Techniques for performing entangling qubit gates and related systems and methods

Abstract

Techniques are described for coupling superconducting qubits, and for performing entangling operations on superconducting qubits, which simplify the control hardware for qubits while providing for high fidelity operations. Superconducting qubits may be coupled together via a tunable coupler that may be controlled to adjust the energy of one or more states of the coupler, allowing the extent to which states of the coupler couple to states of the qubits to be adjusted. For instance, the coupler may be controlled to turn the coupling between the qubits on and off. Moreover, entangling gates between the qubits may be performed by driving the coupler and / or the qubits between their different states.

Description

TECHNIQUES FOR PERFORMING ENTANGLING QUBIT GATES AND RELATED SYSTEMS AND METHODSTECHNICAL FIELD

[0001] This disclosure relates to manipulating qubits in a quantum computing system, and in particular for controlling qubits to perform entangling (e.g., two-qubit) gates.BACKGROUND

[0002] Quantum computing platforms promise to provide solutions to many computationally intractable problems. In a quantum computing platform, information is stored in quantum bits or “qubits,” and the power of the platform generally increases with the number of qubits that can be independently and simultaneously controlled. In quantum computing platforms comprising qubits such as trapped ions or neutral atoms, directed electromagnetic waves (e.g., microwaves, optical beams) implement independent qubit manipulations, while platforms comprising qubits such as electron dots or superconducting circuits use guided RF or microwave beams.SUMMARY

[0003] According to some aspects, the techniques described herein relate to a system including: a plurality of qubits including a first qubit, a second qubit and a third qubit; a plurality of couplers each coupled to multiple qubits of the plurality of qubits, the plurality of couplers including a first coupler coupled to the first qubit and to the second qubit, and a second coupler coupled to the second qubit and to the third qubit; a microwave drive line; and at least one controller configured to: receive a microwave signal from the microwave drive line; generate a first pulse envelope signal; generate a first microwave pulse by applying the first pulse envelope signal to the microwave signal; apply the first microwave pulse to the first coupler; generate a second pulse envelope signal; generate a second microwave pulse by applying the second pulse- 1 -ACTIVE 705838443v1envelope signal to the microwave signal; and apply the second microwave pulse to the second coupler.

[0004] According to some aspects, the techniques described herein relate to a method including: generating, by at least one digital logic controller, a first microwave pulse by applying a first pulse envelope signal to a microwave signal; applying, by the at least one digital logic controller, the first microwave pulse to a first coupler, the first coupler coupled to a first qubit and to a second qubit; generating, by the at least one digital logic controller, a second microwave pulse by applying a second pulse envelope signal to the microwave signal; and applying, by the at least one digital logic controller, the second microwave pulse to a second coupler, the second coupler coupled to the second qubit and to a third qubit.

[0005] The foregoing apparatus and method embodiments may be implemented with any suitable combination of aspects, features, and acts described above or in further detail below. These and other aspects, embodiments, and features of the present teachings can be more fully understood from the following description in conjunction with the accompanying drawings.BRIEF DESCRIPTION OF DRAWINGS

[0006] Various aspects and embodiments will be described with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing.

[0007] FIG. 1 is a schematic diagram of an example quantum computing system, according to some embodiments;

[0008] FIG. 2 depicts a qubit-coupler-qubit system, according to some embodiments;

[0009] FIG. 3 shows a schematic diagram of the states of the qubit-coupler-qubit system shown in FIG. 2, according to some embodiments;

[0010] FIG. 4 depicts the relationship between transition energy and coupler flux bias for three illustrative state transitions, according to some embodiments;- 2 -ACTIVE 705838443v1

[0011] FIGs. 5A-5D depict illustrative energy level diagrams of a qubit-coupler- qubit system, according to some embodiments;

[0012] FIG. 6 depicts a fluxonium qubit, according to some embodiments;

[0013] FIG. 7A depicts an illustrative implementation of a qubit-coupler-qubit system, according to some embodiments;

[0014] FIGs. 7B and 7C depict illustrative circuits that can each be used as a subcircuit within the system of FIG. 7A, according to some embodiments;

[0015] FIG. 8 is a flowchart of a method of performing an entangling gate, according to some embodiments;

[0016] FIG. 9 is a schematic of an illustrative quantum computing system in which entangling gates may be performed between qubits using a shared microwave source, according to some embodiments;

[0017] FIGS. 10A-C are schematic diagrams of two-qubit gate pulse sequences to drive an example quantum computing system, according to some embodiments;

[0018] FIG. 11 is a schematic diagram of an example quantum computing system, according to some embodiments;

[0019] FIG. 12A is a schematic diagram of an example qubit coupling system comprising multiple modes, according to some embodiments;

[0020] FIG. 12B-12F are schematic diagrams of energy levels associated with example qubit coupling systems, according to some embodiments;

[0021] FIG. 13A depicts a first illustrative implementation of a qubit-coupler-qubit system with a dual-mode coupler, according to some embodiments;

[0022] FIG. 13B depicts a second illustrative implementation of a qubit-coupler- qubit system with a dual -mode coupler, according to some embodiments;

[0023] FIGs. 13C-13E depict illustrative circuits that can each be used as a subcircuit within the systems of FIGs. 13A and 13B, according to some embodiments;

[0024] FIGS. 14A-B are schematic diagrams of example quantum computing systems comprising coupler interconnects between quantum processor chips, according to some embodiments;

[0025] FIGS. 14C-D are schematic diagrams of example quantum computing systems comprising coupler interconnects between quantum processor chips, according to some embodiments; and- 3 -ACTIVE 705838443v1

[0026] FIG. 15 illustrates an example of a computing system environment on which aspects of the disclosure may be implemented.DETAILED DESCRIPTION

[0027] Qubits can be implemented in superconducting circuits that are engineered to exhibit two or more discrete quantum states at different levels of energy.Superconducting qubits typically include one or more non-linear devices, such as Josephson junctions, so that only desired transitions between quantum states can be stimulated. Superconducting circuits also have the advantage of being non-dissipative at low temperatures.

[0028] There are several different types of superconducting qubits that exhibit distinct energy states such that two of the states can be mapped to the logical quantum states 10) and 11). For instance, a charge qubit exhibits states that correspond to different discrete amounts of charge in a small superconducting area, whereas a flux qubit exhibits energy states that correspond to different persistent current states around a superconducting loop.

[0029] One type of superconducting qubit is the fluxonium qubit. Fluxonium qubits offer inherent noise protection against environmental noise sources, allowing for longer coherence times compared to other superconducting qubits like the transmon. For this reason, a fluxonium qubit is considered a promising building block for low-error quantum computers. However, maintaining long coherence times for fluxonium qubits, as well as the high fidelity of logic gates, can be challenging to implement in a system comprising a large number of coupled fluxonium qubits.

[0030] In addition, control of superconducting qubits conventionally requires a great deal of physical overhead, both in physical space and in thermal load. The large number of cables required to route signals between qubits and room temperature, in addition to the cooling requirements of the cables, likely imposes physical space limits on the potential size of quantum processors of thousands of qubits. Yet, by most estimates, hundreds of thousands to millions of qubits will be needed to perform practically useful quantum computations.- 4 -ACTIVE 705838443v1

[0031] The inventors have recognized and appreciated techniques for coupling superconducting qubits, and for performing entangling operations on superconducting qubits, which simplify the control hardware for qubits while providing for high fidelity operations. In particular, the superconducting qubits may be coupled together via a tunable coupler that may be controlled to adjust the energy of one or more states of the coupler, allowing the extent to which states of the coupler couple to states of the qubits to be adjusted. For instance, the coupler may be controlled to turn the coupling between the qubits on and off. Moreover, entangling gates between the qubits may be performed by driving the coupler and / or the qubits between their different states.

[0032] According to some embodiments, the superconducting qubits and the coupler are implemented as superconducting circuits that exhibit different flux states. For instance, the superconducting qubits may be flux qubits, such as fluxonium qubits, and the coupler may be a superconducting circuit that includes a superconducting loop through which a magnetic flux is threaded and around which a persistent current may be generated. In this example, the energy of one or more states of the coupler may be adjusted by adjusting the flux through the coupler, which can increase or decrease an effective coupling between qubits that are coupled together via the coupler. For example, where two states are far apart in energy, the coupling between these states may be different compared with a situation in which these states have been manipulated to be much closer together in energy.

[0033] Entangling gates may be applied to the qubits by driving either or both qubits and / or the coupler between states of the qubit-qubit-coupler system. This approach has an advantage that the flux qubits may be biased to their so-called ‘sweet spot’ and this bias maintained during the entangling gate. In other approaches, a flux qubit may be driven away from the sweet spot during gates, which may compromise qubit coherence and / or reduce fidelity of the gates. By adjusting the flux bias of the coupler, but not the qubits, these challenges may be avoided.

[0034] In addition, the inventors have recognized and appreciated techniques for driving the qubit(s) and / or the coupler during entangling gates that greatly simplify the control hardware. Conventional microwave control is expensive and cumbersome as it requires advanced microprocessors to generate high-frequency microwave signals and- 5 -ACTIVE 705838443v1bulky coaxial cables to deliver these signals without loss. There is also a practical limit to the number of wires that can be connected to the superconducting circuits, as the heat dissipation and / or load from these wires can exceed the cooling capacity of the cryostat. The techniques described herein utilize digital logic to produce microwave pulses from a shared microwave signal, which can be configured to drive desired transitions between the coupler and / or qubit(s) energy state(s). In this approach, a single microwave source may be shared by many, or even all, qubits and couplers, leading to much simpler control hardware.

[0035] This above-described combination of a signal that adjusts the flux bias of a coupler (which is referred to herein as a baseband drive or a baseband signal), and a microwave signal that drives qubits and / or couplers between energy states may be referred to herein as Baseband- and Microwave-Activated Phase (BMAP). According to some embodiments, the baseband signals may be generated by superconducting digital logic, which generates an analog signal that adjusts the flux bias of a coupler, whereas the microwave signals may be generated by superconducting digital logic that applies an envelope to a shared microwave source signal according to digital values, and produces an analog microwave pulse that drives the qubit(s) and / or coupler between energy states.

[0036] Following below are more detailed descriptions of various concepts related to, and embodiments of, techniques for coupling superconducting qubits and for performing entangling operations on superconducting qubits. It should be appreciated that various aspects described herein may be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only. In addition, the various aspects described in the embodiments below may be used alone or in any combination, and are not limited to the combinations explicitly described herein.

[0037] FIG. 1 shows an example quantum computing system 100 comprising qubits 102, which includes multiple qubits with at least some interconnected by a coupler as described herein. The qubits 102 are arranged within housing 104, and send and receive signals to / from a control instrumentation system 108 via a digital signal interface 106.- 6 -ACTIVE 705838443v1

[0038] As referred to herein, a “qubit” includes any multi-level quantummechanical system capable of being controlled within a quantum computing system, such as quantum computing system 100. The quantum states of the qubit may for instance include electronic states, polarization states, vibrational states, rotational states, or spin states. As referred to herein, a “superconducting qubit” includes any superconducting electronic circuit that may be operated as a multi-level quantummechanical system, such as a charge qubit (e.g., a transmon), a flux qubit (e.g., a fluxonium qubit), or a phase qubit. The term “qubit” refers to a physical implementation of a quantum system, and so for example “superconducting qubit” refers to a superconducting circuit that may be operated as a qubit.

