Performance of entangling gates between fluxonium qubits coupled together using a resonator coupler

Abstract

Techniques and apparatus for performing entangling gates between two fluxonium qubits, coupled together using a resonator coupler, are disclosed. A microwave pulse is emitted such that the resonator coupler is driven from the ground state to the first excited state and back down when the two fluxonium qubits are in their first excited states, and such that the resonator coupler is maintained in the ground state when one or both of the fluxonium qubits are in energy states other than their first excited states. High quality materials, such as Tantalum, are used to fabricate the given system in order to suppress potential incoherent errors. In addition, by tuning circuit parameters of the given system, such entangling gates are designed to be microwave-activated, and trend towards maximal entanglement fidelity of the two fluxonium qubits for a duration of a given gate.

Description

PERFORMANCE OF ENTANGLING GATES BETWEEN FLUXONIUM QUBITS COUPLED TOGETHER USING A RESONATOR COUPLER BACKGROUND

[0001] Quantum computing utilizes the laws of quantum physics to process information. Quantum physics is a theory that describes the behavior of reality at the fundamental level. It is currently the only physical theory that is capable of consistently predicting the behavior of microscopic quantum objects like photons, molecules, atoms, and electrons.

[0002] A quantum computer is a device that utilizes quantum physics to allow one to write, store, process and read out information encoded in quantum states, e.g., the states of quantum objects. A quantum object is a physical object that behaves according to the laws of quantum physics. The state of a physical object is a description of the object at a given time.

[0003] In quantum physics, the state of a two-level quantum system, or simply, a qubit, is a list of two complex numbers whose squares sum up to one. Each of the two numbers is called an amplitude, or quasi-probability, and their squared absolute values are probabilities that a measurement of the qubit results in zero or one. A fundamental and counterintuitive difference between a probabilistic bit (e.g., a classical zero or one bit) and the qubit is that a probabilistic bit represents a lack of information about a two-level classical system, while a qubit contains maximal information about a two-level quantum system.

[0004] Quantum computers are based on such quantum bits (qubits), which may experience the phenomena of “superposition” and “entanglement.” Superposition allows a quantum system to be in multiple states at the same time. For example, whereas a classical computer is based on bits that are either zero or one, a qubit may be both zero and one at the same time, with different probabilities assigned to zero and one. Entanglement is a strong correlation between quantum systems, such that the quantum systems are inextricably linked even if separated by great distances.

[0005] A quantum algorithm comprises a reversible transformation acting on qubits in a desired and controlled way, followed by a measurement on one or multiple qubits. For example, if a system has two qubits, a transformation may modify four numbers; with three qubits this becomes eight numbers, and so on. As such, a quantum algorithm acts on a list of numbers exponentially large as dictated by the number of qubits. To implement a transform, the transform may be decomposed into small operations acting on a single qubit, or a pair of qubits, as an example. Such small operations may be called quantum gates and a specific arrangement of the quantum gates implements a quantum circuit.

[0006] There are different types of qubits that may be used in quantum computers, each having different advantages and disadvantages. For example, some quantum computers may include qubits built from superconductors, trapped ions, semiconductors, photonics, etc. Each may experience different levels of interference, errors and decoherence. Also, some may be more useful for generating particular types of quantum circuits or quantum algorithms, while others may be more useful for generating other types of quantum circuits or quantum algorithms. Also, costs, run-times, error rates, availability, etc. may vary across quantum computing technologies. BRIEF DESCRIPTION OF THE DRAWINGS

[0007] FIG.1 illustrates a hardware layout of a quantum hardware device that is configured to couple fluxonium hardware components that implement fluxonium qubits together using a resonator coupler, and that may then be used to perform entangling gates between said fluxonium qubits, according to some embodiments.

[0008] FIG.2 illustrates examples of two-qubit entangling gates that may be performed using a quantum hardware device such as that which is shown in FIG. 1, according to some embodiments.

[0009] FIG. 3 illustrates a process of performing an entangling gate between two fluxonium qubits, coupled together using a resonator coupler, according to some embodiments.

[0010] FIG.4A illustrates an energy state diagram for two fluxonium qubits and a resonator coupling element that are configured in a hardware layout such as that which is shown in FIG.1, wherein a corresponding drive frequency for performing an entangling gate between said two fluxonium qubits is additionally depicted in the figure, according to some embodiments.

