Control of quantum computing circuit devices

Abstract

Systems, methods, and apparatus for multiplexed control and readout of quantum computing systems that can include qubits, couplers, and other related quantum computing circuit devices. Numerous examples of superconducting quantum computing systems are described that include some, or all, of these quantum computing circuit devices integrated into a superconducting quantum circuit that can be interfaced by a classical control system. In one example, a superconducting circuit is described. The superconducting circuit includes: a superconducting device including a superconducting loop interrupted by one or more Josephson junctions; a first microwave resonator inductively coupled to the superconducting loop of the superconducting device; a second microwave resonator capacitively coupled to the superconducting device, where the first and second microwave resonators each have a different fundamental frequency; and a microwave transmission line evanescently coupled to each of the first and second microwave resonators.

Description

Attorney Docket No.56113-0516WO1 CONTROL OF QUANTUM COMPUTING CIRCUIT DEVICES CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 601,140, titled “CONTROL OF QUANTUM COMPUTING CIRCUIT DEVICES”, filed on Nov.20, 2023, which is hereby incorporated by reference in its entirety. TECHNICAL FIELD

[0002] This disclosure relates generally to systems, methods, and apparatus for controlling quantum computing circuit devices. BACKGROUND

[0003] This disclosure relates to quantum computing. Classical computers have memories made up of bits, where each bit can represent either a zero or a one. Quantum computers maintain sequences of quantum bits, called qubits, where each quantum bit can represent a zero, one, or any quantum superposition of zeros and ones. Quantum computers operate by setting qubits in an initial state and controlling the qubits, e.g., according to a sequence of quantum logic gates. Superconducting quantum computers utilize superconducting qubits that can be realized in a superconducting circuit. SUMMARY

[0004] This disclosure describes systems, methods, and apparatus for multiplexed control and readout of quantum computing systems that can include qubits, couplers, and other related quantum computing circuit devices. Numerous examples of superconducting quantum computing systems are described herein that include some, or all, of these devices integrated into a superconducting quantum circuit that can be interfaced with a classical control system.

[0005] In general, the superconducting qubits and couplers described herein include a superconducting loop interrupted by one or more Josephson junctions. Hence, a magnetic flux bias (Φ௫) through the superconducting loop is tunable, leading to a tunable qubit (^^୯) or coupler (^^ୡ) frequency for the qubit or coupler. For example, when a pair of qubits interact with each other via a coupler, the qubit and coupler frequencies can be tuned to implement a two-qubit quantum gate, e.g., an imaginary swap (iSWAP) gate, a controlled-Z (CZ) gate, and others. Tuning the qubit frequencies can bring two different states of the qubits on resonance with each other, e.g., computational states |01^ and |10^. Tuning the coupler frequency can adjust the effective two-qubit coupling strength between the qubits to performAttorney Docket No.56113-0516WO1 resonant Rabi cycles (or flops) between the two different states, e.g., between thecomputational states |01^ → െ^^|10^ and |10^ → െ^^|01^ which implements an iSWAP gate(where ^^ ൌ √െ1 is the imaginary unit).

[0006] The superconducting loop of a superconducting qubit or coupler may also include an inductor (or multiple inductors) connected in series with the Josephson junction(s) to increase the effective Josephson inductance of the Josephson junction(s). These situations can be appropriate, for example, to implement a qubit or coupler with phase states as basis states, e.g., such that the ratio of the Josephson energy (^^^^ to the charging energy (^^^^ is aboutequal to one or less ^^^ / ^^^ ≲ 1. In cases where the superconducting loop includes multipleJosephson junctions, a qubit or coupler may also include a shunt capacitor (or multiple shunt capacitors) connected in parallel with a Josephson junction to increase the effective junction capacitance of the Josephson junction. These situations can be appropriate, for example, to implement a qubit or coupler with charge states as basis states by reducing the charging energy, e.g., such that the ratio of the Josephson energy to the charging energy is significantlygreater than one ^^^ / ^^^ ≫ 1.

[0007] In general, series inductors, shunt capacitors, as well as shunt inductors, can be connected to the superconducting loop to alter the quantum properties of the qubit or coupler, e.g., such that the ratio of the Josephson energy to the charging energy is about equal to one ^^^ / ^^^~1, implement particular basis states, flatten energy levels, increase tunability, reduce sensitivity to noise (e.g., charge or magnetic field fluctuations), among other properties. Examples of these types of superconducting qubits and couplers include, but are not limited to, single Josephson junction flux qubits (e.g., rf-SQUIDs), double Josephson junction (2JJ) flux qubits (e.g., dc-SQUIDs), triple Josephson junction (3JJ) flux qubits, C-shunted 3JJ flux qubits, fluxonium qubits, tunable transmon qubits (e.g., C-shunted dc-SQUIDs), xmon qubits, gatemon (gmon) qubits, or any other qubit or coupler having at least one superconducting loop interrupted by at least one Josephson junction.

[0008] The superconducting quantum computing systems disclosed herein facilitate multiplexed control and readout of one or more superconducting qubits using appropriately designed microwave resonators coupled to the qubit(s). Particularly, each superconducting qubit is respectively coupled to: (i) a first microwave resonator, referred to as a “flux resonator”, for controlling the qubit; and (ii) a second microwave resonator, referred to as a “readout resonator”, for reading out the qubit. The flux resonator is inductively coupled to the superconducting loop of the qubit to tune the qubit’s frequency via AC magnetic flux biasing.Attorney Docket No.56113-0516WO1 The readout resonator is capacitively coupled to the qubit to perform dispersive readout via AC electric charge biasing. In general, the flux resonator is sufficiently positively detuned from the qubit such that the fast oscillations in the biasing magnetic flux average out to an effective AC Stark shift, e.g., represented as a rotating-wave approximation (RWA) in the Hamiltonian picture. This effect exploits the inherent nonlinearity exhibited by the Josephson junction(s) of the superconducting loop, which may be understood as a homodyning of the flux resonator on the qubit.

[0009] The flux and readout resonators are generally detuned from one another, corresponding to different fundamental frequencies, while being coupled to a common microwave transmission line, also referred to as a “feedline”. Hence, the flux and readout resonators can be driven independently on the same feedline, enabling multiple qubits to be multiplexed for simultaneous control and readout. The flux and readout resonators can also be driven at each qubit’s qubit frequency to implement one-qubit quantum gates, e.g., Pauli gates (e.g., X, Y, and Z gates), phase shift gates, rotation gates, and Hadamard gates.

[0010] The circuit topologies can be extended to include one or more superconducting couplers each mediating interactions between two or more superconducting qubits to perform two-qubit (or higher) quantum gates, e.g., controlled gates (e.g., CX, CY, CZ, and CNOT gates), swap gates, and Toffoli gates. For example, each coupler can be capacitively coupled to two or more neighboring qubits (e.g., nearest-neighbor (NN) coupling) and the two or more neighboring qubits may be capacitively coupled to one another (e.g., next-nearest- neighbor (NNN) coupling). The superconducting loop of each coupler can be inductively coupled to a respective flux resonator for controlling the coupler via AC magnetic flux biasing, e.g., tuning the coupler’s coupler frequency and thus the effective coupling strength(s) between the two or more neighboring qubits.

[0011] With the fundamental frequencies of each flux and readout resonator detuned from one another, the qubits and couplers can be simultaneously controlled via multiplexed control signals on the common feedline, e.g., to implement a quantum circuit including a sequence of quantum gates for performing a quantum processing algorithm. Using microwave resonators of sufficiently large quality factor, e.g., microwave transmission line resonators of various lengths and boundary conditions, the superconducting quantum computing systems allow multiplexing of a large number of superconducting qubits and couplers, e.g., five, ten, fifteen, twenty, twenty-five, fifty, one hundred, five hundred, one thousand, ten thousand or more qubits and couplers. In general, the number of qubits and couplers that can be multiplexedAttorney Docket No.56113-0516WO1 may be limited by the overall bandwidth of the system and / or the quality factors of the chosen resonators.

[0012] Although superconducting qubits and couplers are described in detail, the systems, methods, and apparatus disclosed herein can be applied to any superconducting device (or other quantum computing circuit device) that utilizes a superconducting loop with a tunable magnetic flux bias. Particularly, a flux resonator can be inductively coupled to the superconducting loop of the superconducting device to control (or otherwise manipulate) the superconducting device via AC magnetic flux biasing. Additionally, a readout resonator can be capacitively coupled to the superconducting device to readout (or otherwise manipulate) the superconducting device, e.g., via AC electric charge biasing. Such superconducting devices can include, but are not limited to, rf-SQUIDs, dc-SQUIDs, microwave amplifiers, microwave filters, fault current limiters, kinetic inductance detectors (KIDs), among others.

[0013] Particular embodiments of the subject matter described in this specification can be implemented so as to realize one or more of the following advantages.

[0014] The disclosed superconducting quantum computing systems provide a means of controlling multiple superconducting qubits, couplers, and other quantum computing circuit devices entirely with alternating current (AC) signals that utilizes the full bandwidth of the quantum computing system. Multiplexing can be difficult or infeasible using direct current (DC) signals – the simultaneous control of separate objects along a single information channel typically involves the time and / or frequency domain. However, achieving AC multiplexing has been a stubborn challenge, particularly in the superconducting quantum computing systems, e.g., due to the need for circuit device tunability, which has generally relied on DC control and biasing.

[0015] This specification provides systems, methods, and apparatus for time and / or frequency-division multiplexing for simultaneous control and readout that achieves device tunability over wide operational bandwidths in the microwave frequency range, e.g., bandwidths of about 0.1 gigahertz (GHz) to 20 GHz. For example, each superconducting qubit and coupler in the superconducting quantum computing systems disclosed herein can be independently tuned (on the same channel) with AC signals having different frequencies spaced apart by about 10 megahertz (MHz) to 100 MHz. Over the microwave frequency range, this can allow about two hundred to two thousand (or more) qubits and couplers to be multiplexed with one another.

[0016] AC multiplexing has several other advantages over DC besides higher information density, tunability, and controllability. For example, DC circuits generally involve significantAttorney Docket No.56113-0516WO1 crosstalk amongst devices due to shared grounds in the circuits. This can cause unwanted current to flow to and / or from neighboring devices which can reduce fidelity, increase error, reduce coherence times (e.g., relaxation and dephasing times), among other undesirable effects.

[0017] Moreover, AC multiplexing can significantly reduce the density of wiring involved in superconducting (and other) circuits, as well as eliminate the need for Galvanic contacts altogether, allowing the superconducting quantum circuits to be scaled up more practically. Particularly, the superconducting quantum circuit devices described herein can be coupled to one another inductively and / or capacitively without the need for direct physical contacts.

[0018] Another advantage of AC multiplexing is the avoidance of 1⁄ ^^ noise (aka pink orfractional noise) which can be significant at low frequencies. Hence, the superconducting quantum computing systems, circuits, and devices described herein can operate with reduced error and at higher performance compared with conventional superconducting quantum computing systems.

[0019] The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. INCORPORATION BY REFERENCE

[0020] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and / or take precedence over any such contradictory material. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] FIG.1A is a top view depicting an example of a superconducting subcircuit including a superconducting qubit evanescently coupled to flux and readout resonators.

[0022] FIG.1B shows a lumped-element circuit model of the superconducting subcircuit depicted in FIG.1A.

[0023] FIG.1C is a top view depicting an example of a superconducting circuit including multiple ones of the superconducting subcircuit depicted in FIG.1A.Attorney Docket No.56113-0516WO1

[0024] FIG.2A is a top view depicting another example of a superconducting subcircuit including a superconducting qubit evanescently coupled to flux and readout resonators.

[0025] FIG.2B shows a lumped-element circuit model of the superconducting subcircuit depicted in FIG.2A.

[0026] FIG.2C is a top view depicting another example of a superconducting circuit including multiple ones of the superconducting subcircuit depicted in FIG.2A.

[0027] FIG.3A is a plot of qubit frequency versus magnetic flux bias for a superconducting qubit.

[0028] FIG.3B is a plot of magnetic flux bias versus time for a superconducting qubit coupled with a flux resonator.

[0029] FIG.3C is a plot of experimental data showing coupling efficiency of a superconducting qubit versus flux resonator drive power and qubit drive frequency.

[0030] FIG.4A is a schematic diagram depicting an example of a superconducting circuit including multiple superconducting subcircuits.

[0031] FIG.4B is a plot of coupling efficiency into different flux resonators versus frequency of an input control signal.

[0032] FIG.5 is a schematic diagram depicting another example of a superconducting circuit including multiple superconducting subcircuits having superconducting qubits and couplers coupled to one another.

[0033] FIG.6A is a schematic diagram depicting an example of a superconducting quantum computer including a superconducting circuit and a control system configured for multiplexed control can readout of the superconducting circuit.

[0034] FIG.6B is a schematic diagram depicting an example of a quantum circuit that can be implemented by the superconducting quantum computer depicted in FIG.6A.

[0035] FIG.7A is a flow chart of an example process for multiplexed control and readout of a superconducting circuit including multiple superconducting qubits and couplers.

[0036] FIG.7B is a flow chart of an example process for multiplexed control of multiple superconducting qubits and couplers in a superconducting circuit.

[0037] FIG.7C is a flow chart of an example process for multiplexed readout of multiple superconducting qubits in a superconducting circuit.

[0038] Like reference numbers and designations in the various drawings indicate like elements.Attorney Docket No.56113-0516WO1 DETAILED DESCRIPTION

[0039] Quantum algorithms can be represented by quantum circuits that include a set of universal instructions, e.g., universal quantum gates. An example of a universal gate set includes the Hadamard gate, S gate, T gate, and the two-qubit entangling controlled-not (CNOT) gate. Another example universal gate set includes a single three-qubit Fredkin gate and the Hadamard gate. Yet another example universal gate set includes representations of the Fermionic simulation (fSim) gate set which include imaginary swap (iSWAP) and controlled-Z (CZ) gates as notable examples. The fSim gate set contains excitation- conserving two-qubit quantum gates which can be efficiently implemented by superconducting qubits as they are photon-conserving.

[0040] For the near-term application of quantum computers, one major focus is to minimize the experimental resources necessary for building a useful quantum circuit from elementary quantum gates. One of the main objectives of these quantum circuits is implementing quantum algorithms with error correction, e.g., using iSWAP and / or CZ gates that have the ability to realize the error correcting surface code. This specification provides systems, methods, and apparatus for multiplexing quantum computing systems, particularly superconducting quantum computing systems, that can significantly reduce the experimental resources and complexity involved in controlling qubits, couplers, and other related quantum computing circuit devices, e.g., amplifiers, filters, fault current limiters, kinetic inductance detectors (KIDs), among others.

[0041] As a particular example, the systems, methods, and apparatus disclosed herein can simultaneously control, address, and / or read out multiple superconducting qubits (and couplers) using a single transmission line that is frequency-division multiplexed to each superconducting device. This can increase the density of qubits (and couplers) on a single quantum chip, quantum processor, or other quantum hardware while eliminating large arrays of peripheral circuitry, e.g., separate input and / or output transmission lines for each device. Using purely AC control signals, the systems, methods, and apparatus can significantly increase the information density and tunability available to quantum computing systems while also reducing common sources of error such as pink noise and crosstalk from shared grounds.