[0039] Qubits 102 can be initialized and brought into superpositions in a controlled manner to perform quantum computation. A single-qubit gate can be used to apply a quantum operation that changes the state of a single qubit. A qubit can also be entangled with one or more other qubits such that the qubits form an entangled state. A multi -qubit gate (e.g., a two-qubit gate) can be used to apply a quantum operation that changes the states of qubits at its input, for example, to bring the qubits into a particular entangled state or to otherwise change the states of the qubits. When a qubit is measured, the wave function associated with the qubit collapses into one of the states in the superposition according to a probability that is based on the wave function. Combinations of quantum logic gates and measurement enable the realization of quantum algorithms, which can be specified using one or more collections of interconnected quantum gates (also called “quantum circuit programs” which can represent a high-level specification of operations performed on a physical device, but are generally different from the actual physical implementation of qubit circuits or other quantum device circuitry). In some systems, multiple physical qubits can be treated as representing a single logical qubit or error- corrected qubit, and quantum algorithms can be performed with respect to the logical qubits or error-corrected qubits.

[0040] In the example of FIG. 1, qubits 102 may include circuitry to connect the qubits and couplers to external signal pathways. Such circuitry may include transmission lines, resonators, filters, charge antennas, and flux antennas, for example. The qubits 102- 7 -ACTIVE 705838443v1can be operated as a quantum processor (i.e., a quantum chip) that is housed in a housing 104 (e.g., a cryogenically cooled chamber).

[0041] In the example of FIG. 1, a digital signal interface 106 is configured to receive digital control signals for performing qubit control and readout, as described in more detail below. A control and instrumentation system 108 can provide signals over the digital signal interface 106. In some implementations, the control and instrumentation system 108 can bypass the digital signal interface 106 and provide control signals directly to the qubits 102. The control and instrumentation system 108 includes a memory 110 that acts as a storage medium and may store information related to the control and instrumentation system 108 and the qubits 102. For example, the memory 110 may store a program specification (e.g., a quantum circuit program) specifying one or more algorithms comprising quantum operations.

[0042] In the example of FIG. 1, the digital signal interface 106 serves as a secondary control and instrumentation system to perform tasks that may require or benefit from real-time processing where a small signal latency between the quantum processor and the instrumentation system may be desirable (e.g., to perform real-time error decoding for fault-tolerant quantum computing). The control and instrumentation system 108 may indirectly (e.g., via the digital signal interface 106) or directly perform quantum operations on one or more of the qubits in the qubits 102 by applying coupling and transformation operations to a plurality of quantum states associated with the qubits 102, according to the program specification. For example, a radio-frequency (RF) pulse may be applied to one or more of the qubits 102 by or via the digital signal interface to alter the state of one or more of the qubits, such as producing a superposition of states in a qubit, or by driving the qubit from one state to another.

[0043] Referring to FIG. 1, the quantum processor implemented by the qubits 102 can include qubits implemented as an on-chip superconducting circuit. The qubits of the quantum processor can include a variety of types of electrical circuits. For example, the electrical circuits can include superconducting material with insulating barriers, which may be in the form of a Josephson junction. The circuits can be implemented as superconducting quantum interference devices (SQUIDs) (either direct current (DC), or radio-frequency (RF) SQUIDs), transmon qubits comprising Josephson junctions- 8 -ACTIVE 705838443v1shunted by capacitive elements, and / or fluxonium qubits comprising Josephson junctions shunted by inductive and capacitive elements.

[0044] In some implementations of a quantum computing system, the individual qubit circuits of the qubits are fluxonium qubit circuits, which can be implemented as inductive loops containing a single, small Josephson junction connected to a large inductor, such as one formed by an array of multiple large Josephson junctions or formed from a particular material with kinetic inductance (e.g., granular aluminum, disordered superconducting nitrides). In fluxonium qubits, the qubit circuit is shunted by a capacitance and its properties (e.g., its operating frequency) are set by specific circuit parameters (e.g., inductance, capacitance, etc.). Some properties of fluxonium qubits also can be tuned in-situ by an external magnetic flux applied to one or more inductive loops associated with the fluxonium qubits. A fluxonium qubit circuit can have exceptionally long coherence times compared to the typical gate times for superconducting qubits, especially when operated at low frequencies. In some examples, operation at low frequencies (e.g., frequencies around or below 1 GHz), also referred to as baseband control, can be used to facilitate high-fidelity single- and multi-qubit gate operations, which can simplify the quantum system by removing high-frequency signal generators and cables. Operation at high frequencies (e.g., frequencies around or above 1GHz) is referred to as micro wave control.

[0045] The superconducting phase (pta node i in a superconducting circuit is the phase of the collective quantum wavefunction of the Cooper pairs at that node. It is a macroscopic quantum variable that encapsulates the quantum mechanical properties of the superconducting state. The Hamiltonian of a superconducting circuit can be described in terms of superconducting phases at each node, superconducting phase differences between nodes, or the sum of superconducting phases of nodes. For example, a transmon superconducting qubit, formed by a Josephson junction with a Josephson energy Ej shunted by a capacitor with capacitance C, can be described by the following Hamiltonian: H = ~ Ej cos <p where <p is a quantum operator describing thesuperconducting phase difference across the Josephson junction. In this case, the transmon qubit quantum state, or mode as defined below, is associated with the- 9 -ACTIVE 705838443v1superconducting phase difference between the two nodes, which are connected via a Josephson junction shunted by a capacitor.

[0046] A fluxonium qubit circuit comprises a Josephson junction with the Josephson energy Ej shunted by a capacitance with charging energy Ec, and an inductance with inductive energy EL. The Hamiltonian H of a fluxonium qubit circuit can be written asHere, h is the Planck constant and <pext= 2 zrext / <b0is the reduced magnetic flux threading the superconducting loop, which is interrupted by the Josephson junction and the inductor. The Cooper pair number operator n is associated with the number of Cooper electron pairs that have tunneled across the Josephson junction, and the superconducting phase operator <p is associated with phase difference across the Josephson junction. In some implementations of the fluxonium qubit circuit, Ej ranges from 1 GHz to 10 GHz, Ecranges from 0. 1 GHz to 2 GHz and ELranges from 0. 1 GHz to 2 GHz to ensure that the operation frequency (which is associated with the energy difference between the ground state |0> and the first excited state 11>) is around or below 1 GHz.

[0047] In the example of FIG. 1, the control and instrumentation system 108 and the digital signal interface 106 is configured for signal generation and data acquisition. Operating a quantum processor (e.g., the qubits 102) may be performed by using control electronics to generate the signals that control and probe the quantum processor, which can comprise elements such as qubits, couplers, and resonators. The control and instrumentation system 108 may include waveform generators that can generate low frequency signals (e.g., < 1 GHz) to provide baseband control. Such waveform generators may be used to directly drive the qubits.

[0048] In some embodiments, the qubits 102 are flux qubits and the control and instrumentation system 108 and the digital signal interface 106 is configured to control the magnitude of a magnetic flux threaded through a flux qubit (also referred to herein as the magnitude of the flux bias of the flux qubit). In some embodiments, the qubits 102- 10 -ACTIVE 705838443v1are flux qubits, and the control and instrumentation system 108 and the digital signal interface 106 are configured to independently control a plurality of magnetic flux biases that are threaded through respective different superconducting loops within the flux qubits. In some embodiments, control of an external magnetic flux comprises providing a baseline DC current signal that is fixed, in addition to providing a time-dependent current signal that modulates the baseline DC current signal. For instance, an antenna may be mutually inductively coupled to a superconducting loop of the flux qubit, and a current signal may be provided to this antenna to adjust the magnetic flux threaded through the superconducting loop of the flux qubit.

[0049] According to some embodiments, controlling the flux bias of a flux qubit 102 by the control and instrumentation system 108 and the digital signal interface 106 comprises directing a flux bias signal (e.g., a current signal) along a flux bias line that is inductively coupled to a superconducting loop of the flux qubit (e.g., inductively coupled via an antenna).

[0050] In some embodiments, the control and instrumentation system 108 and the digital signal interface 106 may include digital and analog components, in which the analog components are configured to generate an analog flux bias signal based on digital data supplied by the digital components. Generating an analog flux bias signal in this way may include generating a digital signal and converting the digital signal to an analog signal (e.g., via a digital to analog converter (DAC)), and / or may comprise generating an analog flux bias signal based on one or more digital values.

[0051] The control and instrumentation system 108 may include microwave sources that generate high-frequency signals (e.g., > 1 GHz) to provide microwave control. Such microwave sources may be used to directly drive the qubits or couplers or mixed with baseband waveform generators to shape the microwave pulses. The control and instrumentation system 108 may also include high-frequency sources (5-10 GHz) to generate readout signals at the resonator frequencies or pump tones for parametric amplifiers. In addition, direct current (DC) sources may be used to statically flux bias the quantum circuit elements.

[0052] In some embodiments, the control and instrumentation system 108 is configured to direct microwave signals (e.g., microwave pulses) to a qubit 102 (and in- 11 -ACTIVE 705838443v1some cases to multiple of qubits 102) to drive a level transition of the qubit. The control and instrumentation system 108 may be configured to drive such a transition by applying a microwave pulse that stimulates Rabi oscillations between two states having a transition frequency that corresponds to the frequency of the microwave pulse (or which is detuned therefrom). For example, the control and instrumentation system 108 may be configured to direct a microwave pulse through one or more drive lines (also called charge lines) that are capacitively or inductively coupled to the qubit, and which drive a transition between states. The control and instrumentation system 108 may also be configured to direct a microwave pulse through one or more drive lines that are capacitively or inductively coupled to a coupler that couples together two qubits, as described further below.

[0053] In some embodiments, the control and instrumentation system 108 includes a microwave source that produces a microwave signal at a single frequency. As described above, one of the advantages of the techniques described herein may be to reduce the complexity of the microwave electronics needed to control the states of superconducting qubits. One way in which this complexity may be reduced is to utilize a microwave source at a single frequency for driving entangling (e.g., two-qubit) gates on multiple qubits (e.g., multiple qubit pairs). It will be appreciated that, in practice, the microwave signal may not necessarily have a bandwidth that is precisely zero, though such a signal may nonetheless be considered a single-frequency microwave signal so long as the bandwidth is sufficiently small.

[0054] According to some embodiments, the digital signal interface 106 may include digital and analog components, wherein the analog components generate an analog microwave signal based on digital data supplied by the control and instrumentation system 108. Generating an analog microwave signal in this way may include generating a digital signal and converting the digital signal to an analog signal (e.g., via a digital to analog converter (DAC)), and / or may comprise generating an analog micro wave signal based on one or more digital values.

