Multi-mode coupler for controlling the interaction between qubits

Abstract

The device comprises a first and a second qubit, and a qubit coupler coupled between the first and second qubits. The second qubit comprises a first and a second mode, with the first mode configured to store data. The qubit coupler comprises a first and a second mode and operates in a first or a second state. In the first state, for suppressing the interaction between the first and second qubits, the first qubit is exchange-coupled to the first mode of the qubit coupler, and the second mode of the second qubit is exchange-coupled to the second mode of the qubit coupler. In the second state, when a control signal is applied to the qubit coupler, the first modes of the first and second qubits are exchange-coupled to both the first and second modes of the qubit coupler to enable the interaction between the first and second qubits for entangling gate operations.

Description

Technical Field

[0001] The present disclosure generally relates to superconducting quantum computing, and more particularly to superconducting quantum systems and devices implemented using superconducting quantum bits (qubits). A superconducting quantum computing system is implemented using a circuit quantum electrodynamics (QED) device that utilizes the quantum dynamics of electromagnetic fields in a superconducting circuit containing superconducting qubits to generate and process quantum information. Generally, a superconducting qubit is implemented using components such as a superconducting tunnel junction (e.g., Josephson junction), an inductor, or a capacitor, or a combination thereof, and behaves as a quantum mechanical anharmonic (nonlinear) oscillator with quantized states when cooled to extremely low temperatures. The qubit can be substantially operated as a two-level system using the ground state and the first excited state of the qubit due to the anharmonicity imparted by a nonlinear inductor element of the qubit (e.g., Josephson inductance) that allows the ground state and the first excited state of the qubit to be uniquely addressed at the transition frequency of the qubit without significantly hindering higher excited states of the qubit.

Background Art

[0002] Various types of quantum information processing operations can be implemented using a superconducting quantum processor comprising a plurality of superconducting qubits, which can be coherently controlled, placed in a quantum superposition state (e.g., via a single-gate operation), exhibit a quantum interference effect, and be entangled with each other (e.g., via an entangling-gate operation). The fidelity of the quantum gate operation may be adversely affected by unwanted crosstalk (e.g., residual static ZZ interaction) between adjacent superconducting qubits. For example, unwanted crosstalk between superconducting qubits may cause the transition frequency of one superconducting qubit to depend on the state of one or more neighboring superconducting qubits. Such unwanted crosstalk becomes an increasingly problematic issue as the number of superconducting qubits increases and the integration density increases and the quantum processor is scaled. Summary of the Invention

[0003] Exemplary embodiments of the present disclosure include a quantum device configured to control the interaction (e.g., ZZ interaction) between coupled superconducting qubits.

[0004] The exemplary embodiments include a device comprising a first superconducting qubit, a second superconducting qubit, and a superconducting qubit coupler. The second superconducting qubit comprises a first mode and a second mode, and the first mode is configured to store quantum data. The superconducting qubit coupler is coupled between the first superconducting qubit and the second superconducting qubit. The superconducting qubit coupler comprises a first mode and a second mode, and is configured to operate in one of a first state and a second state in response to a control signal being applied to the superconducting qubit coupler. In the first state of the superconducting qubit coupler, the first superconducting qubit is exchange-coupled to the first mode of the superconducting qubit coupler and the second mode of the second superconducting qubit is exchange-coupled to the second mode of the superconducting qubit coupler in order to suppress the interaction between the first superconducting qubit and the first mode of the second superconducting qubit. In the second state of the superconducting qubit coupler, the first superconducting qubit and the first mode of the second superconducting qubit are exchange-coupled to both the first mode and the second mode of the superconducting qubit coupler in order to enable the interaction between the first superconducting qubit and the first mode of the second superconducting qubit and to perform a entangling gate operation in response to the control signal being applied to the superconducting qubit coupler.

[0005] Conveniently, in an example where the superconducting qubit coupler is in a state where mode-selective exchange coupling is performed in which the first and second superconducting qubits are exchange-coupled to different modes of the superconducting qubit coupler, the coupling of the first and second superconducting qubits using a superconducting qubit coupler having two completely different excitation modes enables simultaneous suppression of both the static ZZ interaction and the exchange interaction between the first superconducting qubit and the second superconducting qubit. In such mode-selective exchange coupling, the interaction between the first superconducting qubit and the second superconducting qubit becomes substantially zero. Further, the mode-selective exchange coupling functions to suppress the spectator effect between the superconducting qubit coupler and the remote superconducting qubit.

[0006] Another exemplary embodiment includes a system comprising a quantum processor and a control system. The quantum processor comprises an array of superconducting qubits. The control system is configured to generate control signals for controlling the quantum processor. The array of superconducting qubits comprises a first superconducting qubit, a second superconducting qubit, and a superconducting qubit coupler. The second superconducting qubit comprises a first mode and a second mode, and the first mode is configured to store quantum data. The superconducting qubit coupler is coupled between the first superconducting qubit and the second superconducting qubit. The superconducting qubit coupler comprises a first mode and a second mode, and is configured to operate in one of a first state and a second state in response to a control signal being applied to the superconducting qubit coupler. In the first state of the superconducting qubit coupler, the first superconducting qubit is exchange-coupled to the first mode of the superconducting qubit coupler and the second mode of the second superconducting qubit is exchange-coupled to the second mode of the superconducting qubit coupler to suppress the interaction between the first superconducting qubit and the first mode of the second superconducting qubit. In the second state of the superconducting qubit coupler, the first superconducting qubit and the first mode of the second superconducting qubit are exchange-coupled to both the first mode and the second mode of the superconducting qubit coupler to enable the interaction between the first superconducting qubit and the first mode of the second superconducting qubit and to perform a entangling gate operation in response to a control signal being applied to the superconducting qubit coupler.

[0007] Another exemplary embodiment includes a computer program product for calibrating a crosstalk gate process. The computer program product comprises one or more computer-readable storage media and program instructions collectively stored on the one or more computer-readable storage media. The program instructions comprise program instructions for calibrating parameters of a control pulse configured to control the operation of a superconducting qubit coupler to perform a crosstalk gate operation between a first superconducting qubit and a second superconducting qubit coupled to the superconducting qubit coupler. The program instructions for calibrating parameters of the control pulse comprise program instructions for performing an iterative process. The iterative process includes one or more iterations of performing an amplitude calibration process and performing a frequency detuning calibration process. The amplitude calibration process is performed to calibrate the amplitude of the control pulse such that, in response to applying a control pulse having a calibrated amplitude to the superconducting qubit coupler, the superconducting qubit coupler is rotated from the ground state to the first excited state and back to the ground state. The frequency detuning calibration process is performed to calibrate the detuning of the frequency of the control pulse relative to the operating frequency of the superconducting qubit coupler to achieve a target conditional rotation of the first superconducting qubit and the second superconducting qubit in response to applying a control pulse having a calibrated amplitude and a calibrated frequency detuning to the superconducting qubit coupler.

[0008] Conveniently, an exemplary entanglement gate calibration process provides a precise and efficient method for calibrating control pulses for two-qubit conditional gate operations, and the calibration process is configured to separately calibrate the control pulse shapes to achieve a complete 2π X-axis rotation with high accuracy through the superconducting qubit coupler regardless of the detuning between the control pulse and the mode frequency of the superconducting qubit coupler. The pulse detuning is thus separately calibrated to achieve the desired conditional Z-rotation of the first and second superconducting qubits. A process of repeatedly iterating separate detuning and amplitude calibration stages allows the entanglement gate calibration process to gradually achieve higher fidelity.

[0009] In the following detailed description of exemplary embodiments, which will be read in conjunction with the accompanying figures, other embodiments are described.

Brief Description of the Drawings

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DETAILED DESCRIPTION OF THE INVENTION

[0011] Exemplary embodiments of the present disclosure will now be described in more detail with respect to a quantum device configured to control the interaction (e.g., ZZ interaction) between coupled superconducting qubits. It should be understood that the various features shown in the accompanying drawings are schematic diagrams that are not drawn to scale. Moreover, the same or similar reference numbers are used throughout the drawings to represent the same or similar features, elements, or structures, and thus, detailed descriptions of the same or similar features, elements, or structures are not repeated for each of the drawings. Further, as used herein, the term "exemplary" means "serving as an example, instance, or illustration." Any embodiment or design described herein as "exemplary" should not be construed as being preferred or advantageous over other embodiments or designs.

[0012] Furthermore, the phrase "configured to" as used with a circuit, structure, element, component, or the like that performs one or more functions or otherwise provides some functionality is intended to encompass embodiments in which the circuit, structure, element, component, or the like is implemented in hardware, software, and / or a combination thereof, and in an implementation that includes hardware, where the hardware can include quantum circuit elements (e.g., qubits, coupler circuitry, etc.), individual circuit elements (e.g., transistors, inverters, etc.), programmable elements (e.g., application specific integrated circuit (ASIC) chips, field programmable gate array (FPGA) chips, etc.), processing devices (e.g., central processing unit (CPU), graphics processing unit (GPU), etc.), one or more integrated circuits, and / or combinations thereof. Thus, by way of example only, when a circuit, structure, element, component, etc. is defined as being configured to provide a particular functionality, it is intended to cover embodiments that include an element, processing device, or integrated circuit, or a combination thereof, that enables the circuit, structure, element, component, etc. to perform the particular functionality when in an operational state (e.g., being connected or otherwise deployed in a system, being powered on, receiving an input, or producing an output, or a combination thereof), without limiting to embodiments where the circuit, structure, element, component, etc. is in a non-operational state (e.g., not being connected or otherwise deployed in a system, not being powered on, not receiving an input, or not producing an output, or a combination thereof), or a partial operational state.

[0013] As is known in the art, quantum computing provides a computing paradigm that utilizes the fundamental principles of quantum mechanics to perform calculations. Quantum computing algorithms and applications are defined using quantum circuits. A quantum circuit is a computational routine that defines coherent quantum operations to be performed on quantum data stored in qubits, in combination with operations performed using classical computing. Quantum circuits are utilized to abstractly define complex algorithms and applications that are executable on a quantum computer. In a quantum computer, primitive operations include gate operations (e.g., single-qubit gate operations, two-qubit gate operations, multi-qubit gate operations (e.g., three or more qubits)) applied to qubits to perform quantum computing operations for a given application. A quantum circuit enables a quantum computer to receive classical data, perform quantum operations based on the received data, and output a classical solution.

[0014] A single qubit can have a basic state of |0〉 or |1〉, or a linear combination of such basic states known as a superposition state. As is known in the art, the state of a qubit can be graphically represented as a point on a unit sphere (radius = 1), called the Bloch sphere, as illustrated in FIG. 1. In particular, FIG. 1 illustrates an exemplary Bloch sphere 100 in which the basic states |0〉 or |1〉 of a qubit are represented along the Z-axis of the Bloch sphere 100, where the point 101 on the positive Z-axis represents the ground state |0〉 and the point 102 on the negative Z-axis represents the first excited state |1〉 of the qubit. The superposition state |ψ〉 of a qubit can be represented as a point on the Bloch sphere as follows,

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[0015] The state of a qubit can be changed by applying single - qubit gate operations to the qubit. Single - qubit gate operations rotate the current state of the qubit around, for example, the X - axis, Y - axis, or Z - axis, or a combination thereof, according to a given gate operation. A rotation around the Z - axis changes the angle Φ. Further, a qubit can be entangled with the states of two or more qubits, and thus can be controlled using entanglement gate operations to generate a combined state of two or more qubits that contains more information than the individual states of the qubits.

[0016] As described above, a superconducting quantum computing system can be implemented using superconducting qubits. For example, a superconducting transmon (transmission - line shunted type plasma oscillator) qubit is one type of superconducting qubit that includes a superconducting tunnel junction device (e.g., a Josephson junction) connected in parallel with a capacitor. A Josephson junction functions as a non - linear inductor that forms an anharmonic oscillator with individually addressable energy levels (e.g., two lowest energy levels corresponding to the ground state |0〉 and the first excited state |1〉) when shunted by a capacitor. Transmon qubits are widely used in superconducting quantum computing because they have a relatively simple structure that provides a sufficient coherence time, allows coupling to other superconducting circuit elements, and facilitates qubit readout, etc. To control and read out the qubit, the structure can be designed to capacitively couple or inductively couple the transmon qubit to other circuit elements including microwave and flux drive lines, readout resonators, and couplers.

[0017] Furthermore, the transmon qubit can be designed to have a relatively high anharmonic spectrum, with a relatively high frequency separation between the computational and non-computational states, enabling efficient use of the superconducting transmon qubit as a two-level quantum system. In particular, as is known in the art, the operating frequency (alternatively, the transition frequency) of a superconducting qubit is the frequency corresponding to the energy difference between the qubit's ground state |0〉 and its first excited state |1〉. In the case of a superconducting qubit, higher energy levels are available for a given qubit (e.g., |2〉, |3〉, etc.), but the quantum system is designed to utilize each superconducting qubit as a basic two-level system for quantum computing, insulating the two lowest energy levels of the superconducting qubit (i.e., the ground state |0〉 and the first excited state |1〉), and ignoring the higher energy states. The term "anharmonicity" as used herein refers to the difference between (i) the frequency (f01) for the transition of the qubit from the ground state |0〉 to the first excited state |1〉 and (ii) the frequency (f12) for the transition of the qubit from the first excited state |1〉 to the second excited state |2〉.

[0018] Another type of superconducting qubit that can be used to implement a quantum system includes a multi-mode two-junction superconducting qubit architecture, as schematically illustrated in FIGS. 2A and 2B. More specifically, FIG. 2A is a schematic lumped-element circuit diagram of a multi-mode two-junction superconducting qubit 200, alternatively referred to as a superconducting tunable coupler qubit (TCQ). The multi-mode two-junction superconducting qubit 200 includes a shunt capacitor 221 and two capacitive shunt-type superconducting tunnel junctions (e.g., Josephson junctions) connected in series between a first node N1 and a second node N2 of the multi-mode two-junction superconducting qubit 200. The first capacitive shunt-type Josephson junction includes a first Josephson junction 222 and a first capacitor 223, and the second capacitive shunt-type Josephson junction includes a second Josephson junction 224 and a second capacitor 225. The shunt capacitor 221 has a capacitance CS, the first Josephson junction 222 has a Josephson energy EJ1, the second Josephson junction 224 has a Josephson energy EJ2, the first capacitor 223 has a capacitance C1, and the second capacitor 225 has a capacitance C2. The two capacitive shunt-type superconducting Josephson junctions are coupled to an intermediate node N3.

[0019] Next, FIG. 2B schematically illustrates a planar circuit configuration of the multi-mode two-junction superconducting qubit 200 of FIG. 2A, according to an exemplary embodiment of the present disclosure. More specifically, FIG. 2B schematically illustrates a planar multi-mode two-junction superconducting qubit 201 including a first superconducting pad 201-1, a second superconducting pad 201-2, and a third (intermediate) superconducting pad 201-3. The first Josephson junction 222 is coupled between the first and third superconducting pads 201-1 and 201-3, and the second Josephson junction 224 is coupled between the second and third superconducting pads 201-2 and 201-3.