[0042] This specification is organized as follows: Section I introduces the subject matter described in this specification; Section II provides examples of superconducting circuits; Section III provides a quantum mechanical treatment of a superconducting subcircuit; SectionAttorney Docket No.56113-0516WO1 VI provides additional examples of superconducting circuits; and Section V provides examples of superconducting quantum computer systems.

[0043] These features and other features are described in more detail below. I. Introduction

[0044] In general, this specification describes systems, methods, and apparatus for multiplexed control and readout of superconducting circuits 10 including one or more superconducting subcircuits 100, where the superconducting subcircuit(s) 100 provide a discrete, modular component of the larger superconducting circuit 10. Examples of simulated superconducting circuits 10A, 100A, 10B, and 100B and their corresponding lumped-element circuit models are shown in FIGs.1A-2C. To provide improved contrast throughout the drawings, electrically conducting material is depicted in white while electrically insulating material is depicted in gray. In general, the superconducting circuits 10 and 100 described herein can be manufactured using standard fabrication techniques for integrated circuits and semiconductor devices, such as lithography techniques (e.g., optical and e-beam lithography), etching techniques, and evaporation techniques (e.g., shadow evaporation). For example, a superconducting circuit 10 or 100 can include a patterned superconducting layer disposed on a substrate, where the superconducting layer corresponds to the conducting material and the substrate corresponds to the insulating material. The superconducting layer can be deposited on the substrate via sputtering or chemical vapor deposition (CVD) and subsequently patterned using the abovementioned fabrication techniques to form the superconducting circuit 10 or 100.

[0045] Examples of conducting material that can be used for the superconducting layer include, but are not limited to, niobium tin (Nb3Sn), niobium-titanium (Nb-Ti), niobium- titanium-nitride (NbTiN), yttrium barium copper oxide (YBCO), niobium (Nb), aluminum (Al), tin (Sn), lead (Pb), mercury (Hg), rhenium (Re), lanthanum (La), or any appropriate conducting material that exhibits a superconducting phase (e.g., type I or type II phase) below a particular critical temperature (^^^). In most cases, the critical temperature is within about 20 kelvin (K) or less of absolute zero, thus the superconducting circuits 10 and 100 described herein generally operate at cryogenic temperatures in a vacuum (e.g., within a cryostat), unless a room temperature or near-room temperature superconductor were to be discovered. Examples of insulating material (e.g., dielectric material) that can be used for the substrate include, but are not limited to, silicon (Si), silica (SiO2), silicon nitride (SiN), titanium nitrideAttorney Docket No.56113-0516WO1 (TiN), tantalum oxide (Ta2O5), strontium titanate (SrTiO3), aluminum oxide (Al2O3) or any appropriate insulating material that is suitable for cryogenic applications.

[0046] In general, a superconducting subcircuit 100 can include: (i) a superconducting device 110 including a superconducting loop 112 interrupted by one or more Josephson junctions 113; (ii) a first microwave resonator 120 inductively coupled to the superconducting loop 112; and (iii) a second microwave resonator 130 capacitively coupled to the superconductive device 110. The first 120 and second 130 microwave resonators can be evanescently coupled (e.g., inductively coupled, capacitively coupled, or both) to a microwave transmission line 150. For brevity, the first microwave resonator 120 is referred to as a “flux resonator”, the second microwave resonator 130 is referred to as a “readout resonator”, and the microwave transmission line 150 is referred to as a “feedline”. The superconducting device 110 can be a superconducting qubit, a superconducting coupler (in which case the readout resonator 130 may be omitted), or other quantum computing circuit device (e.g., an rf- or dc-SQUID).

[0047] For reference, a superconducting loop 112 generally refers to any superconducting wire or pathway in a superconducting circuit 100 or 10 that enables persistent flow of a supercurrent, such that the phase of the superconducting condensate satisfies a periodic boundary condition. In other words, the total phase shift of the condensate around the superconducting loop 112 is an integer multiple of 2^^. Thus, a superconducting loop 112 can generally be realized as any closed path in a superconducting circuit 100 or 10, such as circles, ellipses, squares, rectangles, trapezoids, triangles, polygons, and other paths that enclose an area. As is described in more detail below, the net magnetic flux Φ௫threading such a superconducting loop 112 can be tuned to control various circuit SinceJosephson junctions 113 and inductors permit persistent electric current a superconducting loop 112 can include any number of these elements connected in series as they do not break the periodic boundary condition. Moreover, any number of shunt capacitors and / or shunt inductors can be connected in parallel with these elements without breaking the periodic boundary condition.

[0048] For two circuit devices to be inductively and / or capacitively coupled to each other, the two circuit devices should be within about one decay length of each other’s evanescent electromagnetic fields, referred to more broadly as “evanescently coupled”. For inductive coupling, these are the magnetic fields which are generated in regions of a circuit device with relatively high electric current flow, e.g., loop antennas, superconducting loops, short circuits. Inductive coupling between two circuit devices implies that an electric current through one circuit device induces a magnetic flux in the other circuit device, and vice versa. ForAttorney Docket No.56113-0516WO1 capacitive coupling, these are the electric fields which are generated in regions of a circuit device with relatively high charge accumulation, e.g., patch antennas, capacitor electrodes, open circuits. Capacitive coupling between two circuit devices implies that an electric charge on one circuit device induces a voltage in the other circuit device, and vice versa. Combinations of both inductive and capacitive coupling can also be implemented between two circuit devices depending on the relative configuration and / or design of the circuit devices.

[0049] In general, an evanescent field is an oscillating electric and / or magnetic field that exists in the immediate vicinity of a source, such as an interface between two different media, e.g., a conducting and insulating material, but decays exponentially with distance from the source. Evanescent coupling, e.g., inductive and / or capacitive coupling, occurs when two circuit devices are brought close enough that the evanescent fields of one overlap with the other, allowing energy to transfer between the two without direct physical contact. Distances between two circuit devices typically range from about ^^ / 100 to ^^ / 10 depending on the design and size of the devices. Here, ^^ is the wavelength of the electromagnetic fields and ranges from about 15 millimeters (cm) to 15 centimeters (cm) in a microwave frequency spectrum of about 2 gigahertz (GHz) to 20 GHz. Hence, distances between two evanescently coupled circuit devices typically on the order of microns for strong coupling and millimeters for moderate or weak coupling.

[0050] In the example superconducting subcircuits 100 described herein, the flux 120 and readout 130 resonators are microwave transmission line resonators (aka microwave waveguide resonators). Transmission line resonators are described in detail as they have a number of advantages such as straightforward fabrication in planar superconducting circuits, relatively large quality factors, and fundamental frequencies that can be accurately tuned (e.g., by adjusting the lengths and / or boundary conditions of the transmission line resonators). However, other types of resonators can also be utilized for the flux 120 and readout 130 resonators. For example, different types of coplanar waveguide resonators, microstrip resonators, stripline resonators, ring resonators (e.g., loop, racetrack, square, or rectangular, etc.), two-dimensional (2D) resonators (e.g., disk, square, or rectangular, etc.), and three- dimensional (3D) resonators (e.g., cavities of various shapes such as cubes, cuboids, spheres, ellipsoids, cylinders, etc.) can also be utilized for the flux 120 and readout 130 resonators.

[0051] For clarity, a brief review of transmission line resonators is provided. Device properties such as the total length ^^, the capacitance per unit length ^^^, and the inductanceAttorney Docket No.56113-0516WO1 per unit length ^^^of a transmission line resonator are design and material parameters that influence its operating characteristics. Generally, increasing the total length of a transmission line resonator reduces its fundamental frequency (e.g., by supporting longer wavelengths), while decreasing the total length of a transmission line resonator increases its fundamental frequency (e.g., by supporting shorter wavelengths). For reference, the total capacitance andinductance of a transmission line resonator are ^^^ ൌ ^^^^^ and ^^^ ൌ ^^^^^, respectively.

[0052] In general, a transmission line resonator can be described, at least approximately, by a one-dimensional wave equation of the form: ∂ଶΦ^^^, ^^^ ∂ଶΦ^^ ^(1) ^^^, ^^

[0053] where Φ^^^, ^^^ is thewavespeed within the resonator, ^^ is a spatial coordinate along the length of the resonator, and ^^ isthe time coordinate. The characteristic impedance of the resonator is ^^^ ൌ ^^^^ / ^^^.

[0054] Typical values of the wave speed are ^^ ଼^~1.3 ൈ 10 m / s, about a third of the speed oflight in vacuum. For example, the flux 120 and readout 130 resonators can each have a wavespeed of at least about 0.5 ൈ 10଼ m / s, 0.75 ൈ 10଼ m / s, 1 ൈ 10଼ m / s, 1.25 ൈ 10଼ m / s,1.5 ൈ 10଼ m / s, 1.75 ൈ 10଼ m / s, 2 ൈ 10଼ m / s, or more. The flux 120 and readout 130resonators can each have a wave speed of at most about 2 ൈ 10଼ m / s, 1.75 ൈ 10଼ m / s,1.5 ൈ 10଼ m / s, 1.25 ൈ 10଼ m / s, 1 ൈ 10଼ m / s, 0.75 ൈ 10଼ m / s, 0.5 ൈ 10଼ m / s, or less.Typical values of the characteristic impedance are ^^^~50 Ω. For example, the flux 120 and readout 130 resonators can each have a characteristic impedance of at least about 25 Ω, 30 Ω, 35 Ω, 40 Ω, 45 Ω, 50 Ω, 55 Ω, 60 Ω, 65 Ω, 70 Ω, 75 Ω, or more. The flux 120 and readout 130 resonators can each have a characteristic impedance of at most about 75 Ω, 70 Ω, 65 Ω, 60 Ω, 55 Ω, 50 Ω, 45 Ω, 40 Ω, 35 Ω, 30 Ω, 25 Ω, or less.

[0055] The solution to Eq. (1) can be expressed in terms of a set of harmonic (or normal) modes (^^) that oscillate at respective resonant angular frequencies (^^^). These correspond to the stable standing wave excitations of the resonator: ^ (2) Φ^^^, ^^^ ൌ ^ ^^^^^^^Φ^^^^^, ^ୀ^

[0056] with Φ^ ^ ൌ െ^^^ଶ Φ^ of themode. The wavevector ^^^ ൌby theboundary ି^the local current ^^ ൌ െ^^^ ^^௫Φ or the local voltage ^^ ൌ Φ^ tozero at each end of the0,^^. An open circuit boundary condition implies ^^ ൌ 0Attorney Docket No.56113-0516WO1 while a short circuit boundary condition implies ^^ ൌ 0. Eq. (2) is the sum over independentharmonic modes of the resonator with ^^ ൌ 0 being the fundamental mode having acorresponding fundamental frequency ^^^ / 2^^. The resonator frequencies generally increase as ^^^ ^ such that ^^^ is the lowest angular frequency that can excite the resonator.remainder of the discussion, it is assumed that the microwave resonators operate in their fundamental modes ^^ ൌ 0. When operating in their fundamental modes, theflux 120 and readout 130 resonators are well described by lumped-element circuit models, see FIGs.1B and 2B for example, which provides an intuitive circuit-based description. In general, the flux 120 and readout 130 resonators described herein can be configured to operate over the entire microwave spectrum corresponding to the frequency range from about 100 megahertz (MHz) to 300 gigahertz (GHz). However, this is often limited by practical constraints. Particularly, the useful frequency range is generally bounded from above by the superconducting gap ∆^ 1.764^^^^^^ of the conducting material (e.g., about 82 GHz foraluminum) andfrom below by the thermal occupation ^^^^^ of the condensate at its absolute temperature ^^^^ (e.g., about 100 MHz at 10 millikelvin (mK)). This corresponds to the approximate bandwidth of the system for operation in the quantum regime. Here, ^^^is the Boltzmann constant.

[0058] In some implementations, the flux 120 and readout 130 resonators described herein are configured to operate within a bandwidth from about 4 GHz to 20 GHz (e.g., about 5 GHz to 15 GHz) where microwave electronics is well developed. For the superconducting circuits 10 and 100 describe herein, this bandwidth can support hundreds, to thousands, to tens of thousands of multiplexed resonators 120 and 130 for qubit control and readout due to their quality factors achievable in the superconducting circuits 10 and 100, e.g., about 10ଷ, 10ସ, 10ହ, 10^, 10^, or more.

[0059] Further details relating to microwave resonators, as well as various other aspects of superconducting circuits and circuit quantum electrodynamics (e.g., dispersive readout), is provided in the review paper by Blais, Alexandre, et al. “Circuit Quantum Electrodynamics,” Reviews of Modern Physics 93.2 (2021): 025005. II. Examples of Superconducting Circuits and Subcircuits

[0060] FIGs.1A-1C depict examples of a superconducting subcircuit 100A that can be utilized for multiplexed control and readout of a superconducting qubit 110 either alone, or in a larger superconducting circuit 10A that includes multiple such superconducting subcircuitsAttorney Docket No.56113-0516WO1 100A, see FIG.1C for example. FIG.1A is a top view depicting the superconducting subcircuit 100A. The superconducting subcircuit 100A includes: (i) a superconducting qubit 110; (ii) a flux resonator 120A; (iii) a readout resonator 130; (iv) a ground plane 140 (or “ground”); and (v) a feedline 150. FIG.1B shows a lumped-element circuit model of the superconducting subcircuit 100A depicted in FIG.1A. For ease of description, reference will be made to both FIGs.1A and 1B when describing the structure and operating principles of the superconducting subcircuit 100A. FIG.1C is a top view depicting the superconducting circuit 10A that includes multiple ones of the superconducting subcircuit 100A-1 through 100A-5.

[0061] At a glance, a notable feature of the superconducting circuits 100A and 10A is the common feedline 150 for propagating both control 612 and readout 614 signals to and from the superconducting qubit(s) 110. Particularly, the feedline 150 includes a first port 152 and a second port 154 each positioned at a respective end of the feedline 150. The feedline 150 is configured to propagate microwave signals between its first 152 and second 154 ports. The first 152 and second 154 ports can each be utilized as an input / output (IO) port. For example, the first 152 and / or second 154 ports can receive multiplexed control signals 612 from a control system 200 that interact with the superconducting subcircuit(s) 100. The first 152 and / or second 154 ports can then transmit multiplexed readout signals 614 to the control system 200 in response to the control signals 612, e.g., for executing a sequence of quantum gate operations on the superconducting qubit(s) 110 followed by a measurement operation on the superconducting qubit(s) 110. An example of a superconducting quantum computer system 600 that can perform such quantum gate operations is described in Section V. In general, the readout signals 614 are transmitted and / or reflected signals resulting from the interaction of the control signals 612 with the superconducting subcircuit(s) 100, where the transmitted and / or reflected signals have differing amplitudes and / or phases relative to the control signals 612 that provide information about the state of superconducting qubit(s) 110.