[0055] In some embodiments, the digital signal interface 106 comprises one or more digital devices, which may include a general purpose computing device and / or digital logic devices such as ASICs or FPGAs, and / or may include low temperature- 12 -ACTIVE 705838443v1digital logic devices such as one or more cryo-CMOS, adiabatic quantum flux parametron (AQFP), single-flux quantum (SFQ), and / or quantum flux parametron (QFP) devices. In some embodiments, the digital signal interface 106 comprises one or more mixers implemented in low temperature digital logic, such as AQFP, which are configured to mix an oscillator current and a shaping current to produce a microwave pulse, which is switched on and off based on one or more digital current inputs. The generated microwave pulse(s) may be directed through one or more drives lines to one or more of the qubits 102.

[0056] In some embodiments, the digital signal interface 106 comprises one or more superconducting digital logic circuits. Such circuits process information using superconducting elements, such as Josephson junctions. Several Josephson-junction- based superconducting logic families exist, including rapid single-flux quantum (RSFQ) logic, energy-efficient RSFQ (ERSFQ) logic, energy-efficient SFQ (eSFQ) logic, dynamic SFQ (DSFQ) logic, reciprocal quantum logic (RQL), inductor-resistor RSFQ (LR-RSFQ) logic, low-voltage RSFQ (LV-RSFQ) logic, quantum flux parametron (QFP) logic, adiabatic quantum flux parametron (AQFP) logic, and nSQUID (a SQUID with negative mutual inductance) based logic. In some embodiments, the digital signal interface 106 comprises superconducting digital logic circuits (e.g., adiabatic quantum flux parametrons (AQFPs)) to generate control pulses either directly on the quantum processor chip or on a chip that is located next to or above the quantum processor chip (e.g., in a bump-bonded multi -chip stack). AQFPs are a family of superconducting digital logic circuits that can have very small energy dissipation due to their adiabatic nature. Benefits of an AQFP may include its low power dissipation, classical reversibility, compatibility with low-frequency baseband control (because the pulses can be slow), and clean high-frequency spectral content (which can reduce undesirable level transitions to other quantum states, sometimes referred to as leakage). AQFP is a member of the single flux quantum (SFQ) family of superconducting device technology.

[0057] In order to acquire data from the quantum processor, digitizers may convert the analog readout pulses to digital data. The data may then be demodulated with field- programmable gate arrays (FPGAs) and / or in software. The control and instrumentation system 108 may include a controller box containing FPGAs that interfaces to a- 13 -ACTIVE 705838443v1conventional computer to execute measurement and calibration programs and handle user inputs. Some or all of these control and instrumentation capabilities can also be implemented within the digital signal interface 106, either in part or as a whole, to reduce overall signal latency in the system.

[0058] Additional sub-systems of the quantum computing system can be connected to the quantum processor and / or to one another.

[0059] FIG. 2 depicts a qubit-coupler-qubit system, according to some embodiments. In the example of FIG. 2, system 200 comprises qubit 202 (e.g., a superconducting qubit) that is coupled to another qubit 208 (e.g., a superconducting qubit) through a coupler 206. Arrow 204 represents the coupling of the quantum states of the qubit 202 and the qubit 208, according to an effective coupling strength.

[0060] The effective coupling strength is associated with a rate at which quantum state population between a first quantum state (e.g., of a first superconducting circuit) and a second quantum state (e.g., of a second superconducting circuit) are transferred. The quantum state population of a quantum state is equal to the square of the absolute amplitude of the quantum state’s wave function and characterizes the probability that measurement of a qubit associated with the quantum state will yield the quantum state. Thus, a larger quantum state population of a given quantum state translates to a larger probability of measuring a qubit to be in the given quantum state. In some examples, the wave functions of different quantum states can interfere (e.g., destructively or constructively) with one another, and thus alter the quantum state population of the quantum states. In some quantum operations that involve three or more quantum states, quantum state population may be transferred between a first quantum state and a third quantum state, via a second quantum state, without placing substantial quantum state population in the second quantum state during the transfer. The effective coupling strength may depend at least in part on an inherent coupling strength that can depend on the superconducting circuits and the capacitances or inductances that connect them, for example. The effective coupling strength may also depend on two or more other effective coupling strengths (e.g., a first effective coupling strength, between a first and a second qubit, and a second effective coupling strength, between the second and a third qubit, can result in a third effective coupling strength, between the first and third qubits).- 14 -ACTIVE 705838443v1In superconducting circuits, the flow of electric charge can be quantized and thus associated with quantum states each with respective quantum state populations. Since the flow of the electric charge results in a magnetic flux, the magnetic flux of a superconducting circuit may also be quantized and associated with quantum states.

[0061] According to some embodiments, coupler 206 comprises one or more modes through which the qubits 202 and 208 can interact. In a quantum system, a mode refers to a distinct quantum state or a set of quantum states that the system can occupy. In some quantum systems, modes can be represented as a set of eigenstates of the Hamiltonian of the quantum system. Each mode is associated with a specific quantum operator or variable, which is associated with a physical quantity, such as charge across a capacitor, magnetic flux in an inductor, or phase difference across a Josephson junction. For example, the mode of a resonator formed by a linear inductor L and a capacitor C,N can be defined as the eigenstates of the following Hamiltonian: H = —2<J> + —2, where N is a quantum operator corresponding charge difference across the capacitor and is a quantum operator corresponding magnetic flux in the inductor.

[0062] A computational subspace refers to the subset of the entire Hilbert space of a quantum system that is used for performing quantum computations. It is the space spanned by the computational basis states, which are the states that represent logical information, i.e. bits, in a quantum system. For instance, when qubit and coupler modes exist in the system, the computational subspace includes only those states where the qubits are in their ground state or first excited state ( 10) or 11)) and the couplers are in their ground state (| 0».

[0063] A system of components including qubits and / or couplers can be represented in terms of the underlying quantum states of the components. For instance, the state of a system comprising two qubits connected by a coupler having two modes can be represented as | QB1 QB2 CPLR), where QB1 and QB2 are the states of qubit mode 1 and qubit mode 2, respectively, and CPLR is the state of the coupler that provides coupling between the qubits. For instance, the state of a mode of qubit 202 may be represented as QB1, the state of a mode of the qubit 208 may be represented as QB2 and the state of the coupler 206 may be represented as CPLR. For instance, where each- 15 -ACTIVE 705838443v1of the qubit 202, qubit 208 and coupler 206 exhibit energy states that are treated as basis states 10), 11) or 12), the state of the system 200 when the qubit 202 is in state 11), the qubit 208 is in state |0) and the coupler 206 is in state |0) may be written as | 100).

[0064] FIG. 3 shows a schematic diagram of the states of the qubit-coupler-qubit system shown in FIG. 2, according to some embodiments. In the example of FIG. 3, two qubit modes 302 and 308 interact through the mode of a coupler 306. Two modes of the qubit-coupler-qubit system 306a and 306b are depicted, which differ in the state of the coupler; in both, the qubits are in their |1) states.

[0065] QB1 and QB2’s higher excited states — the 1210) and 1120) states - can interact through a coupler-excited state 1111). By tuning the energy of | 111), the effective coupling strength between |210) and 1120) can be adjusted. For example, if | 111) is brought closer to 1210) and 1120), the effective qubit-qubit interaction through | 111) becomes stronger (e.g., may effectively turn on the coupling). On the other hand, if |111) is brought further from |210) and | 120), the effective qubit-qubit interaction strength through 1111) decreases. For instance, adjusting 1111) in this manner may effectively turn off the coupling if the effective coupling strength through | 111) is reduced to be negligibly small. Additionally, or alternatively, adjusting | 111) in this manner may effectively turn off the coupling if the effective coupling strength through | 111) is canceled out by the direct coupling between 1210) and 1120) (e.g., direct coupling arising from the direct coupling capacitance between two fluxonium qubits).

[0066] Prior to performing an entangling gate, the qubits 202 and 208, and the coupler 206, can be arranged in a state referred to herein as an idle operating regime. An idle operating regime refers to the state when the qubits are not undergoing any entangling gate operations. During this operating regime, the flux biases of fluxonium qubits may be set to half-integer magnetic flux quantum values, specifically <bext= / 2n+l\ . . . . h , , . , , i i( — — I <P0, where n is an integer and <P0= — where h. is Planck s constant and e is the charge of an electron. This choice of flux bias allows the fluxonium qubit to exhibit long coherence times due to its operation frequency being insensitive (in first-order) to external magnetic flux. Additionally, during the idle operating regime, a coupler that mediates the interactions between the qubits may be biased to the point where the net- 16 -ACTIVE 705838443v1coupling between the qubits represented by arrow 204 is suppressed or turned off, allowing for isolation between the qubits.

[0067] Subsequently, the coupler 206 may be biased so that the coupling between the qubits 202 and 208 is active or turned on, thereby allowing entangling gates to be performed between the qubits.

[0068] In some cases, the states of the qubits and the coupler may interact with one another (and possibly other components within the same system), leading to changes in the energy level of one or more of these states. The intrinsic quantum states of a system without considering any interactions between the system components may be referred to as “bare” states, whereas the quantum states of the system that take into account any interaction of the qubits and couplers may be referred to as “dressed” states. In other words, bare states describe the state of a qubit or coupler that is isolated from other components. On the other hand, dressed states describe the state of the system including any interaction of the qubits and couplers. These dressed states incorporate the shifts and modifications in the energy level of states, and reflect the properties of the qubit and coupler states resulting from such interactions. Herein, the bare state representation and dressed state representation are distinguished by putting tilde on the state, e.g., 1110) vs 1110), where 1110) refers to the bare state and 1110) refers to the dressed state.

[0069] FIGs. 5A-5D depict illustrative energy level diagrams of the qubit-coupler- qubit system 200 shown in FIG. 2, according to some embodiments. Due to the coupling between the qubits and the coupler, for example through capacitive interactions between the qubit and coupling circuits, the qubit states and coupler state are hybridized. This hybridization results in level repulsion, or energy level shifts, resulting in the dressed states as illustrated in FIGs. 5A-5D. In system 200, entangling gates (e.g., the conditional phase (CPHASE) gate) may be performed on the qubits by driving state transitions from the computational states 1110), |100), |010) and |000); the relevant transitions for these four cases are shown in FIGs. 5A, 5B, 5C and 5D, respectively. The state transitions can be driven by sending microwave pulses through the flux bias lines or charge lines of qubits or / and couplers, as described above, and may be referred to herein as microwave activated phase (MAP) transitions. The labels of “LOW,” “MID” and “HIGH” in FIGs. 5A-5D correspond to the different transition energies shown in FIG. 4,- 17 -ACTIVE 705838443v1which illustrates the frequency tunability of the MAP state transitions. In particular, FIG. 4 shows how the transition frequencies of these illustrative state transitions can be adjusted by adjusting the flux bias through the coupler. For example, the “LOW” state transition frequency can be varied between 3 GHz to 4 GHz by adjusting the coupler flux between 0.5<boand 0.6<bo.

