, photonic qubits, spin-based qubits, and the like. Additionally, the term “bare” may modify calibration parameters (e.g., qubit / coupler frequencies, coupling rates, and the like) for a quantum device (e.g., qubits and / or qubit couplers) when the quantum device is calibrated in isolation (e.g., other neighboring quantum devices are currently “turned off.”). The term “dressed” may modify such calibration parameters for a quantum device when the quantum device is calibrated with neighboring quantum devices being “turned on”.

[0019] Furthermore, the following discussion assumes that the QCS is operated in an “analog” mode, as compared to a mode that implements discrete quantum logic gates on the qubits (e.g., without regard to the qubit architecture). Throughout, the use of discrete quantum logic gate may be referred to as “digital quantum computing.” Digital quantum computing may be in contrast to “analog quantum computing,” which involves allowing an initial state encoded in the qubits to evolve via a Hamiltonian of interest. The evolution of the Hamiltonian may be simulated by controlling the coupling strength between the qubits via the tuning of the qubit couplers, which facilitates the coupling (or interaction) strength between the qubits. Because such “analog” modes may be implemented to perform quantum simulations of the evolution of a unitary operator (e.g., a unitary based on the many-body Hamiltonian of interest) operating on the qubits, a QCS operated in an analog mode may be referred to as a “quantum simulator.”

[0020] The development of quantum simulators in various platforms has opened an experimental avenue toward answering the theoretical question of thermalization. Such questions seek to reconcile the unitarity of quantum evolution with the emergence of statistical mechanics in constituent subsystems. A particularly interesting setting for studying thermalization is that in which a quantum system is swept through a critical point. Varying the sweep rate can allow foraccessing dramatically different paths through phase space and correspondingly distinct coarsening behavior. In systems with complex phase diagrams involving multiple phases, such effects have been theoretically predicted to cause deviations from the Kibble-Zurek (KZ) mechanism. The LZ mechanism states that the correlation length ξ of the final state follows a universal power-law scaling with the ramp time ^^^.

[0021] While technical advancements in quantum simulation have enabled the observation of various thermalization-related phenomena, the analog nature of these systems has also imposed challenges with respect to their experimental versatility. Studying thermalization dynamics may require advanced state characterization and preparation of initial states across different regions of the eigenspectrum, both of which are difficult without universal quantum control. While digital quantum processors are in principle suitable for such tasks, implementing Hamiltonian evolution may require a high number of digital gates, making large-scale Hamiltonian simulation infeasible under current gate error rates.

[0022] The embodiments provide a pipeline for implementing a hybrid analog-digital quantum simulator. Such a hybrid analog-digital quantum simulator may be implemented via a QCS (e.g., the QCS 100 of FIG.1). In particular, the embodiments provide a calibration method for the quantum devices (e.g., the qubits and qubit couplers) that are employed in a quantum simulation performed by the QCS. As noted above, superconducting transmon qubits are assumed (although the embodiments are not limited to transmon qubits) in the implementation of the QCS. The qubits are coupled by tunable couplers in a square lattice. The quantum simulator supports universal entangling gates with pairwise interaction between qubits, and high-fidelity analog simulation of a U(1) symmetric (many-body) spin-Hamiltonian when all couplers are activated at once. The low analog evolution error, which was previously difficult to achieve with transmon qubits due to correlated cross-talk effects, is enabled by a scalable calibration scheme that is provided by the embodiments.

[0023] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments may be provide a calibration method for the quantum devices (e.g., the qubits and qubit couplers) that are employed in a quantum simulation performed by the QCS. Quantum Computing Systems

[0024] FIG.1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can beimplemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0025] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as any of the operations described herein. 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 (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spin- based qubits, and the like. The superconducting qubits may be located in a cryostat to cool the qubits to superconducting temperatures (e.g., less than about 3 Kelvin). However, aspects of the present disclosure are not limited to superconducting qubits. In some examples, any suitable qubit structure may be used without deviating from the scope of the present disclosure, such as photonic qubits, trapped ion qubits, spin qubits, neutral atom qubits, quantum dot qubits, molecular qubits, or other qubits.

[0026] 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.

[0027] 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 more control 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 112may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0028] 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.

[0029] 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.

[0030] In some implementations, 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 to the 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.

[0031] In some embodiments, 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 4x4 qubits, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 caninteract 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.

[0032] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit t. 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.

[0033] 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.