[0062] The superconducting qubit 110 is inductively coupled 21-I to the flux resonator 120A and capacitively coupled 31-C to the readout resonator 130. The superconducting qubit 110 is a tunable transmon qubit with a substantially rectangular geometry. The superconducting qubit 110 includes a rectangular superconducting loop 112 which is identified as a dotted line in FIG.1A and a hatched box in FIG.1B. The superconducting loop 112 is interrupted by a first Josephson junction 113-1 and a second Josephson junction 113-2, forming a dc-SQUID loop. If unperturbed by an external signal, electrons (more specifically Cooper pairs) will flow indefinitely (or almost indefinity) around the dc-SQUID loop 112 due to the zero (orAttorney Docket No.56113-0516WO1 near zero) electrical resistance of the superconducting subcircuit 100A. Note, although tunable transmon qubits are described in detail herein, various different types of superconducting qubits 110 can be utilized in a superconducting subcircuit 100. As an example, the superconducting qubit 110 can be implemented as a fluxonium qubit by replacing one of the Josephson junctions 113-1 or 113-2 with a series inductor. As another example, an additional, third Josephson junction 113 can be introduced into the superconducting loop 112 to implement a C-shunted 3JJ flux qubit. In general, the superconducting qubit 110 can be any qubit archetype that includes at least one superconducting loop interrupted by at least one Josephson junction.

[0063] In most cases, the dc-SQUID loop 112 is symmetric, having (within suitable tolerances) identical device parameters for each Josephson junction 113-1 and 113-2. A few notable advantages of symmetric dc-SQUID loops are outlined in Section III. Nevertheless, asymmetric dc-SQUID loops can also be implemented if desired where each Josephson junction 113-1 and 113-2 has different device parameters, e.g., due to having different barrier widths and / or barrier potentials. The device parameters of the Josephson junctions 113-1 and 113-2 depend on the particular materials and dimensions employed for the superconducting subcircuit 100A. Generally, as long as the electric current through the dc-SQUID loop 112 is below the critical current (^^^) of each Josephson junction 113-1 and 113-2, the superconducting qubit 110 operates in the quantum regime. The Josephson energy ^^^ൌ Φ^^^^ / 2^^ of each Josephson junction 113-1 and 113-2 is proportional to the rate of tunnellingof Cooper pairs across the Josephson junction 113-1 or 113-2. Here, Φ^ ൌ ℎ / ^2^^^ is themagnetic flux quantum, ℎ is the Planck constant, and ^^ is the elementary charge of the electron.

[0064] In some implementations, each Josephson junction 113-1 and 113-2 can have a junction capacitance (^^^) in a range from about 5 femtofarad (fF) to 50 fF. For example, each Josephson junction 113-1 and 113-2 can have a junction capacitance of at least about 5 fF, 10 fF, 15 fF, 20 fF, 25 fF, 30 fF, 35 fF, 40 fF, 45 fF, 50 fF, or more. Each Josephson junction 113-1 and 113-2 can have a junction capacitance of at most about 50 fF, 45 fF, 40 fF, 35 fF, 30 fF, 25 fF, 20 fF, 15 fF, 10 fF 5 fF, or less. In some implementations, each Josephson junction 113-1 and 113-2 can have critical current (^^^) in a range from about 20 nanoamps (nA) to 100 nA. For example, each Josephson junction 113-1 and 113-2 can have a critical current of at least about 20 nA, 30 nA, 40 nA, 50 nA, 60 nA, 70 nA, 80 nA, 90 nA, 100 nA orAttorney Docket No.56113-0516WO1 more. Each Josephson junction 113-1 and 113-2 can have a critical current of at most about 100 nA, 90 nA, 80 nA, 70 nA, 60 nA, 50 nA, 40 nA, 30 nA, 20 nA or less.

[0065] A top and bottom portion of the dc-SQUID loop 112 are separated by the pair of Josephson junctions 113-1 and 113-2. The bottom portion of loop 112 is electrically connected (e.g., short-circuited) to the ground plane 140. The ground plane 140 maintains a relatively large quantity of free electrons at an approximately constant (e.g., zero) potential. The ground plane 140 can be used as the reference potential for the superconducting subcircuit 100A and any other devices (e.g., a control system 200) interfacing with the superconducting subcircuit 100A. The top portion of the dc-SQUID loop 112 is electrically connected to a rectangular capacitor electrode 114 that maintains a relatively large quantity of charge within the superconducting qubit 110.

[0066] The capacitor electrode 114 is separated from the ground plane 140 by a rectangular loop 115 of insulating material which forms a shunt capacitor 14-C for the superconducting qubit 110. The width of the rectangular loop 115 can be adjusted to tune the shunt capacitance (^^ௌ^ of the shunt capacitor 14-C. The shunt capacitor 14-C is electrically connected to the dc-SQUID loop 112, in parallel with the Josephson junctions 113-1 and 113-2, which increases the sum total junction capacitance ^^ஊ ൌ ^^^^ ^ ^^^ଶ ^ ^^ௌ of the Josephsonjunctions 113-1 and 113-2. In somethe shunt capacitance can be in a range from about 50 fF to 200 fF. For example, the shunt capacitor 14-C can have a shunt capacitance at least about 50 fF, 60 fF, 70 fF, 80 fF, 90 fF, 100 fF, 110 fF, 120 fF, 130 fF, 140 fF, 150 fF, 160 fF, 170 fF, 180 fF, 190 fF, 200 fF, or more. The shunt capacitor 14-C can have a shunt capacitance at most about 200 fF, 190 fF, 180 fF, 170 fF, 160 fF, 150 fF, 140 fF, 130 fF, 120 fF, 110 fF, 100 fF, 90 fF, 80 fF, 70 fF, 60 fF, 40 fF, or less.

[0067] A large value for the shunt capacitance can significantly decrease the charging energy^^^ ൌ ^^ଶ / 2^^ஊ of the superconducting qubit 110 relative the sum total Josephson energy ^^^ஊ ൌ^^^^ ^ ^^^ଶ, such that the ratio of the sum total Josephson energy to the charging energy issignificantly greater than one, ^^^ஊ / ^^^ ≫ 1. Thus, the charge degree of freedom ofsuperconducting qubit 110 is highly delocalized, thereby reducing its sensitivity to charge fluctuations and increasing its coherence time(s), e.g., relaxation (^^^) and / or dephasing (^^ଶ) times. In some implementations, the ratio of the sum total Josephson energy to the charging energy (^^^ஊ / ^^^) can be in a range from about 20 to 80. For example, the ratio of the sum total Josephson energy to the charging energy can be at least 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, or more. The ratio of the sum total Josephson energy to the charging energy canAttorney Docket No.56113-0516WO1 be at most about 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, or less. In some implementations, e.g., aluminum-based transmons, the superconducting qubit 110 can have coherence times in a range from about 50 microseconds (µs) to 300 µs. For example, the superconducting qubit 110 can have coherence time(s) of at least about 50 µs, 75 µs, 100 µs, 125 µs, 150 µs, 175 µs, 200 µs, 225 µs, 250 µs, 275 µs, 300 µs, or more.

[0068] Referring now to the flux resonator 120A. The flux resonator 120A is a short- circuited, quarter-wavelength (^^ / 4), microwave transmission line resonator. In this case, thewavevector and phase of the fundamental mode is ^^^^^ ൌ ^^ / 2 and ^^^ ൌ െ^^ / 2. Thiscorresponds to a fundamental frequency of ^^^ / 2^^ ൌ 1 / ൫4^^^^^^^൯ ൌ ^^^ / ^4^^^. As the nameimplies, the fundamental mode of a quarter-wavelength resonator supports a quarter of awavelength (^^^ ൌ 4^^) due to the opposing short circuit and open circuit boundary conditions.Thus, when described as a lumped LC oscillator in a lumped-element circuit model (see FIG.1B), the flux resonator 120A has an effective capacitance of ^^^ ൌ 2^^^ / ^^ and an effectiveinductance of ^^^ ൌ 2^^^ / ^^, such that the flux resonator 120A resonates at an angularfrequency of ^^^ ൌ ^^^ ൌ 1 / ^^^^^^^. In some implementations, the flux resonator 120A’sfundamental frequencybe in a range from about 6 GHz to 12 GHz. For example, the flux resonator 120A’s fundamental frequency can be at least about 6 GHz, 6.5 GHz, 7 GHz, 7.5 GHz, 8 GHz, 8.5 GHz, 9 GHz, 9.5 GHz, 10 GHz, 10.5 GHz, 11 GHz, 11.5 GHz, 12 GHz, or more. The flux resonator 120A’s fundamental frequency can be at most about 12 GHz, 11.5 GHz, 11 GHz, 10.5 GHz, 10 GHz, 9.5 GHz, 9 GHz, 8.5 GHz, 8 GHz, 7.5 GHz, 7 GHz, 7.5 GHz, 6 GHz, or less.

[0069] In more detail, the flux resonator 120A includes a thin strip 122 of conducting material that is separated from ground 140 by adjacent strips of insulating material such that the transport and / or oscillation of electrons is confined to the strip 122. The strip 122 follows a tortuous, serpentine path in a folded region 123 to increase the total length of the flux resonator 120A to a specified length, and thereby a specified fundamental frequency for the flux resonator 120A.

[0070] A short-circuited end 124-SC of the flux resonator 120A is electrically connected to the ground plane 140. This boundary condition fixes the local voltage to zero at the short- circuited end 124-SC. Thus, the amplitude of the local electric current is generally maximized at the short-circuited end 124-SC since the spatial profile of the flux resonator 120A’sfundamental mode is |^^௫^^^| ൌ 1 here. The bottom portion of the dc-SQUID loop 112 isinductively coupled 21-I to the flux resonator 120A near the short-circuited end 124-SC toAttorney Docket No.56113-0516WO1 exploit this property. As shown in FIGs.1A-1B, the flux resonator 120A is inductively coupled to the dc-SQUID loop 112 between the pair of Josephson junctions 113-1 and 113-2. However, in other implementations, the flux resonator 120A may be inductively coupled to the dc-SQUID loop 112 across one of the Josephson junctions 113-1 or 113-2 and / or to the top portion of the dc-SQUID loop 112. An open-circuited end 126-OC of the flux resonator 120A fixes the local electric current to zero at the open-circuited end 124-OC. Thus, the amplitude of the local electric charge is generally maximized at the open-circuited end 126-OC since the spatial profile of the flux resonator 120A’s fundamental mode is |^^^| ൌ 1 here.The feedline 150 is capacitively coupled 25-C to the flux resonator 120A near the open- circuited end 126-OC to exploit this property.

[0071] In general, the flux resonator 120A is configured to bias a magnetic flux (Φ௫) through the dc-SQUID loop 112 to tune the superconducting qubit 110’s qubit frequency, or otherwise control and / or address the superconducting qubit 110. Assuming the geometric self-inductance of the dc-SQUID loop 112 is negligible, the magnetic flux bias generatedthrough the dc-SQUID loop 112 is equal to Φ௫ ൌ ^^^ / ^^^^Φ^, where ^^ is the mutualinductance between the flux resonatorthe dc-SQUID loop 112, and Φ^is the local magnetic flux at the short-circuited end 124-SC. In some implementations, the mutual inductance can be in a range from about 5 picohenry (pH) to 20 pH. For example, the mutual inductance can be at least about 5 pH, 6 pH, 7 pH, 8 pH, 9 pH, 10 pH, 11 pH, 12 pH, 13 pH, 14 pH, 15 pH, 16 pH, 17 pH, 18 pH, 19 pH, 20 pH, or more. The mutual inductance can be at most about 20 pH, 19 pH, 18 pH, 17 pH, 16 pH, 15 pH, 14 pH, 13 pH, 12 pH, 11 pH, 10 pH, 9 pH, 8 pH, 7 pH, 6 pH, 5 pH, or less.

[0072] Referring now to the readout resonator 130. The readout resonator 130 is also a short- circuited, quarter-wavelength (^^ / 4), microwave transmission line resonator. Thus, when described as a lumped LC oscillator in a lumped-element circuit model (see FIG.1B), the readout resonator 130 has an effective capacitance of ^^୰and an effective inductance of ^^୰,such that the readout resonator 130 resonates at an angular frequency of ^^୰ ൌ 1 / ^^^୰^^୰. Insome implementations, the readout resonator 130’s fundamental frequency (^^୰ / 2^^) can be in a range from about 4 GHz to 8 GHz. For example, the readout resonator 130’s fundamental frequency can be at least about 4 GHz, 4.5 GHz, 5 GHz, 5.5 GHz, 6 GHz, 6.5 GHz, 7 GHz, 7.5 GHz, 8 GHz, or more. The readout resonator 130’s fundamental frequency can be at most about 8 GHz, 7.5 GHz, 7 GHz, 6.5 GHz, 6 GHz, 5.5 GHz, 5 GHz, 4.5 GHz, 4 GHz, or less.Attorney Docket No.56113-0516WO1

[0073] The readout resonator 130 includes a thin strip 132 of conducting material that is separated from the ground plane 140 by adjacent strips of insulating material such that transport and / or oscillation of electrons is confined to the strip 132. The strip 132 follows a tortuous, serpentine path in a folded region 133 to increase the total length of the readout resonator 130 to a specified length, and thereby a specified fundamental frequency for readout resonator 130. As shown in FIG.1A, in this example, the readout resonator 130 islonger than the flux resonator 120 and therefore has a lower fundamental frequency ^^୰ ^ ^^^.

[0074] An open-circuited end 134-OC of the readout resonator 130 is configured as a rectangular patch antenna (aka a gate electrode) to maximize the accumulation of local electric charge at the open-circuited end 134-OC (and thus the gate capacitance). The patch antenna 134-OC is separated from the ground plane 140 by a rectangular loop 135 of insulating material. The patch antenna 134-OC is capacitively coupled 31-C to the capacitor electrode 114 of the superconducting qubit 110. A short-circuited end 136-SC of the readout resonator 130 is electrically connected to the ground plane 140. This boundary condition fixes the local voltage to zero at the short-circuited end 136-SC. Thus, the amplitude of the local electric current is generally maximized at the short-circuited end 136-SC. The feedline 150 is inductively coupled 35-I to the readout resonator 130 near the short-circuited end 136- SC to exploit this property.

[0075] In general, the readout resonator 130 is configured to bias an electric charge (^^^) on the superconducting qubit 110 to perform dispersive readout, or otherwise control and / or address the superconducting qubit 110. The electric charge bias ^^^generated on thesuperconducting qubit 110 is equal to ^^^ ൌ ൫^^^ / ^^୰൯^^୰, where ^^^ is the gate capacitancebetween the readout resonator 130 and the qubit 110, and ^^୰is the local electric charge in the patch antenna 134-implementations, the gate capacitance can be in a range from about 1 fF to 10 fF. For example, the gate capacitance can be at least about 1 fF, 2 fF, 3 fF, 4 fF, 5 fF, 6 fF, 7 fF, 8 fF, 9 fF, 10 fF, or more. The gate capacitance can be at most about 10 fF, 9 fF, 8 fF, 7 fF, 6 fF, 5 fF, 4 fF, 3 fF, 2 fF, 1 fF, or less.