[0070] In each of the cases shown in FIGs. 5A-5D, entangling gates may be performed by driving a particular one of the eight transitions. Because of the depicted level repulsion, the transition frequencies of these transitions may be different, allowing only particular transitions to be driven, which therefore changes the state of one or both qubits in a manner that is dependent on their initial state . In each of FIGs . 5 A-5D, the energies of the dressed and undressed states are shown as illustrative examples, and are not intended to limit any embodiments to having particular energies for any states, or any particular relative differences between the dressed and undressed states.

[0071] FIG. 4 depicts how illustrative transition frequencies of MAP transitions can be adjusted by adjusting the flux bias threaded through the coupler, according to some embodiments. For example, in the example of FIG. 4, the “LOW”, “MID”, and “HIGH” MAP state frequencies can be varied by adjusting the coupler flux between 0.50and 0.6<bo. The “LOW”, “MID”, and “HIGH” MAP transition frequencies may, for instance, correspond to the transitions |210) to |110>, | 120) to | 110) and |111> to 1110), respectively. As shown, when the coupler’s excited state 1111) is far detuned away from the qubits’ higher excited states 1210) and 1120), the effective interaction strength between the 1210) and 1120) states is suppressed, effectively turning off coupling between these states. Consequently, driving MAP transitions at this bias point results in little or no entangling operation. To activate entangling operations, the | 111) state needs to be brought closer to the 1210) and 1120) states (“Coupling-on point”), as described above in relation to FIG. 3, so that the 1111) state and 1210) and 1120) states can be strongly hybridized with sufficient level repulsion. This allows the states to be selectively driven from 1110) to 1210), 1111) or 1120) without exciting other computational states: 1100), 1010), and 1000). Similarly, selective driving is possible from 1100) to 1110) or 1200), from 1010) to 1011) or 1020), and from 1000) to 1001). Importantly, the frequencies of these MAP transitions are tunable by adjusting the- 18 -ACTIVE 705838443v1coupler frequency. This allows the MAP transition frequencies of multiple qubit pairs to be aligned to a single value, even if device parameters, such as the Josephson energy Ej of a Josephson junction in a qubit and / or coupler, deviate from the target due to fabrication variations. By achieving this frequency alignment, Cphase gates can be implemented on any of multiple qubit pairs using a single shared microwave tone. When this shared microwave tone is applied, the activation or deactivation of two-qubit gates can be controlled through baseband coupler flux pulses.

[0072] Entangling gates that may be performed in this manner include a controlled phase (CPhase) gate. The CPhase gate is a type of two-qubit gate that applies a phase shift on the state of one of the qubits depending on the state of the other qubit. This conditional phase shift can be implemented by driving the transition from a computational state, such as 1110), to a non-computational state, such as 1210), 1120) or | 111). Non-computational states are quantum states that lie outside the computational subspace and are not treated as states that are intended to represent logical information or store long-term quantum information. However, they may be used during the execution of one or more quantum gates as resources to induce two-qubit interaction. The controlled-Z (CZ) gate is a specific type of the CPhase gate that applies a conditional phase shift of 180 degrees.

[0073] Another example entangling gate that may be performed by driving selected transitions as shown in FIGs. 5A-5D is the controlled-not (CNOT) gate. A CNOT gate is a two-qubit gate that applies an X gate on one of the qubits (the target qubit) depending on the state of the other qubit (the control qubit). In some examples, the CNOT gate can be implemented by combining the CZ gate with Hadamard gates, which are a type of single-qubit gate. For example, a CNOT gate may be applied between a control qubit and a target qubit by applying a Hadamard gate to the target qubit, applying a CZ gate (where either the control qubit or the target qubit may act as the CZ control qubit), and applying another Hadamard gate to the target qubit.

[0074] According to some embodiments, when performing an entangling gate, flux and microwave pulses applied to the system can result in unwanted single-qubit operations, such as X, Y, or Z rotations, or a combination of these rotations, on either or both qubits (or other qubits in the system). Undesired operations can be reversed by- 19 -ACTIVE 705838443v1applying single-qubit gates before, during, or after the entangling gate process. These single-qubit operations can be performed virtually by adjusting the phases of the microwave pulses applied to the system without adding new pulses. In some cases, undesired Z operations can be echoed out by applying pi pulses (e.g., X(TI) or Y(TI)) to the qubits between the entangling gate processes. Alternatively, these same undesired single-qubit operations can be utilized to implement a different type of two-qubit gate. For instance, by combining the undesired operations from the CZ gate and extra singlequbit gates before, during, or after the two-qubit gate process, the CNOT gate can be implemented.

[0075] According to some embodiments, the qubits 202 and 208 are flux qubits, such as the fluxonium qubit, depicted in FIG. 6. In the example of FIG. 6, the fluxonium qubit 600 comprises a superconducting loop with a capacitor 621, a Josephson junction 622, and an inductor 623 arranged in parallel with one another. A flux bias 625 is threaded through the superconducting loop, and may be independently controlled by a suitable flux controller (such as provided by control and instrumentation system 108 and / or digital signal interface 106) which generates a flux bias signal in flux bias lines that are inductively coupled to each loop, as described above. In addition, the fluxonium qubit 600 may be driven by a suitable microwave controller (such as provided by control and instrumentation system 108 and / or digital signal interface 106) which generates a microwave signal in drive lines that are inductively or capacitively coupled to the fluxonium qubit 600.

[0076] FIG. 7A depicts an illustrative implementation of the qubit-coupler-qubit system 200 shown in FIG. 2, according to some embodiments. In system 700, the qubits and the coupler are implemented as superconducting circuits. In the example of FIG. 7A, fluxonium qubits 702 and 704 are connected to a coupler 705. The qubits 702 and 704 may each be implemented, for example, as the fluxonium qubit 600 shown in FIG. 6. In FIG. 7A, the coupler 705 comprises a sub-circuit 706 that is connected to two nodes 714 and 716. Node 714 is connected to capacitor 720 that is connected to ground 722, and node 716 is connected to capacitor 724 that is connected to ground 726. The subcircuit 706 has a coupler mode associated with a phase difference (<p714— <p716) across node 714 and node 716. Capacitors 710 are distributed in the circuit between the nodes- 20 -ACTIVE 705838443v1and the qubits and determine the capacitive coupling strengths between the qubit modes and the coupler modes. The capacitance values of the capacitors 710 may be different from each other.

[0077] In some embodiments, rather than the sub-circuit 706 being arranged capacitively coupled to ground as shown in FIG. 7A (i.e., floating), the sub-circuit is instead directly coupled to ground. In this configuration, nodes 714 and 716 can share the same phase (<p714= <p7ie), and the coupler mode is associated with the phase relative to ground 722 or ground 726.

[0078] FIGs. 7B and 7C depict illustrative circuits that can each be used as the subcircuit 706. FIG. 7B depicts a sub-circuit 730 comprising single or multiple Josephson junctions 734 connected in series with single or multiple Josephson junctions 736 shunted by a capacitor 732. A magnetic flux 791 is threaded through the lower superconducting loop (i.e., bounded by the Josephson junction(s) 734 and the Josephson junction(s) 736). FIG. 7C depicts a sub-circuit 740 that is connected to ground 748 and comprises a set of one or more Josephson junctions 742 connected in series to single or multiple Josephson junctions 746 shunted by a capacitor 744. In addition, a with a magnetic flux 792 is threaded through the left superconducting loop (i.e., bounded by the Josephson junction(s) 742 and the Josephson junction(s) 746).

[0079] More generally, the sub-circuit 706 may comprise a first circuit portion that includes two or more Josephson junctions connected in series over a first closed path without any capacitors in the first closed path and a second circuit portion including two or more Josephson junctions connected in series over a second closed path without any capacitors in the second closed path. FIGs. 7A-7C provide non-limiting, illustrative examples of how such a coupler may be implemented.

[0080] FIG. 8 is a flowchart of a method of performing an entangling gate, according to some embodiments. Method 800 may be performed by system 100, which includes one or more instances of the qubit-coupler-qubit system shown in FIG. 2. For the purposes of explanation, a single instance of the qubit-coupler-qubit system 200 will be assumed below, although in general the method may be performed for any number of such systems.- 21 -ACTIVE 705838443v1

[0081] Method 800 includes act 802 in which the qubit 202 and qubit 208 are flux- biased at, or at approximately at, 0.5<b0- the so-called “sweet spot.” Moreover, the flux bias of the coupler 206 is tuned so that the coupling between qubit 202 and qubit 208 is sufficiently low to be turned off. In some embodiments, the flux bias of the coupler 206 may be adjusted by directing a flux bias signal (e.g., a current signal) along a flux bias line that is inductively coupled to a superconducting loop of the coupler (e.g., inductively coupled via an antenna). For instance, the coupler may be implemented as shown in FIG. 7A, using either of the sub-circuits 730 or 740 shown in FIGs. 7B and 7C, and the flux bias line may be inductively coupled to the superconducting loop of the sub-circuit.

[0082] In act 804, the flux bias of the coupler 206 is adjusted by directing a flux bias signal (e.g., a current signal) along a flux bias line that is inductively coupled to a superconducting loop of the coupler (e.g., inductively coupled via an antenna). In act 804, the flux bias of the coupler is adjusted so that a transition between states of the coupler is resonant with a transition between states of the qubits, thereby producing a coupling between the qubits via the coupler.

[0083] For example, as shown in FIG. 3, if the transition frequency between the 11) and 12) states of the qubits, and the transition frequency between the 10) and 11) states of the coupler, are resonant, then the 1210), 1120) and 1111) states will be resonant, thereby producing a coupling between the qubits via the coupler.

[0084] Due to the above-described level hybridization between the states as shown in FIGs. 5A-5D, transitions between states of the qubits and the coupler may be driven selectively by applying a microwave pulse through one or more charge lines that are capacitively coupled to qubit 202, qubit 208 or coupler 206. Alternatively, the transitions may be driven through one or more flux lines that are inductively coupled to qubit 202, qubit 208 or coupler 206. Act 806 comprises applying any number of such microwave pulses to the qubit 202, qubit 208 and / or coupler 206 to perform desired operations, including entangling gate operations. In some embodiments, act 806 comprises performing an entangling gate by applying a microwave pulse to the coupler 206.- 22 -ACTIVE 705838443v1

[0085] In act 808, the flux bias of the coupler is adjusted so that a transition between states of the coupler is no longer resonant with a transition between states of the qubits, thereby reducing (or removing) the coupling between the qubits via the coupler.

[0086] FIG. 9 is a schematic of an illustrative quantum computing system in which entangling gates may be performed between qubits using a shared microwave source, according to some embodiments. System 900 is an example of a quantum computing system comprising multiple instances of system 200 or system 700, and further includes control hardware configured to adjust the coupling strength between qubits (e.g., by controlling the flux through a coupler) and to perform entangling gates between qubits.