[0034] FIG.1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure. Quantum Simulations

[0035] FIG.2 shows a schematic view for a quantum simulation pipeline (QSP) 200, according to various embodiments. That is, QSP 200 is a pipeline for implementing a hybrid analog-digital quantum simulator. The hybrid analog-digital quantum simulator enables the performance of quantum simulations as discussed herein. Such a hybrid analog-digital quantum simulator may be implemented via a QCS (e.g., the QCS 100 of FIG.1). In particular, the embodiments provide a calibration method for the quantum devices (e.g., the qubits and qubit couplers) employed in the quantum simulations enabled via QSP 200. As noted above, superconducting transmon qubits are assumed (although the embodiments are not limited to transmon qubits) in the implementation of the QCS. The qubits are coupled by tunable couplers in a square lattice. The quantum simulator supports universal entangling gates with pairwiseinteraction between qubits, and high-fidelity analog simulation of a U(1) symmetric (many-body) spin-Hamiltonian when all couplers are activated at once. The low analog evolution error, which was previously difficult to achieve with transmon qubits due to correlated cross-talk effects, is enabled by a scalable calibration scheme that is provided by the embodiments.

[0036] Throughout, the terms “bare” and “dressed” are used when referring to calibration parameters of the quantum devices (e.g., qubits and qubit couplers). Such calibration parameters may include but are not otherwise limited to frequencies (^^^^) and coupling rates (^^^^). Dressed values for calibration parameters (e.g., dressed quantities 216) are indicated with tildes over the quantities (e.g., ^^^^^ and ^^^^^), whereas the bare quantities are referenced without the tilde (e.g., ^^^^). Bare calibration quantities typically refer to the calibration quantities for quantum devices that are subject to “bare” states, while dressed calibration quantities typically refer to the calibration quantities for quantum devices that are subject to “dressed” states. A quantum device (e.g., a qubit) is subject to bare states when the quantum device is being operated in isolation. That is, the bare states are the unperturbed quantum states of the qubit when the qubit is operated in isolation. Thus, the bare quantities for a qubit are the calibration parameters for when the qubit is operated in isolation. When neighboring quantum devices are also operated, the quantum states of the qubit may become “hybridized” (or perturbed). When hybridized (e.g., for instance when couplers and / or neighboring qubits are turned on), the values for calibration parameters for the qubit may shift. These shifted values for the calibration parameters are referred to as dressed quantities. That is, the bare frequency of a qubit is the frequency of the qubit when no other (at least nearby) qubits are turned on (or operated). The dressed frequency of a qubit may be the frequency of the qubit when the eigenstates of the qubit become hybridized with other nearby qubits that are turned on. For a qubit coupler example, when calibrating a coupler that couples two qubits, in isolation, the coupler may have a bare frequency (e.g., ^^^^^) of 10 MHz. When all the couplers are turned on, the coupler may have a dressed frequency (e.g., ^^^^^^) of 12 MHz.

[0037] QSP 200 includes the calibration of the qubits and qubits couplers (e.g., quantum devices). The calibration of the quantum devices includes a two-qubit spectroscopy 210 step and a device model 220 step that is based on measurements of the two-qubit spectroscopy 210 step. QSP 200 also includes a digital preparation of the initial state 230 to be evolved in the simulation. The device model 220 enables implementing the (many-body) spin-Hamiltonian that simulates the evolution of the initial states via the high-fidelity (HiFi) evolution 230 step. The calibration of the quantum devices (e.g., the two-qubit spectroscopy 210 step and the device model step 220) is used in the high-fidelity analog evolution 240 step (e.g., which simulates the evolution of the initial stateprepared in the digital preparation of the initial state 230 step) and an input. The evolved state is measured in the characterization of the evolved state 250 step. Note that the following terms and notations are used throughout. The two-qubit spectroscopy 210 step generates “dressed” quantities 216 as output, where ^^^^^ indicates a “dressed” coupling rate and ^^^^^ indicates “dressed” qubit frequencies. The device model 220 step shows qubit eigenenergies 224 and coupler eigenenergies 226 (e.g., with respect to an approximate potential well for each). Note that five eigenenergies (or levels) as shown for each. There may be more levels than shown for each in practice. At least for qubits, computation is usually performed in the two eigenstates (e.g., computational states) corresponding to the first two levels (or superpositions of the two computational states), while the higher levels corresponds to leakage states.

[0038] As shown in FIG.2, the two-qubit spectroscopy 210 step includes two types of spectroscopy measurements: swap spectroscopy measurements 212 a single-photon spectroscopy measurements 214. The combination of the swap spectroscopy measurements 212 and the single- photon measurements 214 generates the dressed quantities 216, e.g., the dressed coupling rates ^^^^^ and the dressed qubit / coupler frequencies ^^^^^. The device model 220 step transforms the dressed quantities 216 into “bare” quantities 222, where ^^^^^ indicates “bare” coupler frequencies and ^^^^^ indicates “bare” qubit frequencies.