[0011] FIG.4B illustrates a duration of time and a Gaussian-shaped envelope of a microwave pulse used to perform an entangling gate between two fluxonium qubits, coupled together using a resonator coupler, according to some embodiments.

[0012] FIG.4C illustrates an example of the population of a single energy state of an energy state diagram, such as that which is shown in FIG.4A, during performance of an entangling gate between two fluxonium qubits, according to some embodiments.

[0013] FIG.5A illustrates an energy state diagram, such as that which is also shown in FIG. 4A, wherein various transition frequencies that are not the drive frequency for performing an entangling gate between two fluxonium qubits are determined, such that they are detuned from proximity to the frequency of the drive, according to some embodiments.

[0014] FIG. 5B illustrates relationships between respective ones of the detuned transition frequencies shown in FIG.5A, according to some embodiments.

[0015] FIG. 5C illustrates parasitic, always-on ^^^^ interaction strengths and an inherited anharmonicity of the resonator coupler, both with respect to hybridization between the two fluxoniums and the resonator coupler, according to some embodiments.

[0016] FIG.6A illustrates an energy state diagram, such as that which is also shown in FIGs. 4A and 5A, wherein various other transition frequencies are evaluated in order to further optimize performance of a two-qubit entangling gate between fluxonium qubits, according to some embodiments.

[0017] FIG. 6B illustrates usage of simulation techniques to further optimize a two-qubit entangling gate towards maximal gate fidelity, according to some embodiments.

[0018] FIG. 7 illustrates a visual representation of simulation techniques used to determine and refine circuit parameters for a quantum hardware device that is to be configured to perform two-qubit entangling gates between fluxonium qubits that are coupled together using a resonator coupler, according to some embodiments.

[0019] FIG.8 illustrates a process of determining and refining circuit parameters for a quantum hardware device that is to be configured to perform two-qubit entangling gates between fluxonium qubits that are coupled together using a resonator coupler, according to some embodiments.

[0020] FIG.9 is a block diagram illustrating an example quantum hardware device that may be configured to execute quantum gates (e.g., entangling gates) and perform readout measurements following the execution of those quantum gates, according to some embodiments.

[0021] FIG.10 is a block diagram illustrating an example classical computing device that may be used in at least some embodiments.

[0022] While embodiments are described herein by way of example for several embodiments and illustrative drawings, those skilled in the art will recognize that embodiments are not limited to the embodiments or drawings described. It should be understood, that the drawings and detailed description thereto are not intended to limit embodiments to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope as defined by the appended claims. The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description or the claims. As used throughout this application, the word “may” is used in a permissive sense (i.e., meaning having the potential to), rather than the mandatory sense (i.e., meaning must). Similarly, the words “include,” “including,” and “includes” mean including, but not limited to. When used in the claims, the term “or” is used as an inclusive or and not as an exclusive or. For example, the phrase “at least one of x, y, or z” means any one of x, y, and z, as well as any combination thereof.DETAILED DESCRIPTION

[0023] The present disclosure relates to methods and apparatus for simulating circuit parameters for a quantum hardware device that couples two fluxonium qubits together using a resonator coupling element. Such simulations may be used to tune, refine, and / or optimize said circuit parameters prior to fabrication of the quantum hardware device, in order to provide higher gate fidelity and faster gate performance upon fabrication. The present disclosure additionally relates to hardware layouts of quantum hardware devices that are configured to perform two-qubit entangling gates between fluxonium qubits, which are coupled together using a resonator coupler.

[0024] Within the noisy intermediate-scale quantum (NISQ) hardware regime, quantum error correction and / or mitigation techniques are faced with the difficult task of developing methods for correcting single and multi-qubit errors, logical errors, and / or additional quantum processing unit (QPU) specific qubit coherence time concerns, crosstalk noise levels, etc. Current quantum error correction and / or mitigation techniques have limited numbers of errors that they may compensate for within a given quantum task. Therefore, ensuring that known sources of coherent and / or incoherent error that may be suppressed, prior to fabrication of a given quantum hardware device, are treated through simulation and / or optimization techniques provides a larger bandwidth to correct for other unforeseen errors and / or errors that may not be as successfully pre-treated (e.g., errors occurring post-fabrication of the given quantum hardware device), and which occur during execution of various quantum gates of the given quantum computation.