[0020] In this configuration, the superconducting pads 201-1, 201-2, and 201-3 include planar capacitor electrodes that form a coplanar parallel plate capacitor structure. For example, in the exemplary configuration of FIG. 2B, the first and second superconducting pads 201-1 and 201-2 include the planar capacitor electrodes of the shunt capacitor 221 of FIG. 2A. Further, the first and third superconducting pads 201-1 and 201-3 include the planar capacitor electrodes of the first capacitor 223 of FIG. 2A, and the second and third superconducting pads 201-2 and 201-3 include the planar capacitor electrodes of the second capacitor 225 of FIG. 2A. The first, second, and third superconducting pads 201-1, 201-2, and 201-3 respectively correspond to the first, second, and third nodes N1, N2, and N3 of FIG. 2A.

[0021] The multi-mode two-junction superconducting qubit 201 includes two excitation modes having entirely different frequencies and spatial symmetries. More specifically, the multi-mode two-junction superconducting qubit 201 includes two entirely different excitation modes corresponding to symmetric and anti-symmetric combinations of excitations associated with the two junctions, and the two entirely different normal modes are: (i) a low-frequency "bright" mode (referred to herein as the A mode) having a non-zero dipole moment, and (ii) a high-frequency "dark" mode (referred to herein as the B mode) having no dipole moment and not coupling to an external field. FIG. 2C schematically illustrates the two entirely different excitation modes of the multi-mode two-junction superconducting qubit 201 of FIG. 2B, with the "bright" mode depicted as the A mode 202 and the "dark" mode depicted as the B mode 204.

[0022] As schematically shown in FIG. 2C, the A mode 202 has a charge pattern in which, at any given point in time, the first and second superconducting pads 201-1 and 201-2 have opposite charges and the third (intermediate) superconducting pad 201-3 has a net zero charge. On the other hand, the B mode 204 has a charge pattern in which, at any given point in time, the first and second superconducting pads 201-1 and 201-2 have the same charge and the third (intermediate) superconducting pad 201-3 has a charge opposite to the charges on the first and second superconducting pads 201-1 and 201-2. The A and B modes have different frequencies. For example, the mode frequency fB of the B mode 204 is greater than the mode frequency fA of the A mode 202, allowing individual excitation of one mode with respect to the other mode.

[0023] To implement an extensible quantum computing system (e.g., a quantum processor) using superconducting qubits, it is important to, for example, enable independent qubit control for high-fidelity single qubit gate operations and to minimize or eliminate crosstalk and unwanted ZZ interactions between coupled or adjacent qubits to implement high-fidelity entangling gates (e.g., two-qubit gates) between superconducting qubits. As described above, an entangling gate is an operation in which an external field (e.g., a microwave signal) is applied to a quantum processor to create an entangled state between two or more distinct qubits. For example, a controlled phase gate (referred to herein as a CPHASE gate) between two qubits is one type of entangling gate in which one qubit (e.g., the target qubit) acquires a phase shift if and only if both qubits are in their first excited state.

[0024] The ZZ interaction is a type of long - range interaction between two qubits or modes where the excitation of one qubit or mode causes a shift in the transition frequency of the other qubit or mode. In certain examples, the ZZ interaction enables a state - dependent shift in the qubit frequency equivalent to a state - dependent phase shift, thus providing a means to entangle two different qubits and create, for example, a CPHASE gate. The ZZ interaction is sometimes referred to as a long - range coupling or represented as chi or 2 - chi coupling. On the other hand, in some examples, the unwanted ZZ interaction between two superconducting qubits can be a source of crosstalk that can reduce the fidelity of gate operations.

[0025] For example, the static ZZ interaction is a type of ZZ interaction that exists between two qubits or modes in the absence of any external RF drive. This "always - on" interaction contains unwanted crosstalk and can be detrimental to the qubit quantum system by inhibiting independent control of each qubit and creating unwanted entanglement. In particular, the static ZZ coupling can occur between neighboring qubits, where the state of one qubit affects the qubit frequency of the adjacent qubit, and the static ZZ coupling can lead to gate errors. In this regard, the static ZZ interaction is a type of unwanted crosstalk between neighboring qubits and can negatively impact the fidelity of single - qubit gate operations, calibrated under the assumption that the qubit frequencies are fixed.

[0026] Another source of crosstalk between neighboring qubits involves the exchange interaction that occurs when a microwave pulse is applied to one qubit (such as for single - qubit gate operations), and such an exchange interaction between neighboring qubits can potentially affect another neighboring qubit if it exists. For example, if an exchange interaction exists between two qubits, applying a microwave pulse to one qubit can, to the extent possible, excite the other qubit, which is a form of crosstalk that can be detrimental.

[0027] Furthermore, another type of crosstalk is called the "spectator effect", where the state of an irrelevant qubit (e.g., a "spectator") represents a state that affects the fidelity / performance of a gate operation between two other qubits. For example, a two - qubit gate operation can be implemented by driving a coupling element that connects the two qubits. In such an example, the coupling element can have its own operating frequency (e.g., transition frequency) calibrated to perform a high - fidelity two - qubit gate operation. If there is a static ZZ coupling between the coupling element and the spectator qubit, the state of the spectator qubit can affect the transition frequency of the coupling element in such a way as to reduce the fidelity of the calibrated two - qubit gate operation.

[0028] To implement high-fidelity gate operations, exemplary embodiments of the present disclosure regarding techniques for implementing an extensible quantum computing architecture with superconducting qubits coupled using a multi-mode qubit coupler are discussed herein. Exemplary qubit coupling techniques as described herein are configured to suppress unwanted crosstalk between coupled superconducting qubits (e.g., suppressing the static ZZ interaction between coupled superconducting qubits and suppressing the exchange interaction between coupled superconducting qubits when performing a single gate operation on one of the coupled qubits), and suppressing the spectator affects between the multi-mode qubit coupler and neighboring qubits (e.g., spectator qubits).

[0029] FIG. 3 schematically illustrates a quantum device comprising superconducting qubits coupled by a multimode coupler circuit according to an exemplary embodiment of the present disclosure. In particular, FIG. 3 schematically illustrates a quantum device 300 comprising a first superconducting qubit 310 (or first qubit 310), a second superconducting qubit 320 (or second qubit 320), and a superconducting multimode coupler circuit 330 (or multimode coupler circuit 330). In the exemplary configuration, the first and second qubits 310 and 320 comprise computational qubits, and the multimode coupler circuit 330 is configured to mediate interactions (e.g., ZZ interactions) between the first and second qubits 310 and 320 during different gate operations. In some embodiments, the first and second qubits 310 and 320 comprise different types of superconducting qubits. For example, in some embodiments, the first qubit 310 comprises a transmon qubit, and the second qubit 320 comprises a multimode two-junction superconducting qubit having a circuit architecture as shown in FIG. 3. In some embodiments, the first qubit 310 can be implemented using other types of superconducting qubits, such as a superconducting fluxonium qubit. In some embodiments, the first qubit 310 and the second qubit 320 comprise (non-adjustable) fixed-frequency qubits, while in other embodiments, the first and second qubits 310 and 320 can be adjustable qubits, such as adjustable transition frequencies.

[0030] Furthermore, in some embodiments, the multimode coupler circuit 330 includes a multimode two-junction superconducting qubit having a circuit architecture as shown in FIG. 3 and is capacitively coupled to the first and second qubits 310 and 320. The multimode coupler circuit 330 is configured to control the exchange interaction between the first and second qubits 310 and 320 in order to perform entanglement gate operations, as well as to suppress crosstalk between the first and second qubits 310 and 320 during idle times (e.g., suppressing the static ZZ interaction), or to suppress crosstalk between the first and second qubits 310 and 320 when single-qubit gate operations are being performed on one or both of the first and second qubits 310 and 320 (e.g., suppressing the ZZ interaction). In other embodiments, the multimode coupler circuit 330 can be implemented using any suitable superconducting multimode qubit framework that includes a combination of two strongly interacting non-harmonic oscillators, where the composite quantum system is characterized by multiple excitation modes that exhibit strong long-range coupling between them, i.e., an excitation in one mode can strongly shift the transition frequency of another mode.

[0031] The quantum device 300 further includes a plurality of control lines (e.g., transmission line resonators), including but not limited to qubit drive lines 312 and 322, qubit readout lines 314 and 324, and coupler drive line 332. In some embodiments, qubit drive lines 312 and 322 are coupled (e.g., capacitively coupled) to first and second qubits 310 and 320, respectively. In some embodiments, qubit drive lines 312 and 322 are configured to apply control signals (e.g., microwave control pulse signals) to independently change the states of the respective first and second qubits 310 and 320 (e.g., single qubit gate operations). As is known in the art, the state of a qubit can be changed by applying a microwave control signal (e.g., a control pulse) whose center frequency is equal to the transition frequency of the qubit (denoted as f01), and the transition frequency f01 corresponds to the energy difference between the ground state |0〉 and the excited state |1〉 of the qubit. Further, the axis of rotation about a given axis of the Bloch sphere 100 (e.g., the X-axis, the Y-axis, or any axis in the X-Y plane), and the amount (angle) of such rotation are based on the phase of the microwave control signal, as well as the amplitude and duration of the microwave control signal, respectively.

[0032] Furthermore, in some embodiments, readout lines 314 and 324 are coupled to first and second qubits 310 and 320 respectively using known techniques (e.g., dispersive readout). In some embodiments, readout lines 314 and 324 comprise transmission line readout resonators (e.g., coplanar transmission lines) configured to have resonance frequencies that are detuned from the respective transition frequencies of the first and second qubits 310 and 320. The coupling of readout lines 314 and 324 to the respective first and second qubits 310 and 320 results in a shift in the resonance frequency of the transmission line readout resonator depending on the states of the respective first and second qubits 310 and 320. A change in the resonance frequency of the transmission line readout resonator coupled to a given qubit is utilized to determine the readout state of the given qubit, e.g., the ground state |0〉 or the excited state |1〉, and in the case of readout, a superposition state of a given qubit is projected onto one of the ground or excited states of the qubit as is known in the art.

[0033] Coupler drive line 332 is coupled (e.g., capacitively coupled) to multimode coupler circuit 330. In some embodiments, multimode coupler circuit 330 comprises two distinct normal excitation modes (e.g., distinct A and B excitation modes having distinct frequencies and spatial symmetries as shown in FIG. 2C), and coupler drive line 332 is configured to apply control signals to multimode coupler circuit 330 to control mode-selective exchange coupling between first and second qubits 310 and 320 and the different excitation modes of multimode coupler circuit 330.

[0034] For example, as will be described in more detail below, the multimode coupler circuit 330 includes a first mode and a second mode, and is configured to operate in a first state or a second state in response to a drive control signal applied to the multimode coupler circuit 330. In a first state of the multimode coupler circuit 330 (e.g., a ground state |0〉 in which a control pulse is not applied to the coupler drive line 332), the first qubit 310 is exchange-coupled to a first mode (e.g., an A mode) of the multimode coupler circuit 330, the second qubit 320 is exchange-coupled to a second mode (e.g., a B mode) of the multimode coupler circuit 330, and the interaction between the first and second qubits 310 and 320 is suppressed.

[0035] On the other hand, in a second state of the multimode coupler circuit 330, a properly calibrated control pulse is applied to the coupler drive line 332 to drive a second mode (e.g., a B mode) of the multimode coupler circuit 330 through a full 2π rotation (e.g., from the ground state |0〉 to the first excited state |1〉 and back to the ground state |0〉), whereby the first and second qubits 310 and 320 are exchange-coupled to both the first and second modes of the multimode coupler circuit 330, enabling the interaction between the first and second qubits 310 and 320 and performing a entangling gate operation.

[0036] In embodiments where the first qubit 310 or the second qubit 320 or both include a frequency tunable qubit (e.g., a flux tunable transmon qubit, or a fluxonium qubit, etc.), it should be noted that the quantum device 300 should comprise flux - bias control lines coupled (e.g., inductively coupled) to the respective tunable qubits. In such embodiments, the flux - bias control lines should be configured to apply a flux - bias control signal to the tuning structure of the qubit in order to tune the operating frequency of the tunable qubit, as required for a given application. For example, a flux - tunable transmon qubit can be implemented by replacing the transmon Josephson junction with a superconducting quantum interference device (SQUID) that forms a superconducting loop (referred to as a SQUID loop) through which an external magnetic flux can be threaded to change the substantial Josephson energy of the transmon qubit and thus change the transition frequency of the transmon qubit. In this regard, the flux - bias control line should be configured to apply an external magnetic flux to the SQUID loop of the transmon qubit in order to tune the transition frequency of the transmon qubit.

[0037] The quantum device 300 of FIG. 3 can be implemented using different circuit architectures according to exemplary embodiments, which will be described in further detail hereinafter in conjunction with FIGS. 4, 5, and 6. For example, FIG. 4 schematically illustrates a circuit diagram of a quantum device comprising superconducting qubits coupled by a superconducting multimode coupler circuit according to an exemplary embodiment of the present disclosure. More specifically, FIG. 4 is a schematic circuit diagram (e.g., a lumped element circuit diagram) of a superconducting quantum circuit 400 comprising a superconducting transmon qubit (or first data qubit 410), a superconducting multimode qubit 420 (or second data qubit 420), a superconducting multimode qubit coupler 430, a first coupling capacitor 440, a second coupling capacitor 442, and a third coupling capacitor 450. The superconducting multimode qubit coupler 430, as well as the coupling capacitors 440 and 442, collectively comprise an exemplary embodiment of the multimode coupler circuit 330 of FIG. 3.

[0038] The superconducting transmon qubit 410 comprises a superconducting Josephson tunnel junction 412 (with a critical current ICT1) and a shunt capacitor 414 (with a capacitance CT1), which are connected in parallel between a first node N1 and a second node N2 of the superconducting transmon qubit 410. The Josephson tunnel junction 412 comprises a small junction capacitance, which is omitted from FIG. 4 for ease of illustration. The capacitance CT1 of the shunt capacitor 414 is large compared to the junction capacitance of the Josephson tunnel junction 412.

[0039] In some embodiments, as shown in FIG. 4, the superconducting multi-mode qubit 420 and the superconducting multi-mode qubit coupler 430 each have a multi-mode two-junction superconducting qubit architecture similar to the multi-mode two-junction superconducting qubit 200 of FIG. 2A. In particular, the superconducting multi-mode qubit coupler 430 includes a shunt capacitor 431 and two capacitive shunt-type superconducting tunnel junctions (e.g., Josephson junctions) connected in series between the first node N1 and the second node N2 of the superconducting multi-mode qubit coupler 430. The first capacitive shunt-type Josephson junction includes a Josephson junction 432 and a capacitor 433, and the second capacitive shunt-type Josephson junction includes a Josephson junction 434 and a capacitor 435. The shunt capacitor 431 has a capacitance CS1, the Josephson junction 432 has a critical current IC1, the Josephson junction 434 has a critical current IC2, the capacitor 433 has a capacitance C1, and the capacitor 435 has a capacitance C2. The two capacitive shunt-type superconducting Josephson junctions are coupled to an intermediate node N3 of the superconducting multi-mode qubit coupler 430.