[0087] In the example of FIG. 9, system 900 includes qubits 948A, 948B and 948C, and couplers 950A, 950B and 950C that couple together pairs of qubits as shown. For example, the combination of qubit 948A, coupler 950A and qubit 948B may represent system 200 in FIG. 2, or system 700 shown in FIG. 7A. It may be presumed that a number of additional couplers and qubits may also be present in system 900, as represented by the ellipses in FIG. 9.

[0088] In the example of FIG. 9, the controller 960 is configured to generate an analog signal that adjusts the flux bias of a given coupler. In particular, signal 932A adjusts the flux bias of the coupler 950A, the signal 932B adjusts the flux bias of the coupler 950B, and signal 932C adjusts the flux bias of the coupler 950C. In some embodiments, the analog signals 932A, 932B and 932C may be baseband signals. In some embodiments, the controller 960 may be implemented by superconducting digital logic, such as AQFP logic.

[0089] In the example of FIG. 9, controllers 942A, 942B and 942C are each coupled to a shared microwave line 924 and are configured to produce a microwave pulse based on received signals 926A, 926B and 926C, respectively. In some embodiments, the signals 926A, 926B and 926C may be pulse envelope signals that indicate when (and optionally, at what amplitude) a microwave signal should be produced from the controllers 942A, 942B and 942C, respectively. The microwave pulses produced may be directed to one of the qubits and / or one of the couplers as shown to drive the qubit and / or coupler between energy states. As described above, by generating microwave pulses from a shared microwave signal, a single microwave- 23 -ACTIVE 705838443v1source may be shared by many, or even all, qubits and couplers, leading to much simpler control hardware.

[0090] FIG. 9 is an example of such an approach in that system 900 may be implemented in digital logic coupled to a single microwave source. For example, the signals 926A, 926B and 926C may be generated by a suitable superconducting digital logic controller (not shown in FIG. 9), and controller 960 may be implemented by the same, or by a different, superconducting digital logic controller. As a result, a single (or at least, fewer) microwave signals may be supplied to the low temperature stage of the system, which utilizes such a signal in combination with digital hardware to control the coupling between qubits and perform entangling gates between the qubits, though a process as described above in relation to FIG. 8.

[0091] According to some embodiments, system 900 may be configured to perform entangling gates between qubits by driving one or more of the couplers 950A, 950B or 950C, and / or one or more of the qubits 948A, 948B or 948C. An advantage of driving the couplers instead of the qubits is that the microwave pulse applied to the coupler activates the MAP transitions specifically within one qubit pair that is coupled to the driven coupler. In contrast, driving the qubits directly can undesirably cause the high- frequency drive to activate the MAP transitions on multiple qubit pairs that are coupled to the driven qubit (e.g., it can activate four qubit pairs if the qubit is coupled to four nearest neighbors).

[0092] According to some embodiments, controllers 942A, 942B and 942C each is, or comprises, a microwave mixer (which may include a digital mixer, an analog mixer, or a hybrid analog and digital mixer). For instance, each of controllers 942A, 942B and 942C may consist of, or may comprise, an in-phase and quadrature (I / Q) mixer that is configured to generate a microwave pulse by mixing the coherent microwave tone sent through the shared line 924 with pulsed low frequency signal 926A, 926B or 926C, respectively. This mixing effectively allows the signals 926A, 926B and 926C to switch the microwave drives applied to the qubits and / or couplers on and off, thereby avoiding errors from inadvertent microwave driving of qubits or couplers when they are not intended to interact with the high-frequency (microwave) drive. In some embodiments, each of controllers 942A, 942B and 942C may be configured to perform frequency up-- 24 -ACTIVE 705838443v1conversion and / or down-conversion of the microwave line 924 (e.g., by up to a few hundred MHz) to fine-tune frequencies of microwave pulses applied to each qubit and / or coupler to thereby avoid driving undesired transitions. These undesired transitions can include high-order parasitic leakage transitions or two-level defects that are in resonance with the MAP transition frequency.

[0093] In some embodiments, one or more of the controllers 942A, 942B and 942C may be configured to generate shaped microwave drives that are applied to the qubits or / and couplers to perform gates. Shaped drives may in at least some cases suppress non-adiabatic gate errors. In some cases, one or more of the controllers may be configured to perform fast adiabatic pulse shaping to generate a microwave pulse from the shared microwave signal line 924. According to some embodiments, one or more of the controllers 942A, 942B and 942C may be configured to generate microwave drives that are shaped to be symmetric, with equal positive and negative areas resulting in a net pulse area of zero. This net-zero pulse shaping can help suppress the effects of pulse distortion that come from various factors, including hardware imperfections.

[0094] In some embodiments, the controller 960 may be configured to generate shaped baseband flux pulses to the couplers. Shaped flux pulses may in at least some cases suppress non-adiabatic gate errors, such as leakages from computational states to non-computational states. According to some embodiments, the controller 960 may be configured to generate baseband flux pulses that are shaped to be symmetric, with equal positive and negative areas resulting in a net pulse area of zero. This net-zero pulse shaping can help suppress the effects of pulse distortion that come from various factors, including hardware imperfections.

[0095] In some implementations, machine learning techniques such as reinforcement learning can be utilized to optimize the shapes of control pulses for single and two-qubit gates. The cost function includes various types of gate errors such as leakage errors, coherent errors, and incoherent errors. Gate errors can be quantified by measuring the outputs of quantum circuits where the expected ideal outcomes (assuming no errors in the applied gates) can be predicted using classical computers. The optimizer controls pulse parameters to determine the overall pulse shape, as well as parameters for the extra corrective single-qubit gate operations that cancel out unwanted single-qubit- 25 -ACTIVE 705838443v1rotations induced by the applied control pulse. Optimizing the pulse shape may optimize the spectral distribution of the pulse to minimize inadvertent driving of nearest undesired level transitions.

[0096] FIGs. 10A-10C depict examples of various approaches to generating signals for adjusting the flux bias of a coupler, and for driving a qubit and / or the coupler to perform an entangling gate, according to some embodiments. For example, the system 900 may be operated to produce the depicted coupler flux bias signals 1001, 1002, 1003, 1004 and 1005 by operating controller 960, and to produce the depicted microwave drives 1011, 1012 and 1013 by operating one of the controllers 942A, 942B or 942C.

[0097] In the example of FIGs. 10A and 10B, two-qubit gate pulse sequences are depicted for implementing entangling gates (e.g., Cphase gates), also referred to herein as Baseband and Microwave Activated Phase (BMAP) gates. As described above, The flux pulse applied to the coupler circuits controls the effective coupling strength between the qubits’ higher excited states (e.g., |210) and 1120)) by adjusting the energies of the coupler excited state (e.g., 1111)). Throughout the gate operation, the flux biases of fluxonium qubits may be set to half-integer magnetic flux quantum values to maintain its high coherence times. Additional flux pulses can be applied to the qubits to compensate for any unwanted flux crosstalk induced by the flux pulse applied to the coupler circuits. The microwave pulse applied to the qubits and / or coupler circuits drives one or more of the MAP transitions. The pulsed microwave drive 1011 can be delivered to the coupler and / or to qubit(s) through either the flux or charge lines of the qubits and / or coupler. Instead of generating a microwave pulse as shown in FIG. 10A (e.g., by shaping a common microwave signal as described in the example of FIG. 9 above), a continuous microwave tone as shown in FIG. 10B can instead be applied to a qubit and / or coupler, which eliminates the need for components for shaping microwave signals. This approach may be feasible because the coupler can address any two-qubit gate errors coming from always-on microwave signals by effectively turning off the qubit-qubit interactions.

[0098] FIG. 10C shows an example pulse sequence implementing multiple CPhase gates in a resource-efficient manner by sharing a common microwave drive 1016 for multiple qubit pairs. Microwave drives 1013, 1014 and 1015 may be generated by shaping the common microwave drive 1016 using the flux bias signals 1003, 1004 and- 26 -ACTIVE 705838443v11005, respectively, as a shaping envelope. The resulting flux bias signals may be applied to respective couplers to control the effective coupling strengths for each qubit pair. The amplitude, shape, and duration of each flux bias signal may be selected to optimize the performance of the two-qubit gate on each qubit pair. The microwave pulse or tone applied through the shared common microwave drive line drives one or more of the MAP transitions, implementing a CPhase gate on each qubit pair. Because the number of microwave sources and the cables routing from the source to the chip can be significantly reduced by sharing the microwave drive line, this scheme may be referred to as a resource-efficient Baseband and Microwave Activated Phase (BMAP) gate.

[0099] As described above, some quantum computer systems may include a control and instrumentation system that generates and manipulates control waveforms, part or all of which can be housed in a dilution refrigerator and co-located with the qubits and couplers. Such a system may be referred to as a cold control system. For example, a cold control system may be implemented by superconducting digital logic circuits, a type of electronic circuitry that utilize superconducting materials to possibly enable energyefficient and high-performance digital signal processing. Examples of suitable superconducting digital logic circuits are described above.

[0100] In superconducting digital logic circuits, information is typically stored, processed, and transferred in the form of a single quantum of magnetic flux, also referred to as single flux quantum (SFQ), in various superconducting loops. In Rapid Single Flux Quantum (RSFQ) logic and Reciprocal Quantum Logic (RQL), which are other SFQ logic approaches related to AQFP logic, a Josephson junction is “flipped” (i.e., switched into a dissipative voltage state) during the operation. Once the junction is flipped, one does not have control over the dynamics, whereas in AQFP dynamics may be better controlled. One or more AQFP circuits can be combined into a digital-to-analog converter (DAC) that takes digital input and converts it into baseband flux pulses to control the quantum system, e.g., couplers. The pulse may be transferred to the coupler by mutually inductively coupling the AQFP and the coupler circuit.

[0101] In some examples, firmware protocols can be used to control AQFP circuits to perform multiplexing, set trim SQUIDs, etc. In general, software infrastructure can be- 27 -ACTIVE 705838443v1used to compile a set of gates into AQFP architecture control pulses so that timing, amplitude, and idling are properly scheduled and synchronized.

[0102] FIG. 11 illustrates one implementation of such a cold control system 1100 for the implementation of the BMAP gate, based on AQFP circuits. System 1100 may for instance form at least part of the digital interface 106 shown in FIG. 1, wherein the qubit or coupler 1140 forms part of the qubits 102.

[0103] In the example of FIG. 11, AQFP -based mixer 1120 is configured to mix a digital input signal 1102 with a continuous micro wave tone from a micro wave drive line 1101, thereby generating a microwave pulse according to the digital input. For example, the digital input could specify whether the microwave pulse is active or inactive, with a 0 meaning no output, and a 1 meaning the mixer should output a microwave signal. More complex methods of digital control may also be envisioned and implemented by AQFP mixer 1120. A filter and / or tunable coupling element 1130 (e.g., DC SQUID or RF SQUID) or trimming circuit is arranged to mediate the coupling between the AQFP mixer 1120 and the qubit or coupler 1140 in order to control the amplitude of the microwave pulse applied to the qubit or coupler 1140. In some embodiments, AQFP mixer 1120, and / or the filter and / or tunable coupling element 1130, may represent an implementation of part or all of a controller 942A, 942B or 942C shown in FIG. 9.