[0076] FIGs.2A-2C depict additional examples of a superconducting subcircuit 100B that can be utilized for multiplexed control and readout of a superconducting qubit 110 either alone, or in a larger superconducting circuit 10B that includes multiple such superconducting subcircuits 100A, see FIG.2C for example. FIG.2A is a top view depicting the superconducting subcircuit 100B. The superconducting subcircuit 100B includes: (i) a superconducting qubit 110; (ii) a flux resonator 120B; (iii) a readout resonator 130; (iv) aAttorney Docket No.56113-0516WO1 ground plane 140 (or “ground”); and (v) a feedline 150. FIG.2B shows a lumped-element circuit model of the superconducting subcircuit 100B depicted in FIG.2A. For ease of description, reference will be made to both FIGs.2A and 2B when describing the superconducting subcircuit 100B. FIG.2C is a top view depicting the superconducting circuit 10B that includes multiple ones of the superconducting subcircuit 100B-1 through 100B-5.

[0077] The superconducting subcircuit 100B in FIGs.2A-2C is configured similarly to the superconducting subcircuit 100A in FIGs.1A-1C except the flux resonator 120B is an open- circuited, half-wavelength (^^ / 2), microwave transmission line resonator (as opposed to the quarter-wave flux resonator 120A). The discussion presented above for superconducting circuits 100A and 10A holds equally well for superconducting circuits 100B and 10B, and therefore will not be repeated. The structure and operating principles of the half-wave flux resonator 120B, however, are described in detail below.

[0078] For the flux resonator 120B, the wavevector and phase of its fundamental mode are ^^^^^ ൌ ^^ and ^^^ ൌ 0. This corresponds to a fundamental frequency of ^^^ / 2^^ ൌ1 / ൫2^^^^^^^൯ ൌ ^^^ / ^2^^^. As the name implies, the fundamental mode of a half-wavelengtha half of a wavelength (^^^ ൌ 2^^) due to the two open circuit boundaryconditions. Thus, when described as a lumped LC oscillator in a lumped-element circuit model (see FIG. 2B), the flux resonator 120B has an effective capacitance of ^^^ ൌ ^^^ / ^^ andan effective inductance of ^^^ ൌ ^^^ / ^^, such that the flux resonator 120B resonates at anangular frequency of ^^^ ൌ ^^^ ൌ 1 / ^^^^^^^. In some implementations, the flux resonator120B’s fundamental frequency (^^^ / 2^^) can be in a range from about 6 GHz to 12 GHz. For example, the flux resonator 120B’s fundamental frequency can be at least about 6 GHz, 6.5 GHz, 7 GHz, 7.5 GHz, 8 GHz, 8.5 GHz, 9 GHz, 9.5 GHz, 10 GHz, 10.5 GHz, 11 GHz, 11.5 GHz, 12 GHz, or more. The flux resonator 120B’s fundamental frequency can be at most about 12 GHz, 11.5 GHz, 11 GHz, 10.5 GHz, 10 GHz, 9.5 GHz, 9 GHz, 8.5 GHz, 8 GHz, 7.5 GHz, 7 GHz, 7.5 GHz, 6 GHz, or less.

[0079] In more detail, the flux resonator 120B includes a thin strip of conducting material that is folded into a first piece 122-1 and a second piece 122-2 which run adjacent to each other. The respective end 126-OC1 and 126-OC2 of each piece 122-1 and 122-2 is terminated with an open boundary condition that fixes the local electric current to zero at that end 126- OC1 or 126-OC2. The open-circuited end 126-OC2 of the second piece 122-2 is capacitively coupled to the feedline 150. Alternatively, or in addition, the open-circuited end 126-OC1 of the first piece 122-1 may be capacitively coupled to the feedline 150.Attorney Docket No.56113-0516WO1

[0080] Each piece 122-1 and 122-2 follows a tortuous, serpentine path in a folded region 123 to increase the total length of the flux resonator 120B to a specified length, and thereby a specified fundamental frequency for the flux resonator 120B. The pieces 122-1 and 122-2 meet in the center of the flux resonator 120B to form a loop antenna 124 proximate the superconducting qubit 110. The loop antenna 124 includes a first region 125-1 of insulating material in the shape of an oval and a second region 125-2 of insulating material in the shape of a rectangle. The first region 125-1 is positioned between the two pieces 122-1 and 122-2 to establish the geometry and conductive pathway of the loop antenna 124.

[0081] The amplitude of the local electric current is generally maximized at the center of the flux resonator 120B, in the loop antenna 124, since the spatial profile of the flux resonator120B’s fundamental mode is |^^௫^^^| ൌ 1 here. The loop antenna 124 is inductively coupled21-I to the dc-SQUID loop 112 relatively strongly. Particularly, the second region 125-2 is positioned above the first region 125-1 near the bottom portion of the dc-SQUID loop 112 which enhances the inductive coupling between the two, e.g., increasing the mutual inductance (^^).

[0082] Table 1 below lists target and simulated design parameters relating to the superconducting subcircuit 100B shown in FIGs.2A-2C. Table 1: Superconducting Subcircuit 100B Design Parameter Target Simulation Q bit fr n (^^ / 2^^) 400 GHz ~ 405 GHznon-exhaustive list as many different configurations and coupling schemes can beAttorney Docket No.56113-0516WO1 implemented. For example, flux resonator 120A can be a short-circuited, half-wavelength microwave transmission line resonator with two short-circuited ends. In this case, the short- circuited end 124-SC of the flux resonator 120A can remain inductively coupled to the dc- SQUID loop 112 and the open-circuited end 126-OC can instead be short-circuited, allowing the flux resonator 120A to be inductively coupled to the feedline 150. As another example, the readout resonator 130 can be an open-circuited, half-wavelength microwave transmission line resonator with two open-circuited ends. In this case, the patch antenna 134-OC of the readout resonator 130 can remain capacitively coupled to the superconducting qubit 110 and the short-circuited end 136-SC can instead be open-circuited, allowing the readout resonator 130 to be capacitively coupled to the feedline 150. III. Quantum Mechanical Treatment of a Superconducting Subcircuit

[0084] To help implement the systems, methods, and apparatus described herein, a detailed quantum mechanical treatment of a superconducting subcircuit 100 including a superconducting qubit 110 coupled to a flux 120 and readout 130 resonator is provided below.

[0085] The magnetic flux (Φ^) in the flux resonator 120 can be represented in the “black-boxquantization” as Φ^ ൌ ^ℏ^^^ / 2^^^^ற^ ^ ^^^^^, where ^^^ற^ and ^^^^are the creation and annihilation operators, respectively, for photons in the fundamental mode of the flux resonator 120. These operators satisfy the commutation relation ^^^^^,^^^ற^ ^ ൌ 1. Analogously, the electric charge (^^୰)in the readout resonator 130 can be represented in the black-box quantization as ^^୰ൌ^^^ℏ / 2^^ற^^^^^୰ െ ^^^୰^, where ^^^ற୰ and ^^^୰are the creation and annihilation operators, respectively, for photons in the fundamental mode of the readout resonator 130. These operators satisfy the commutation relation ^^^^୰,^^^ற୰^ ൌ 1.

[0086] Thus, in general, the total Hamiltonian (^^) of the superconducting qubit 110 coupled to the flux 120 and readout 130 resonators can be represented, at least approximately, as: ^^ ൌ ℏ^^^^^^ற^ ^^^^ ^ ℏ^^ றଶ ୰^^^୰^^^୰ ^ 4^^^൫^^ െ ^^^ / 2^^൯ െ ^^^^Φ௫^ cos^^^ െ ^^^^, (3)

[0087] and,(4) ^^^^Φ௫^ ൌ ^^^ஊcos ൬Φ^^1 ^ ^^ଶ tanଶ ൬^, ^ Φ^

[0088] where ^^ ൌoperator of theAttorney Docket No.56113-0516WO1 across the Josephson junctions 113-1 and 113-2 that provides the phase operator of thesuperconducting qubit 110, ^^^ ൌ ^^ tan^^^Φ௫ / Φ^^ is the phase offset, and ℏ ൌ ℎ / 2^^ is thereduced Planck constant. Here, the periodic boundary condition around the dc-SQUID loop 112 has been invoked such that the differences in phase across the Josephson junctions 113-1and 113-2 is equal to ^^^ െ ^^ଶ ൌ 2^^Φ௫ / Φ^ ^mod 2^^).

[0089] The phase offset ^^^can generally be ignored for time-independent magnetic fluxbiases Φ^ ൌ 0. However, one of the main contributions of this disclosure is showing how atime- magnetic flux bias Φ^ ് 0 can be effectively utilized for qubit control,particularly multiplexed qubitit is advantageous to employ a symmetric dc-SQUID loop 112 such that the junction asymmetry is ^^ ൌ 0 and ^^^ ൌ 0. This simplifies theeffective Josephson energy to ^^^^Φ௫^ ൌ ^^^ஊcos ^^^Φ௫ / Φ^^. A larger value for the junctionasymmetry can also lead to athe superconducting qubit 110.

[0090] The charge number and phase operators of the superconducting qubit 110 can be converted to canonical form after applying the following transformations: ^ ^^ ^^ ସ (5) ^^ ^ஊ^^^^^ற

[0091] and,^ (6) 2^ ସ ^^^ ^^^^

[0092] where ^^^றand ^^^ are the photonsin the superconducting qubit 110. These operators satisfy the commutation relation ^^^^, ^^^ற൧ ൌ1. The photon number states |0^and |1^are used as the computational states of the superconducting qubit 110, corresponding to the superconducting qubit 110’s ground and excited states, respectively. The photon number states |2^, |3^, and so on, are typically referred to as non-computational states – transitions from computational states to a non- computational state are undesirable and referred to as leakage.

[0093] Now consider that the magnetic flux bias oscillates at relatively high frequencies. As a general application of the rotating-wave approximation (RWA), the fluctuations in the biasing magnetic flux (Φ௫) average out, which results in an AC Stark effect on the bare states of the superconducting qubit 110. Particularly, after using the small angle approximation on the cosine functions (to fourth order) and applying the RWA, the Hamiltonian of Eq. (3) reduces to:Attorney Docket No.56113-0516WO1 (7)

[0094] ^^^ ^^ qubitshift from the flux resonator 120, and ^^ is the resonator- qubit coupling strength between the readout resonator 130 and the superconducting qubit 110. The AC Stark shift and resonator-qubit coupling strength are given as: (8)

[0095] and,^^^^^ ^ / ସ ^^^^^(9)

[0096] where ^^ ଶ^ ൌ ℎ / ^^ is thetheresonators 120 and 130 interacting with the superconducting qubit 110 through zero-point fluctuations have been absorbed into their bare frequencies. Also, an anharmonicity ^^^ൌെℏ^^ଶ / ^^^ of the flux resonator 120 has been ignored as this term is negligibly^^^ / ^^ ൌ^ℏ^^ / ^^^^ଶ~10ି^^. From a lumped-element circuit point of view, see FIGs.1B and 2B for example, the superconducting qubit 110 can be understood as a nonlinear LC oscillator where the nonlinear inductance arising from the Josephson junctions 113-1 and 113-2 is coupled to the flux resonator 120.

[0097] As shown in Eq. (7), the AC Stark shift is proportional to the number of photons in the flux resonator 120. For example, resonantly driving the flux resonator 120 at itsfundamental frequency produces a net shift on the bare qubit frequency of ^^^୯ ൌ ^^୯ െ ^^^^,where ^^ is the mean photon number in the flux resonator 120, and ^^^୯is the shifted qubit frequency. Thus, the superconducting qubit 110 can be described as a quantum Duffing oscillator (QDO) with an AC-tunable qubit frequency.

[0098] In some implementations, the AC Stark shift (^^ / 2^^) can be in a range from about 500 hertz (Hz) to 5 kilohertz (kHz). For example, the AC Stark shift can be at least about 500 Hz, 750 Hz, 1000 Hz, 1250 Hz, 1500 Hz, 1750 Hz, 2000 Hz, 2250 Hz, 2500 Hz, 2750 Hz, 3000 Hz, 3250 Hz, 3500 Hz, 3750 Hz, 4000 Hz, or more. In some implementations, the superconducting qubit 110 can have a (bare) qubit frequency (^^୯ / 2^^^ in a range from about 2 GHz to 6 GHz. For example, the superconducting qubit 110 can have a bare qubitAttorney Docket No.56113-0516WO1 frequency of at least about 2 GHz, 2.5 GHz, 3 GHz, 3.5 GHz, 4 GHz, 4.5 GHz, 5 GHz, 5.5 GHz, 6 GHz, or more. The superconducting qubit 110 can have a (bare) qubit frequency of at most about 6 GHz, 5.5 GHz, 5 GHz, 4.5 GHz, 4 GHz, 3.5 GHz, 3 GHz, 2.5 GHz, 2 GHz, or less. In some implementations, the superconducting qubit 110’s qubit frequency can be tunable over a frequency range of about 300 MHz or more. For example, the superconducting qubit 110’s qubit frequency can be tunable over a frequency range of about 300 MHz, 325 MHz, 350 MHz, 375 MHz, 400 MHz, 425 MHz, 450 MHz, 475 MHz, 500 MHz, or more.

[0099] FIGs.3A and 3B show the AC Stark effect from a semi-classical perspective where the magnetic flux in the flux resonator 120 is treated classically. FIG.3A is a plot 300A of qubit frequency ℏ^^୯^Φ௫^ versus magnetic flux bias (Φ௫), depicting its quadratic-like functional dependence. FIG.3B is a plot 300B of the magnetic flux bias Φ௫^^^^ versus time ^^, depicting its sinusoidal oscillatory behavior. As shown in FIG.3A, when the oscillations in the biasing magnetic flux (Φ௫) are relatively fast, the qubit frequency averages to (^^^୯) which may be understood as a type a homodyning of the flux resonator 120 on the superconducting qubit 110.

[0100] Now consider a control signal 612 generated at the first 152 and / or 154 second port of the feedline 150, where the control signal 612 oscillates at the flux resonator 120’s fundamental frequency. In general, the mean photon number (^^) in the flux resonator 120 is proportional to the squared amplitude (aka the intensity) of the control signal 612. Thus, it can also be useful to describe the AC Stark effect in terms of the drive power (^^) into the fluxresonator 120 as ℏ^^^୯ ൌ ℏ^^୯ െ ^^^^. Here, ^^ is a time constant related to the light-matterinteraction that is given, at least approximately, as: (10 ^^ ଶ ) ^^ ^^ ^^ ൌ^^^^ ൬ ^ ^^ ^ஊ,

[0101] where ^^^ ൌ ^^^⁄ ^^^ is the ^ / ^^^is the linewidth of the flux^^^^is thethe flux resonator 120 and the feedline 150, and ^^^୫୪is the characteristic impedance of the feedline 150. For the example circuit parameters described herein, e.g., with quality factors of about 10ସto 10ହfor the flux 120 and readout 130 resonators, the time constant can be in a range from about 100 yoctoseconds (ys) to 10 zeptosecond (zs).