[0025] As opposed to previous quantum hardware implementations of coupling two fluxonium qubits together using another coupling element, such as a tunable transmon coupling element, the present disclosure relates to capacitively coupling a first and a second fluxonium qubit to a resonator coupler, and driving the resonator coupler between its ground state and its first excited state, conditional on respective energy states of the two fluxonium qubits, and using a single microwave pulse, directed onto the resonator coupler, in order to perform a given two-qubit entangling gate between said fluxonium qubits. Furthermore, as opposed to previous implementations which could not overcome inherent fidelity and / or latency limitations due to the use of lesser quality materials, such as Aluminum, for fabrication of a tunable transmon coupling element, the present disclosure relates to fabricating the resonator coupler using a high-quality material, such as Tantalum. By capacitively coupling two fluxonium qubits to a resonator coupler, fabricating the given quantum hardware device using high quality materials, and by engineering circuit parameters of the corresponding hardware layout such that sources of coherent errors destructively interfere with one another, the present disclosure enables high fidelity, fast entangling gates that may be performed.

[0026] In addition, as related to the description herein, it may be understood that quantum hardware, such as quantum hardware devices, may be used to implement quantum computers, and / or various components of quantum computers (e.g., quantum processing units / cores, routing spaces, magic state distillation factories, other components used to perform logical quantum computations, etc.). For example, a given quantum hardware device may resemble “building blocks” of a quantum computer, such as a grid (e.g., a one-dimensional grid, a two-dimensional grid, etc.) of qubits that may be initialized in various ways in order to form various components of a quantum computer, such as topological quantum codes. Quantum hardware devices may be further configured such that single qubit gates, multi-qubit gates, and / or other operations of quantum circuits may be performed between qubits of the quantum hardware devices (according to a given physical qubit connectivity graph of the quantum hardware device which details which physical qubits are connected to respective other physical qubits via edges). Quantum hardware devices may also comprise and / or be connected to various control devices (e.g., microwave pulse generators, devices for temperature, magnetic, and / or other environmental controls pertaining to local environments of the physical qubits, etc.) that may be used to maintain and / or transform various properties of the qubits and / or other physical components of a given quantum computer.

[0027] Moreover, as related to the description herein, a qubit may refer to both a logical bit (e.g., a one or a zero with some probability) and to one or more physical components used to construct the given qubit based, at least in part, on the type of qubit technology being applied. For example, a fluxonium qubit may be constructed using at least a superconducting material and a non-superconducting material in which the non-superconducting material is located in between sections of superconducting material (see also description pertaining to fluxonium hardware components herein). With regard to this understanding, it should also be understood that quantum hardware may therefore be used to implement physical qubits, in ways such as those as described above, that may again be combined in various ways to implement one or more logical qubits such that logical quantum operations may be performed using said physical elements of said quantum hardware. Further examples of interactions between hardware layouts of quantum hardware devices and associated classical measurement and control devices are discussed with regard to at least FIG.9 herein.

[0028] FIG. 1 illustrates a hardware layout of a quantum hardware device 100 that is configured to couple fluxonium hardware components that implement fluxonium qubits together using a resonator coupler, and that may then be used to perform entangling gates between said fluxonium qubits, according to some embodiments.

[0029] In some embodiments, a set of fluxonium hardware components may be arranged in parallel, as shown in FIG.1, in order to implement a fluxonium qubit. For example, fluxonium (A) 102 includes an inductor, a capacitor, and a Josephson junction, arranged in parallel with one another. In another example, fluxonium (B) 106 includes another inductor, capacitor, and Josephson junction, arranged in parallel with one another. Fluxonium hardware components configured as shown by fluxonium (A) 102 and fluxonium (B) 106 in FIG.1 may then be used to implement respective fluxonium qubits.

[0030] As additionally explained with regard to energy state diagram 300 in FIG. 4A, computational basis states of a given fluxonium qubit that may be implemented using hardware components of quantum hardware device 100 may be logically defined as |0^ and |1^, and may correspond to a ground state and a first excited state of energy states of an energy state diagram for fluxonium.