[0104] In the example of FIG. 11, AQFP -based DAC 1150 is configured to generate a baseband flux pulse given the digital input signal 1102. For example, the digital input could specify whether the baseband flux pulse is active or inactive, with a 0 meaning no output, and a 1 meaning the mixer should output a baseband flux signal. More complex methods of digital control may also be envisioned and implemented by AQFP -based DAC 1150, such as converting a digital waveform supplied as part of (or all of) digital input signal 1102 to an analog signal. The baseband flux pulse is delivered to a qubit or coupler 1140. In the case of a coupler 1140, the baseband flux pulse signal thereby allows the system to switch the qubit-qubit effective coupling on and off. A filter and / or tunable coupling element 1160 (e.g., DC SQUID or RF SQUID) or trimming circuit is arranged to mediate the coupling between the AQFP DAC 1150 and the qubit or coupler circuit 1140 in order to control the amplitude of the baseband flux pulse signal applied to the qubit or coupler 1140.- 28 -ACTIVE 705838443v1

[0105] Also in the example of FIG. 11, a filter and / or tunable coupling element 1110 is arranged to filter the microwave signal from the microwave drive line 1101 before providing the filtered microwave signal to the AQFP mixer 1120.

[0106] Each of the elements depicted in FIG. 11 can be capacitively, inductively, or galvanically coupled to one another along the dashed arrowed lines.

[0107] As an alternative to the coupler described above and shown in FIGs. 2, 3, 5A-5D and 7A, which exhibits a single mode, a dual-mode coupler may also be utilized in the systems, and with the techniques described herein. Configurations in which the coupler is a dual-mode coupler (which may also be referred to as a dual-mode coupling circuit) are described below.

[0108] FIG. 12A depicts a schematic diagram of a system 1210 that includes a dual -mode coupler that couples together two qubits. In the example of FIG. 12A, dualmode coupler 1212 comprises two modes 1214 and 1218, with mode 1214 providing positive effective coupling 1216 and the other mode 1218 providing negative effective coupling 1220 between qubit modes 1213 and 1222. The coupler mode that provides positive effective coupling is referred to as the plus mode, and the coupler mode that provides negative effective coupling is referred to as the minus mode. Illustrative circuits that can provide dual-mode coupling capabilities are illustrated in FIGS. 13A and 13B.

[0109] In the example of FIG. 12A, the effective couplings 1216 and 1218 can be tuned by adjusting the frequencies of the plus and minus modes. The overall coupling provided by the dual -mode coupler is determined by the sum of these two effective couplings, allowing for control over the total coupling, including the ability to switch the coupling on and off. One advantage of a dual-mode coupler over a single-mode coupler is its ability to completely turn off the overall coupling through destructive interference between the plus and minus couplings. This capability can provide greater flexibility in physical layout and circuit design while improving performance, such as suppressing errors caused by parasitic couplings with neighboring spectator qubits.

[0110] FIG. 12B depicts an energy level diagram of illustrative system 1210 in which two qubit modes interact through the two modes of a dual-mode coupler 1232. Specifically, QB1 and QB2’s higher excited states — the 12100) and 11200) states - can- 29 -ACTIVE 705838443v1interact through two coupler-excited states - the 11110) and 11101) states. 11110) (| 1101)) can either provide positive or minus effective qubit-qubit coupling. By tuning the energies of either 11110) and 11101) the effective coupling strength between 12100) and 11200 ) can be adjusted. For example, if 11110) is brought closer to 12100) and 11200), the effective qubit-qubit interaction through |1110) becomes stronger (coupling ON operation). On the other hand, if 11110) is brought further from 12100) and 11200), the effective qubit-qubit interaction strength through |1110) decreases. If the effective coupling strengths through 11110) and 11101) are equal, but have opposite signs, they cancel each other and the interaction between 12100) and 11200) can be negligibly small (coupling OFF operation).

[0111] FIG. 12C shows an exemplary energy level diagram of the illustrative system 1210 in which two qubit modes interact through the two modes of a coupler. Due to the coupling between the qubits and the coupler, for example through capacitive interactions between the qubit and coupling circuits, the qubit states and coupler states are hybridized. This hybridization results in level repulsion, or energy level shifts of the dressed states as illustrated in FIG. 12C. In the illustrative system 1210, entangling gates, such as conditional phase (CPHASE) gates, can be performed on the qubits by driving level transitions from the computational state 11100). These level transitions can be driven by sending microwave pulses through the flux bias lines or charge lines of qubits or / and couplers and such level transitions are referred to herein as microwave activated phase (MAP) transitions. The applied microwave pulses drive a single period of an oscillation from the computational state 11100) to one of the non-computational states such as 12100), 11200), 11101), and 11110), which are referred to as “MAP states” herein. After a full period of the oscillation, the state 11100) picks up some phase shift. The amount of this phase shift corresponds to the conditional phase shift of the CPhase gate on the qubits.

[0112] In some examples, as illustrated in FIG. 12D, entangling gates, such as CPhase gates, may be implemented by driving a single period of an oscillation from the computational state 11000) to one of the non-computational states such as 12000), 11001), and 11010). In some examples, as illustrated in FIG. 12E, entangling gates, such as CPhase gates, may be implemented by driving a single period of an oscillation- 30 -ACTIVE 705838443v1from the computational state 10100) to one of the non-computational states such as 10200), 10101), and 10110) . In some examples, as illustrated in FIG. 12F, entangling gates, such as CPhase gates, may be implemented by driving a single period of an oscillation from the computational state 10000) to one of the non-computational states such as 10010) and 10001).

[0113] In each of FIGs. 12A-12D, the energies of the dressed and undressed states are shown as illustrative examples, and are not intended to limit any embodiments to having particular energies for any states, or any particular relative differences between the dressed and undressed states.

[0114] FIG. 13A depicts an illustrative implementation of a dual -mode coupler, according to some embodiments. In the example of FIG. 13 A, system 1300 comprises fluxonium qubits 1302 and 1304 connected to a dual mode coupler 1305. The qubits 1302 and 1304 may each be implemented, for example, as the fluxonium qubit 600 shown in FIG. 6. The dual -mode coupling circuit comprises a subcircuit 1306 that is connected to two nodes 1314 and 1316. Node 1314 is connected to capacitor 1320 that is connected to ground 1322, and node 1316 is connected to capacitor 1324 that is connected to ground 1326. A subcircuit 1308 is connected to a node 1318 that has a coupler mode associated a superconducting phase <pi3i8at node 1318. The subcircuit 1306 has a coupler mode associated with a phase difference (<p1314— <Pi3i6) across node 1314 and node 1316. Capacitors 1310 are distributed in the circuit between the nodes and the qubits and determine the capacitive coupling strengths between the qubit modes and the coupler modes. The capacitance values of the capacitors 1310 may be different from each other.

[0115] FIG. 13B depicts an illustrative implementation of a dual-mode coupler, according to some embodiments. In the example of FIG. 13B, the system 1370 comprises fluxonium qubits 1372 and 1374 connected to a dual-mode coupling circuit 1376. The qubits 1372 and 1374 may each be implemented, for example, as the fluxonium qubit 600 shown in FIG. 6. The dual-mode coupling circuit comprises a subcircuit connected between nodes 1384 and 1386. Node 1384 is connected to a sub-circuit 1378 that is shunted by a capacitor 1392 and connected to ground 1390. Node 1386 is connected to a sub-circuit 1382 that is shunted by a capacitor 1394 and connected to- 31 -ACTIVE 705838443v1ground 1396. At least one sub-circuit of the sub-circuits 1378, 1380, and 1382 comprises single or multiple Josephson junctions and the remaining sub-circuits of the sub-circuits 1378, 1380, and 1382 can comprise inductors or single or multiple Josephson junctions. The dual-mode coupling circuit has a first coupler mode that is associated with a superconducting phase difference (<p1384—<Pi386) across nodes 1384 and 1386 and a second coupler mode that is associated with a superconducting phase sum (<p1384+ <p1386) across nodes 1384 and 1386. This distinct characteristic of the modes (one is associated with the phase difference between the nodes, the other associated with the phase sum of the nodes) results in the sign difference between the effective couplings provided by the first and second coupler modes. Capacitors 1398 are distributed in the circuit between the nodes and the qubits and determine the capacitive coupling strengths between the qubit modes and the coupler modes. Additional capacitors 1388 shunting the sub-circuit 1380 can be added to the dual-mode coupling circuit to engineer the level structures of the coupler modes, enhancing the performance of the two-qubit gate. More specifically, adjusting the energy levels of the coupler modes allows modification of the coupler flux bias where the qubit-qubit coupling (“the coupling-off flux bias” or “idle operating regime”) is effectively turned off. For example, the coupling-off flux bias can be changed from 0.25 o to 0 o or 0.5 o, which are first-order flux-insensitive points, making the qubit-qubit coupling less sensitive to flux noise or flux pulse distortion.

[0116] Each of the sub-circuits 1306 and 1308 shown in FIG. 13A, and the subcircuits 1378, 1380 and 1382 shown in FIG. 13B, may be implemented as the illustrative sub-circuits 730 and 740 shown in FIGs. 7B and 7C, or as any of the sub-circuits 1337, 1350 or 1360 depicted in FIGs. 13C-13E. Each of these sub-circuits may be implemented as any of the four depicted sub-circuits 730, 740, 1337, 1350 or 1360, although preferably at least one of sub-circuits 730 or 740 is present in system 1300 or in system 1370, since these sub-circuits allow for a flux to be threaded through a superconducting loop within the dual -mode coupler, whereas sub-circuits 1337, 1350 and 1360 do not.

[0117] FIG. 13C depicts a circuit 1337 comprising single or multiple Josephson junctions 1339 connected shunted by a capacitor 1338. FIG. 13D depicts a circuit 1350 that is connected to ground 1356 and comprises a set of one or more Josephson junctions- 32 -ACTIVE 705838443v11352 shunted by a capacitor 1354. FIG. 13E depicts a circuit 1360 that is connected to ground 1366 and comprises a linear inductor 1362 and a capacitor 1364 connected in parallel.

[0118] Irrespective of whether a system includes a single mode coupler or a dualmode coupler, both of which described above, the techniques for baseband and microwave control of qubits via a suitable coupler may be extended to a system with a larger number of qubits. Where the below description of FIGs. 14A-14D refers to a coupler, it will be understood that any such coupler may be implemented as a single mode or dual-mode coupler, and that in some cases a system may comprise both one or more single mode couplers and one or more dual-mode couplers.

[0119] The various embodiments described above may provide a unit cell that can be tiled into a two-dimensional (2D) array of qubits. As one example, FIG. 14A contains a schematic diagram of a quantum computing architecture 1400 comprising chip 1402 on which couplers 1408 (e.g., single mode or dual-mode couplers) are interconnected with qubits 1404 and arranged in a 2D grid. In particular, in contrast to some of the abovedescribed architectures, FIG. 14A is an example of an arrangement in which a qubit may be coupled to multiple other qubits via respective couplers. For example, the central qubit 1404 shown in FIG. 14A is coupled to four other qubits each via a different coupler. The qubit connectivity (e.g., the number of nearest-neighboring qubits coupled to a central qubit) may vary depending, for example, on the target applications or error correction schemes.