[0102] For example, the feedline 150 can have a characteristic impedance of at least about 25 Ω, 30 Ω, 35 Ω, 40 Ω, 45 Ω, 50 Ω, 55 Ω, 60 Ω, 65 Ω, 70 Ω, 75 Ω, or more. The feedlineAttorney Docket No.56113-0516WO1 150 can have a characteristic impedance of at most about 75 Ω, 70 Ω, 65 Ω, 60 Ω, 55 Ω, 50 Ω, 45 Ω, 40 Ω, 35 Ω, 30 Ω, 25 Ω, or less. Likewise, the feedline 150 can have a wavespeed of at least about 0.5 ൈ 10଼ m / s, 0.75 ൈ 10଼ m / s, 1 ൈ 10଼ m / s, 1.25 ൈ 10଼ m / s,1.5 ൈ 10଼ m / s, 1.75 ൈ 10଼ m / s, 2 ൈ 10଼ m / s, or more. The feedline 150 can have a wavespeed of at most about 2 ൈ 10଼ m / s, 1.75 ൈ 10଼ m / s, 1.5 ൈ 10଼ m / s, 1.25 ൈ 10଼ m / s,1 ൈ 10଼ m / s, 0.75 ൈ 10଼ m / s, 0.5 ൈ 10଼ m / s, or less.

[0103] FIG.3C is a plot 300C of experimental data showing coupling efficiency of a superconducting qubit 110 versus flux resonator 120 drive power and qubit drive frequency. Amplitude of the coupling efficiency is in arbitrary units. In this experiment, a flux resonator 120 was inductively coupled to the dc-SQUID loop 112 of the superconducting qubit 110 and had a fundamental frequency of about 7.5 GHz. The flux resonator 120 was coupled to a feedline 150 which was resonantly driven at the flux resonator 120’s fundamental frequency. The flux resonator 120 drive power ranged from about 0 milliwatts (mW) to 0.6 mW. Each vertical slice in the plot 300C is a qubit spectroscopy measurement where the qubit drive frequency of a ^^-pulse excitation was swept from about 3.2 GHz to 3.7 GHz. The maximum amplitude of the coupling efficiency along a vertical slice corresponds to hitting the superconducting qubit 110’s shifted qubit frequency at the particular flux resonator 120 drive power.

[0104] The experimental plot 300C agrees well with the theory outlined above, confirming that the superconducting qubit 110’s qubit frequency decreases monotonically (linearly) with the flux resonator 120 drive power. The slope of the shifted qubit frequency in the experimental plot 300C produces a time constant of about 550 ys which also agrees with the theory. Data points D0-D3 in the plot 300C show the approximate relaxation (^^^) and dephasing (^^ଶ) times of the superconducting qubit 110 at the particular flux resonator 120 drive power and qubit drive frequency. Dropouts in the plot 300C (e.g., at data point D1) where the coupling efficiency abruptly goes to zero along the decreasing line were due to incorrect ^^-pulse amplitudes.

[0105] As an aside, the superconducting subcircuit 100 is typically configured as ^^୯ ^ ^^୰ ^^^^, where the readout resonator 130’s fundamental frequency is greater than the superconducting qubits 110’s qubit frequency, and the flux resonators 120’s fundamental frequency is greater than the readout resonator 130’s fundamental frequency. However,configurations like ^^୰ ^ ^^୯ ^ ^^^ can also be implemented, where the superconductingqubits 110’s qubit frequency is greater than the readout resonator 130’s fundamentalAttorney Docket No.56113-0516WO1 frequency, and the flux resonators 120’s fundamental frequency is greater than the superconducting qubits 110’s qubit frequency.

[0106] These configurations of the superconducting subcircuit 100 allow the flux resonator 120 to tune the superconducting qubit 110 as described above without “reading out” the state of the superconducting qubit 110. For example, the dispersive coupling (^^^) between the superconducting qubit 110 and the flux resonator 120 is typically much less than the linewidth of the flux resonator 120, ^^ଶ^^ (11) ^^^^ ൫ଶ ≪ ^^^,^^^ െ ^^୯൯

[0107] such that the superconducting qubit 110 and the flux resonator 120 are substantially unentangled.

[0108] Conversely, these configurations of the superconducting subcircuit 100 allow the readout resonator 130 to perform dispersive readout of the superconducting qubit 110. For example, the dispersive coupling (^^୰) between the superconducting qubit 110 and the readout resonator 130 can be about equal to or greater than the linewidth of the readout resonator 130, ^^ଶ^^ (12) ^^୰^≳ ^^୰,

[0109] such that thesubstantially entangled. Here, ^^୰ ൌ ^^^ ୪ ^ / ^^୰is the linewidth of the readout resonator 130 from inductively coupling with the feedline 150, and ^^୰^is the mutual inductance between the readout resonator 130 and the feedline 150.

[0110] Entanglement of the superconducting qubit 110 and the readout resonator 130 produces dressed fundamental frequencies (^^୰,േ) for the readout resonator 130 of: ^^୰,േ ൌ ^^୰ േ ^^୰, (13)

[0111] where ^^୰,ାis the upper dressed fundamental frequency of the readout resonator 130 corresponding to the superconductingis the lower dressed fundamental frequency of the readout resonator 130 corresponding to the superconducting qubit 110 being in the ground state |0^.

[0112] Now consider a control signal 612 generated at the first port 152 of the feedline 150. A control signal 612 oscillating at the readout resonators 130’s upper dressed fundamental frequency (^^୰,ା) is substantially transmitted to the second port 154 if the superconducting qubit 110 is in the excited state, and substantially reflected back to the first port 152 if the superconducting qubit 110 is in the ground state. Likewise, a control signal 612 oscillating atAttorney Docket No.56113-0516WO1 the readout resonator 130’s lower dressed fundamental frequency (^^୰,ି) is substantially transmitted to the second port 154 if the superconducting qubit 110 is in the ground state, and substantially reflected back to the first port 152 if the superconducting qubit 110 is in the excited state. Alternatively, a control signal 612 oscillating at the readout resonator 130’s bare fundamental frequency (^^୰) produces a phase shift between the control signal 612 and the transmitted signal at the second port 154 of: ^^േ ൌ േarctan ^2^^୰ / ^^୰^, (14)

[0113] where ^^ାis a positive phase if the superconducting qubit 110 is in the excited state, and ^^ିis a negative phase shift if the superconducting qubit 110 is in the ground state. Thephase shift between the control signal 612 and the reflected signal at the first port 154 is ^^ ^^^േ, such that the reflected and transmitted signals are 180 degrees out of phase with each other.

[0114] Note that the above holds when the control signal 612 is generated at the second port 154, but with the roles of the first 152 and ports 154 switched. Thus, measuring the amplitude and / or phase of the transmitted and / or reflected signals, i.e., the readout signals 614, at the first 152 and / or second 154 port of the feedline 150 reveals information about the state of the superconducting qubit 110. On the other hand, when a control signal 612 oscillates at the superconducting qubits 110’s qubit frequency, e.g., to realize a one-qubit quantum gate, the phase shift of the readout resonator 130 only negligibly depends on the state of the superconducting qubit 110. This results in negligible entanglement between the superconducting qubit 110 and the readout resonator 130, and consequently negligible measurement-induced dephasing on the superconducting qubit 110. IV. Additional Examples of Superconducting Circuits

[0115] FIG.4A is a schematic diagram depicting another example of a superconducting circuit 10C including multiple superconducting subcircuits 100-1 through 100-5. Each superconducting subcircuit 100 includes a respective superconducting qubit 110, a respective flux resonator 120 (e.g., flux resonator 120A or 120B) inductively coupled to the superconducting qubit 110, and a respective readout resonator 130 capacitively coupled to the superconducting qubit 110. Each flux 120 and readout 130 resonator is also evanescently coupled to a common feedline 150, in this case capacitively and inductively coupled to the feedline 150, respectively.Attorney Docket No.56113-0516WO1

[0116] In general, the resonators 120 and 130 each have a different fundamental frequency and therefore can be individually driven on the feedline 150. Transmission line resonators can be detuned from one another by using different lengths, e.g., by varying the number and / or lengths of folds in the resonators. Folding transmission line resonators in this manner can be an effective means to tune fundamental frequencies while conserving circuit area. Consequently, the first port 152 of the feedline 150 can receive multiplexed control signals 612 from a control system 200 that each addresses multiple superconducting qubits 110 via respective resonators 120 and / or 130. A multiplexed readout signal 614 can then be transmitted to the control system 200 from the second port 154 of the feedline 150 that includes information about the respective state of each superconducting qubit 110 that was addressed.

[0117] FIG.4B is a plot 400 of coupling efficiency into different flux resonators 120-1 through 120-4 versus drive frequency of a control signal generated on a feedline 150 evanescently coupled to each of the flux resonators 120. Amplitude of the coupling efficiency is in arbitrary units. Each flux resonator 120-1 through 120-4 is tuned to a different fundamental frequency of about 4.2 GHz, 5.8 GHz, 6.5 GHz, or 7.0 GHz, and has a relatively narrow linewidth of about 50 MHz. Hence, the flux resonators 120-4 through 120-4 can be independently driven via frequency-division multiplexing on the feedline 150 to control respective superconducting qubits 110, superconducting couplers 110*, or other quantum circuit devices described herein. As mentioned previously, due to the large available bandwidth for modern microwave electronics and narrow linewidths achieved by microwave resonators, hundreds, to thousands, to tens of thousands of such devices may be multiplexed together.

[0118] FIG.5 is a schematic diagram depicting another example of a superconducting circuit 10D. The superconducting circuit 10D includes multiple superconducting subcircuits 100B and 100B* that each include a respective superconducting qubit 110 or superconducting coupler 110*. The superconducting circuit 10D is an example of superconducting circuit that can implement two-qubit quantum gates, e.g., iSWAP gates, CZ gates, among others, using multiplexed control and readout. Here, the superconducting qubits 110-1 and 110-2 are each capacitively coupled 11*-C to the coupler 110* (e.g., nearest-neighbor (NN) coupling) and capacitively coupled 11-C to each other (e.g., next-nearest-neighbor (NNN) coupling).

[0119] The superconducting coupler 110* can be configured similarly to any of the superconducting qubits 110 described herein (e.g., the transmon qubits 110 of FIGs.1A-2C), but without the need for a respective readout resonator 130. In general, the superconductingAttorney Docket No.56113-0516WO1 coupler 110* mediates the effective two-qubit coupling strength (^^^ଶ) between the superconducting qubits 110-1 and 110-2 – the state of the superconducting coupler 110* is not read out after performing a quantum gate operation. In most implementations, the superconducting coupler 110* remains in its ground state during the execution of a quantum gate operation assuming appropriate detuning and control signals, e.g., control signals that avoid coherent tones resonant with the superconducting coupler 110*. Particularly, in the strong dispersive region, the superconducting coupler 110*’s coupler frequency (^^ୡ) ispositively detuned from the qubit frequencies ^^ୡ ^ ^^୯^,ଶ and the qubit-coupler couplingstrength between the superconducting coupler 110* and each superconducting qubit 110-1and 110-2 is dispersive ^^ୡ^,ଶ ≪ |^^ୡ െ ^^୯^,ଶ|.

[0120] Analogously to the superconducting qubits 110, the coupler frequency is AC-tunable via resonant drive of the superconducting coupler 110*’s flux resonator 120-3, thereby tuning the effective two-qubit coupling strength ^^^ଶ^^^^ୡ^ between the superconducting qubits 110-1 and 110-2. In some implementations, the two-qubit coupling strength is tunable through zero^^^ଶ ൌ 0, such that the interaction between the superconducting qubits 110-1 and 110-2 canbe “turned off” at a particular flux resonator 120 drive power (^^^). This can be advantageous for monolithic quantum computing systems. For example, the flux resonator 120-3 can be resonantly driven at ^^^to “idle” the two superconducting qubits 110-1 and 110-2 before performing a gate operation involving them, facilitating precise control of qubit interactions and quantum gate sequencing. Hence, the superconducting circuit 10D provides a means for tunable coupling schemes with multiplexed control and readout, e.g., for performing high- fidelity two-qubit quantum gates entirely with AC signals.

[0121] Consider an iSWAP gate as an example. In this case, one or both of the flux resonators 120-1 and 120-2 can be resonantly driven via the feedline 150 to tune the qubitfrequencies ^^^୯^ ൌ ^^^୯ଶ, bringing the computational states |01^ and |10^ on resonance witheach other. Simultaneously, the flux resonator 120-3 can be resonantly driven via the feedline 150 to tune the coupler frequency, such that the two-qubit coupling strength is a particular nonzero value ^^^ଶ^^^^ୡ^, e.g., corresponding to a specified gate length ^^^. To perform resonantRabi cycles (or flops) between the states |01^ → െ^^|10^ and |10^ → െ^^|01^, the two-qubitcoupling strength can be set to a ^^-pulse excitation |^^^ଶ| ൌ ^^ / ^2^^^^. Here, ^^ ൌ √െ1 is theimaginary unit. At the completion of the iSWAP gate, the readout resonators 130-1 and 130-2 may both be resonantly driven via the feedline 150 to determine the respective state of each superconducting qubit 110.Attorney Docket No.56113-0516WO1

[0122] Note, in some implementations, a superconducting coupler 110* may mediate interactions between more than two superconducting qubits 110, e.g., three, four, five, six, or more superconducting qubits 110, using the coupling scheme outlined in FIG.5. Alternatively, or in addition, using the same coupling scheme, a superconducting qubit 110 can interact with multiple other superconducting qubits 110 with respective superconducting couplers 110* mediating such interactions, see FIG.6B for example. Further details relating to tunable coupling schemes for transmon qubits is provided by Yan, Fei, et al., “Tunable coupling scheme for implementing high-fidelity two-qubit gates,” Physical Review Applied 10.5 (2018): 054062. V. Examples of Superconducting Quantum Computer Systems

[0123] FIG.6A is a schematic diagram depicting an example of a superconducting quantum computer system 600. The superconducting quantum computer 600 includes a superconducting circuit 10 which can be implemented as a quantum processor, quantum chip, quantum hardware, or other quantum system. The superconducting circuit 10 includes: (i) multiple superconducting subcircuits 100-1 through 100-N that each include a respective superconducting qubit 110, and (ii) one or more superconducting subcircuits 100*-1 through 100*-M that each include a respective superconducting coupler 110*. Each superconducting subcircuit 100 includes a respective flux 120 and readout 130 resonator evanescently coupled to its respective superconducting qubit 110 and the feedline 150. Each superconducting subcircuit 100* includes a respective flux resonator 120 evanescently coupled to its respective superconducting coupler 110* and the feedline 150. In some implementations, the superconducting circuit 10 can include five, ten, fifteen, twenty, twenty-five, fifty, one hundred, two hundred, three hundred, four hundred, five hundred, or more superconducting subcircuits 100 and 100*.

[0124] As mentioned above, the superconducting qubits 110 and couplers 110* can be coupled to one another in various ways for implementing two-qubit and higher-order quantum gates. The quantum computer 600 further includes a (classical) control system 200 coupled to the superconducting circuit 10 via control lines 602 and 604 connected to first 152 and second 154 ports, respectively, of the feedline 150. In general, the control system 200 can operate at room (or ambient) temperature due to performing classical computations, while the superconducting circuit 10 operates at cryogenic temperatures to enable superconducting quantum computation. For example, the superconducting circuit 10 can be positioned and cooled within a cryostat and the control system 200 can be positioned outside the cryostat.Attorney Docket No.56113-0516WO1 The superconducting circuit 10 and control system 200 can then communicate with one another via microwave signals over control lines 602 and 604.