[0039] Stated another way, the QSP 200 combines analog evolution (e.g., the high-fidelity evolution 240 step) with digital gates for extensive state preparation (e.g., the digital preparation of the initial state 230 step) and state characterization (e.g., the characterization of evolved state 250 step). The two-qubit spectroscopy 210 step and the device model 220 step provide a schematic representation of the scalable analog calibration protocol of the embodiments. The two-qubit spectroscopy 210 step includes swap spectroscopy measurements 212 and single-photon spectroscopy measurements 214, which are employed to extract dressed coupling rates ^^^^^ and dressed qubit frequencies ^^^^^. The two two-qubit circuits to perform the two sets of spectroscopy measurements are also shown in FIG.2, where the two-qubit gate indicated as ^^^is an analog unitary-evolution operator. The device model 220 step converts the dressed quantities 216 to the bare quantities 222 (or bare frequencies). The bare quantities 222 allow for establishing a device Hamiltonian of the full system. The device Hamiltonian is finally projected to the spin Hamiltonian 224.

[0040] Operating coupled qubits (e.g., transmon qubits) as a high-fidelity analog quantum system (e.g., a QCS that performs a quantum simulation such as high-fidelity simulation 240 of QSP 200) requires precise knowledge of the many-body spin Hamiltonian ^^ௌ224. As notedabove, the spin Hamiltonian 224 depends on the bare quantities 222, e.g., the bare qubit frequencies ^^^^^^ and the bare coupler frequencies ^^^^ೕ^, where i and j serve as indices for the individual qubits and qubit couplers in the quantum processor device. However, experimental calibration may only be capable of resolving “dressed“ frequencies which - unlike the bare frequencies - change from local (isolated) calibration measurements to full scale experiments due to hybridization with neighboring qubits and couplers. Given this difficulty, past experiments either suffered from large errors or resorted to multi-parameter learning protocols that are difficult to scale up.

[0041] The two-qubit spectroscopy 210 step and the device model 220 step shown in FIG. 2 schematically demonstrate the scalable calibration protocol of the embodiments. The calibration protocol of the embodiments achieves low error by explicitly calibrating the bare frequencies (e.g., the bare quantities 222). As shown in FIG.2, the calibration protocol begins with the two-qubit calibration 210 step which provides spectroscopy measurements for pairs of qubits. Via the spectroscopy measurements, the dressed quantities 216 are extracted. In the next step of the calibration protocol (i.e., the device model 220 step) an extensive modeling of the underlying quantum device physics is employed to transform the dressed quantities 216 to the bare quantities 222 (e.g., bare frequencies ^^^^^^ and ^^^^ೕ^). Then, a projection technique is applied to project ahigh-dimension device Hamiltonian (e.g., ^^ௗ ^^^^^^^, ^^^^ೕ^^ into the spin Hamiltonian 222 (e.g.,^^ ^^^^^^^, ^^^^ೕ^^^, given by eq (1):couplings, respectively. The latter is significantly smaller than the qubit anharmonicity η ≫ g. Thisrestricts the photon occupation number of each qubit to ^^^ ൌ 0,1. ^^^ ,^^^ are Pauli operators actingin this subspace. Utilizing local measurements and accurate modeling of the underlying device physics (e.g., as modeling in the device modeling 220 step) makes this approach more scalable than fitting a large number of parameters. The Hamiltonian in eq (1) is in the universality class of an XY model with on-site z-fields. A natural consequence of the hybridization in the system is that the spin Hamiltonian 224 contains not only nearest-neighbor hopping, but also density-densityinteractions and next-nearest neighbor terms, which scale as ^^ଶ / ^^ and are typically 5-10 times smaller than g. The Analog Calibration Protocol

[0043] This section describes additional details of the scalable analog calibration framework (or protocol) of the embodiments. The calibration protocol enables a cycle error per qubit of 0.1% or less. In order to achieve a scalable protocol with such reduced error rates, the embodiments perform pairwise calibration measurements (e.g., the two-qubit spectroscopy 210 step of FIG.2) - specifically the single-photon spectroscopy measurements 212 and the swap spectroscopy measurements 214. These spectroscopy measurements enable accurately setting theeffective dressed coupling (e.g., ^^^) and dressed qubit frequencies (e.g., ^^^^ഢ ) in each qubit pair.That is, the two-qubit spectroscopy 210 step is applied to each pair ofFor each pair of qubits, the dressed quantities 216 are generated based on the spectroscopy measurements (e.g., the swap spectroscopy measurements 212 and the single-photon spectroscopy measurements 214). A challenge in analog calibration (e.g., the type of calibration performed by the embodiments) that contrasts its digital counterpart is that these dressed quantities 216 in the pairwise scenario change drastically when all couplers are turned on in the fully-coupled global case. Therefore, the embodiments include the device model 220 step. The service model 220 step includes modeling of the device physics to accurately convert the dressed quantities 216 to the bare quantities 222 (e.g., to the bare qubit frequencies (e.g., ^^^^^^) and the coupler frequencies (e.g., ^^^^ೕ^), which may not significantly change from the local calibration measurements to the full-scale experiments. Hamiltonian of the Device Model