[0040] Similarly, the superconducting multi-mode qubit 420 includes a shunt capacitor 421 and two capacitive shunt-type superconducting tunnel junctions (e.g., Josephson junctions) connected in series between the first node N1 and the second node N2 of the superconducting multi-mode qubit 420. The first capacitive shunt-type Josephson junction includes a Josephson junction 422 and a capacitor 423, and the second capacitive shunt-type Josephson junction includes a Josephson junction 424 and a capacitor 425. The shunt capacitor 421 has a capacitance CS2, the Josephson junction 422 has a critical current IC3, the Josephson junction 424 has a critical current IC4, the capacitor 423 has a capacitance C3, and the capacitor 425 has a capacitance C4. The two capacitive shunt-type superconducting Josephson junctions are coupled to an intermediate node N3 of the superconducting multi-mode qubit 420.

[0041] As further shown in FIG. 4, the first coupling capacitor 440 is configured to capacitively couple the first node N1 of the superconducting transmon qubit 410 to the first node N1 of the superconducting multimode qubit coupler 430. Further, the second coupling capacitor 442 is configured to capacitively couple the second node N2 of the superconducting transmon qubit 410 to the second node N2 of the superconducting multimode qubit coupler 430. Further, the third coupling capacitor 450 is configured to capacitively couple the third node N3 of the superconducting multimode qubit 420 to the third node N3 of the superconducting multimode qubit coupler 430.

[0042] In the exemplary circuit configuration of FIG. 4, the superconducting transmon qubit 410 and the superconducting multimode qubit 420 are configured as "data qubits" for storing quantum data, while the superconducting multimode qubit coupler 430 includes a multimode qubit configured as a "coupler qubit" for coupling the data qubits and controlling the interaction between the data qubits. As described above, thanks to the multimode two-junction superconducting qubit architecture, the superconducting multimode qubit 420 and the superconducting multimode qubit coupler 430 each have two completely different excitation modes, such as the A mode and the B mode, as shown in FIG. 2C. The superconducting transmon qubit 410 and the superconducting multimode qubit 420 are directly coupled to different modes of the superconducting multimode qubit coupler 430.

[0043] For example, in some embodiments, the superconducting transmon qubit 410 is directly coupled to the A-mode of the superconducting multimode qubit coupler 430 by virtue of capacitive coupling of each first node N1 through the first coupling capacitor 440 and capacitive coupling of each second node N2 through the second coupling capacitor 442. Further, the superconducting multimode qubit 420 is directly coupled to the B-mode of the superconducting multimode qubit coupler 430 by virtue of capacitive coupling of each third node N3 through the coupling capacitor 450. More specifically, in the exemplary embodiment shown in FIG. 4, the B-mode of the superconducting multimode qubit 420 is directly coupled to the B-mode of the superconducting multimode qubit coupler 430 by virtue of capacitive coupling of each third node N3 through the third coupling capacitor 450. Further, the A-mode of the superconducting multimode qubit 420 is configured as a data mode utilized for storing quantum data, and thus, in the exemplary configuration of FIG. 4, the A-mode is not directly coupled to the superconducting multimode qubit coupler 430.

[0044] Further, in the exemplary configuration of FIG. 4, the superconducting multimode qubit coupler 430 is configured to operate in a first state (referred to herein as the “off” state) and a second state (referred to herein as the “on” state) to control the interaction between coupled data qubits such as, for example, the superconducting transmon qubit 410 and the superconducting multimode qubit 420. In some embodiments, the operating state of the superconducting multimode qubit coupler 430 is controlled by applying a microwave control pulse to the superconducting multimode qubit coupler 430 (via a coupler drive line 332 of FIG. 3 not specifically shown in FIG. 4) to control the state of the B-mode of the superconducting multimode qubit coupler 430.

[0045] For example, in the "off" state, the B mode of the superconducting multi-mode qubit coupler 430 is in the ground state |0〉, and the microwave control pulse is not applied to the superconducting multi-mode qubit coupler 430. In the "off" state, the superconducting multi-mode qubit coupler 430 performs mode-selective exchange coupling between the first and second data qubits 410 and 420, as well as different excitation modes of the superconducting multi-mode qubit coupler 430 to create a state of essentially zero interaction (e.g., suppressed static ZZ interaction) between the first and second data qubits 410 and 420. More specifically, when the superconducting multi-mode qubit coupler 430 is in the "off" state, due to different charge patterns of the A and B modes of the superconducting multi-mode qubit coupler 430 (as shown in, for example, FIG. 3C), the superconducting transmon qubit 410 is exchange-coupled only to the A mode of the superconducting multi-mode qubit coupler 430, and the B mode of the superconducting multi-mode qubit 420 is exchange-coupled only to the B mode of the superconducting multi-mode qubit coupler 430. In this regard, in the "off" state, the residual long-term interaction (e.g., static ZZ interaction) between the first and second data qubits 410 and 420 is substantially suppressed because the first and second data qubits 410 and 420 are exchange-coupled to different modes of the superconducting multi-mode qubit coupler 430.

[0046] On one hand, to perform the entangling gate operation, the superconducting multi-mode qubit coupler 430 is temporarily turned "on" by applying a microwave control pulse (with an appropriately calibrated frequency, duration, envelope amplitude, etc.) to drive the B mode of the superconducting multi-mode qubit coupler 430 through a complete 2π rotation around the X axis of the Bloch sphere from the ground state |0〉 to the first excited state |1〉 and back to the ground state |0〉 again. The microwave control pulse changes the energies of the A and B modes of the superconducting multi-mode qubit coupler 430 and drives the B mode of the superconducting multi-mode qubit coupler 430 in a manner that essentially separates the mode-selective couplings of the first and second data qubits 410 and 420 into the respective A and B modes of the superconducting multi-mode qubit coupler 430, thereby enabling the superconducting multi-mode qubit coupler 430 to perform an entangling gate operation (e.g., a CPHASE gate operation) on the ZZ interaction between the first and second data qubits 410 and 420.

[0047] More specifically, driving the B mode of the superconducting multi-mode qubit coupler 430 causes a state-dependent Stark shift in both the data modes of the first and second data qubits 410 and 420 (e.g., the plasmon mode of the superconducting transmon qubit 410 and the A mode of the superconducting multi-mode qubit 420), such that the data modes of the first and second data qubits 410 and 420 have a long-term coupling (e.g., a ZZ coupling) to the B mode of the superconducting multi-mode qubit coupler 430. The state-dependent Stark shift of the data modes is equivalent to a microwave-activated ZZ coupling between the data modes of the first and second data qubits 410 and 420 that mediates the entangling gate operation between the first and second data qubits 410 and 420. For example, in the case of a CPHASE gate operation, a conditional Z rotation can be achieved when both of the data modes of the first and second data qubits 410 and 420 are in the excited state (e.g., the ket|1〉 state).

[0048] Note that the state-dependent Stark shift is a type of AC Stark shift utilized in the coupling configuration of FIG. 4 to mediate a long-term interaction (e.g., a ZZ interaction) between first and second data qubits 410 and 420 for implementing entangling gate operations. In general, the AC Stark shift is based on the AC Stark effect, where the transition frequency of a qubit can be shifted to some extent by applying an electromagnetic (EM) tone to the qubit. The amount of the AC Stark shift of the qubit transition frequency is based on, for example, the intensity (e.g., amplitude) of the EM tone and the amount of detuning between the frequency of the EM tone and the transition frequency of the qubit. In the exemplary embodiment of FIG. 4, a state-dependent Stark shift is realized by applying a microwave pulse having a given frequency to drive the B mode of a superconducting multimode qubit coupler 430 (having a given transition frequency), causing an energy exchange coupling between the superconducting multimode qubit coupler 430 and the first and second data qubits 410 and 420. Depending on the states of the first and second data qubits 410 and 420, a shift in the transition frequency of the superconducting multimode qubit coupler 430 can occur depending on the data qubit states, and such a state-based shift in the transition frequency of the superconducting multimode qubit coupler 430 then changes the intensity of the AC Stark shift felt by the first and second data qubits 410 and 420. The resulting state-dependent Stark shift is utilized to implement a controlled Z gate between the first and second data qubits 410 and 420.

[0049] The exemplary superconducting quantum circuit 400 shown in FIG. 4 can be implemented using planar microwave circuit elements formed on a substrate (e.g., a silicon substrate) using state-of-the-art semiconductor fabrication techniques and materials. For example, various components such as superconducting qubits (e.g., Josephson tunnel junction devices, inductors, capacitors), interconnects, coupling circuit devices, coupler drive lines, qubit drive lines, and state readout lines comprise lithographically defined patterns of superconducting material formed on a semiconductor substrate. The circuit elements can be formed using various types of superconducting materials suitable for a given application, including but not limited to basic metals such as niobium (Nb), aluminum (Al), tantalum (Ta), and compounds such as titanium nitride (TiN), niobium nitride (NbN), niobium titanium nitride (NbTiN). A Josephson tunnel junction device comprises two superconducting electrodes separated by a thin insulating barrier. For example, a Josephson tunnel junction device for a superconducting qubit can comprise an Al-AlOx-Al three-layer tunnel junction fabricated using a double-angle evaporation technique or other suitable fabrication techniques.

[0050] FIG. 5 schematically illustrates a planar circuit configuration of the superconducting quantum circuit 400 of FIG. 4 according to an exemplary embodiment of the present disclosure. More specifically, FIG. 5 schematically illustrates a planar superconducting quantum circuit 500 comprising a superconducting transmon qubit 510 (or first data qubit 510), a superconducting multimode qubit 520 (or second data qubit 520), a superconducting multimode qubit coupler 530, a first superconducting coupling capacitor 540, a second superconducting coupling capacitor 542, and a third superconducting coupling capacitor 550.

[0051] The superconducting transmon qubit 510 includes a Josephson tunnel junction device 512 and a shunt capacitor 514. In some embodiments, the shunt capacitor 514 has a coplanar parallel plate capacitor structure including a first superconducting pad 514-1 (or first electrode) and a second superconducting pad 514-2 (or second electrode), and has a shunt capacitance CT1 corresponding to the shunt capacitor 414 (in FIG. 4). The Josephson tunnel junction device 512 includes first and second superconducting electrodes respectively coupled to the first and second superconducting pads 514-1 and 514-2 of the shunt capacitor 514. In some embodiments, as schematically illustrated in FIG. 5, the Josephson tunnel junction device 512 is disposed between the first and second superconducting pads 514-1 and 514-2 of the shunt capacitor 514.

[0052] The superconducting multimode qubit 520 includes a first superconducting pad 521-1, a second superconducting pad 521-2, a third superconducting pad 521-3, a fourth superconducting pad 521-4, a fifth superconducting pad 521-5, a first Josephson tunnel junction device 522, and a second Josephson tunnel junction device 524. The first and second superconducting pads 521-1 and 521-2 include electrodes of a coplanar parallel plate capacitor structure corresponding to the shunt capacitor 421 (in FIG. 4) having a capacitance CS2. The first Josephson tunnel junction device 522 is coupled to the first and third superconducting pads 521-1 and 521-3 and is disposed between the first and third superconducting pads 521-1 and 521-3. The second Josephson tunnel junction device 524 is coupled to the second and third superconducting pads 521-2 and 521-3 and is disposed between the second and third superconducting pads 521-2 and 521-3.

[0053] The first and third superconducting pads 521-1 and 521-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 423 (FIG. 4) that provides a capacitance C3 across the Josephson tunnel junction device 522. Similarly, the second and third superconducting pads 521-2 and 521-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 425 (FIG. 4) that provides a shunt capacitance C4 across the Josephson tunnel junction device 524. Further, in some embodiments, as shown in FIG. 5, the fourth and fifth superconducting pads 521-4 and 521-5 are connected to both ends of the third superconducting pad 521-3 corresponding to the third node N3 of the superconducting multimode qubit 520. The fourth and fifth superconducting pads 521-4 and 521-5 are configured to mediate capacitive coupling to the B mode of the superconducting multimode qubit 520, as discussed above.

[0054] The superconducting multimode qubit coupler 530 includes a first superconducting pad 531-1, a second superconducting pad 531-2, a third superconducting pad 531-3, a fourth superconducting pad 531-4, a first Josephson tunnel junction device 532, and a second Josephson tunnel junction device 534. The first and second superconducting pads 531-1 and 531-2 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 431 (FIG. 4) having a capacitance CS1. The first Josephson tunnel junction device 532 is coupled to the first and third superconducting pads 531-1 and 531-3 and is disposed between the first and third superconducting pads 531-1 and 531-3. The second Josephson tunnel junction device 534 is coupled to the second and third superconducting pads 531-2 and 531-3 and is disposed between the second and third superconducting pads 531-2 and 531-3.

[0055] The first and third superconducting pads 531-1 and 531-3 comprise electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 433 (FIG. 4) that provides a shunt capacitance C1 across the Josephson tunnel junction device 532. Similarly, the second and third superconducting pads 531-2 and 531-3 comprise electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 435 (FIG. 4) that provides a shunt capacitance C2 across the Josephson tunnel junction device 534. Further, in some embodiments, as shown in FIG. 5, a fourth superconducting pad 531-4 is connected to an end of the third superconducting pad 531-3 corresponding to the third node N3 of the superconducting multimode qubit coupler 530. The fourth superconducting pad 521-4 is configured to enable and mediate capacitive coupling to the B mode of the superconducting multimode qubit coupler 530, as discussed above.

[0056] The first and second superconducting coupling capacitors 540 and 542 are configured to enable and mediate capacitive coupling of the first data qubit 510 to the A mode of the superconducting multimode qubit coupler 530, and the third superconducting coupling capacitor 550 is configured to enable and mediate capacitive coupling of the B mode of the second data qubit 520 to the B mode of the superconducting multimode qubit coupler 530. More specifically, as shown in the exemplary embodiment of FIG. 5, the first superconducting coupling capacitor 540 (corresponding to the first coupling capacitor 440 of FIG. 4) comprises a superconducting planar transmission line, such as a coplanar waveguide, configured to capacitively couple a first superconducting pad 514-1 of the shunt capacitor 514 of the superconducting transmon qubit 510 to a first superconducting pad 531-1 (e.g., node N1) of the superconducting multimode qubit coupler 530. The first superconducting coupling capacitor 540 comprises (i) a first end overlapping a portion of the first superconducting pad 514-1 of the shunt capacitor 514 of the superconducting transmon qubit 510, and (ii) a second end overlapping a portion of the first superconducting pad 531-1 of the superconducting multimode qubit coupler 530. The amount of coupling capacitance (e.g., CC1) provided by the first superconducting coupling capacitor 540 depends at least in part on the amount of overlap and spacing between the ends of the first superconducting coupling capacitor 540 and the first superconducting pads 514-1 and 531-1.