[0125] By controlling the superconducting circuit 10 with the control system 200, the quantum computer 600 can implement quantum circuits, see FIG.6B for example, that can execute quantum processing algorithms. Such quantum processing algorithms can include, but are not limited to, Shor’s algorithm, Grover’s algorithm, the Deutsch-Jozsa algorithm, Simon’s algorithm, the quantum phase estimation (QPE) algorithm, error correction (e.g., surface code error correction), among others. The quantum computer 600 can also be utilized for quantum supremacy experiments that use a two-dimensional qubit array, e.g., quantum supremacy experiments that aim to entangle every pair of qubits in the qubit array using a circuit depth such that no classical computer can conveniently calculate the amplitude of each computational basis without requiring an exponential number of qubit computational steps. An example of such a quantum supremacy experiment is provided by Frank Arute, et al. “Quantum supremacy using a programmable superconducting processor,” Nature 574.7779 (2019): 505-510, which utilized the “Sycamore” quantum processor to demonstrate the advantage of quantum hardware over classical hardware for a particular computational problem.

[0126] In general, a quantum processing algorithm proceeds by the control system 200 initializing the superconducting qubits 110 in a selected initial state (e.g., the ground state) and then applying a sequence of quantum gate operations to the superconducting qubits 110, in which the quantum gate operations can include: (i) one-qubit quantum gate operations such as Pauli gates (e.g., X, Y, and Z gates), Hadamard gates, phase shift gates, and rotation gates; (ii) two-qubit quantum gate operations such as controlled gates (e.g., CX, CY, CZ, CNOT, and CPHASE gates), two-qubit interaction gates (e.g., XX, YY, ZZ, and XY gates), and swap gates (e.g., SWAP, iSWAP,√SWAP, and√iSWAP gates); and (iii) quantum gate operations involving three or more superconducting qubits 110 such as Toffoli and Fredkin gates.

[0127] An example of a quantum processing algorithm being realized by an example quantum circuit 800 is shown in FIG.6B. The control system 200 can carry out quantum gate operations by applying various control signals 612 to the superconducting qubits 110 and couplers 110*. At the conclusion of the quantum processing algorithm, the control system 200 measures the final states of the superconducting qubits 110 using a quantum observable such as Z, which is determined from the readout signals 614.Attorney Docket No.56113-0516WO1

[0128] In the described example of FIG.6A, the control system 200 includes one or more classical (i.e., non-quantum) control devices 230-1 through 230-P. For example, the control device(s) 230 can include digital-to-analog converters (DACs), analog-to-digital converters (ADC), linear controllers (e.g., proportional-integral-derivative (PID) controllers), nonlinear controllers, signal generators, function generators, pitch generators, arbitrary waveform generators, digital pattern generators, signal analyzers, spectrum analyzers, oscillators, mixers, multiplexers, demultiplexers, amplifiers, attenuators, filters, circulators, or any other microwave electronics appropriate to implement the selected control and measurement operations on the superconducting circuit 10.

[0129] In general, the control system 200 is configured to generate multiplexed control signals 612 that include multiple coherent tones within the bandwidth of the superconducting circuit 10. For example, the control system 200 can generate multiplexed control signals that include 2, 3, 4, 5, 10, 25, 50, 100, 500, 100, 1000, or more multiplexed tones. Each tone has a respective frequency that can correspond to one of the following:

[0130] (i) a flux resonator 120’s fundamental frequency for tuning a respective superconducting qubit 110 or coupler 110*, e.g., to perform a multi-qubit quantum gate involving the superconducting qubit 110 or coupler 110*;

[0131] (ii) a readout resonator 130’s fundamental frequency (bare or dressed fundamental frequency), e.g., to read out a respective superconducting qubit 110; or

[0132] (iii) a superconducting qubit 110’s qubit frequency (bare or shifted qubit frequency), e.g., to initialize the superconducting qubit 110 or perform a one-qubit quantum gate on the superconducting qubit 110.

[0133] The control system 200 can generate each tone with a respective amplitude, phase, and / or pulse width as appropriate to implement these abovementioned operations.

[0134] The control system 200 is further configured to receive multiplexed readout signals 614 and correlate them with the multiplexed control signals 612 to determine information about the superconducting circuit 10, e.g., to determine the respective state of each super conducting qubit 110 at the completion of a sequence of quantum gate operations. Note that the control system 200 generally references and / or biases the control 612 and readout 614 signals against a (common) ground plane 140 of the superconducting circuit 10 that provides a common reference potential.

[0135] FIG.6B shows a schematic diagram of an example quantum circuit 800 that can execute a quantum processing algorithm. The quantum circuit 800 is realized by the superconducting circuit 10 via operations of the control system 200. The control system 200Attorney Docket No.56113-0516WO1 may implement quantum circuits using any number of the superconducting qubits 110, ranging from two of the superconducting qubits 110 to all the superconducting qubits 110 included the superconducting circuit 10. The control system 200 may also implement multiple quantum circuits in parallel, e.g., using specifically designed sequences of control signals 612.

[0136] In the described example of FIG.6B, at least a portion of the superconducting circuit 10 includes five superconducting qubits 110-1, 110-2, 110-3, 110-4, and 110-5 interacting via four superconducting couplers 110*-A, 110*-B, 110*-C, and 110*-D to realize the quantum circuit 800. The flux 120 and readout 130 resonators of the superconducting qubits 110 and couplers 110* are omitted in FIG.6B for clarity. The superconducting qubits 110 and couplers 110* of the superconducting circuit 10 are manipulated and measured by the control system 200 using appropriate control signals 612 and readout signals 614.

[0137] Particularly, the control system 200 initializes 810 an initial state |^^୧୬^^ of the superconducting circuit 10, in this case all the superconducting qubits 110 in their respectiveground state |^^୧୬^^ ൌ |00 … 0^, and evolves (controls) 710 its state over a sequence of timesteps to reach a final state |^^^୧୬^. During the evolving, the control system 200 implements one-qubit 812 and two-qubit 822 quantum gates at each time step (^^^) by applying variouscontrol signals 612 to the superconducting circuit 10 for a total period of time ^^ ൌ ∑^ ^^^ .Each time step is generally equal to the gate length of the quantum gates beingat the time step, which can vary between time steps. For example, each time can at most about 50 nanoseconds (ns), 45 ns, 40 ns, 35 ns, 30 ns, 25 ns, 20 ns, 15 ns, 10, ns or less. In most implementations, the control system 200 controls 710 the superconducting circuit 10 for a total period of time less than the coherence time(s) of the superconducting qubits 110 to avoid nonunitary processes resulting from decoherence.

[0138] The sequence of quantum gates corresponds to a sequence of ^^ ൌ 1,2, … ,^^ unitarytransformations |^^^୧୬^ ൌ ^^ே^^ேି^ …^^^|^^୧୬^^. In this example, each unitary transformation(^^^) is a combination of one-qubit quantum gates 812 or a two-qubit quantum gate 822. However, the controlcan also perform one-qubit 812 and two-qubit 822 quantum gates simultaneously such that a unitary transformation (^^^) is a combination of one or more one-qubit quantum gates 812 and one or more two-qubit quantum gates 822. In some implementations, the control system 200 can implement higher-order quantum gates on the superconducting circuit 10, e.g., three-qubit quantum gates, four-qubit quantum gates, five- qubit quantum gates, and so on. Hence, more generally, each unitary transformation can be aAttorney Docket No.56113-0516WO1 one-qubit quantum gate, a multi-qubit quantum gate, a combination of multiple one-qubit quantum gates, a combination of multiple multi-qubit quantum gates, or a combination of one or more one-qubit quantum gates and one or more multi-qubit quantum gates.

[0139] After the sequence of time steps, the control system 200 performs a measurement (readout) 720 of the final state |^^^୧୬ୟ୪^ of the superconducting circuit 10, e.g., by applying ameasurement operator ^^|^^^୧୬ୟ୪^ such as ^^ ൌ ^^ on the superconducting qubits 110, whichcan be read out by control system 200 via readout signals 614.

[0140] FIG.7A is an example process 700 for multiplexed control and readout of the superconducting circuit 10. The process 700 will be described as being performed by the control system 200 of the quantum computer 600 to implement a quantum circuit, e.g., the quantum circuit 600 of FIG.6B.

[0141] The control system 200 controls the superconducting qubits 110 and couplers 110* ofthe superconducting circuit 10 over a sequence of time steps ^^^, ^^ଶ, … , ^^ே (710).

[0142] FIG.7B is a flow chart of the process 710 for controlling the superconducting qubits 110 and couplers 110* over the sequence of time steps.

[0143] At each time step (^^^), the control system performs one or more quantum gate processes 712 and / or 714. Gate process 712 implements a multi-qubit quantum gate at the time step, while g ate process 714 implements a one-qubit quantum gate at the time step. In general, the length of the time step is equal to the gate length of the quantum gates, i.e., the pulse widths of the corresponding control signals used to implement the quantum gates.

[0144] For each multi-qubit quantum gate implemented at the time step:

[0145] The control system 200 generates a respective control signal for the respective flux resonator 120 of each superconducting qubit 110 and coupler 110* involved in the muti-qubit quantum gate (712). Each flux resonator 120 control signal oscillates at the respective flux resonator 120’s fundamental frequency to resonantly drive the flux resonator 120. As described elsewhere herein (e.g., FIG.5), the respective amplitude of each flux resonator 120 control signal corresponds to an appropriate shift of the respective superconducting qubit 110 or coupler 110*’s frequency for implementing the multi-qubit quantum gate with the gate length. The control system 200 may also specify the phases of the flux resonator 120 control signals, e.g., to implement particular magnetic flux and electric charge quadratures in the flux resonators 120.

[0146] For each one-qubit quantum gate implemented at the time step:Attorney Docket No.56113-0516WO1

[0147] The control system 200 generates a respective control signal for the respective superconducting qubit 110 involved in the one-qubit quantum gate (712). The qubit control signal oscillates at the superconducting qubit 110’s qubit frequency to resonantly drive the superconducting qubit 110. The amplitude and phase of the qubit control signal is an appropriate combination for implementing the one-qubit quantum gate with the gate length. For example, control system 200 can specify the amplitude and phase of the qubit control signal to perform a specific rotation (e.g., X, Y, or Z rotations) of the superconducting qubit 110 on the Bloch sphere.

[0148] The control system 200 multiplexes each of the control signals at the time step into a multiplexed control signal (713). Note that control system 200 may generate the multiplexed control signal outright (e.g., using a function or arbitrary waveform generator). Alternatively, the control system 200 may generate each individual control signal separately (e.g., using a respective signal generator) and then multiplex them (e.g., using a multiplexer) into the multiplexed control signal.

[0149] The control system 200 then transmits the multiplexed control signal at the time step to the superconducting circuit 10 (714).

[0150] In some implementations, the control system 200 may receive, from the superconducting circuit 10, a multiplexed readout signal at the time step in response to the multiplexed control signal at the time step. In these cases, the control system 200 can demultiplex the multiplexed readout signal and use the information to, for example, determine the success of the one or more gate processes 712 and / or 714 at the time step (or other information relating to the state of the superconducting circuit 10 at the time step).

[0151] Returning to FIG.7A, the control system 200 reads out the superconducting qubits 110 after the sequence of time steps (720).

[0152] FIG.7C is a flowchart of the process 720 for reading out the superconducting qubits 110 after the sequence of time steps.

[0153] For each superconducting qubit 110 that was controlled over the sequence of time steps:

[0154] The control system 200 generates a respective control signal for the respective readout resonator 130 of the superconducting qubit 110 (721). In the example process 720, the readout resonator 130 control signal oscillates at the readout resonator 130’s fundamental frequency to resonantly drive the readout resonator 130. As mentioned elsewhere herein, the control system 200 can determine the state of the superconducting qubit 110 from a phase difference measurement using such driving. This is typically the most straightforwardAttorney Docket No.56113-0516WO1 measurement approach. However, the readout resonator 130 control signal can also oscillate at one of the readout resonator 130’s dressed frequencies, e.g., for amplitude measurements, or somewhere in between its fundamental and dressed frequencies for phase and / or amplitude measurements.

[0155] The control system 200 multiplexes each of the control signals into a multiplexed control signal (722).

[0156] The control system 200 transmits the multiplexed control signal to the superconducting circuit 200 (724).

[0157] The control system 200 receives, from the superconducting circuit 10, a multiplexed readout signal in response to the multiplexed control signal (725).

[0158] The control system 200 then demultiplexes the multiplexed readout signal into multiple readout signals that include a respective readout signal for each superconducting qubit 110 (725). The respective readout signal for each superconducting qubit 110 oscillates at the respective readout resonator 130’s fundamental frequency. Note that the control system 200 may receive and analyze the readout signals in the multiplexed readout signal together (e.g., using a spectrum analyzer). Alternatively, the control system 200 may demultiplex the multiplexed readout signal into separate readout signals (e.g., using a demultiplexer) and then analyze them individually (e.g., using a respective signal analyzer).

[0159] For each superconducting qubit 110 being read out:

[0160] The control system 200 calculates a respective phase difference between: (i) the respective readout resonator 130 control signal for the superconducting qubit 110, (ii) and the respective readout signal for the superconducting qubit 110 (726). For example, the control system 200 can perform heterodyne detection to calculate the phase difference (as well as amplitude change) between the two signals using a signal analyzer that includes an amplifier, mixer, and local oscillator.

[0161] The control system 200 determines a respective state of the superconducting qubit 110 based on the respective phase difference (727). For example, depending on the configuration of the superconducting circuit 10, a positive or negative phase difference may correspond to the ground state |0^ or the excited state |1^ of the superconducting qubit 110.

[0162] Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. TheAttorney Docket No.56113-0516WO1 term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0163] Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively, or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0164] 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 are possible.

[0165] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. TheAttorney Docket No.56113-0516WO1 apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0166] A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper.

[0167] A 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 computer program can be deployed to be executed on one computer or on multiple 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.

[0168] The processes and logic flows described in this specification can be performed by one or more programmable computers, operating with one or more processors, as appropriate, executing one or more computer programs to perform functions by operating on input 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.Attorney Docket No.56113-0516WO1

[0169] For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, 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.

[0170] Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a random-access memory, or quantum systems suitable for transmitting quantum data, e.g., photons, or combinations thereof.

[0171] The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, 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 computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices.

[0172] Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., rf-SQUID or dc-SQUID), among others.

[0173] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operationsAttorney Docket No.56113-0516WO1 on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors.

[0174] In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements. Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics, and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography).

[0175] During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature.

[0176] In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.Attorney Docket No.56113-0516WO1

[0177] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0178] Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification. OTHER EMBODIMENTS

[0179] In addition to the embodiments described above, the following numbered embodiments are also innovative:

[0180] 1. A superconducting circuit comprising: a superconducting device comprising a superconducting loop interrupted by one or more Josephson junctions; a first microwave resonator inductively coupled to the superconducting loop of the superconducting device; a second microwave resonator capacitively coupled to the superconducting device, wherein the first and second microwave resonators each have a different fundamental frequency; and a microwave transmission line evanescently coupled to each of the first and second microwave resonators.