[0044] The device model 220 step models both the qubits and couplers in a tunable coupler architecture as Kerr oscillators, with 4 or 5 levels in each transmon, depending on the number of photons involved in the Hamiltonian term of interest. In order to ensure high accuracy, the embodiments account for not only coupling terms between neighboring qubits and couplers, but also diagonal pathways, including between couplers. FIG.3 shows a device Hamiltonian 300 employed in the device model, according to various embodiments. The device Hamiltonian 300 (^^ௗ) includes five terms labeled as equations (2)-(6). Eq (2) shows a single qubit term of the device Hamiltonian 300. Eq (3) shows a single coupler term of the device Hamiltonian 300. Eq (4) shows a qubit-qubit coupling term of the device Hamiltonian 300. Eq (5) shows a qubit-coupler coupling term of the device Hamiltonian 300. Eq (6) shows a coupler-coupler coupling term of the device Hamiltonian 300.

[0045] In the three coupling terms (e.g., equations (4)-(6)) of the device Hamiltonian 300,the operators ^^^ ൌ ^^^ற ^ ^^ ^ / 2 and ^^^ are the effective coupling efficiencies between transmons,including both direct (i.e., ^^^) and in-direct (i.e., ^^^) capacitive contributions. Note that the indirectcontribution should not be confused with contributions due to virtual exchange interactions, which are included indirectly when the couplers are projected out when the device Hamiltonian 300 is projected onto a spin Hamiltonian (e.g., the spin Hamiltonian of eq (1)). Also note that the indirect capacitive couplings (i.e., ^^^) are “dressed” capacitive couplings.

[0046] FIG.4A shows coupling pathways modeled in the device Hamiltonian 300 of FIG. 3, according to various embodiments. More specifically, FIG.4 shows a “patch” of four qubits and four qubit couplers arranged in a 2D array of qubits and qubit couplers. A compass 402 is provided for orientation around the patch. The four qubits of the parch are labeled as follows: ^^^(in the NW corner of the patch), ^^ଶ(in the NE corner of the patch), ^^ଷ(in the SE corner of the patch), and ^^ସ(in the SW corner of the patch). The four qubit couplers of the patch are labeled as: ^^^ଶ(coupling ^^^and ^^ଶ), ^^ଶଷ(coupling ^^ଶand ^^ଷ), ^^ଷସ(coupling ^^ଷand ^^ସ), and ^^ସ^(coupling ^^ସand ^^^). Four (“dressed”) coupling pathway types 404: qubit-qubit coupling pathways (labeled as q-q), qubit-coupler coupling pathways (labeled as q-c), and coupler-coupler coupling pathways (labeled as c-c).

[0047] Note that in the device model employed in the device model 220 step of FIG.2), the couplings are capacitive couplings. In addition to capacitive coupling between neighboring qubits and neighboring qubit couplers, there are also diagonal next-nearest-neighbor couplings. Asymmetry in the underlying structure of the qubits causes a difference in the couplings along the NW-SE and NE-SW diagonals. That is, each qubit of the four qubits is capacitively coupled to each of the other three qubits (via a q-q coupling), each of the four qubit couplers is capacitively coupled to each of the other three qubit couplers (via a c-c coupling), and each of the four qubits is capacitively coupled to each of the four qubit couplers (via a q-c coupling).

[0048] In the device Hamiltonian 300, the couplings are expressed in terms of (“dressed”)coupling efficiencies ^^^ . More specifically, there are separate flavors (or types) of couplingefficiencies for the three types of coupling pathways. In the device Hamiltonian 300, the qubit- qubit coupling coupling efficiencies are indicated as ^^^^భ,^మ, where ^^^and ^^ଶare indices indicatingthe two coupled qubits. The qubit-coupler coupling efficiencies are indicated as ^^^^భ,^భ, where ^^^and ^^ଶare indices indicating the qubit coupled to the qubit coupler. The coupler-coupler coupling coupling efficiencies are indicated as ^^^^భ,^మ, where ^^^and ^^ଶare indices indicating the two coupled qubit couplers.