[0057] Similarly, a second superconducting coupling capacitor 542 (corresponding to the second coupling capacitor 442 in FIG. 4) comprises a superconducting planar transmission line, such as a coplanar waveguide, configured to capacitively couple a second superconducting pad 514-2 of a shunt capacitor 514 of a superconducting transmon qubit 510 to a second superconducting pad 531-2 (e.g., node N2) of a superconducting multimode qubit coupler 530. The second superconducting coupling capacitor 542 comprises (i) a first end overlapping a portion of the second superconducting pad 514-2 of the shunt capacitor 514 of the superconducting transmon qubit 510, and (ii) a second end overlapping a portion of the second superconducting pad 531-2 of the superconducting multimode qubit coupler 530. The amount of coupling capacitance (e.g., CC2) provided by the second superconducting coupling capacitor 542 depends at least in part on the amount of overlap and spacing between the ends of the second superconducting coupling capacitor 542 and the second superconducting pads 514-2 and 531-2.

[0058] Furthermore, a third superconducting coupling capacitor 550 (corresponding to the third coupling capacitor 450 in FIG. 4) comprises a first superconducting pad 552 (or first coupling electrode) and a second superconducting pad 554 (or second coupling electrode), which are connected by a superconducting planar transmission line 556 (e.g., a coplanar waveguide). The first superconducting pad 552 is capacitively coupled to a fourth superconducting pad 521-4 of a superconducting multimode qubit 520. The second superconducting pad 554 is capacitively coupled to a fourth superconducting pad 531-4 of a superconducting multimode qubit coupler 530. In this exemplary configuration, a third node N3 of the superconducting multimode qubit 520 is capacitively coupled to a third node N3 of the superconducting multimode qubit coupler 530 (e.g., a direct coupling of the B mode of the second data qubit 520 to the superconducting multimode qubit coupler 530). The amount of coupling capacitance (e.g., CC3) provided by the third superconducting coupling capacitor 550 depends at least in part on (i) the amount of capacitive coupling between the superconducting pads 552 and 524-1, and (ii) the amount of capacitive coupling between the superconducting pads 554 and 531-4.

[0059] FIG. 6 schematically illustrates a planar circuit configuration of the superconducting quantum circuit 400 of FIG. 4 according to another exemplary embodiment of the present disclosure. More specifically, FIG. 6 schematically illustrates a planar superconducting quantum circuit 600 including a superconducting transmon qubit 610 (or a first data qubit 610), a superconducting multimode qubit 620 (or a second data qubit 620), and a superconducting multimode qubit coupler 630. In contrast to the exemplary embodiment of FIG. 5, the planar superconducting quantum circuit 600 of FIG. 6 is configured to provide direct capacitive coupling of the first and second data qubits 610 and 620 to the superconducting multimode qubit coupler 630 without the need for a separate superconducting coupling capacitor.

[0060] The superconducting transmon qubit 610 includes a Josephson tunnel junction device 612 and a shunt capacitor 614. In some embodiments, the shunt capacitor 614 includes a coplanar parallel plate capacitor structure including a first superconducting pad 614-1 (or a first electrode) and a second superconducting pad 614-2 (or a second electrode), and has a shunt capacitance CT1 corresponding to the shunt capacitor 414 of FIG. 4. The Josephson tunnel junction device 612 includes first and second superconducting electrodes respectively coupled to the first and second superconducting pads 614-1 and 614-2 of the shunt capacitor 614. In some embodiments, as schematically illustrated in FIG. 6, the Josephson tunnel junction device 612 is disposed between the first and second superconducting pads 614-1 and 614-2 of the shunt capacitor 614.

[0061] The superconducting multimode qubit 620 includes a first superconducting pad 621-1, a second superconducting pad 621-2, a third superconducting pad 621-3, a fourth superconducting pad 621-4, a fifth superconducting pad 621-5, a first Josephson tunnel junction device 622, and a second Josephson tunnel junction device 624. The first and second superconducting pads 621-1 and 621-2 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 421 (FIG. 4) having a capacitance CS2. The first Josephson tunnel junction device 622 is coupled to the first and third superconducting pads 621-1 and 621-3 and is disposed between the first and third superconducting pads 621-1 and 621-3. The second Josephson tunnel junction device 624 is coupled to the second and third superconducting pads 621-2 and 621-3 and is disposed between the second and third superconducting pads 621-2 and 621-3.

[0062] The first and third superconducting pads 621-1 and 621-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 423 (FIG. 4) that provides a shunt capacitance C3 across the Josephson tunnel junction device 622. Similarly, the second and third superconducting pads 621-2 and 621-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 425 (FIG. 4) that provides a shunt capacitance C4 across the Josephson tunnel junction device 624. Further, in some embodiments, as shown in FIG. 6, the fourth and fifth superconducting pads 621-4 and 621-5 are connected to both ends of a third superconducting pad 621-3 corresponding to a third node N3 of the superconducting multimode qubit 620. The fourth and fifth superconducting pads 621-4 and 621-5 are configured to mediate capacitive coupling to the B mode of the superconducting multimode qubit 620 as discussed above.

[0063] The superconducting multimode qubit coupler 630 includes a first superconducting pad 631-1, a second superconducting pad 631-2, a third superconducting pad 631-3, a fourth superconducting pad 631-4, a first Josephson tunnel junction device 632, and a second Josephson tunnel junction device 634. The first and second superconducting pads 631-1 and 631-2 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 431 (FIG. 4) having a capacitance CS1. The first Josephson tunnel junction device 632 is coupled to the first and third superconducting pads 631-1 and 631-3 and is disposed between the first and third superconducting pads 631-1 and 631-3. The second Josephson tunnel junction device 634 is coupled to the second and third superconducting pads 631-2 and 631-3 and is disposed between the second and third superconducting pads 631-2 and 631-3.

[0064] The first and third superconducting pads 631-1 and 631-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 433 (FIG. 4) that provides a shunt capacitance C1 across the Josephson tunnel junction device 632. Similarly, the second and third superconducting pads 631-2 and 631-3 include electrodes of a coplanar parallel plate capacitor structure corresponding to a shunt capacitor 435 (FIG. 4) that provides a shunt capacitance C2 across the Josephson tunnel junction device 634.

[0065] Furthermore, in some embodiments, as shown in FIG. 6, a fourth superconducting pad 631-4 is connected to an end of a third superconducting pad 631-3 corresponding to a third node N3 of a superconducting multimode qubit coupler 430. The fourth superconducting pads 621-4 and 631-4 are configured to enable and mediate a direct capacitive coupling of the B mode of the superconducting multimode qubit 620 to the B mode of the superconducting multimode qubit coupler 630. Moreover, as further shown in FIG. 6, the superconducting transmon qubit 610 is directly capacitively coupled to the A mode of the superconducting multimode qubit coupler 630 as a result of (i) a direct capacitive coupling of the respective first superconducting pads 614-1 and 631-1, and (ii) a direct capacitive coupling of the respective second superconducting pads 614-2 and 631-2 of the superconducting transmon qubit 610 and the superconducting multimode qubit coupler 630.

[0066] The various superconducting circuit elements of the superconducting quantum circuits 400, 500, and 600 of FIGS. 4, 5, and 6 can be configured to achieve desired operating characteristics. In some embodiments, the superconducting multimode qubit coupler that couples the first and second data qubits should be configured with A and B modes having frequencies close to the transition frequencies of the first and second data qubits. Further, the transition frequencies of the first and second data qubits should be slightly different. To implement the entangling gate, the superconducting multimode qubit coupler should be driven with microwave control signals having frequencies close to the transition frequencies of the first and second data qubits.

[0067] For example, referring to FIG. 4, in a non-limiting, exemplary embodiment, the superconducting transmon qubit 410 comprises a fixed-frequency transmon qubit configurable to have a transition frequency fQ = 4.29 GHz and a given anharmonicity, and such operating parameters are achieved, for example, based on the selection of the critical current ICT1 of the Josephson tunnel junction device 412 and the capacitance CT1 of the shunt capacitor 414. Further, in a non-limiting, exemplary embodiment, the superconducting multi-mode qubit 420 can be configured to have an A-mode frequency (fA) of fA = 4.2 GHz and a B-mode frequency (fB) of fB = 5.54 GHz, as well as respective anharmonicities, and such operating parameters are achieved, for example, based on the selection of the critical currents IC3 and IC4 of the Josephson tunnel junction devices 422 and 424, and the capacitances CS2, C3, and C4 of the respective capacitors 421, 423, and 425.

[0068] Moreover, in a non-limiting, exemplary embodiment, the superconducting multi-mode qubit coupler 430 can be configured to have an A-mode frequency (fA) of fA = 4.1 GHz and a B-mode frequency (fB) of fB = 5.82 GHz, as well as respective anharmonicities, and such operating parameters are achieved, for example, based on the selection of the critical currents IC1 and IC2 of the Josephson tunnel junction devices 432 and 434, and the capacitances CS1, C1, and C2 of the respective capacitors 431, 433, and 435. Further, the coupling capacitance is selected to achieve a desired level of exchange coupling between the first data qubit 410 and the superconducting multi-mode qubit coupler 430, and between the second data qubit 420 and the superconducting multi-mode qubit coupler 430.

[0069] Using these exemplary parameters, computer simulations were performed, indicating that a significantly low static ZZ coupling (e.g., as low as 1.6 kHz) between the superconducting transmon qubit 410 and the A mode (data mode) of the superconducting multimode qubit 420 by the "off" state superconducting multimode qubit coupler 430 is achievable. Further, the computer simulations indicated that when the superconducting multimode qubit coupler 430 is put into the "on" state via a calibrated microwave pulse, (i) a relatively high long-term coupling (e.g., as high as 22.4 MHz) between the superconducting transmon qubit 410 and the B mode of the superconducting multimode qubit coupler 430 is achievable, and (ii) a relatively high long-term coupling (e.g., as high as 29.6 MHz) between the A mode (data mode) of the superconducting multimode qubit 420 and the B mode of the superconducting multimode qubit coupler 430 is achievable. In this regard, the long-term coupling (e.g., ZZ coupling) between the B mode of the superconducting multimode qubit coupler 430 and the data modes of the first and second data qubits 410 and 420 is approximately 10,000 times greater than the static ZZ coupling between the data modes of the first and second data qubits 410 and 420.

[0070] Furthermore, as described above, exemplary quantum circuits as discussed herein are configured to suppress the effects of spectator qubits. For example, FIG. 7 schematically illustrates a quantum device 700 comprising an array of superconducting qubits and multimode qubit couplers configured to suppress crosstalk between a spectator quantum bit and an adjacent multimode qubit coupler, according to an exemplary embodiment of the present disclosure. More specifically, the quantum device 700 includes a first superconducting multimode qubit 710, a superconducting transmon qubit 720, a second superconducting multimode qubit 730, a first superconducting multimode qubit coupler 740, and a second superconducting multimode qubit coupler 750. The first superconducting multimode qubit 710 and the superconducting transmon qubit 720 are coupled together through the first superconducting multimode qubit coupler 740. Further, the superconducting transmon qubit 720 and the second superconducting multimode qubit 730 are coupled together through the second superconducting multimode qubit coupler 750.

[0071] In the exemplary configuration of FIG. 7, the second superconducting multimode qubit 730 can function as a spectator qubit that can potentially adversely affect the transition frequency of the first superconducting multimode qubit coupler 740 through the static ZZ coupling 760, and thus potentially adversely affect the fidelity of the entanglement gate operation performed between the first superconducting multimode qubit 710 and the superconducting transmon qubit 720. Similarly, the first superconducting multimode qubit 710 can function as a spectator qubit that can potentially adversely affect the transition frequency of the second superconducting multimode qubit coupler 750 through the static ZZ coupling, and thus potentially adversely affect the fidelity of the entanglement gate operation performed between the second superconducting multimode qubit 730 and the superconducting transmon qubit 720. The exemplary quantum device 700 is still configured to suppress crosstalk between a given superconducting multimode qubit and an adjacent multimode qubit coupler that is not directly coupled to the given superconducting multimode qubit.

[0072] For example, assume that the superconducting transmon qubit 720, the second superconducting multimode qubit coupler 750, and the second superconducting multimode qubit 730 have the same operating characteristics as the superconducting transmon qubit 410, the superconducting multimode qubit coupler 430, and the second superconducting multimode qubit 420 (FIG. 4) as discussed above. Further assume that the first superconducting multimode qubit coupler 740 is configured to have an A-mode frequency fA = 4.0 GHz and a B-mode frequency fB = 5.72 GHz. Using these exemplary operating parameters, computer simulation indicates a long-term coupling as low as about 0.004 kHz between the A-mode (data mode) of the second superconducting multimode qubit 730 and the B-mode of the first superconducting multimode qubit coupler 740, which is negligibly small and thereby significantly suppresses the spectator effect.

[0073] It should be appreciated that exemplary quantum devices (e.g., FIG. 7) as discussed herein are configured to perform, such as suppressing the spectator effect, such as suppressing the static ZZ between a multimode qubit and a remote multimode qubit coupler. In essence, suppression of the spectator effect is achieved by mode-selective exchange coupling between a multimode qubit coupler and a transmon qubit. Only the A-mode of the multimode qubit coupler has an exchange coupling to the transmon qubit, which blocks any exchange or long-term (ZZ) coupling between the B-mode of the multimode qubit coupler connected to a particular transmon. This insulation then blocks any exchange or long-term coupling between the B-mode of a given multimode qubit coupler and any mode of a spectator multimode qubit connected to another multimode qubit coupler.

[0074] FIG. 5 and FIG. 6 schematically illustrate planar superconducting quantum circuits 500 and 600 comprising two superconducting data qubits coupled by a superconducting multi-mode qubit coupler. However, it should be understood that superconducting quantum devices such as quantum processors can be implemented using an array of superconducting qubits comprising dozens, hundreds, thousands, or more coupled superconducting qubits, depending on the application. For example, FIG. 8 schematically illustrates an array of superconducting qubits coupled by a superconducting multi-mode quantum bit coupler circuit according to an exemplary embodiment of the present disclosure. More specifically, FIG. 8 schematically illustrates a multi-qubit array 800 comprising superconducting multi-mode qubits 810-1, 810-2, 810-3, 810-4, and 810-5, and superconducting quadrupole transmon qubits 820-1 and 820-2, and a plurality of superconducting multi-mode qubit couplers 830-1, 830-2, 830-3, 830-4, 830-5, and 830-6. In the exemplary configuration of FIG. 8, the data qubits are capacitively coupled directly to the superconducting multi-mode qubit couplers 830-1, 830-2, 830-3, 830-4, 830-5, and 830-6 (without using separate superconducting coupling capacitors).

[0075] FIG. 8 illustrates an exemplary layout of a unit cell with a heavy hexagonal lattice structure as known in the art. The heavy hexagonal lattice layout is particularly suitable for superconducting qubit architectures to minimize frequency collisions and crosstalk between qubits (the term "heavy" represents an exemplary layout where qubits are placed at both the vertices and edges of the hexagonal lattice). In the exemplary embodiment of FIG. 8, the heavy hexagonal lattice structure is facilitated by the use of superconducting quadrupole transmon qubits 820-1 and 820-2, each of which enables dipole-dipole interactions and couplings with at least three superconducting multi-mode qubit couplers as shown.