[0181] 2. The superconducting circuit of embodiment 1, wherein the superconducting device is a superconducting qubit.

[0182] 3. The superconducting circuit of embodiment 2, wherein the superconducting qubit’s qubit frequency is in a range from 2 gigahertz (GHz) to 6 GHz.

[0183] 4. The superconducting circuit of any of embodiments 1-3, wherein the first microwave resonator’s fundamental frequency is greater than the second microwaveAttorney Docket No.56113-0516WO1 resonator’s fundamental frequency, and the second microwave resonator’s fundamental frequency is greater than the superconducting qubit’s qubit frequency.

[0184] 5. The superconducting circuit of any of embodiments 1-4, wherein the first microwave resonator is configured to bias a magnetic flux through the superconducting loop.

[0185] 6. The superconducting circuit of any of embodiments 1-5, wherein the second microwave resonator is configured to bias an electric charge on the superconducting device.

[0186] 7. The superconducting circuit of any of embodiments 1-6, wherein the first microwave resonator is capacitively coupled to the microwave transmission line.

[0187] 8. The superconducting circuit of any of embodiments 1-7, wherein the second microwave resonator is inductively coupled to the microwave transmission line.

[0188] 9. The superconducting circuit of any of embodiments 1-8, wherein the first microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

[0189] 10. The superconducting circuit of any of embodiments 1-8, wherein the first microwave resonator is an open-circuited, half-wavelength, microwave transmission line resonator.

[0190] 11. The superconducting circuit of embodiment 10, wherein the first microwave resonator comprises a loop antenna inductively coupled to the superconducting loop.

[0191] 12. The superconducting circuit of any of embodiments 1-11, wherein the second microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

[0192] 13. The superconducting circuit of embodiment 12, wherein the second microwave resonator comprises a patch antenna capacitively coupled to the superconducting device.

[0193] 14. The superconducting circuit of any of embodiments 1-13, wherein the one or more Josephson junctions are a plurality of Josephson junctions.

[0194] 15. The superconducting circuit of embodiment 14, wherein the superconducting circuit further comprises a ground plane, the superconducting device further comprises a capacitor electrode capacitively coupled to the ground plane, and the superconducting loop is electrically connected to the ground plane and the capacitor electrode of the superconducting device.

[0195] 16. The superconducting circuit of any of embodiments 14-15, wherein the superconducting loop of the superconducting device is a dc-SQUID loop.Attorney Docket No.56113-0516WO1

[0196] 17. The superconducting circuit of any of embodiments 1-16, wherein each Josephson junction has a junction capacitance in a range from 5 femtofarad (fF) to 50 fF, and a critical current in a range from 20 nanoamps (nA) to 100 nA.

[0197] 18. The superconducting circuit of any of embodiments 15-17, wherein a shunt capacitance between the ground plane and the capacitor electrode of the superconducting device is in a range from 50 fF to 200 fF.

[0198] 19. The superconducting circuit of any of embodiments 1-18, wherein a mutual inductance between the first microwave resonator and the superconducting loop of the superconducting device is in a range from 5 picohenry (pH) to 20 pH, and a gate capacitance between the second microwave resonator and the superconducting device is in a range from 1 fF to 10 fF.

[0199] 20. The superconducting circuit of any of embodiments 1-19, wherein the first microwave resonator’s fundamental frequency is in a range from 6 GHz to 12 GHz, and the second microwave resonators’ fundamental frequency is in a range from 4 GHz to 8 GHz.

[0200] 21. A method for multiplexed control and readout of the superconducting circuit of any of embodiments 1-20, the method comprising: controlling the superconducting device over a sequence of time steps, comprising, at one or more of the time steps: generating, at an input of the microwave transmission line, a first control signal oscillating at the first microwave resonator’s fundamental frequency; and reading out the superconducting device after the sequence of time steps, comprising: generating, at the input of the microwave transmission line, a second control signal oscillating at the second microwave resonator’s fundamental frequency; and receiving, at an output of the microwave transmission line, a readout signal oscillating at the second microwave resonator’s fundamental frequency.

[0201] 22. The method of embodiment 21 when also dependent on embodiment 2, wherein at each time step, an intensity of the first control signal at the time step is proportional to a shift of the superconducting qubit’s qubit frequency.

[0202] 23. The method of embodiment 22, wherein at each time step, the first control signal at the time step shifts the superconducting qubit’s qubit frequency by 300 megahertz (MHz) or more.

[0203] 24. The method of any of embodiments 22-23, wherein controlling the superconducting qubit over the sequence of time steps further comprises, at one or more of the time steps: generating, at the input of the microwave transmission line, a third control oscillating at the superconducting qubit’s qubit frequency.Attorney Docket No.56113-0516WO1

[0204] 25. The method of embodiment 24, wherein at each time step, the third control signal at the time step implements a one-qubit quantum gate on the superconducting qubit.

[0205] 26. The method of embodiment 25, wherein at each time step, the one-qubit quantum gate at the time step is a Pauli gate, a Hadamard gate, a phase shift gate, or a rotation gate.

[0206] 27. The method of any of embodiments 24-26, wherein generating, at the input of the microwave transmission line, the first and third control signals comprises, at each time step: multiplexing the first and third control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

[0207] 28. The method of any of embodiments 22-27, wherein reading out the superconducting qubit after the sequence of time steps further comprises: calculating a phase difference between: (i) the second control signal, and (ii) the readout signal; and determining a state of the superconducting qubit after the sequence of time steps based on the phase difference.

[0208] 29. The method of any of embodiments 21-28, wherein each time step has a length of 30 nanoseconds (ns) or less.

[0209] 30. An apparatus comprising: the superconducting circuit of any of embodiments 1- 20; and a control system comprising: one or more control devices; and one or more control lines electrically coupled to the one or more control devices and the superconducting circuit, wherein the control system is configured, during use of the apparatus, to perform the method of any of embodiments 21-29.

[0210] 31. A superconducting circuit comprising: a plurality of superconducting devices each comprising a superconducting loop interrupted by one or more Josephson junctions; a plurality of microwave resonators each inductively coupled to the superconducting loop of a corresponding one of the plurality of superconducting devices, wherein the plurality of microwave resonators each have a different fundamental frequency; and a microwave transmission line evanescently coupled to each of the plurality of microwave resonators.

[0211] 32. The superconducting circuit of embodiment 31, wherein each microwave resonator is configured to bias a magnetic flux through the superconducting loop of the respective superconducting device.

[0212] 33. The superconducting circuit of any of embodiments 31-32, wherein each microwave resonator is capacitively coupled to the microwave transmission line.Attorney Docket No.56113-0516WO1

[0213] 34. The superconducting circuit of any of embodiments 31-33, wherein each microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

[0214] 35. The superconducting circuit of any of embodiments 31-33, wherein each microwave resonator is an open-circuited, half-wavelength, microwave transmission line resonator.

[0215] 36. The superconducting circuit of embodiment 35, wherein each microwave resonator comprises a loop antenna inductively coupled to the superconducting loop of the respective superconducting device.

[0216] 37. The superconducting circuit of any of embodiments 31-36, wherein for each superconducting device, the one or more Josephson junctions are a plurality of Josephson junctions.

[0217] 38. The superconducting circuit of embodiment 37, wherein the superconducting circuit further comprises a ground plane, each superconducting device further comprises a capacitor electrode capacitively coupled to the ground plane, and the superconducting loop of each superconducting device is electrically connected to the ground plane and the capacitor electrode of the superconducting device.

[0218] 39. The superconducting circuit of any of embodiments 37-38, wherein the superconducting loop of each superconducting device is a dc-SQUID loop.

[0219] 40. The superconducting circuit of any of embodiments 31-39, wherein each Josephson junction has a junction capacitance in a range from 5 fF to 50 fF, and a critical current in a range from 20 nA to 100 nA.

[0220] 41. The superconducting circuit of any of embodiments 38-40, wherein a shunt capacitance between the ground plane and the capacitor electrode of each superconducting device is in a range from 50 fF to 200 fF.

[0221] 42. The superconducting circuit of any of embodiments 31-41, wherein a mutual inductance between each microwave resonator and the superconducting loop of the respective superconducting device is in a range from 5 pH to 20 pH.

[0222] 43. The superconducting circuit of any of embodiments 31-32, wherein each microwave resonator’s fundamental frequency is in a range from 6 GHz to 12 GHz.

[0223] 44. The superconducting circuit of any of embodiments 31-43, wherein the plurality of microwave resonators is a plurality of first microwave resonators, the superconducting circuit further comprises a plurality of second microwave resonators each capacitively coupled to a corresponding superconducting device in a subset of the plurality ofAttorney Docket No.56113-0516WO1 superconducting devices, the plurality of second microwave resonators each have a different fundamental frequency, and the microwave transmission line is evanescently coupled to each of the plurality of second microwave resonators.

[0224] 45. The superconducting circuit of embodiment 44, wherein each second microwave resonator is configured to bias an electric charge on the respective superconducting device.

[0225] 46. The superconducting circuit of any of embodiments 44-45, wherein each second microwave resonator is inductively coupled to the microwave transmission line.

[0226] 47. The superconducting circuit of any of embodiments 44-45, wherein each second microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

[0227] 48. The superconducting circuit of embodiment 47, wherein each second microwave resonator comprises a patch antenna capacitively coupled to the respective superconducting device.

[0228] 49. The superconducting circuit of any of embodiments 44-48, wherein a gate capacitance between each second microwave resonator and the respective superconducting device is in a range from 1 fF to 10 fF.

[0229] 50. The superconducting circuit of any of embodiments 44-49, wherein each second microwave resonators’ fundamental frequency is in a range from 4 GHz to 8 GHz.

[0230] 51. The superconducting circuit of any of embodiments 44-50, wherein the subset of the plurality of superconducting devices is an improper subset.

[0231] 52. The superconducting circuit of any of embodiments 44-50, wherein the subset of the plurality of superconducting devices is a proper subset.

[0232] 53. The superconducting circuit of embodiment 52, wherein each superconducting device in a complement of the subset is capacitively coupled to a respective plurality of superconducting devices in the subset.

[0233] 54. The superconducting circuit of embodiment 53, wherein each plurality of superconducting devices in the subset is a respective pair of superconducting devices in the subset.

[0234] 55. The superconducting circuit of any of embodiments 53-54, wherein for each plurality of superconducting devices in the subset, each superconducting device in the plurality is capacitively coupled to one another.

[0235] 56. The superconducting circuit of any of embodiments 53-55, wherein each superconducting device in the subset is a superconducting qubit, and each superconducting device in the complement of the subset is a superconducting coupler.Attorney Docket No.56113-0516WO1

[0236] 57. The superconducting circuit of embodiment 56, wherein for each superconducting qubit: the respective first microwave resonator’s fundamental frequency is greater than the respective second microwave resonator’s fundamental frequency, and the respective second microwave resonator’s fundamental frequency is greater than the superconducting qubit’s qubit frequency.

[0237] 58. The superconducting circuit of any of embodiments 56-57, wherein for each superconducting coupler, the respective first microwave resonator’s fundamental frequency is greater than the superconducting coupler’s coupler frequency.

[0238] 59. The superconducting circuit of any of embodiments 56-58, wherein for each superconducting coupler, the superconducting coupler’s coupler frequency is greater than the qubit frequency of each superconducting qubit in the respective plurality of superconducting qubits.

[0239] 60. The superconducting circuit of embodiment 59, wherein for each superconducting coupler, a respective qubit-coupler strength between the superconducting coupler and each superconducting qubit in the respective plurality of superconducting qubits is dispersive.

[0240] 61. A method for multiplexed control and readout of the superconducting circuit of any of embodiments 31-60, the method comprising: controlling the plurality of superconducting devices over a sequence of time steps, comprising, at one or more of the time steps for each superconducting device: generating, at an input of the microwave transmission line, a respective control signal oscillating at the respective microwave resonator’s fundamental frequency.

[0241] 62. The method of embodiment 61, wherein generating, at the input of the microwave transmission line, the respective control signal for each superconducting device comprises, at each time step: multiplexing each of the control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

[0242] 63. The method of any of embodiments 61-62 when also dependent on embodiment 44, wherein the control signals are first control signals, and the method further comprises: reading out each superconducting device in the subset after the sequence of time steps, comprising, for each superconducting device in the subset: generating, at the input of the microwave transmission line, a respective second control signal oscillating at the respective second microwave resonator’s fundamental frequency; and receiving, at an output of the microwave transmission line, a respective readout signal oscillating at the respective second microwave resonator’s fundamental frequency.Attorney Docket No.56113-0516WO1

[0243] 64. The method of embodiment 63, wherein generating, at the input of the microwave transmission line, the respective second control signal for each superconducting device in the subset comprises: multiplexing each of the second control signals into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

[0244] 65. The method of embodiment 64, wherein receiving, at the output of the microwave transmission line, the respective readout signal for each superconducting device in the subset comprises: receiving a multiplexed readout signal at the output of the microwave transmission line; and demultiplexing the multiplexed readout signal into each of the readout signals.

[0245] 66. The method of any of embodiments 63-65 when also dependent on embodiment 56, wherein for each superconducting qubit, a respective intensity of the respective first control signal is proportional to a respective shift of the superconducting qubit’s qubit frequency, and for each superconducting coupler, a respective intensity of the respective first control signal is proportional to a respective shift of the superconducting coupler’s coupler frequency.

[0246] 67. The method of embodiment 66, wherein at each time step, the first control signals at the time step implement one or more multi-qubit quantum gates on the superconducting qubits.

[0247] 68. The method of embodiment 67, wherein at each time step, each multi-qubit quantum gate at the time step is a controlled gate, a two-qubit interaction gate, a swap gate, a Toffoli gate, or a Fredkin gate.

[0248] 69. The method of any of embodiments 66-68, wherein controlling the plurality of superconducting devices over the sequence of time steps further comprises, at one or more of the time steps for each superconducting qubit: generating, at the input of the microwave transmission line, a respective third control signal oscillating at the superconducting qubit’s qubit frequency.

[0249] 70. The method of embodiment 69, wherein at each time step, the third control signals at the time step implement one or more one-qubit quantum gates on the superconducting qubits.

[0250] 71. The method of embodiment 70, wherein at each time step, each one-qubit quantum gate at the time step is a Pauli gate, a Hadamard gate, a phase shift gate, or a rotation gateAttorney Docket No.56113-0516WO1

[0251] 72. The method of any of embodiments 69-71, wherein generating, at the input of the microwave transmission line, the respective first control signal for each superconducting device and the respective third control signal for each superconducting qubit comprises, at each time step: multiplexing each of the first and third control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

[0252] 73. The method of any of embodiments 66-72, wherein reading out each superconducting qubit after the sequence of time steps further comprises, for each superconducting qubit: calculating a respective phase difference between: (i) the respective second control signal, and (ii) the respective readout signal; and determining a respective state of the superconducting qubit after the sequence of time steps based on the respective phase difference.

[0253] 74. The method of any of embodiments 61-73, wherein each time step has a length of 30 nanoseconds (ns) or less.