[0049] FIG.4B indicates the multiple types of coupling efficiencies that are employed for each of the coupling types in the device model, according to various embodiments. More specifically, FIG.4B shows the relationships between the “dressed” coupling efficiencies (indicated as ^^^) and “bare” coupling efficiencies (indicated as k). There are three separate types (or flavors) of qubit-qubit coupling efficiencies 410. As shown in FIG.4B, the three types of qubit-qubit coupling efficiencies 410 are distinguished by the relative positioning of the qubits. Notably, the geometry of the qubits break the symmetry between the northwest-southeast and northeast-southwest directions. There are three types of qubit-coupler coupling efficiencies 410, and two types of coupler-couple coupling efficiencies 430. Calibration Experiments

[0050] FIG.5A shows an overview of the calibration steps 500, according to various embodiments. In order to calibrate the bare qubit and coupler frequencies for a given set of applied biases, some embodiments employ three types of calibration experiments: Ramsey spectroscopy, two-qubit swap spectroscopy, and two-qubit single-photon spectroscopy. The first column of the table in FIG.5A indicates the calibration spectroscopy measurements. The second row indicates the frequency configuration for the corresponding calibration measurements. The top row of the table of FIG.5A indicates the idle configuration with coupling (e.g., the couplers turned “off” suchthat the dressed coupling rate ^^^ ൌ 0). The bottom two rows indicate the interaction configurations.More specifically, the middle row indicates the configurations of the Ramsey configuration measurements and the bottom row indicates the configurations for both the two-qubit swap spectroscopy measurements and the single-photon spectroscopy measurements. The spectroscopy measurements allow for determining the bare frequencies of the qubits and couplers in the idle configuration (top row), as well as in the interaction configuration (bottom two rows). For each step, the embodiments model a subsystem (third column) to convert the measured dressed frequencies (fourth column) to bare frequencies (fifth column). For the Ramsey spectroscopy measurements, Ramsey spectroscopy measurements are performed for a range of applied qubit bias values, while keeping the couplers turned off and the neighboring qubits detuned, in order to prevent swapping.

[0051] FIG.5B shows details for performing the swap spectroscopy measurements of the various embodiments. More particularly, FIG.5B shows a swap spectroscopy circuit 510 for two qubits. The swap spectroscopy measurements (e.g., implemented via swap spectroscopy circuit 510) are performed on a pairwise level, where neighboring couplers (except the one connecting the pair) are turned off. As shown in the swap spectroscopy circuit 510, the two qubits are prepared in the |10^-product state and the swap rate is measured as a function of detuning between the two qubits. FIG. 5B shows the measured population difference 512 (e.g., ^ ^^^ െ ^^ଶ ^) as a functionof qubit detuning and time. as well as the extracted swap rate 514. The extracted swap rate 514 is from the Fourier Transform (FT) vs. the qubit detuning. As shown in FIG.5B, the detuning value where the extracted swap rate 514 is minimized allows for determining the dressed coupling rate (e.g., ^^^ ) and the difference of the dressed qubit frequencies (e.g., ^^^^భ െ ^^^^మ^.

[0052] That is, the minimum swap rate is employed to determine the effective coupling between the two qubits (e.g., the dressed coupling rate ^^^), and the detuning value at which the swap rate is minimized is equivalent to the difference between the dressed frequencies of the qubits (e.g., ^^^^భ െ ^^^^మ^. Some embodiments use an iterative scheme to calibrate the coupler biasthe target effective coupling.

[0053] FIG.5C shows details for performing the single-photon spectroscopy measurements of the various embodiments. More particularly, FIG.5C shows a single-photon spectroscopy circuit 510 for two qubits. FIG.5C also shows the FT of single-photon spectroscopy measurements. As shown in the single-photon spectroscopy measurements, the observable being measured in the single-photon spectroscopy measurements is ^ X ^ ^^^ ^ Y ^. The average ofthe peak positions in the FT of ^ X ^ ^^^ ^ Y ^ is equal to the average of the sum of the dressedqubit frequencies (e.g., ^^^^^భ ^ ^^^^మ ^ / 2).

[0054] As discussed in with FIG.5B, the swap spectroscopy measurements enable theof the dressed frequencies (for the pair of qubits). The single-photon spectroscopy measurements enable the determination of the sum of the dressed frequencies (for the pair of qubits). Values for the individual dressed frequency for each to the two qubits (e.g., ^^^^భand ^^^^మ) may readily be determined via the sum and difference of the pair of dressed frequencies. As shown in the single-photon spectroscopy circuit 520 of FIG.5C, the qubit pair is prepared in the product state ^|0 ^ ^|1 ^^ ⋅ ^|0 ^^ / √2. The quantity ^ ^^ ^ ^^^ ^ ^^ ^ ismeasured as a function of evolution time. The Fourier Transform (FT) of the signal then reveals theeigenfrequencies of the two-qubit system, and the average of which is determined (e.g., see FT of single-photon spectroscopy measurements of FIG.5C).