[0076] As schematically illustrated in FIG. 8, the superconducting multimode qubits 810-1, 810-2, 810-3, 810-4, and 810-5 each have a layout similar to the exemplary layout of the superconducting multimode qubits 520 and 620 as discussed above with FIGS. 5 and 6. For example, as specifically shown in FIG. 8, the superconducting multimode qubit 810-1 includes a first superconducting pad 811-1, a second superconducting pad 811-2, a third superconducting pad 811-3, a fourth superconducting pad 811-4, a fifth superconducting pad 811-5, a first Josephson tunnel junction device 812, and a second Josephson tunnel junction device 814. The superconducting pads 811-1, 811-2, 811-3, 811-4, and 811-5 include capacitor electrodes of a coplanar parallel plate capacitor structure that form the shunt capacitor of the superconducting multimode qubit 810-1 as discussed above.

[0077] Furthermore, the superconducting multimode qubit couplers 830-1, 830-2, 830-3, 830-4, 830-5, and 830-6 each have a layout similar to that of the superconducting multimode qubit couplers 530 and 630 as discussed above with reference to FIGS. 5 and 6. For example, as specifically shown in FIG. 8, the superconducting multimode qubit coupler 830-1 includes a first superconducting pad 831-1, a second superconducting pad 831-2, a third superconducting pad 831-3, a first Josephson tunnel junction device 832, and a second Josephson tunnel junction device 834. The superconducting pads 831-1, 831-2, and 831-3 include capacitor electrodes of a coplanar parallel plate capacitor structure that form the shunt capacitor of the superconducting multimode qubit coupler 830-1 as discussed above. Further, FIG. 8 schematically illustrates an exemplary embodiment of the superconducting multimode qubit coupler 830-1 in which the end of the superconducting pad 831-3 is capacitively coupled directly to the fourth superconducting pad 811-4 of the superconducting multimode qubit 810-1 without using additional extended superconducting pads such as the fourth superconducting pad 631-4 shown in FIG. 6.

[0078] In the exemplary layout of FIG. 8, the B mode of the superconducting multimode qubit 810-1 is capacitively coupled directly to the B mode of the superconducting multimode qubit coupler 830-1. The B mode of the superconducting multimode qubit 810-2 is capacitively coupled directly to the B mode of the superconducting multimode qubit coupler 830-2. The B mode of the superconducting multimode qubit 810-3 is capacitively coupled directly to the B mode of the superconducting multimode qubit coupler 830-3. The B mode of the superconducting multimode qubit 810-4 is capacitively coupled directly to the B mode of the superconducting multimode qubit coupler 830-4. The B mode of the superconducting multimode qubit 810-5 is capacitively coupled directly to the B modes of the superconducting multimode qubit coupler 830-5 and the superconducting multimode qubit coupler 830-6.

[0079] As described above, in the exemplary layout shown in FIG. 8, each superconducting quadrupole transmon qubit 820-1 and 820-2 is configured to enable dipole-dipole interactions with three superconducting tunable coupler qubits. The superconducting quadrupole transmon qubits 820-1 and 820-2 operate in a manner similar to the superconducting transmon qubits shown in FIGS. 4, 5, and 6, except that the superconducting quadrupole transmon qubits 820-1 and 820-2 each have different arrangements of shunt capacitor pads having a quadrupole charge pattern when excited. For example, as illustrated in FIG. 8, the superconducting quadrupole transmon qubit 820-1 includes a single Josephson tunnel junction device 822 having a Manhattan Josephson junction framework, and first, second, third, and fourth superconducting pads 824-1, 824-2, 824-3, and 824-4 collectively including capacitor electrode pads that form the shunt capacitor of the superconducting quadrupole transmon qubit 820-1.

[0080] The first, second, third, and fourth superconducting pads 824-1, 824-2, 824-3, and 824-4 of the superconducting quadrupole transmon qubit 820-1 are arranged in a rectangular array. The first and second superconducting pads 824-1 and 824-2 are configured to implement the first capacitor electrode of the shunt capacitor and are generally connected to the first electrode of a single Josephson tunnel junction device 822. The third and fourth superconducting pads 824-3 and 824-4 are configured to implement the second capacitor electrode of the shunt capacitor and are generally connected to the second electrode of a single Josephson tunnel junction device 822. In the case of an exemplary quadrupole configuration as shown in FIG. 8, the superconducting quadrupole transmon qubit 820-1 is configured to capacitively couple directly to the respective A modes of three superconducting multimode qubit couplers 830-1, 830-3, and 830-5. For example, the first superconducting pad 831-1 and the second superconducting pad 832-2 of the superconducting multimode qubit coupler 830-1 are capacitively coupled directly to the first superconducting pad 824-1 and the fourth superconducting pad 824-4 of the superconducting quadrupole transmon qubit 820-1, respectively.

[0081] As described above, the heavy hexagonal structure of the multi-qubit array 800 shown in FIG. 8 is facilitated by the use of superconducting quadrupole transmon qubits 820-1 and 820-2 since each superconducting quadrupole transmon qubit can be coupled to three superconducting tunable coupler qubits to implement the unit cell layout of the multi-qubit array 800 comprising seven (7) superconducting data qubits. It should be understood that a larger multi-qubit array can be implemented by adding one or more heavy hexagonal unit cells to the array shown in FIG. 8. For example, in some embodiments, the multi-qubit array 800 can be expanded by coupling one or more of the superconducting multi-mode qubits 810-1, 810-2, 810-3, and 810-4 to their respective superconducting multi-mode qubit couplers (B-mode coupling) to expand the heavy hexagonal structure. In other exemplary embodiments, the multi-qubit array 800 can be expanded by coupling additional superconducting multi-mode qubit couplers to each of the superconducting quadrupole transmon qubits 820-1 and 820-2 (A-mode coupling) to expand the lattice structure, in which case each superconducting quadrupole transmon qubit 820-1 and 820-2 should be coupled to four (4) superconducting multi-mode qubit couplers to implement a heavy square lattice.

[0082] As is known in the art, calibration procedures are periodically performed on a quantum system such as a quantum processor to calibrate various quantum elements such as readout resonators, data qubits, and coupler circuitry so as to enable high-fidelity gate operations (e.g., single-qubit gate operations and entanglement gate operations). For example, determining the resonance frequency of a readout resonator, determining the transition frequency of a qubit, determining the coherence time (T1) of a qubit (where the coherence time T1 of a given qubit represents the time it takes for the qubit state to decay from an excited state to a ground state), determining the transverse relaxation time (T2) of a qubit (or the phase dissipation time, decoherence time of the qubit along the X-Y plane of the Bloch sphere), calibrating the control pulses applied to the qubit to perform single-qubit gate operations, calibrating the control pulses applied to the active coupler circuitry to perform entanglement gate operations, etc. For this purpose, various types of in-situ calibration procedures are periodically performed, for example. The calibration procedures are maintained in a calibration database and will determine various control parameters that are periodically updated, as needed, in units such as seconds, minutes, hours, days, etc., depending on the type and operating characteristics of the quantum elements of the quantum computing system, as well as other factors understood by those skilled in the art.

[0083] Exemplary embodiments of the present disclosure include techniques for calibrating two-qubit gate operations for superconducting qubits coupled by a superconducting multimode qubit coupler. For example, in some embodiments, as described above, the data qubit is made to undergo a state-dependent Stark shift (equivalent to ZZ activated by microwaves), and thereby, a properly calibrated control pulse is used to drive the B mode of the superconducting multimode qubit coupler so as to cause a conditional Z-axis rotation of the data qubit depending on the state of the data qubit, whereby a two-qubit entanglement gate operation (e.g., a CPHASE gate) is performed on the data qubit coupled to the superconducting multimode qubit coupler.

[0084] As part of the two - qubit gate calibration process, for example, the shape and amplitude of the control pulse are calibrated to drive the B - mode of a superconducting multi - mode qubit coupler through a complete 2π rotation from the ground state |0〉 to the first excited state |1〉 and back to the ground state |0〉 again. Further, the calibration process includes repeatedly adjusting the detuning of the control pulse to ensure that while achieving a complete 2π rotation of the B - mode of the superconducting multi - mode qubit coupler, a desired conditional Z - rotation for the data qubit is achieved. When calibrating the control pulse, it is desirable to utilize a control pulse with a short duration and sufficient amplitude to perform gate operations sufficiently faster than the finite coherence time of the qubit. Nevertheless, very short pulses that turn on and off very quickly (e.g., rectangular - wave pulses) typically contain unwanted spectral components that can excite higher qubit states such as the second excited state |2〉 and above, which can lead to low - fidelity gate operations. For example, the increased spectral width of a high - speed pulse can excite the f12 transition and cause errors.

[0085] In this regard, various pulse shaping techniques can be utilized to calibrate a precise pulse shape that suppresses unnecessary spectral components driving transitions to and between higher states, while providing the desired spectral components driving transitions between the ground state |0〉 and the first excited state |1〉. For example, shaped pulses such as Gaussian pulses, cosine pulses (e.g., sum of half-cosine), or hyperbolic secant pulses can be used to suppress f12 and higher transitions while driving the f01 transition. Essentially, such pulse shaping techniques suppress / reduce the transient currents associated with turning the control pulse on and off. Further, the pulse shaping techniques can be used in conjunction with shaped pulses (such as Gaussian pulses, cosine pulses, or hyperbolic secant pulses) to further suppress unwanted state transitions while maintaining the same pulse envelope area (or integral of the pulse envelope), and include DRAG (derivative removal by adiabatic gate) correction pulses. The DRAG correction technique essentially superimposes a main control pulse, which has an envelope that is the derivative of the main control pulse and is configured to further suppress f12 and higher transitions, with a finely tuned phase-shifted component.

[0086] Exemplary techniques for calibrating two-qubit gate operations for superconducting qubits coupled by a superconducting multi-mode qubit coupler are discussed in further detail herein with reference to FIGS. 9, and 10A - 10I. In some embodiments, the two-qubit gate calibration process includes calibrating a hyperbolic secant pulse with DRAG correction to drive a superconducting coupler (e.g., a superconducting multi-mode qubit coupler) to perform an entangled gate operation (e.g., a CPHASE gate) between two superconducting qubits (e.g., a superconducting transmon qubit and a superconducting multi-mode qubit) coupled by the superconducting coupler.

[0087] More specifically, in one exemplary embodiment, an analog control pulse A(t) having a hyperbolic secant pulse shape with DRAG correction is defined as follows:

Number

[0088] Referring now to FIG. 9, a flowchart illustrates a method for calibrating an entanglement gate operation for superconducting qubits coupled by a superconducting multi-mode qubit coupler, according to an exemplary embodiment of the present disclosure. The first step includes starting an entanglement gate calibration process at an appropriate stage of the overall calibration process (block 900). For example, as described above, the calibration process involves different stages of calibration related to calibrating, for example, a readout resonator, qubits, couplers, etc. In some embodiments, the entanglement gate calibration process is initiated following a calibration stage that is performed to, for example, determine the resonance frequencies of the readout resonator coupled to the data qubits and qubit couplers, as well as to determine the transition frequencies of the data qubits and qubit couplers. Note that the entanglement gate calibration process of FIG. 9 is performed for each two-qubit gate circuit in the quantum system because each two-qubit gate circuit has different operating frequencies and characteristics, and thus the control parameters will vary between different two-qubit gate circuits. For ease of explanation, the process flow of FIG. 9 is described in the context of a single two-qubit gate circuit, but it should be understood that the same entanglement gate calibration process should be applied to each two-qubit gate circuit in the quantum system.

[0089] The first step of the entanglement gate calibration process involves initializing the parameters of the control pulse and generating and applying an initial control pulse to the target multi-mode qubit coupler (block 901). For example, in one exemplary embodiment where the control pulse includes a hyperbolic secant pulse with DRAG correction, the parameters A0, σ, A1, pulse duration, and frequency are set to initial values. Note that control pulses for a given superconducting multi-mode qubit coupler can be generated using any known arbitrary waveform generation (AWG) technique that implements IQ amplitude modulation. In particular, an exemplary AWG system comprises a baseband signal generator, a digital-to-analog converter (DAC) stage, an IQ modulation stage, and a phase lock loop system to generate local oscillator (LO) signals (e.g., quadrature LO signals LO_I and LO_Q) for the IQ modulation stage.

[0090] The baseband signal generator generates a digital baseband signal (e.g., a digital IQ signal) under the software control of the gate calibration process, and the digital baseband signal is generated based on target parameters to generate an analog control pulse. The DAC stage converts the digital baseband signal into an analog baseband signal having a baseband frequency (e.g., analog baseband signals I(t) and Q(t)). In particular, the DAC stage operates as an envelope generator for analog IQ envelope pulses (e.g., I(t) and Q(t)) having a hyperbolic secant pulse shape and an overlaid DRAG correction pulse shape. The analog IQ signals are filtered and applied to the IQ signal modulation stage. The IQ modulation stage performs amplitude modulation by modulating the analog IQ signal with an orthogonal LO signal (e.g., an in-phase LO signal (LO_I) and a quadrature-phase LO signal (LO_Q)), and generates a modulated analog control signal having an analog control pulse used to drive a superconducting multimode qubit coupler.

[0091] In the case of a first calibration process in which no previous calibrated pulse was generated, the amplitude parameter A1 (of the DRAG correction pulse envelope) can be initialized to A1 = 0, and the amplitude parameter A0 (of the hyperbolic secant pulse envelope) can be selected based on known operating characteristics of the hardware circuitry that generates the control pulse and applies it to the superconducting multimode qubit coupler (e.g., the amount of signal attenuation applied along the RF path to the control input of the superconducting multimode qubit coupler, the operating characteristics of the IQ mixer of the IQ modulation stage, the amplitude / intensity of the LO signal, etc.). Further, the parameters σ and the pulse duration are first set to target values that define the initial control pulse length. Further, the LO frequency is set to a frequency close to the B-mode frequency of the superconducting multimode qubit coupler such that a given two-qubit gate is calibrated. Note that in the case of the current calibration process in which a previous calibration process has been performed to calibrate the gate control pulse, the current calibration process can utilize some or all of the previously calibrated control pulse parameters as initial parameters.

[0092] The current control pulse is applied to drive the B-mode of a given superconducting multimode qubit coupler, and the state of the B-mode of the given superconducting multimode qubit coupler is measured (block 902) to determine the amount of X rotation obtained in response to the applied control pulse. At this stage of the entanglement gate calibration process, it is desirable to achieve a full 2π rotation of the B-mode about the X-axis of the Bloch sphere (i.e., the transition from the ground state |0〉 to the first excited state |1〉 and back to the ground state |0〉). In some embodiments, the process (block 902) is performed hundreds or thousands of times for the current control pulse to determine whether a full 2π rotation of the B-mode was achieved with high probability based on probability information derived from the collection of measured states. For each iteration, the B-mode of the superconducting multimode qubit coupler is initialized to the ground state |0〉, the current control pulse is applied to drive the B-mode, and the B-mode state is measured, e.g., via a readout system.