[0254] 75. An apparatus comprising: the superconducting circuit of any of embodiments 31- 60; and a control system comprising: one or more control devices; and one or more control lines electrically coupled to the one or more control devices and the superconducting circuit, wherein the control system is configured, during use of the apparatus, to perform the method of any of embodiments 61-74.

[0255] 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.

[0256] 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 beAttorney Docket No.56113-0516WO1 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 into a single software product or packaged into multiple software products.

[0257] 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.

[0258] What is claimed is:

Claims

Attorney Docket No.56113-0516WO1 CLAIMS 1. A superconducting circuit, comprising: a superconducting device comprising a superconducting loop interrupted by one or more Josephson junctions; a first microwave resonator inductively coupled to the superconducting loop of the superconducting device; a second microwave resonator capacitively coupled to the superconducting device, wherein the first and second microwave resonators each have a different fundamental frequency; and a microwave transmission line evanescently coupled to each of the first and second microwave resonators.

2. The superconducting circuit of claim 1, wherein the superconducting device is a superconducting qubit.

3. The superconducting circuit of claim 2, wherein the superconducting qubit’s qubit frequency is in a range from 2 gigahertz (GHz) to 6 GHz.

4. The superconducting circuit of any of claims 1-3, wherein: the first microwave resonator’s fundamental frequency is greater than the second microwave resonator’s fundamental frequency, and the second microwave resonator’s fundamental frequency is greater than the superconducting qubit’s qubit frequency.

5. The superconducting circuit of any of claims 1-4, wherein the first microwave resonator is configured to bias a magnetic flux through the superconducting loop.

6. The superconducting circuit of any of claims 1-5, wherein the second microwave resonator is configured to bias an electric charge on the superconducting device.

7. The superconducting circuit of any of claims 1-6, wherein the first microwave resonator is capacitively coupled to the microwave transmission line.

8. The superconducting circuit of any of claims 1-7, wherein the second microwave resonator is inductively coupled to the microwave transmission line.Attorney Docket No.56113-0516WO1 9. The superconducting circuit of any of claims 1-8, wherein the first microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

10. The superconducting circuit of any of claims 1-8, wherein the first microwave resonator is an open-circuited, half-wavelength, microwave transmission line resonator.

11. The superconducting circuit of claim 10, wherein the first microwave resonator comprises a loop antenna inductively coupled to the superconducting loop.

12. The superconducting circuit of any of claims 1-11, wherein the second microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

13. The superconducting circuit of claim 12, wherein the second microwave resonator comprises a patch antenna capacitively coupled to the superconducting device.

14. The superconducting circuit of any of claims 1-13, wherein the one or more Josephson junctions are a plurality of Josephson junctions.

15. The superconducting circuit of claim 14, wherein: the superconducting circuit further comprises a ground plane, the superconducting device further comprises a capacitor electrode capacitively coupled to the ground plane, and the superconducting loop is electrically connected to the ground plane and the capacitor electrode of the superconducting device.

16. The superconducting circuit of any of claims 14-15, wherein the superconducting loop of the superconducting device is a dc-SQUID loop.

17. The superconducting circuit of any of claims 1-16, wherein each Josephson junction has a junction capacitance in a range from 5 femtofarad (fF) to 50 fF, and a critical current in a range from 20 nanoamps (nA) to 100 nA.

18. The superconducting circuit of any of claims 15-17, wherein a shunt capacitance between the ground plane and the capacitor electrode of the superconducting device is in a range from 50 fF to 200 fF.

19. The superconducting circuit of any of claims 1-18, wherein:Attorney Docket No.56113-0516WO1 a mutual inductance between the first microwave resonator and the superconducting loop of the superconducting device is in a range from 5 picohenry (pH) to 20 pH, and a gate capacitance between the second microwave resonator and the superconducting device is in a range from 1 fF to 10 fF.

20. The superconducting circuit of any of claims 1-19, wherein the first microwave resonator’s fundamental frequency is in a range from 6 GHz to 12 GHz, and the second microwave resonators’ fundamental frequency is in a range from 4 GHz to 8 GHz.

21. A method for multiplexed control and readout of the superconducting circuit of any of claims 1-20, the method comprising: controlling the superconducting device over a sequence of time steps, comprising, at one or more of the time steps: generating, at an input of the microwave transmission line, a first control signal oscillating at the first microwave resonator’s fundamental frequency; and reading out the superconducting device after the sequence of time steps, comprising: generating, at the input of the microwave transmission line, a second control signal oscillating at the second microwave resonator’s fundamental frequency; and receiving, at an output of the microwave transmission line, a readout signal oscillating at the second microwave resonator’s fundamental frequency.

22. The method of claim 21 when also dependent on claim 2, wherein at each time step, an intensity of the first control signal at the time step is proportional to a shift of the superconducting qubit’s qubit frequency.

23. The method of claim 22, wherein at each time step, the first control signal at the time step shifts the superconducting qubit’s qubit frequency by 300 megahertz (MHz) or more.

24. The method of any of claims 22-23, wherein controlling the superconducting qubit over the sequence of time steps further comprises, at one or more of the time steps: generating, at the input of the microwave transmission line, a third control oscillating at the superconducting qubit’s qubit frequency.

25. The method of claim 24, wherein at each time step, the third control signal at the time step implements a one-qubit quantum gate on the superconducting qubit.Attorney Docket No.56113-0516WO1 26. The method of claim 25, wherein at each time step, the one-qubit quantum gate at the time step is a Pauli gate, a Hadamard gate, a phase shift gate, or a rotation gate.

27. The method of any of claims 24-26, wherein generating, at the input of the microwave transmission line, the first and third control signals comprises, at each time step: multiplexing the first and third control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

28. The method of any of claims 22-27, wherein reading out the superconducting qubit after the sequence of time steps further comprises: calculating a phase difference between: (i) the second control signal, and (ii) the readout signal; and determining a state of the superconducting qubit after the sequence of time steps based on the phase difference.

29. The method of any of claims 21-28, wherein each time step has a length of 30 nanoseconds (ns) or less.

30. An apparatus, comprising: the superconducting circuit of any of claims 1-20; and a control system comprising: one or more control devices; and one or more control lines electrically coupled to the one or more control devices and the superconducting circuit, wherein the control system is configured, during use of the apparatus, to perform the method of any of claims 21-29.

31. A superconducting circuit, comprising: a plurality of superconducting devices each comprising a superconducting loop interrupted by one or more Josephson junctions; a plurality of microwave resonators each inductively coupled to the superconducting loop of a corresponding one of the plurality of superconducting devices, wherein the plurality of microwave resonators each have a different fundamental frequency; andAttorney Docket No.56113-0516WO1 a microwave transmission line evanescently coupled to each of the plurality of microwave resonators.

32. The superconducting circuit of claim 31, wherein each microwave resonator is configured to bias a magnetic flux through the superconducting loop of the respective superconducting device.

33. The superconducting circuit of any of claims 31-32, wherein each microwave resonator is capacitively coupled to the microwave transmission line.

34. The superconducting circuit of any of claims 31-33, wherein each microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

35. The superconducting circuit of any of claims 31-33, wherein each microwave resonator is an open-circuited, half-wavelength, microwave transmission line resonator.

36. The superconducting circuit of claim 35, wherein each microwave resonator comprises a loop antenna inductively coupled to the superconducting loop of the respective superconducting device.

37. The superconducting circuit of any of claims 31-36, wherein for each superconducting device, the one or more Josephson junctions are a plurality of Josephson junctions.

38. The superconducting circuit of claim 37, wherein: the superconducting circuit further comprises a ground plane, each superconducting device further comprises a capacitor electrode capacitively coupled to the ground plane, and the superconducting loop of each superconducting device is electrically connected to the ground plane and the capacitor electrode of the superconducting device.

39. The superconducting circuit of any of claims 37-38, wherein the superconducting loop of each superconducting device is a dc-SQUID loop.

40. The superconducting circuit of any of claims 31-39, wherein each Josephson junction has a junction capacitance in a range from 5 fF to 50 fF, and a critical current in a range from 20 nA to 100 nA.Attorney Docket No.56113-0516WO1 41. The superconducting circuit of any of claims 38-40, wherein a shunt capacitance between the ground plane and the capacitor electrode of each superconducting device is in a range from 50 fF to 200 fF.

42. The superconducting circuit of any of claims 31-41, wherein a mutual inductance between each microwave resonator and the superconducting loop of the respective superconducting device is in a range from 5 pH to 20 pH.

43. The superconducting circuit of any of claims 31-32, wherein each microwave resonator’s fundamental frequency is in a range from 6 GHz to 12 GHz.

44. The superconducting circuit of any of claims 31-43, wherein: the plurality of microwave resonators is a plurality of first microwave resonators, the superconducting circuit further comprises a plurality of second microwave resonators each capacitively coupled to a corresponding superconducting device in a subset of the plurality of superconducting devices, the plurality of second microwave resonators each have a different fundamental frequency, and the microwave transmission line is evanescently coupled to each of the plurality of second microwave resonators.

45. The superconducting circuit of claim 44, wherein each second microwave resonator is configured to bias an electric charge on the respective superconducting device.

46. The superconducting circuit of any of claims 44-45, wherein each second microwave resonator is inductively coupled to the microwave transmission line.

47. The superconducting circuit of any of claims 44-45, wherein each second microwave resonator is a short-circuited, quarter-wavelength, microwave transmission line resonator.

48. The superconducting circuit of claim 47, wherein each second microwave resonator comprises a patch antenna capacitively coupled to the respective superconducting device.

49. The superconducting circuit of any of claims 44-48, wherein a gate capacitance between each second microwave resonator and the respective superconducting device is in a range from 1 fF to 10 fF.Attorney Docket No.56113-0516WO1 50. The superconducting circuit of any of claims 44-49, wherein each second microwave resonators’ fundamental frequency is in a range from 4 GHz to 8 GHz.

51. The superconducting circuit of any of claims 44-50, wherein the subset of the plurality of superconducting devices is an improper subset.

52. The superconducting circuit of any of claims 44-50, wherein the subset of the plurality of superconducting devices is a proper subset.

53. The superconducting circuit of claim 52, wherein each superconducting device in a complement of the subset is capacitively coupled to a respective plurality of superconducting devices in the subset.

54. The superconducting circuit of claim 53, wherein each plurality of superconducting devices in the subset is a respective pair of superconducting devices in the subset.

55. The superconducting circuit of any of claims 53-54, wherein for each plurality of superconducting devices in the subset, each superconducting device in the plurality is capacitively coupled to one another.

56. The superconducting circuit of any of claims 53-55, wherein: each superconducting device in the subset is a superconducting qubit, and each superconducting device in the complement of the subset is a superconducting coupler.

57. The superconducting circuit of claim 56, wherein for each superconducting qubit: the respective first microwave resonator’s fundamental frequency is greater than the respective second microwave resonator’s fundamental frequency, and the respective second microwave resonator’s fundamental frequency is greater than the superconducting qubit’s qubit frequency.

58. The superconducting circuit of any of claims 56-57, wherein for each superconducting coupler, the respective first microwave resonator’s fundamental frequency is greater than the superconducting coupler’s coupler frequency.Attorney Docket No.56113-0516WO1 59. The superconducting circuit of any of claims 56-58, wherein for each superconducting coupler, the superconducting coupler’s coupler frequency is greater than the qubit frequency of each superconducting qubit in the respective plurality of superconducting qubits.

60. The superconducting circuit of claim 59, wherein for each superconducting coupler, a respective qubit-coupler strength between the superconducting coupler and each superconducting qubit in the respective plurality of superconducting qubits is dispersive.Attorney Docket No.56113-0516WO1 61. A method for multiplexed control and readout of the superconducting circuit of any of claims 31-60, the method comprising: controlling the plurality of superconducting devices over a sequence of time steps, comprising, at one or more of the time steps for each superconducting device: generating, at an input of the microwave transmission line, a respective control signal oscillating at the respective microwave resonator’s fundamental frequency.

62. The method of claim 61, wherein generating, at the input of the microwave transmission line, the respective control signal for each superconducting device comprises, at each time step: multiplexing each of the control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

63. The method of any of claims 61-62 when also dependent on claim 44, wherein the control signals are first control signals, and the method further comprises: reading out each superconducting device in the subset after the sequence of time steps, comprising, for each superconducting device in the subset: generating, at the input of the microwave transmission line, a respective second control signal oscillating at the respective second microwave resonator’s fundamental frequency; and receiving, at an output of the microwave transmission line, a respective readout signal oscillating at the respective second microwave resonator’s fundamental frequency.

64. The method of claim 63, wherein generating, at the input of the microwave transmission line, the respective second control signal for each superconducting device in the subset comprises: multiplexing each of the second control signals into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.Attorney Docket No.56113-0516WO1 65. The method of claim 64, wherein receiving, at the output of the microwave transmission line, the respective readout signal for each superconducting device in the subset comprises: receiving a multiplexed readout signal at the output of the microwave transmission line; and demultiplexing the multiplexed readout signal into each of the readout signals.

66. The method of any of claims 63-65 when also dependent on claim 56, wherein: for each superconducting qubit, a respective intensity of the respective first control signal is proportional to a respective shift of the superconducting qubit’s qubit frequency, and for each superconducting coupler, a respective intensity of the respective first control signal is proportional to a respective shift of the superconducting coupler’s coupler frequency.

67. The method of claim 66, wherein at each time step, the first control signals at the time step implement one or more multi-qubit quantum gates on the superconducting qubits.

68. The method of claim 67, wherein at each time step, each multi-qubit quantum gate at the time step is a controlled gate, a two-qubit interaction gate, a swap gate, a Toffoli gate, or a Fredkin gate.

69. The method of any of claims 66-68, wherein controlling the plurality of superconducting devices over the sequence of time steps further comprises, at one or more of the time steps for each superconducting qubit: generating, at the input of the microwave transmission line, a respective third control signal oscillating at the superconducting qubit’s qubit frequency.

70. The method of claim 69, wherein at each time step, the third control signals at the time step implement one or more one-qubit quantum gates on the superconducting qubits.

71. The method of claim 70, wherein at each time step, each one-qubit quantum gate at the time step is a Pauli gate, a Hadamard gate, a phase shift gate, or a rotation gate 72. The method of any of claims 69-71, wherein generating, at the input of the microwave transmission line, the respective first control signal for each superconducting device and the respective third control signal for each superconducting qubit comprises, at each time step:Attorney Docket No.56113-0516WO1 multiplexing each of the first and third control signals at the time step into a multiplexed control signal; and generating the multiplexed control signal at the input of the microwave transmission line.

73. The method of any of claims 66-72, wherein reading out each superconducting qubit after the sequence of time steps further comprises, for each superconducting qubit: calculating a respective phase difference between: (i) the respective second control signal, and (ii) the respective readout signal; and determining a respective state of the superconducting qubit after the sequence of time steps based on the respective phase difference.

74. The method of any of claims 61-73, wherein each time step has a length of 30 nanoseconds (ns) or less.

75. An apparatus, comprising: the superconducting circuit of any of claims 31-60; and a control system comprising: one or more control devices; and one or more control lines electrically coupled to the one or more control devices and the superconducting circuit, wherein the control system is configured, during use of the apparatus, to perform the method of any of claims 61-74.

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