[0055] In order to determine the bare qubit and coupler frequencies based on the dressed the relationships of FIG.4B are used. Some embodiments employ separately calibrated coupling efficiencies to model the above calibration experiments with the device Hamiltonian 300 of FIG.3. Some embodiments model not only the two qubits and the coupler involved in pairwise experiments (single qubit involved in the Ramsey spectroscopy measurements), but also the neighboring “padding” qubits and couplers in order to account for their effects. Therefore, these embodiments may start by determining the bare idle frequencies (e.g., ^^^^ௗ^^^, since these may be known to represent the “padding” in the interaction configuration. Having determined all the barefrequencies, the full device Hamiltonian 300 (e.g., ^^ௗ^^^^^^^, ^^^^^^^ can be determined for theglobal system (with all couplers turned onProjection of the Device Hamiltonian onto the Computational Subspace

[0056] Considering that the device Hamiltonian 300 of FIG.3 (e.g., ^^ௗ) involves both qubits and couplers with up to five levels in each (e.g., multiple leakage states in each, see the qubit eigenenergies 224 and the coupler eigenenergies 226 of the device model step 226 of FIG.2), it may be computationally intractable to employ the highly-dimensional device Hamiltonian 300 for computing the time evolution, even at small photon numbers. Moreover, in this form, it is very difficult to map the dynamics of the device Hamiltonian 300 onto physically relevant systems. Therefore, a projection technique may be employed to project the device Hamiltonian 300 into a lower-dimensional spin-Hamiltonian (e.g., ^^ௌ) that acts on the computational subspace of the full Hilbert space of ^^ௗ. To find spin-Hamiltonian terms involving n photons in a system of ^^^qubits, the following spin Hamiltonian may be defined: ^^^^^ ൌ ^|^^ ^ ^ ^^|^^ௗ|^^ ^ ^ ^^|, ^,^where ^|^^ ^^ are ^^^ ൌ ^^^^ A choice of dressed basisstates is motivated byis to simply use the barequbit states, ^|^^ ^^^^^^; however, this would cause the spin-Hamiltonian to have differenteigenenergies from the low-energy spectrum of the device Hamiltonian 300. A second option includes using the ^^^lowest-energy n-photon eigenstates of the device Hamiltonian 300,^|^^ ^^^^^^^. In this second option, the spin-Hamiltonian is guaranteed to have the same lowest n-photon eigenenergies as the device Hamiltonian 300. However, these basis states may be highlydelocalized and may poorly represent the qubits. Hence, some embodiments employ a third option,where the bare qubit states are projected onto the low-energy eigenspace spanned by ^|^^ ^^^^^^^.These projections are not orthonormal, so a singular value decomposition may be performed. In order to arrive at the new dressed basis states, the singular values may be set to 1.

[0057] It can be shown that this third option provides the most localized set of states thatstill preserve the low-energy eigenvalues of the eigenspace spanned by ^|^^ ^^^^^^^. These newbasis states are slightly delocalized on the nearest couplers and qubits, and also may have a weakoverlap with states that have ^^ ^ 2 and ^^ െ 2 photons due to non-rotating wave approximation(RWA) terms. It is noted that typical coupler ramp times of ^ 5 ns are sufficient to ensureadiabatic conversion between the bare qubit states (in which state preparation and measurement is performed) and the dressed basis states that are relevant under analog evolution.

[0058] The spin-Hamiltonian (^^^^^) found from this third option, at least in principle, includes all terms involving ≤ nincluding very long-range interactions; however, theterms drop off rapidly with the photon-photon separation d (typically as ^^^ / ^^^ௗ ∼ 0.1ௗ).Moreover, the terms decay with the number of involved photons in a similar way. Hence, in order to achieve the low error, it is sufficient to include only terms involving up to 2 photons, and where all the involved qubits are a maximum Manhattan distance of 2 sites apart.

[0059] FIG.6A shows a spin-Hamiltonian 600, according to various embodiments. To determine the spin-Hamiltonian 660, the device Hamiltonian 300 of FIG.3 has been projected down onto the lower dimensional computational Hilbertspace, via the third option as discussed above. The spin-Hamiltonian 600 (H) includes five terms. The first three terms are labeled as eq (7), the fourth term is labeled as eq (8), and the fifth term is labeled as eq (9).