[0093] If it is determined that a full 2π rotation in the B mode was not obtained for the current control pulse (negative result in block 903), the software calibration control process proceeds to recalibrate the control pulse (block 904) by adjusting the amplitude A0 of the hyperbolic secant pulse envelope as needed to obtain a full 2π rotation. The iterative process (blocks 902, 903, and 904) thus repeats for each recalibrated control pulse over a range of amplitude A0 values until a full 2π rotation in the B mode of a given superconducting multimode qubit coupling is obtained (positive result in block 903) in response to a given control pulse having a given amplitude A0 value.

[0094] Once the amplitude A0 of the control pulse is properly calibrated to obtain a full 2π rotation of the state of the B mode of a given superconducting multimode qubit coupler (positive result in block 903), the current control pulse is utilized to drive the B mode of the given superconducting multimode qubit coupler to perform a conditional Z rotation operation on the data qubit coupled to the given superconducting multimode qubit coupler (block 905), and the state resulting from the data qubit is measured to determine whether the target conditional Z rotation was obtained in response to the current control pulse (block 906).

[0095] In some embodiments, a process is performed that includes a plurality of iterations of: (i) selecting one data qubit (e.g., a superconducting multimode qubit) as a “target” qubit Q0 and other data qubits (e.g., superconducting transmon qubits) as “control” qubits Q1; (ii) placing the target qubit Q0 in the ground state |0>, the first excited state |1>, or an equal superposition state; (iii) placing the control qubit Q1 in the ground state |0> or the first excited state |1>; (iv) applying a control pulse to drive the B mode of a superconducting multimode qubit coupler; and (v) performing, e.g., a Ramsey experiment to measure the result of the achieved conditional Z rotation. A process for performing a conditional Z rotation gate operation and measuring the conditional Z rotation of a data qubit is thereby implemented. For a proper CPHASE gate operation, when both data qubits are in the excited state, both data qubits will have a Z rotation (e.g., a π rotation) according to the CPHASE gate operation, otherwise neither data qubit will have a Z rotation as a result of the CPHASE gate operation.

[0096] When it is determined that the targeted conditional Z rotation of the data qubit has not been achieved as a result of driving the B mode of the superconducting multi-mode qubit coupler with the current control pulse upon completion of the processes in blocks 905 and 906 (negative result in block 906), the calibration control process proceeds to adjust the "detuning" of the control pulse (block 907). In particular, the term "detuning" represents the difference between the frequency of the control pulse and the B mode frequency of the superconducting multi-mode qubit coupler. In some embodiments, the detuning is implemented by incrementally adjusting the LO frequency applied to the IQ modulation stage to generate a control pulse having a re-calibrated frequency detuned from the B mode frequency of the superconducting multi-mode qubit coupler. The iterative process (blocks 905, 906, and 907) is thus repeated for each re-calibrated control pulse over the detuning range until the desired conditional Z rotation of the data qubit is achieved as a result of driving the B mode of the superconducting multi-mode qubit coupler with a sufficiently detuned control pulse (positive result in block 906).

[0097] To obtain a target conditional Z rotation of a data qubit, when the detuning of the control pulse is sufficiently calibrated, the current control pulse is applied to drive the B mode of a given superconducting multimode qubit coupler, and the state of the B mode of the given superconducting multimode qubit coupler is measured (block 908) to determine the amount of X-axis rotation obtained in response to the applied control pulse. At this stage of the entanglement gate calibration process, it is desirable to achieve a full 2π rotation of the B mode about the X-axis of the Bloch sphere with the newly detuned control pulse. If it is determined that a full 2π X-axis rotation of the B mode was not obtained with the current detuned control pulse (negative result in block 909), the software calibration control process proceeds to recalibrate the control pulse by adjusting, as necessary, the amplitude A0 of the hyperbolic secant pulse envelope and the amplitude A1 of the DRAG correction pulse to achieve a full 2π X-axis rotation with the current detuned control pulse (block 910). The iterative process (blocks 908, 909, and 910) is repeated for each recalibrated control pulse over the range of A0 and A1 amplitude values until a full 2π X-axis rotation of the B mode of the given superconducting multimode qubit coupler is obtained using the recalibrated amplitudes A0 and A1 and the control pulse with the current detuning (positive result in block 909).

[0098] To obtain a full 2π rotation of the B mode of a given superconducting multi-mode qubit coupler, in the current detuning, when the amplitudes A0 and A1 of the control pulses are properly calibrated, a new amplitude-calibrated control pulse is utilized to drive the B mode of the given superconducting multi-mode qubit coupler to perform a conditional Z rotation operation, and the state resulting as the data qubit is measured (block 911) to determine whether the target conditional Z rotation has been obtained in response to the new amplitude-calibrated control pulse with the current detuning. At this point in the entanglement gate calibration process, if the target conditional Z rotation is obtained in response to the new amplitude-calibrated control pulse with the current detuning (positive result in block 911), the entanglement gate calibration process is considered complete and terminated (block 912).

[0099] On the other hand, if the target conditional Z rotation is not obtained in response to the new amplitude-calibrated control pulse with the current detuning (negative result in block 911), and if it is considered that obtaining the target conditional Z rotation is achievable through further iterations of recalibrating the detuning and amplitude of the control pulse (positive determination in block 913), the gate calibration process proceeds to recalibrate the detuning of the control pulse (returns to block 907). Thereafter, the iterative processes of blocks 905, 906, 907, 908, 909, 910, and 911 are repeated for the purpose of obtaining a sufficiently calibrated control pulse, in which both (i) a full 2π X-axis rotation of the B mode of the superconducting multi-mode qubit coupler and (ii) the target conditional Z rotation of the data qubit are obtained in an error state below a predefined error threshold.

[0100] At some points in the entanglement gate calibration process, if it is determined that the target conditional Z rotation of the data qubit is not achievable (negative determination in block 913), the gate calibration control process proceeds to recalibrate the control pulse by adjusting the pulse duration or the pulse transition time σ or both (block 914), and the entanglement gate calibration process is repeated (returns to block 901). For example, in some embodiments, after many iterations of adjusting the amplitudes A0 and A1 and detuning the control pulse in a state where there is not enough convergence towards minimizing the error below a pre-specified error threshold, it may be determined (in block 913) that the target conditional Z rotation of the data qubit is not achievable.

[0101] It should be appreciated that the entanglement gate calibration process of FIG. 9 provides a precise and efficient method for calibrating the control pulse for a two-qubit conditional gate operation. In practice, since the superconducting multi-mode qubit coupler has an anharmonic mode (e.g., anharmonic B mode), the calibration process can be configured to separately calibrate the control pulse shape so that the multi-mode qubit coupler undergoes a full 2π X-axis rotation with high accuracy, regardless of the detuning between the control pulse of the multi-mode qubit coupler and the mode frequency (e.g., B mode frequency). The pulse detuning is thus calibrated separately to achieve the desired conditional Z rotation of the data qubit. The process of repeatedly iterating through separate detuning and amplitude calibration stages allows the entanglement gate calibration process to gradually achieve higher fidelity.

[0102] Figures 10A, 10B, 10C, 10D, 10E, 10F, 10G, 10H, and 10I depict numerical simulation results over time of an exemplary entanglement gate calibration process evaluated using control pulses calibrated based on the process of FIG. 9, according to an exemplary embodiment of the present disclosure. In particular, FIGS. 10A - 10I illustrate the results of a computer simulation of an entanglement gate calibration process (e.g., a two - qubit gate) for an exemplary system comprising a superconducting transmon qubit and a superconducting multimode qubit coupled by a superconducting multimode - qubit coupler (as shown in FIG. 4). For the simulation results, the transmon qubit was designated as the control qubit. Further, the computer simulation was performed using exemplary parameters as discussed above, the superconducting transmon qubit was set to have a transition frequency fQ = 4.29 GHz, the superconducting multimode qubit was set to have an A - mode frequency fA = 4.2 GHz and a B - mode frequency fB = 5.54 GHz, and the superconducting multimode - qubit coupler was set to have an A - mode frequency fA = 4.1 GHz and a B - mode frequency fB = 5.82 GHz. It should be understood that the computer simulation results shown in FIGS. 10A - 10I are presented to illustrate aspects of an exemplary entanglement gate calibration process and should not be taken in any way as limiting with respect to the scope of the claimed subject matter.

[0103] FIG. 10A includes a waveform diagram 1000 of a calibrated control pulse generated using an entangled gate calibration process according to an exemplary embodiment of the present disclosure. In particular, FIG. 10A illustrates an exemplary pulse envelope 1002 of a calibrated control pulse having a pulse duration of 90 nanoseconds (ns), an amplitude A0 = 0.13342 (arbitrary units), a DRAG correction A1 = -3 (dimensionless), σ = 15 ns, and an offset = -5.2 MHz. For ease of illustration, FIG. 10A illustrates a pulse envelope representing the amplitude-modulated envelope (Y-axis) of Rabi oscillations over a period (X-axis) from 0 ns to 90 ns, but omits the actual illustration of the Rabi oscillations that would otherwise be distinguishable at the pulse frequency of the calibrated control pulse. FIG. 10A illustrates an exemplary calibrated control pulse utilized to drive the B-mode of a superconducting multimode qubit coupler to perform a two-qubit entangled gate operation (e.g., a CPHASE gate).

[0104] FIGS. 10B and 10C are timing diagrams illustrating the state projections (along the Z-axis of the Bloch sphere) of the B-mode of a superconducting multimode qubit coupler for different states of a control qubit according to an exemplary embodiment of the present disclosure. In particular, FIG. 10B is a timing diagram 1010 showing a curve 1012 representing the state projection of the B-mode of a superconducting multimode qubit coupler over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multimode qubit coupler with a control qubit in the ground state |0〉 (e.g., a superconducting transmon qubit). Further, FIG. 10C is a timing diagram 1020 showing a curve 1022 representing the state projection of the B-mode of a superconducting multimode qubit coupler over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multimode qubit coupler with a control qubit in the first excited state |1〉 (e.g., a superconducting transmon qubit).

[0105] In FIGS. 10B and 10C, the Y-axis represents the value of the state projection of the B-mode of the superconducting multi-mode qubit coupler (along the Z-axis of the Bloch sphere). The state projection = 1.0 represents the ground state of the B-mode, the state projection = -1.0 represents the first excited state of the B-mode, and the state projection = 0 represents an equal superposition state of the ground state and the first excited state. The simulation results shown in FIGS. 10B and 10C are obtained by applying a calibrated control pulse to the B-mode of the superconducting multi-mode qubit coupler in order to evaluate the ability to place the target qubit (e.g., a superconducting multi-mode qubit) in an equal superposition state of the ground state and the first excited state, and to place the control qubit (e.g., a superconducting transmon qubit) in the ground state |0〉 or the first excited state |1〉, and to drive the B-mode of the coupler through a complete 2π rotation (e.g., from the ground state |0〉 to the first excited state |1〉 and back to the ground state |0〉 again). FIGS. 10B and 10C show that while the superconducting multi-mode qubit coupler transitions through different paths depending on the state of the control qubit, the B-mode returns to the ground state again.

[0106] Next, FIGS. 10D, 10E, 10F, and 10G are timing diagrams illustrating the state projections (along the Z-axis of the Bloch sphere) of data qubits coupled to a superconducting multi-mode qubit coupler resulting from driving the B-mode of the superconducting multi-mode qubit coupler with calibrated control pulses for different states of a control qubit, according to an exemplary embodiment of the present disclosure. In particular, FIG. 10D is a timing diagram 1030 showing a curve 1032 representing the state projection (along the Z-axis of the Bloch sphere) of a target qubit Q0 (e.g., a superconducting multi-mode qubit) over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multi-mode qubit coupler with a control qubit Q1 in the ground state |0〉 (e.g., a superconducting transmon qubit). Further, FIG. 10E is a timing diagram 1040 showing a curve 1042 representing the state projection (along the Z-axis of the Bloch sphere) of the target qubit Q0 over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multi-mode qubit coupler with a control qubit Q1 in the first excited state |1〉.

[0107] Similarly, FIG. 10F is a timing diagram 1050 showing a curve 1052 representing the state projection (along the Z-axis of the Bloch sphere) of the control qubit Q1 over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multi-mode qubit coupler with a control qubit Q1 in the ground state |0〉. Further, FIG. 10G is a timing diagram 1060 showing a curve 1062 representing the state projection (along the Z-axis of the Bloch sphere) of the control qubit Q1 over the duration (from 0 ns to 90 ns) of a calibrated control pulse that drives the B-mode of the superconducting multi-mode qubit coupler with a control qubit Q1 in the first excited state |1〉.

[0108] The simulation results of the state projections as shown in FIGS. 10D, 10E, 10F, and 10G illustrate that when driving the B mode of the superconducting multi-mode qubit coupler with the calibrated control pulse, in each case, the data qubits Q0 and Q1 eventually approach their starting states, which is the desired result. In reality, when both data qubits are in the excited state, only the phase of the data qubits should be changed, so there should be no X or Y rotation of the data qubits when performing the conditional Z gate operation.

[0109] Next, FIG. 10H is a timing diagram 1070 illustrating a curve 1072 representing the conditional Z rotation obtained for the target qubit bit in response to driving the B mode of the superconducting multi-mode qubit coupler with the calibrated control pulse, according to an exemplary embodiment of the present disclosure. The conditional Z rotation can be measured by calculating the X and Y state projections of the target qubit Q0 over time, whereby it can be determined how the phase of the target qubit Q0 will eventually progress while undergoing precession motion around the equator of the Bloch sphere (e.g., the X - Y plane). The phase of the target qubit Q0 is subtracted during the gate operation by the control qubit Q1 in the ground state or the first excited state. This difference is the conditional Z rotation, which is shown in FIG. 10H over a control pulse period of 90 nanoseconds. The simulation results of FIG. 10H show that a conditional Z rotation of π radians can be achieved over a control pulse period of 90 nanoseconds of the calibrated control pulse, which is sufficient for the maximum entanglement gate.

[0110] Next, FIG. 10I is a timing diagram 1080 illustrating a curve 1082 representing the coincidence of the amount of entanglement obtained when a superconducting multi-mode qubit coupler is driven with a calibrated control pulse to perform an entanglement gate operation according to an exemplary embodiment of the present disclosure. The coincidence of the system is calculated to evaluate the amount of entanglement and obtain an upper limit on the gate fidelity. A coincidence value of 1 indicates maximum entanglement, while a deviation from the coincidence value of 1 indicates imperfect entanglement or leakage from the computational basis. For the simulation, the target and control qubits Q0 and Q1 were placed in an equal superposition state of their ground state and first excited state, and the coincidence was determined over the period of the calibrated control pulse. The simulation results shown in FIG. 10I illustrate a final coincidence of 0.9998, indicating a high-fidelity entanglement gate operation.