[0120] In the example of FIG. 14A, the couplers 1408 share a common microwave driveline 1405, through which either a pulsed or continuous microwave tone can be delivered to drive MAP transitions as described above. In some embodiments, each of the couplers 1408 may be arranged as shown in FIG. 11, with each respective coupler 1408 being provided as coupler 1140, and with the common microwave driveline 1405 being provided as the microwave drive line 1101 for each of the couplers 1408. Subsequent to application of a pulsed or continuous microwave tone to a coupler 1408, a two-qubit gate on each qubit pair can be activated or deactivated by dynamically adjusting the flux bias of the coupler (i.e., by applying flux pulses to the couplers). This approach can significantly reduce the number of microwave sources and the amount of- 33 -ACTIVE 705838443v1routing (e.g., number of coax cables) from the signal sources to the chip on which the couplers and qubits are fabricated, enabling quantum computer scale-up with a simplified and cost-effective control hardware setup.

[0121] In some embodiments, a coupler can be made physically long, enabling non-local, long-range connections. Therefore, a quantum computer system architecture may be extended beyond a 2D grid architecture to accommodate more complex three- dimensional (3D) grids with non-local connections. FIG. 14B shows a schematic diagram of a quantum computing architecture 1410 comprising a coupler chip 1416 on which one or more couplers 1418 are arranged, and which serves as an interconnect between a first quantum processor chip 1412 comprising one or more qubits 1414 and a second quantum processor chip 1413 comprising one or more qubits 1411.

[0122] FIG. 14C illustrates an example of a 3D quantum computing device architecture 1420 comprising a first quantum processor chip 1422, a dual -mode coupler chip 1424, and a second quantum processor chip 1426 connected to single interposer layer 1428 by vertical connections 1429. These vertical connections can comprise electrically conducting bonds, for example indium bumps, with thickness on the order of tens of micrometers. These chips could also be part of a multi -chip module housing multiple superconducting circuit layers (such as SFQ or AQFP circuits) used to generate or manipulate control signals for qubits, couplers, and readout resonators.

[0123] FIG. 14D shows an example quantum computing device architecture 1430 comprising a first quantum processor chip 1432 connected to a coupler chip 1434 by a first set of one or more long resonator buses 1438 and a second quantum process chip 1436 connected to the coupler chip by a second set of one or more long resonator buses 1440. Couplers in the coupler chip can be either dual -mode couplers or single-mode couplers. The long resonator buses can have a length on the order of a few millimeters and resonator modes that are far detuned from qubit and coupler modes to prevent frequency collisions during gate operations. Depending on the type of couplers used, one or more electrical connections per interconnection may be used. For example, if the dualmode coupling circuit illustrated in FIG. 13 A is used, single (two) electrical connections is (are) used per interconnection to implement the corresponding coupling capacitance network. Depending on the length of a bus resonator, the coupler and qubit parameters- 34 -ACTIVE 705838443v1may need to be adjusted accordingly to achieve optimal performance; for example, longer bus resonators might use larger coupling capacitance between the bus resonator and the coupler or between the bus resonator and the qubit.

[0124] FIGs. 14C and 14D show circuit schematics of example quantum computing systems in which two quantum processor chips are interconnected through a coupler chip where the couplers can be either dual-mode couplers or single-mode couplers. The coupler chip and quantum processor chips may be linked via long resonator buses (with a length on the order of a few millimeters) where the resonator modes are far detuned from qubit and coupler modes to prevent frequency collisions during gate operations. Depending on the type of couplers used, one or more electrical connections per interconnection may be used. For example, if the dual -mode coupling circuit illustrated in FIG. 13A is used, single (two) electrical connections is (are) used per interconnection to implement the corresponding coupling capacitance network. Depending on the length of a bus resonator, the coupler and qubit parameters may need to be adjusted accordingly to achieve optimal performance; for example, longer bus resonators might use larger coupling capacitance between the bus resonator and the dualmode coupler or between the bus resonator and the qubit.

[0125] As referred to herein, a “qubit” includes any multi-level quantummechanical system capable of being controlled by a quantum information processor. The quantum states of the qubit may for instance include electronic states, polarization states, vibrational states, rotational states, or spin states. As referred to herein, a “superconducting qubit” includes any superconducting electronic circuit that may be operated as a multi-level quantum-mechanical system, such as a charge qubit (e.g., a transmon), a flux qubit (e.g., a fluxonium qubit), or a phase qubit.

[0126] An illustrative implementation of a computer system 1500 that may be used to control a control and instrumentation system, a digital logic, an AQFP-based mixer, etc. to perform any of the techniques described above is shown in FIG. 15. The computer system 1500 may include one or more processors 1510 and one or more non-transitory computer-readable storage media (e.g., memory 1520 and one or more non-volatile storage media 1530). The one or more processors 1510 may control writing data to and reading data from the memory 1520 and the one or more non-volatile storage media- 35 -ACTIVE 705838443v11530 in any suitable manner, as the aspects of the disclosure described herein are not limited in this respect. To perform functionality and / or techniques described herein, the one or more processors 1510 may execute one or more instructions stored in one or more computer-readable storage media (e.g., the memory 1520, storage media, etc.), which may serve as non-transitory computer-readable storage media storing instructions for execution by the one or more processors 1510.

[0127] In connection with techniques described herein, code used to, for example, generate digital data to control generation of a flux bias signal or a microwave pulse, etc. may be stored on one or more computer-readable storage media of computer system 1500. The one or more processors 1510 may execute any such code to perform any of the above-described techniques as described herein. Any other software, programs or instructions described herein may also be stored and executed by computer system 1500. It will be appreciated that computer code may be applied to any aspects of methods and techniques described herein. For example, computer code may be applied to generate digital data to control generation of a flux bias signal or a microwave pulse in response to digital data obtained from reading the state of a fluxonium qubit, etc.

[0128] The various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of numerous suitable programming languages and / or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a virtual machine or a suitable framework.

[0129] In this respect, various inventive concepts may be embodied as at least one non-transitory computer readable storage medium (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, etc.) encoded with one or more programs that, when executed on one or more computers or other processors, implement the various embodiments of the present disclosure. The non- transitory computer-readable medium or media may be transportable, such that the program or programs stored thereon may be loaded onto any computer resource to implement various aspects of the present disclosure as described above.- 36 -ACTIVE 705838443v1

[0130] The terms “program,” “software,” and / or “application” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as described above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present disclosure need not reside on a single computer or processor, but may be distributed in a modular fashion among different computers or processors to implement various aspects of the present disclosure.

[0131] Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0132] Also, data structures may be stored in non-transitory computer-readable storage media in any suitable form. Data structures may have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a non-transitory computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish relationships among information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationships among data elements.

[0133] Having thus described several aspects of at least one embodiment of this disclosure, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. For instance, aspects of the techniques described herein may be combined in any of the following ways:

[0134] Aspect 1. A system comprising: a plurality of qubits including a first qubit, a second qubit and a third qubit; a plurality of couplers each coupled to multiple qubits of the plurality of qubits, the plurality of couplers including a first coupler coupled to the first qubit and to the second qubit, and a second coupler coupled to the second qubit and to the third qubit; a microwave drive line; and at least one controller configured to:- 37 -ACTIVE 705838443v1receive a microwave signal from the microwave drive line; generate a first pulse envelope signal; generate a first microwave pulse by applying the first pulse envelope signal to the microwave signal; apply the first microwave pulse to the first coupler; generate a second pulse envelope signal; generate a second microwave pulse by applying the second pulse envelope signal to the microwave signal; and apply the second microwave pulse to the second coupler.

[0135] Aspect 2. The system of aspect 1, wherein: the at least one controller comprises a first mixer and a second mixer, the first mixer is configured to receive the first pulse envelope signal, to generate the first microwave pulse and to apply the first microwave pulse to the first coupler, and the second mixer is configured to receive the second pulse envelope signal, to generate the second microwave pulse and to apply the second microwave pulse to the second coupler.

[0136] Aspect 3. The system of aspect 2, wherein the at least one controller further comprises superconducting digital logic configured to generate the first pulse envelope signal and provide the first pulse envelope signal to the first mixer, and to generate the second pulse envelope signal and provide the second pulse envelope signal to the second mixer.

[0137] Aspect 4. The system of aspect 2, wherein the first mixer and the second mixer are implemented in superconducting digital logic.

[0138] Aspect 5. The system of aspect 4, wherein the first mixer and the second mixer each comprises one or more adiabatic quantum flux parametron (AQFP), singleflux quantum (SFQ), and / or quantum flux parametron (QFP) devices.

[0139] Aspect 6. The system of aspect 1, wherein the at least one controller is further configured to: generate a third pulse envelope signal; generate a third microwave pulse by applying the third pulse envelope signal to the microwave signal; and apply the third microwave pulse to the first qubit.

[0140] Aspect 7. The system of aspect 6, wherein the at least one controller is further configured to apply the first microwave pulse to the first coupler concurrently with applying the third microwave pulse to the first qubit.- 38 -ACTIVE 705838443v1

[0141] Aspect 8. The system of aspect 1, wherein the at least one controller is further configured to generate the first microwave pulse with an amplitude according to one or more digital values.

[0142] Aspect 9. The system of aspect 1, wherein the plurality of qubits is a plurality of flux qubits, and wherein the plurality of couplers each comprises a superconducting loop with a magnetic flux threaded through the superconducting loop.

[0143] Aspect 10. The system of aspect 9, wherein the at least one controller is further configured to, prior to applying the first microwave pulse to the first coupler, control the magnetic flux to increase a coupling strength between the first qubit and the second qubit via the first coupler.

[0144] Aspect 11. The system of aspect 10, wherein the at least one controller is further configured to, subsequent to applying the first microwave pulse to the first coupler, control the magnetic flux to decrease the coupling strength between the first qubit and the second qubit via the first coupler.

[0145] Aspect 12. The system of aspect 1, wherein the first microwave pulse is configured to perform an entangling gate between the first qubit and the second qubit.

[0146] Aspect 13. The system of aspect 1, wherein the microwave signal is a single-frequency microwave signal at a frequency of at least 1 GHz.

[0147] Aspect 14. The system of aspect 1, wherein the first coupler and the second coupler each comprises a first plurality of Josephson junctions arranged in parallel with at least one capacitor.

[0148] Aspect 15. The system of aspect 14, wherein the first coupler and the second coupler each further comprises a second plurality of Josephson junctions arranged in parallel with the first plurality of Josephson junctions and in parallel with the at least one capacitor.