[0060] Eq (2) shows a single qubit term of the device Hamiltonian 300. Eq (3) shows a single coupler term of the device Hamiltonian 300. Eq (4) shows a qubit-qubit coupling term of the device Hamiltonian 300. Eq (5) shows a qubit-coupler coupling term of the device Hamiltonian 300. Eq (6) shows a coupler-coupler coupling term of the device Hamiltonian 300. Note that ^^^^^^, (e.g., see eq (7)) ^^^^^ ^^^, (e.g., see eq (8)) and ^^^^^^ூ^(e.g., see eq (8)) scale as ^^ଶ / ^^, while ^^^^^^^^(e.g., see eq (9)) scales as ^^ଷ / ^^ଶ, where ^^ is the anharmonicity. The qubits^^, ^^, ^^ are connected (e.g., see FIG. 6B).

[0061] FIG.6B shows the coefficients of the higher-order terms in the spin-Hamiltonian 600 of FIG.6A, according to various embodiments. More specifically, FIG.6B shows four plots, where each of the four plots is a plot of the average coupling coefficient vs. nearest neighboringhopping (g) for a higher-order term in the spin-Hamiltonian 600 of FIG.6B. Each plot also showsthe three qubit (^^, ^^, ^^) connectivity for the term. (coefficient value vs. g). First plot 610 shows aplot for the average coupling coefficient the ^^^^^^ା^term of the spin-Hamiltonian 600. Second plot620 shows a plot for the average coupling coefficient the ^^^^^^^ା^^^^ାଶ ^ ^^^^^^ା^^^^ାଶ^ / 2 term of thespin-Hamiltonian 600. Third plot 630 shows a plot for the average coupling coefficient the^^^^^^^ାଶ ^ ^^^^^^ାଶ^ / 2 term of the spin-Hamiltonian 600. Fourth plot 640 shows a plot for theaverage coupling coefficient the ^^^^ା^^^^ାଶ ^ ^^^ା^^^^ାଶ^ / 2 term of the spin-Hamiltonian 600. Theplots show that the qubits (^^, ^^, ^^) are placed along a connected line. The plots also indicate that thecoefficients of the first three terms (e.g., first plot 610, second plot 620, and third plot 630) scale as scale as ^^ଶ / ^^, where ^^ is the anharmonicity The coefficient of the fourth plot 640 scales as ^^ଷ / ^^ଶ.At ^^ ൌ 2^^ ൈ 10 MHz, the higher-order terms are ^ 1 ൈ 2^^ MHz. In the three latter terms (e.g.,the second plot 620, the third plot 630, and the fourth plot 640), there is an symmetry between the three possible qubit configurations displayed in the corresponding plots. Note that ^^^^ା^^^^ାଶ^^^^ା^^^^ାଶ^ / 2 does not differ on average from ^^^^^^^ାଶ ^ ^^^^^^ାଶ^ / 2.

[0062] The embodiments may find the ^^^lowest-energy n-photon eigenstates of ^^ௗ(e.g., device Hamiltonian 300 of FIG.3), which has a high computational cost for large ^^^. Fortunately, for a given Hamiltonian term involving a certain set of qubits, the effect of other transmons decays quickly with distance and including only the nearest neighboring qubits and couplers may be needed to achieve accuracies on the tens of kHz scale. To find the spin-Hamiltonian terms, some embodiments scan through various subsystems and perform the procedure outlined above for each of them. Methods

[0063] FIG.7 shows a flowchart for a method for a method 700 for operating a quantum computing system, according to various embodiments. The quantum computing system (QCS) may be similar to QCS of FIG.100. Thus, the QCS may include a set of qubits (e.g., the plurality of qubits 120 of FIG.1). Method 700 begins at block 702, where a set of dressed qubit frequencies is determined. Each dressed qubit frequency of the set of dressed qubit frequencies corresponds to a separate qubit pair of the set of qubits. At block 704, a set of bare qubit frequencies is generated based on the set of dressed qubit frequencies. At block 706, a spin- Hamiltonian for the set of qubits is approximated based on the set of bare qubit frequencies.

[0064] In some embodiments, determining the set of dressed qubit frequencies is based on two-qubit calibration measurements.

[0065] The two-qubit calibration measurements may include single-photon swap operations.

[0066] The two-qubit calibration measurements may include swap spectroscopy measurements.

[0067] Generating the set of bare qubit frequencies may be is based on modeling physics of the quantum processor device to convert the set of dressed frequencies to the set of bare frequencies.

[0068] Approximating the spin-Hamiltonian may be based on employing a projection technique on a high-dimension Hamiltonian for the quantum processor device.