[0111] FIG. 11 schematically illustrates a quantum computing system according to an exemplary embodiment of the present disclosure. In particular, FIG. 11 schematically illustrates a quantum computing system 1100 comprising a quantum computing platform 1110, a control system 1120, and a quantum processor 1130. In some embodiments, the quantum computing platform 1110 is configured to perform, for example, an entanglement gate calibration process to calibrate a two-qubit gate system implemented by the quantum processor 1130 and to perform other calibration operations as discussed above with FIG. 9, and implements a software control program such as a gate calibration process 1112. Further, in some embodiments, the control system 1120 comprises a multi-channel arbitrary waveform generator 1122 and a qubit readout control system 1124. The quantum processor 1130 comprises a solid-state semiconductor chip having a superconducting qubit array 1132 and a network 1134 of qubit drive lines, coupler drive lines, and qubit state readout lines, as well as other circuit QED components that may be required for a given application or quantum system configuration.

[0112] In some embodiments, the control system 1120 and the quantum processor 1130 are disposed within a dilution refrigeration system 1140 capable of generating cryogenic temperatures sufficient to operate the components of the control system 1120 for quantum computing applications. For example, the quantum processor 1130 may need to be cooled to near absolute zero K, such as 10 - 15 millikelvin (mK), so that superconducting qubits can exhibit quantum behavior. In some embodiments, the dilution refrigeration system 1140 includes a multi-stage dilution refrigerator capable of maintaining the components of the control system 1120 at different cryogenic temperatures as needed. For example, the quantum processor 1130 may need to be cooled to, for example, 10 - 15 mK, but the circuit components of the control system 1120 may operate at cryogenic temperatures higher than 10 - 15 mK (e.g., cryogenic temperatures in the range of 3K - 4K), depending on the configuration of the quantum computing system.

[0113] In some embodiments, the superconducting qubit array 1132 includes a plurality of superconducting transmon qubits and superconducting tunable coupler qubits, and each pair of superconducting qubits is connected by a respective superconducting qubit coupler using techniques as discussed herein. For example, in some embodiments, the superconducting qubit array 1132 implements the multi-qubit array 800 of FIG. 8. As described above, the superconducting qubit coupler is configured to control the interaction between spatially separated pairs of superconducting qubits to perform entanglement gate operations. The number of superconducting qubits in the qubit array 1132 can be in units such as dozens, hundreds, thousands, or more.

[0114] A network 1134, such as qubit drive lines, coupler drive lines, and qubit state readout lines, is configured to apply microwave control signals to superconducting qubits and coupler circuitry in a superconducting qubit array 1132 to perform various types of gate operations, such as single-gate operations, entangled-gate operations (e.g., CPHASE gate operations), and to read the quantum state of the superconducting qubits. More specifically, as described above, microwave control pulses are applied to the qubit drive lines of respective superconducting qubits to change the quantum state of the superconducting qubits (e.g., change the quantum state of a given qubit between the ground state and the excited state, or change it to a superposition state). In particular, as described above, the qubit drive line for a given superconducting qubit is utilized to change the state of the given superconducting qubit by applying a microwave drive pulse having a center frequency that matches the operating frequency of the given superconducting qubit. Further, when executing a specific quantum information processing algorithm, calibrated control pulses are applied to the coupler drive lines to perform entangled-gate operations between pairs of coupled superconducting qubits.

[0115] Furthermore, as described above, the state readout lines each include a readout resonator coupled to a respective superconducting qubit. The state of a given superconducting qubit can be determined through microwave transmission measurements performed between the readout ports of the readout resonator. The state of the superconducting qubit is read after executing a quantum algorithm. In some embodiments, a dispersive readout operation is performed in which a change in the resonance frequency of a given readout resonator coupled to a given superconducting qubit is utilized to read the state (e.g., ground or excited state) of the given superconducting qubit.

[0116] A network 1134, such as qubit drive lines, coupler drive lines, and qubit state readout lines, is coupled to the control system 1120 through a suitable hardware input / output (I / O) interface that couples I / O signals between the control system 1120 and the quantum processor 1130. For example, the hardware I / O interface can comprise various types of hardware and components such as RF cables, wiring, RF elements, optical fibers, heat exchangers, filters, amplifiers, insulators, and the like.

[0117] In some embodiments, the multi-channel arbitrary waveform generator (AWG) 1122 and other suitable microwave pulse signal generators are configured to generate microwave control pulses that are applied to the qubit drive lines and coupler drive lines to control the operation of superconducting qubits and associated qubit coupler circuitry when performing various gate operations to execute a given specific quantum information processing algorithm. In some embodiments, the multi-channel AWG 1122 comprises a plurality of AWG channels that control respective superconducting qubits within the superconducting qubit array 1132 of the quantum processor 1130. In some embodiments, each AWG channel comprises a baseband signal generator, a digital-to-analog converter (DAC) stage, a filter stage, a modulation stage, an impedance matching network, and a phase lock loop system for generating a local oscillator (LO) signal (e.g., quadrature LO signals LO_I and LO_Q) for respective modulation stages of each AWG channel.

[0118] In some embodiments, the multi-channel AWG 1122 comprises an orthogonal AWG system configured to process orthogonal signals, which include in-phase (I) signal components and quadrature-phase (Q) signal components. The baseband signal generator in each AWG channel is configured to receive baseband data (e.g., from a quantum computing platform) as an input and generate digital orthogonal signals I and Q representing the input baseband data. In this process, the baseband data input to the baseband signal generator for a given AWG channel is separated into two orthogonal digital components, an in-phase (I) baseband component and a quadrature-phase (Q) baseband component. The baseband signal generator for a given AWG channel has to generate the essential digital orthogonal baseband IQ signals that will generate an analog waveform (e.g., a sinusoidal voltage waveform) having a target center frequency configured to operate or otherwise control a given qubit coupled to the output of the given AWG channel.

[0119] The DAC stage for a given AWG channel is configured to convert a digital baseband signal (e.g., the digital IQ signal output from the baseband signal generator) into an analog baseband signal having a baseband frequency (e.g., analog baseband signals I(t) and Q(t)). The filter stage for a given AWG channel is configured to filter the IQ analog signal components output from the DAC stage, thereby generating a filtered analog IQ signal. The modulation stage for a given AWG channel is configured to perform analog IQ signal modulation (e.g., single-sideband (SSB) modulation) by mixing the filtered analog signals I(t) and Q(t) output from the filter stage having orthogonal LO signals (e.g., in-phase LO signal (LO_I) and quadrature-phase LO signal (LO_Q)) to generate and output an analog RF signal (e.g., a single-sideband modulated RF output signal).

[0120] In some embodiments, the qubit readout control system 1124 is configured to apply a microwave tone to a given readout resonator line of a given superconducting qubit to perform a readout operation to read the state of the given superconducting qubit, and to process the readout signal generated by the readout resonator line to determine the state of the given superconducting qubit using techniques known to those skilled in the art.

[0121] The quantum computing platform 1110 comprises software and a hardware platform with various software layers configured to perform various functions including, but not limited to, generating and implementing various quantum applications using an appropriate quantum programming language, configuring and implementing various quantum gate operations, compiling a quantum program into a quantum assembly language, implementing and utilizing an appropriate quantum instruction set architecture (ISA), performing calibration operations for calibrating quantum circuit elements and gate operations, etc. Further, the quantum computing platform 1110 is configured to (i) generate a digital control signal converted into an analog microwave control signal by the control system 1120 for controlling the operation of the quantum processor 1130 when executing a given quantum application, and (ii) acquire and process a digital signal received from the control system 1120 representing the processing result generated by the quantum processor 1130 when executing various gate operations for a given quantum application, and comprises a hardware architecture such as a processor, a memory, etc. configured to control the execution of the quantum application and interface with the control system 1120.

[0122] In some exemplary embodiments, the quantum computing platform 1110 of the quantum computing system 1100 is implemented using any suitable computing system architecture (such as, for example, as shown in FIG. 12) configured to execute a method for supporting quantum computing operations by executing computer-readable program instructions embodied on a computer program product that includes a computer-readable storage medium (or media) having such computer-readable program instructions for causing a processor to implement the control methods discussed herein.

[0123] A computer-readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer-readable storage medium may be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of the computer-readable storage medium includes a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disc (DVD), a memory stick, a floppy disk, a mechanically encoded device such as a punch card or raised structure in a groove having instructions recorded thereon, and any suitable combination of the foregoing. As used herein, a computer-readable storage medium should not be construed as being a transitory signal per se, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagating through a waveguide or other transmission medium (e.g., an optical pulse through an optical fiber cable), or an electrical signal transmitted through a wire.

[0124] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to respective computing / processing devices or to an external computer or external storage device via a network, such as, for example, the Internet, a local area network, a wide area network, or a wireless network, or a combination thereof. The network can comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers, or edge servers, or a combination thereof. Each computing / processing device's network adapter card or network interface receives the computer-readable program instructions from the network and transfers the computer-readable program instructions for storage on a computer-readable storage medium within each computing / processing device.

[0125] Computer-readable program instructions for performing the operations as discussed herein may be in any combination of one or more programming languages, including assembly instructions, instruction set architecture (ISA) instructions, machine language instructions, machine-dependent instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuit devices, or source code or object code written in any combination of object-oriented programming languages such as Smalltalk(R), C++, or the like, and procedural programming languages such as the "C" programming language or similar programming languages. The computer-readable program instructions may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer, or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, electronic circuitry, including programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA), can execute the computer-readable program instructions by utilizing the state information of the computer-readable program instructions to customize the electronic circuitry for performing the various operations as discussed herein.

[0126] Computer-readable program instructions may be provided to a processor of a computer or other programmable data processing apparatus to produce a machine, such that the instructions, which are executed by the processor of the computer or other programmable data processing apparatus, create means for performing the functions / acts discussed herein. These computer-readable program instructions may also be stored in a computer-readable storage medium that includes a product having instructions for performing exemplary computing operations as discussed herein, and may be configured to cause a computer, programmable data processing apparatus, or other devices, or combinations thereof, to function in a particular manner.

[0127] Computer-readable program instructions may also be loaded onto a computer, other programmable apparatus, or other devices to produce a computer-executed process such that the instructions, which are executed on the computer, other programmable apparatus, or other devices, implement the functions / acts discussed herein by causing a series of operational steps to be performed on the computer, other programmable data processing apparatus, or other devices.

[0128] The block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block diagram can represent a module, segment, or portion of instructions that include one or more executable instructions for performing the specified logical function. In some alternative implementations, the functions noted in the blocks may be performed out of the order noted in the figures. For example, two blocks shown in succession may in fact be performed as one step that executes simultaneously, substantially simultaneously, in a partially or fully time-overlapped manner, or the blocks may sometimes be executed in the reverse order depending on the functionality involved. It will also be noted that each block of the block diagrams, and combinations of blocks in the block diagrams, can be executed by a special-purpose hardware-based system that performs the specified function or acts, or that executes a combination of special-purpose hardware and computer instructions.

[0129] These concepts are illustrated with reference to FIG. 12, which schematically illustrates an exemplary architecture of a computing node capable of hosting a quantum computing platform, according to an exemplary embodiment of the present disclosure. For example, FIG. 12 schematically illustrates an exemplary architecture of a computing node 1200 capable of hosting a quantum computing platform 1110 (FIG. 11), according to an exemplary embodiment of the present disclosure. More particularly, FIG. 12 illustrates a computing node 1200 comprising a computer system / server 1212 operable in a number of other general purpose or special purpose computing system environments or configurations. Examples of well-known computing systems, environments, or configurations suitable for use with computer system / server 1212, or combinations thereof, include personal computer systems, server computer systems, thin clients, thick clients, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputer systems, mainframe computer systems, and distributed cloud computing environments including any of the above systems or devices, and the like, but are not limited thereto.

[0130] The computer system / server 1212 can be described in the general context of computer system-executable instructions, such as program modules executed by a computer system. Generally, program modules can include routines, programs, objects, components, logic, data structures, etc. that perform particular tasks or implement particular abstract data types. The computer system / server 1212 may be practiced in a distributed cloud computing environment where tasks are performed by remote processing devices linked through a communications network. In a distributed cloud computing environment, program modules may be located in both local and remote computer system storage media including memory storage devices.

[0131] In FIG. 12, the computer system / server 1212 in the computing node 1200 is shown in the form of a general-purpose computing device. The components of the computer system / server 1212 can include, but are not limited to, one or more processors or processing units 1216, a system memory 1228, and a bus 1218 that couples various system components including the system memory 1228 to the processor 1216.

[0132] The bus 1218 represents one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures can include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus.

[0133] The computer system / server 1212 typically includes various computer system-readable media. Such media can be any available media accessible by the computer system / server 1212, including both volatile and non-volatile media, and both removable and non-removable media.

[0134] System memory 1228 can include computer system-readable media in the form of volatile memory, such as random access memory (RAM) 1230 or cache memory 1232, or both. The computer system / server 1212 can further include other removable / non-removable, volatile / non-volatile computer system storage media. By way of example only, storage system 1234 can be provided for reading and writing to a non-removable non-volatile magnetic medium (not shown, but typically referred to as a "hard drive"). Although not shown, a magnetic disk drive for reading and writing to a removable non-volatile magnetic disk (e.g., a "floppy (R) disk"), and an optical disk drive for reading and writing to a removable non-volatile optical disk such as a CD-ROM, DVD-ROM, or other optical media can be provided. In such examples, each can be connectable to bus 1218 by one or more data media interfaces. As depicted and described herein, memory 1228 can include at least one program product having a set of program modules (e.g., at least one) configured to execute the functions of embodiments of the present invention.

[0135] Program / utility 1240 having a set (at least one) of program modules 1242 may be stored in memory 1228, as well as an operating system, one or more application programs, other program modules, and program data, by way of example and not limitation. Each of the operating system, one or more application programs, other program modules, and program data, or some combination thereof, may include an implementation of a networking environment. Program modules 1242 generally execute the functions or methods of, or both, embodiments of the present disclosure as described herein.

[0136] The computer system / server 1212 can also communicate with one or more external devices 1214 such as a keyboard, a pointing device, a display 1224, one or more devices that enable a user to interact with the computer system / server 1212, or any device that enables the computer system / server 1212 to communicate with one or more other computing devices (e.g., a network card, a modem, etc.), or a combination thereof. Such communication can be carried out via an input / output (I / O) interface 1222. Further, the computer system / server 1212 can communicate with one or more networks such as a local area network (LAN), a general wide area network (WAN), or a public network (e.g., the Internet), or a combination thereof, via a network adapter 1220. As depicted, the network adapter 1220 communicates with other components of the computer system / server 1212 via a bus 1218. Although not shown, it should be understood that other hardware components or software components or both may be used with the computer system / server 1212. Examples include, but are not limited to, microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, SSD drives, and data archive storage systems.

[0137] In some embodiments, the quantum computing system 1100 of FIG. 11 can be implemented in the cloud (quantum as a service (QaaS)) to provide quantum computing facilities as an on-demand service. This disclosure includes a detailed description of cloud computing, but the implementation forms of the teachings recited herein are not limited to cloud computing environments. Rather, the embodiments of this disclosure can be implemented with any other type of computing environment known currently or developed later.

[0138] Cloud computing is a service delivery model that enables convenient on-demand network access to a shared pool of configurable computing resources (such as networks, network bandwidth, servers, processing, memory, storage, applications, virtual machines, and services) that can be rapidly provisioned and released with minimal management effort or interaction with the service provider. This cloud model can include at least five characteristics, at least three service models, and at least four deployment models.