[0149] Aspect 16. The system of aspect 1, wherein the plurality of qubits include a fourth qubit, and wherein the plurality of couplers include a third coupler coupled to the second qubit and to the fourth qubit.- 39 -ACTIVE 705838443v1

[0150] Aspect 17. A method comprising: generating, by at least one digital logic controller, a first microwave pulse by applying a first pulse envelope signal to a microwave signal; applying, by the at least one digital logic controller, the first microwave pulse to a first coupler, the first coupler coupled to a first qubit and to a second qubit; generating, by the at least one digital logic controller, a second microwave pulse by applying a second pulse envelope signal to the microwave signal; and applying, by the at least one digital logic controller, the second microwave pulse to a second coupler, the second coupler coupled to the second qubit and to a third qubit.

[0151] Aspect 18. The method of aspect 17, further comprising, prior to applying the first microwave pulse to the first coupler, controlling a flux bias of the first coupler and thereby increasing a coupling strength between the first qubit and the second qubit.

[0152] Aspect 19. The method of aspect 18, further comprising, subsequent to applying the first microwave pulse to the first coupler, controlling the flux bias of the first coupler and thereby increasing a coupling strength between the first qubit and the second qubit.

[0153] Aspect 20. The method of aspect 18, wherein controlling the flux bias of the first coupler comprises applying a first signal having a frequency below 1 GHz to the first coupler, and wherein the first microwave pulse has a frequency at or above 1 GHz.

[0154] Aspect 21. The method of aspect 17, wherein the first coupler and the second coupler each comprises a superconducting loop with a magnetic flux threaded through the superconducting loop.

[0155] Aspect 22. The method of aspect 17, wherein the first coupler is capacitively coupled to the first qubit and capacitively coupled to the second qubit.

[0156] Aspect 23. The method of aspect 17, wherein the at least one digital logic controller comprises: a first digital logic controller configured to receive the first pulse envelope signal, to generate the first microwave pulse and to apply the first microwave pulse to the first coupler; and a second digital logic controller configured to receive the second pulse envelope signal, to generate the second microwave pulse and to apply the second microwave pulse to the second coupler.- 40 -ACTIVE 705838443v1

[0157] Aspect 24. The method of aspect 17, wherein the at least one digital logic controller comprises one or more adiabatic quantum flux parametron (AQFP), singleflux quantum (SFQ), and / or quantum flux parametron (QFP) devices.

[0158] Aspect 25. The method of aspect 17, further comprising applying a third microwave pulse to the first qubit concurrently with applying the first microwave pulse to the first coupler.

[0159] Aspect 26. The method of aspect 17, wherein the first microwave pulse performs an entangling gate between the first qubit and the second qubit.

[0160] Aspect 27. The method of aspect 17, wherein the first coupler and the second coupler each comprises a first plurality of Josephson junctions arranged in parallel with at least one capacitor.

[0161] Aspect 28. The method of aspect 27, wherein the first coupler and the second coupler each further comprises a second plurality of Josephson junctions arranged in parallel with the first plurality of Josephson junctions and in parallel with the at least one capacitor.

[0162] Aspect 29. The method of aspect 17, wherein the at least one digital logic controller is a superconducting digital logic controller.

[0163] Aspect 30. The system of aspect 1, wherein the first qubit is a first fluxonium qubit, the second qubit is a second fluxonium qubit and the third qubit is a third fluxonium qubit.

[0164] Aspect 31. The method of aspect 17, wherein the first qubit is a first fluxonium qubit, the second qubit is a second fluxonium qubit and the third qubit is a third fluxonium qubit.

[0165] Such alterations, modifications, and improvements are intended to be part of this disclosure and are intended to be within the spirit and scope of the disclosure. Further, though advantages of the present disclosure are indicated, it should be appreciated that not every embodiment of the technology described herein will include every described advantage. Some embodiments may not implement any features described as advantageous herein and in some instances one or more of the described- 41 -ACTIVE 705838443v1features may be implemented to achieve further embodiments. Accordingly, the foregoing description and drawings are by way of example only.

[0166] Aspects of the above-described embodiments of the technology described herein can be implemented in any of numerous ways. For example, aspects of the embodiments may be implemented using hardware, software, or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art by names such as CPU chips, GPU chips, microprocessor, microcontroller, or co-processor. Alternatively, a processor may be implemented in custom circuitry, such as an ASIC, or semi-custom circuitry resulting from configuring a programmable logic device. As yet a further alternative, a processor may be a portion of a larger circuit or semiconductor device, whether commercially available, semi-custom or custom. As a specific example, some commercially available microprocessors have multiple cores such that one or a subset of those cores may constitute a processor. Though, a processor may be implemented using circuitry in any suitable format.

[0167] Various aspects of the present disclosure may be used alone, in combination, or in a variety of arrangements not specifically described in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0168] Also, aspects of the disclosure may be embodied as a method, of which examples have been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0169] Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of- 42 -ACTIVE 705838443v1one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

[0170] The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value. The term “substantially equal” may be used to refer to values that are within ±20% of one another in some embodiments, within ±10% of one another in some embodiments, within ±5% of one another in some embodiments, and yet within ±2% of one another in some embodiments.

[0171] The term “substantially” may be used to refer to values that are within ±20% of a comparative measure in some embodiments, within ±10% in some embodiments, within ±5% in some embodiments, and yet within ±2% in some embodiments. For example, a first direction that is “substantially” perpendicular to a second direction may refer to a first direction that is within ±20% of making a 90° angle with the second direction in some embodiments, within ±10% of making a 90° angle with the second direction in some embodiments, within ±5% of making a 90° angle with the second direction in some embodiments, and yet within ±2% of making a 90° angle with the second direction in some embodiments.

[0172] Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

[0173] What is claimed is:- 43 -ACTIVE 705838443v1

Claims

CLAIMS1. A system comprising: a plurality of qubits including a first qubit, a second qubit and a third qubit; a plurality of couplers each coupled to multiple qubits of the plurality of qubits, the plurality of couplers including a first coupler coupled to the first qubit and to the second qubit, and a second coupler coupled to the second qubit and to the third qubit; a microwave drive line; and at least one controller configured to: receive a microwave signal from the microwave drive line; generate a first pulse envelope signal; generate a first microwave pulse by applying the first pulse envelope signal to the microwave signal; apply the first microwave pulse to the first coupler; generate a second pulse envelope signal; generate a second microwave pulse by applying the second pulse envelope signal to the micro wave signal; and apply the second microwave pulse to the second coupler.

2. The system of claim 1, wherein: the at least one controller comprises a first mixer and a second mixer, the first mixer is configured to receive the first pulse envelope signal, to generate the first microwave pulse and to apply the first microwave pulse to the first coupler, and the second mixer is configured to receive the second pulse envelope signal, to generate the second microwave pulse and to apply the second microwave pulse to the second coupler.

3. The system of claim 2, wherein the at least one controller further comprises superconducting digital logic configured to generate the first pulse envelope signal and provide the first pulse envelope signal to the first mixer, and to generate the second pulse envelope signal and provide the second pulse envelope signal to the second mixer.- 44 -ACTIVE 705838443v14. The system of claim 2, wherein the first mixer and the second mixer are implemented in superconducting digital logic.

5. The system of claim 4, wherein the first mixer and the second mixer each comprises one or more adiabatic quantum flux parametron (AQFP), single-flux quantum (SFQ), and / or quantum flux parametron (QFP) devices.

6. The system of claim 1, wherein the at least one controller is further configured to: generate a third pulse envelope signal; generate a third microwave pulse by applying the third pulse envelope signal to the micro wave signal; and apply the third microwave pulse to the first qubit.

7. The system of claim 6, wherein the at least one controller is further configured to apply the first microwave pulse to the first coupler concurrently with applying the third microwave pulse to the first qubit.

8. The system of claim 1, wherein the at least one controller is further configured to generate the first microwave pulse with an amplitude according to one or more digital values.

9. The system of claim 1, wherein the plurality of qubits is a plurality of flux qubits, and wherein the plurality of couplers each comprises a superconducting loop with a magnetic flux threaded through the superconducting loop.

10. The system of claim 9, wherein the at least one controller is further configured to, prior to applying the first microwave pulse to the first coupler, control the magnetic flux to increase a coupling strength between the first qubit and the second qubit via the first coupler.

11. The system of claim 10, wherein the at least one controller is further configured to, subsequent to applying the first microwave pulse to the first coupler, control the- 45 -ACTIVE 705838443v1magnetic flux to decrease the coupling strength between the first qubit and the second qubit via the first coupler.

12. The system of claim 1, wherein the first micro wave pulse is configured to perform an entangling gate between the first qubit and the second qubit.

13. The system of claim 1, wherein the microwave signal is a single-frequency microwave signal at a frequency of at least 1 GHz.

14. The system of claim 1, wherein the first coupler and the second coupler each comprises a first plurality of Josephson junctions arranged in parallel with at least one capacitor.

15. The system of claim 14, wherein the first coupler and the second coupler each further comprises a second plurality of Josephson junctions arranged in parallel with the first plurality of Josephson junctions and in parallel with the at least one capacitor.

16. The system of claim 1, wherein the first qubit is a first fluxonium qubit, the second qubit is a second fluxonium qubit and the third qubit is a third fluxonium qubit.

17. A method comprising : generating, by at least one digital logic controller, a first microwave pulse by applying a first pulse envelope signal to a microwave signal; applying, by the at least one digital logic controller, the first microwave pulse to a first coupler, the first coupler coupled to a first qubit and to a second qubit; generating, by the at least one digital logic controller, a second microwave pulse by applying a second pulse envelope signal to the microwave signal; and applying, by the at least one digital logic controller, the second microwave pulse to a second coupler, the second coupler coupled to the second qubit and to a third qubit.- 46 -ACTIVE 705838443v118. The method of claim 17, further comprising, prior to applying the first microwave pulse to the first coupler, controlling a flux bias of the first coupler and thereby increasing a coupling strength between the first qubit and the second qubit.

19. The method of claim 18, further comprising, subsequent to applying the first microwave pulse to the first coupler, controlling the flux bias of the first coupler and thereby increasing a coupling strength between the first qubit and the second qubit.

20. The method of claim 18, wherein controlling the flux bias of the first coupler comprises applying a first signal having a frequency below 1 GHz to the first coupler, and wherein the first micro wave pulse has a frequency at or above 1 GHz.

21. The method of claim 17, wherein the at least one digital logic controller comprises: a first digital logic controller configured to receive the first pulse envelope signal, to generate the first microwave pulse and to apply the first microwave pulse to the first coupler; and a second digital logic controller configured to receive the second pulse envelope signal, to generate the second microwave pulse and to apply the second microwave pulse to the second coupler.

22. The method of claim 17, further comprising applying a third microwave pulse to the first qubit concurrently with applying the first microwave pulse to the first coupler.

23. The method of claim 17, wherein the first microwave pulse performs an entangling gate between the first qubit and the second qubit.

24. The method of claim 17, wherein the first coupler and the second coupler each comprises a first plurality of Josephson junctions arranged in parallel with at least one capacitor.- 47 -ACTIVE 705838443v125. The method of claim 17, wherein the first qubit is a first fluxonium qubit, the second qubit is a second fluxonium qubit and the third qubit is a third fluxonium qubit.- 48 -ACTIVE 705838443v1

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