[0069] Each qubit of the set of qubits may be a superconducting qubit

[0070] Each qubit of the set of qubits may be a transmon qubit.

[0071] The method may further include employing the spin-Hamiltonian to perform a simulation of an evolution of a quantum system that corresponds to the spin-Hamiltonian.

[0072] The evolution of the quantum system may includes thermalization dynamics in a quantum magnet.

[0073] Other embodiments are directed to a quantum computing system (QCS). The QCS includes a (quantum) processor that includes a set of qubits. The QCS also includes one or more memory devices. The one or more memory devices store computer-readable instructions that when executed by the one or more quantum processors cause the one or more processors to perform operations for operating the QCS. The operations include determining a set of dressed qubit frequencies. Each dressed qubit frequency of the set of dressed qubit frequencies corresponds to a separate qubit pair of the set of qubits. A set of bare qubit frequencies is generated based on the set of dressed qubit frequencies. A spin-Hamiltonian for the set of qubits is approximated based on the set of bare qubit frequencies. Additional Embodiments

[0074] 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 / computingsystems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0075] 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 / 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.

[0076] 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.

[0077] 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, inaddition 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.

[0078] 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..

[0079] 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.

[0080] 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.

[0081] 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.

[0082] 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.

[0083] 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.

[0084] 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; 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.

[0085] 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.

[0086] 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.

[0087] 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.

[0088] 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.

Claims

WHAT IS CLAIMED IS:

1. A method for operating a quantum computing system (QCS) that comprises a quantum processor device that includes a set of qubits and a set of qubit couplers, the method comprising: determining a set of dressed qubit frequencies, wherein each dressed qubit frequency of the set of dressed qubit frequencies corresponds to a separate qubit pair of the set of qubits; generating a set of bare qubit frequencies based on the set of dressed qubit frequencies; and approximating a spin-Hamiltonian for the set of qubits based on the set of bare qubit frequencies.

2. The method of claim 1, wherein determining the set of dressed qubit frequencies is based on two-qubit calibration measurements.

3. The method of claim 2, wherein the two-qubit calibration measurements include single- photon swap operations. 4 The method of 2, wherein the two-qubit calibration measurements include swap spectroscopy measurements.

5. The method of claim 1, wherein generating the set of bare qubit frequencies is based on modeling physics of the quantum processor device to convert the set of dressed frequencies to the set of bare frequencies.

6. The method of claim 1, wherein approximating the spin-Hamiltonian is based on employing a projection technique on a high-dimension Hamiltonian for the quantum processor device.

7. The method of claim 1, wherein each qubit of the set of qubits is a superconducting qubit.

8. The method of claim 1, wherein each qubit of the set of qubits is a transmon qubit..

9. The method of claim 1, further comprising: employing the spin-Hamiltonian to perform a simulation of an evolution of a quantum system that corresponds to the spin-Hamiltonian.

10. The method of claim 9, wherein the evolution of the quantum system includes thermalization dynamics in a quantum magnet.

11. A quantum computing system (QCS) comprising: a quantum processor that includes a set of qubits; one or more memory devices, the one or more memory devices storing computer- readable instructions that when executed by the one or more quantum processors cause the one or more processors to perform operations for operating the QCS, the operations comprising: determining a set of dressed qubit frequencies, wherein each dressed qubit frequency of the set of dressed qubit frequencies corresponds to a separate qubit pair of the set of qubits; generating a set of bare qubit frequencies based on the set of dressed qubit frequencies; and approximating a spin-Hamiltonian for the set of qubits based on the set of bare qubit frequencies.

12. The QCS of claim 11, wherein determining the set of dressed qubit frequencies is based on two-qubit calibration measurements.

13. The QCS of claim 12, wherein the two-qubit calibration measurements include single- photon swap operations. 14 The QCS of 12, wherein the two-qubit calibration measurements include swap spectroscopy measurements.

15. The QCS of claim 11, wherein generating the set of bare qubit frequencies is based on modeling physics of the quantum processor device to convert the set of dressed frequencies to the set of bare frequencies.

16. The QCS of claim 11, wherein approximating the spin-Hamiltonian is based on employing a projection technique on a high-dimension Hamiltonian for the quantum processor device.

17. The QCS of claim 11, wherein each qubit of the set of qubits is a superconducting qubit.

18. The QCS of claim 11, wherein each qubit of the set of qubits is a transmon qubit..

19. The QCS of claim 11, further comprising: employing the spin-Hamiltonian to perform a simulation of an evolution of a quantum system that corresponds to the spin-Hamiltonian.

20. The QCS of claim 19, wherein the evolution of the quantum system includes thermalization dynamics in a quantum magnet.

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