[0139] The characteristics are as follows.

[0140] On-demand self-service: Cloud consumers can unilaterally provision computing capabilities such as server time and network storage automatically as needed without the need for human interaction with the service provider.

[0141] Broad network access: The capabilities are available over the network and accessed through standard mechanisms that promote use by heterogeneous thin or thick client platforms (such as mobile phones, laptops, and PDAs).

[0142] Resource pooling: The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, with dynamic allocation and reallocation of various physical and virtual resources occurring in response to requests. There is a sense of location independence in that consumers generally have no control or knowledge of the exact location of the provided resources and can specify locations at a higher level of abstraction (such as country, state, or data center).

[0143] Rapid elasticity: The ability is provided quickly and elastically, and in some cases automatically, to scale out rapidly and is able to be released quickly to scale in. To the user, the available ability for provision often appears limitless and can be purchased in any quantity at any time.

[0144] Measured services: The cloud system automatically controls and optimizes resource usage (e.g., storage, processing, bandwidth, and active user accounts) by leveraging metering capabilities at some levels of abstraction suitable for the type of service. The resource utilization rate can be monitored, controlled, and reported, bringing transparency to both the provider and the user of the services utilized.

[0145] The service model is as follows.

[0146] Software as a Service (SaaS): The ability provided to the user is to use the provider's application running on the cloud infrastructure. The application is accessible from various client devices through a thin-client interface such as a web browser (e.g., web-based email). The user does not manage or control the underlying cloud infrastructure, including the network, servers, operating system, storage, or, in some cases, the individual application capabilities, except for limited user-specific application configuration settings.

[0147] Platform as a Service (PaaS): The capabilities provided to users are to deploy user-developed or acquired applications, created using programming languages and tools supported by the provider, onto the cloud infrastructure. Users do not manage or control the underlying cloud infrastructure, which includes the network, servers, operating system, or storage, and instead exercise control over the deployed applications and, in some cases, the environmental configuration in which the applications are hosted.

[0148] Infrastructure as a Service (IaaS): The capabilities provided to users are to offer other basic computing resources such as processing, storage, networking, and any software that may include the operating system and applications that users can deploy and run. Users do not manage or control the underlying cloud infrastructure and exercise control over the operating system, storage, deployed applications, and, in some cases, limited control over selected networking components (e.g., host firewalls).

[0149] The deployment models are as follows.

[0150] Private cloud: The cloud infrastructure is operated solely for an organization. The cloud infrastructure may be managed by the organization or a third party and may be on-premises or off-premises.

[0151] Community cloud: The cloud infrastructure is shared by several organizations and supports a specific community that has shared concerns (e.g., mission, security requirements, policies, and compliance considerations). The cloud infrastructure may be managed by the organization or a third party and may be on-premises or off-premises.

[0152] Public cloud: The cloud infrastructure is made available to the general public or large industry groups and is owned by an organization that sells cloud services.

[0153] Hybrid cloud: The cloud infrastructure remains a single entity but is a composition of two or more clouds (private, community, or public) tied together by standard or proprietary technologies that enable data and application portability (e.g., cloud bursting for load balancing between clouds).

[0154] Cloud computing environments are service-oriented, focusing on statelessness, loose coupling, modularity, and semantic interoperability. At the center of cloud computing is an infrastructure that includes a network of interconnected nodes.

[0155] Referring now to FIG. 13, an exemplary cloud computing environment 1300 is depicted. As shown, cloud computing environment 1300 includes one or more cloud computing nodes 1350 with which local computing devices used by cloud consumers, such as, for example, a personal digital assistant (PDA) or cellular telephone 1354A, desktop computer 1354B, laptop computer 1354C, or automotive computer system 1354N, or a combination thereof, may communicate. The nodes 1350 can communicate with one another. The nodes 1350 may be physically or virtually grouped in one or more networks (not shown), such as private, community, public, or hybrid clouds as described above, or a combination thereof. This enables cloud computing environment 1300 to provide infrastructure, platform, software, or a combination thereof, as a service such that a cloud consumer need not maintain resources on a local computing device. The types of computing devices 1354A - N shown in FIG. 13 are intended to be exemplary only, and it is understood that computing nodes 1350 and cloud computing environment 1300 can communicate with any type of computerized device via any type of network or network addressable connection (e.g., using a web browser) or both.

[0156] Referring now to FIG. 14, a set of functional abstractions provided by cloud computing environment 1300 (FIG. 13) are shown. It is intended that the components, layers, and functions shown in FIG. 14 are exemplary only and that embodiments of the invention are not limited thereto. As depicted, the following layers and corresponding functions are provided.

[0157] The hardware and software layer 1460 includes hardware and software components. Examples of hardware components include mainframe 1461, RISC (Reduced Instruction Set Computer) architecture-based server 1462, server 1463, blade server 1464, storage device 1465, and network and network components 1466. In some embodiments, the software components include network application server software 1467 and database software 1468.

[0158] The virtualization layer 1470 provides an abstraction layer that can provide examples of virtual entities such as virtual server 1471, virtual storage 1472, virtual network 1473 including a virtual private network, virtual applications and operating systems 1474, and virtual clients 1475.

[0159] In one example, the management layer 1480 can provide the functions described below. Resource provisioning 1481 dynamically procures computing resources and other resources that are utilized to perform tasks within a cloud computing environment. Metering and pricing 1482 performs cost tracking when resources are utilized within a cloud computing environment and bills or invoices for the amount of these resources used. In one example, these resources can include application software licenses. Security performs authentication of cloud users and tasks and protects data and other resources. The user portal 1483 provides access to the cloud computing environment to users and system administrators. Service level management 1484 distributes and manages cloud computing resources so that the required service levels are met. Service level agreement (SLA) planning and execution 1485 pre-allocates and procures cloud computing resources for which future requirements are anticipated, in accordance with the SLA.

[0160] The workload layer 1490 provides examples of functions that can be utilized in a cloud computing environment. Examples of workloads and functions that can be provided from this layer include mapping and navigation 1491, software development and lifecycle management 1492, virtual classroom education delivery 1493, data analytics processing 1494, transaction processing 1495, and various functions 1496 for implementing the quantum computing platform of a quantum computing system as discussed herein with FIG. 11. In some embodiments, the hardware and software layer 1460 should include various software and hardware systems, such as those of a quantum computing system 1100 (FIG. 11) as discussed herein, for implementing or otherwise supporting various workloads and functions 1496, for example, for performing quantum computing operations and executing quantum computing applications.

[0161] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen in order to best explain the principles of the embodiments, the practical application, or a technical improvement found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

1. A device comprising: a first superconducting qubit; a second superconducting qubit having a first mode and a second mode, wherein the first mode is configured to store quantum data; a superconducting qubit coupler coupled between the first superconducting qubit and the second superconducting qubit; wherein the superconducting qubit coupler has a first mode and a second mode and is configured to operate in one of a first state and a second state in response to a control signal being applied to the superconducting qubit coupler; in the first state of the superconducting qubit coupler, the first superconducting qubit is exchange-coupled to the first mode of the superconducting qubit coupler and the second mode of the second superconducting qubit is exchange-coupled to the second mode of the superconducting qubit coupler to suppress interaction between the first superconducting qubit and the first mode of the second superconducting qubit; in the second state of the superconducting qubit coupler, the first superconducting qubit and the first mode of the second superconducting qubit are exchange-coupled to both the first mode and the second mode of the superconducting qubit coupler to enable interaction between the first superconducting qubit and the first mode of the second superconducting qubit and to perform a entanglement gate operation in response to the control signal being applied to the superconducting qubit coupler; a device.

2. The device of claim 1, wherein the control signal is configured to drive the second mode of the superconducting qubit coupler to perform the entanglement gate operation.

3. The device of claim 1 or 2, wherein the control signal is configured to drive the second mode of the superconducting qubit coupler through a rotation from a ground state to a first excited state and back to the ground state to perform the entanglement gate operation.

4. The device according to any one of claims 1 to 3, wherein in the first state, the second mode of the superconducting qubit coupler is in a ground state where no control signal is applied to the superconducting qubit coupler.

5. The device according to any one of claims 1 to 4, wherein the first superconducting qubit includes a superconducting transmon qubit.

6. The device according to claim 5, wherein the superconducting transmon qubit includes a superconducting quadrupole transmon qubit.

7. The superconducting qubit coupler is a first superconducting pad, a second superconducting pad, a third superconducting pad disposed between the first superconducting pad and the second superconducting pad, a first superconducting tunnel junction device coupled to the first superconducting pad and the third superconducting pad and located between the first superconducting pad and the third superconducting pad, and a second superconducting tunnel junction device coupled to the second superconducting pad and the third superconducting pad and located between the second superconducting pad and the third superconducting pad The device according to any one of claims 1 to 6, comprising.

8. The first superconducting qubit is a first node coupled to the first superconducting pad of the superconducting qubit coupler, a second node coupled to the second superconducting pad of the superconducting qubit coupler comprising, The second superconducting qubit is a first superconducting pad, a second superconducting pad, a third superconducting pad disposed between the first superconducting pad and the second superconducting pad, a first superconducting tunnel junction device coupled to the first superconducting pad and the third superconducting pad and located between the first superconducting pad and the third superconducting pad, and a second superconducting tunnel junction device coupled to the second superconducting pad and the third superconducting pad and located between the second superconducting pad and the third superconducting pad comprising, The third superconducting pad of the second superconducting qubit is coupled to the third superconducting pad of the superconducting qubit coupler, The device according to claim 7.

9. A system comprising a quantum processor comprising an array of superconducting qubits, and a control system configured to generate a control signal for controlling the quantum processor comprising, The array of superconducting qubits is a first superconducting qubit, a second superconducting qubit having a first mode and a second mode, wherein the first mode is configured to store quantum data, the second superconducting qubit A superconducting qubit coupler coupled between the first superconducting qubit and the second superconducting qubit comprising the superconducting qubit coupler comprising a first mode and a second mode and configured to operate in one of a first state and a second state in response to a control signal being applied to the superconducting qubit coupler in the first state of the superconducting qubit coupler, the first superconducting qubit is exchange-coupled to the first mode of the superconducting qubit coupler and the second mode of the second superconducting qubit is exchange-coupled to the second mode of the superconducting qubit coupler to suppress the interaction between the first superconducting qubit and the first mode of the second superconducting qubit in the second state of the superconducting qubit coupler, the first superconducting qubit and the first mode of the second superconducting qubit are exchange-coupled to both the first mode and the second mode of the superconducting qubit coupler to enable the interaction between the first superconducting qubit and the first mode of the second superconducting qubit and to perform an entanglement gate operation in response to the control signal being applied to the superconducting qubit coupler system

10. the control signal is configured to drive the second mode of the superconducting qubit coupler through a rotation from the ground state to the first excited state and back to the ground state to perform the entanglement gate operation in the first state, the second mode of the superconducting qubit coupler is in a ground state where no control signal is applied to the superconducting qubit coupler The system according to claim 9

11. The system according to claim 9 or 10, wherein the first superconducting qubit comprises a superconducting transmon qubit

12. the first superconducting qubit comprises a superconducting quadrupole transmon qubit the array of superconducting qubits a second superconducting qubit coupler coupled to the superconducting quadrupole transmon qubit a third superconducting qubit coupler coupled to the superconducting quadrupole transmon qubit a third superconducting qubit coupled to the second superconducting qubit coupler a fourth superconducting qubit coupled to the third superconducting qubit coupler The system according to any one of claims 9 to 11, further comprising

13. The system according to any one of claims 9 to 12, wherein the array of superconducting qubits has a hexagonal lattice structure.

14. The superconducting qubit coupler a first superconducting pad, a second superconducting pad, a third superconducting pad disposed between the first superconducting pad and the second superconducting pad, a first superconducting tunnel junction device coupled to the first superconducting pad and the third superconducting pad and located between the first superconducting pad and the third superconducting pad, a second superconducting tunnel junction device coupled to the second superconducting pad and the third superconducting pad and located between the second superconducting pad and the third superconducting pad The system according to any one of claims 9 to 13, comprising

15. The first superconducting qubit a first node coupled to the first superconducting pad of the superconducting qubit coupler, a second node coupled to the second superconducting pad of the superconducting qubit coupler comprising The second superconducting qubit a first superconducting pad, a second superconducting pad, a third superconducting pad disposed between the first superconducting pad and the second superconducting pad, a first superconducting tunnel junction device coupled to the first superconducting pad and the third superconducting pad and located between the first superconducting pad and the third superconducting pad, a second superconducting tunnel junction device coupled to the second superconducting pad and the third superconducting pad and located between the second superconducting pad and the third superconducting pad comprising wherein the third superconducting pad of the second superconducting qubit is coupled to the third superconducting pad of the superconducting qubit coupler. The system according to claim 14.

16. A computer program product for calibrating a entanglement gate process, the computer program product comprising one or more computer-readable storage media and program instructions collectively stored on the one or more computer-readable storage media, the program instructions Program instructions for calibrating parameters of a control pulse configured to control the operation of a superconducting qubit coupler to perform an entanglement gate operation between a first superconducting qubit and a second superconducting qubit coupled to the superconducting qubit coupler comprising the program instructions for calibrating the parameters of the control pulse performing an amplitude calibration process for calibrating the amplitude of the control pulse such that, in response to applying the control pulse having a calibrated amplitude to the superconducting qubit coupler, the superconducting qubit coupler is rotated from a ground state to a first excited state and back to the ground state, and performing a frequency detuning calibration process for calibrating the detuning of the frequency of the control pulse with respect to the operating frequency of the superconducting qubit coupler such that, in response to applying the control pulse having the calibrated amplitude and a calibrated frequency detuning to the superconducting qubit coupler, a target conditional rotation of the first superconducting qubit and the second superconducting qubit is achieved comprising program instructions for performing an iterative process including one or more iterations of a computer program product

17. The computer program product according to claim 16, wherein the program instructions for performing the amplitude calibration process comprise program instructions for calibrating the amplitude of a hyperbolic secant pulse

18. The computer program product according to claim 16 or 17, wherein the program instructions for performing the amplitude calibration process comprise program instructions for calibrating the amplitude of the hyperbolic secant pulse and the amplitude for removing the derivative by an adiabatic gate correction pulse superimposed on the hyperbolic secant pulse

19. The computer program product according to any one of claims 16 to 18, wherein the superconducting qubit coupler comprises a first mode and a second mode, and the calibrated control pulse is applied to the second mode of the superconducting qubit coupler to calibrate the entanglement gate operation

20. The program instructions for calibrating the parameters of the control pulse Program instructions for determining whether the target-conditioned rotations of the first superconducting qubit and the second superconducting qubit can be achieved with further iterations of the amplitude calibration process and the frequency detuning calibration process; Program instructions for adjusting at least one of the pulse duration and the pulse transition time of the control pulse and repeating the iterative process in response to determining that the target-conditioned rotation cannot be achieved; The computer program product according to any one of claims 16 to 19, further comprising.

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