[0031] In some embodiments, a fluxonium, such as fluxonium (A) 102 or fluxonium (B) 106, may be defined by three degrees of freedom in terms of circuit parameters: energy associated with the capacitor (^^^^^), energy associated with the Josephson junction (^^^^), and energy associated with the inductive shunt (^^^) of the given fluxonium. Moreover, control over such circuit parameters allows for a logical mapping to two given energy states of an energy state diagram, enabled by a hardware configuration of fluxonium (A) 102, or fluxonium (B) 106, etc., to be defined as a corresponding fluxonium qubit, along with allowing the qubit’s frequency, the expected lifetime of the qubit’s energy (^^^), and the dephasing time (^^ଶ) of the qubit to be tuned, and, according to some embodiments, tuned independently from one another.

[0032] Furthermore, additional hardware components, such as yet another inductor and capacitor, may be arranged in parallel, as also shown in FIG.1, in order to implement a resonator coupling element such as resonator coupler (C) 104. It may be understood that resonator coupler (C) 104 may have properties of a linear resonator prior to coupling said resonator coupling element to fluxonium (A) 102 and to fluxonium (B) 106, meaning that the “bare” component of resonator coupler (C) 104, prior to implementation into quantum hardware device 100, may be considered to be a linear resonator and, specifically, a harmonic oscillator. As additionally explained herein, however, once resonator coupler (C) 104 is coupled to fluxonium (A) 102 and to fluxonium (B) 106, as shown in quantum hardware device 100, resonator coupler (C) 104 inherits properties of non-linearity due to coupling with said fluxoniums. A quantification of said inherited non-linearity may be referred to herein as non-linearity ^^, and is additionally discussed herein with regard to at least FIGs.5A and 5C.

[0033] In some embodiments, resonator coupler (C) 104 may be fabricated using a high- quality material, such as Tantalum (Ta). A high-quality material for use in fabricating resonator coupler (C) 104 may be defined using a dimensionless figure of merit, ^^, wherein ^^ may refer toa number of oscillation periods that occur before 1⁄ ^^ of the energy associated with resonatorcoupler (C) 104 has decayed away. In some embodiments, Tantalum, or a similar high-quality material that has a ^^ of at least 1,000,000, may be used to fabricate resonator coupler (C) 104. Furthermore, fabrication of resonator coupler (C) 104 using a high-quality material, such as Tantalum, reduces an expected amount of incoherent error that may occur on the entangling gate being performed using hardware components of quantum hardware device 100. In some embodiments, a source of incoherent error may be defined as being due to dissipation of energy into the environment (e.g., dissipation out of a system described by quantum hardware device 100 and into the environment).

[0034] As shown in FIG. 1, two fluxoniums may be coupled together using a resonator coupler, such as in quantum hardware device 100, wherein respective components are capacitively coupled to one another. For example, fluxonium (A) 102 may be capacitively coupled to resonator coupler (C) 104 with a capacitive coupling strength of ^^^,^108, fluxonium (B) 106 may be capacitively coupled to resonator coupler (C) 104 with a capacitive coupling strength of ^^^,^112, and fluxonium (A) 102 may be capacitively coupled to fluxonium (B) 106 with a capacitive coupling strength of ^^^,^110.

[0035] In some embodiments, a Hamiltonian of a system that includes fluxonium (A) 102, resonator coupler (C) 104, and fluxonium (B) 106, wherein said components of the system areconfigured in a hardware layout such as that which is shown in FIG.1, may resemble the following:+^^^,^^^^^^^ + ^^^,^^^^^^^.

[0036] In the above equation4^^^^^,^^^ଶ^ +^^ଶ^ − ^^^^,^ cos൫^^^ + ^^^,^൯ may refer to properties of “bare” fluxoniums Aand B, prior to capacitively coupling with resonator coupler (C) 104. As introduced above, the description herein may be described, when relevant, with respect to the “bare” basis and the “dressed” basis (see also second Hamiltonian description provided below). It should be understood that, when used to describe a “first excited state” herein, for example, the “first excited state” refers to an eigenstate of the full Hamiltonian description, wherein the “first excited state” has a maximal overlap with the corresponding “bare” excitations. Furthermore, and as also introduced above,energies associated with the capacitor (^^^^^), with the Josephson junction (^^^^), and with the inductive shunt (^^^) of the each fluxonium may be tuned according to various hardware design constraints set by a given implementation of quantum hardware device 100, as additionally described herein. Furthermore, ^^^and ^^^may refer to flux for respective fluxoniums. In someembodiments, this parameter may be tuned such that external flux for both fluxoniums are ^^^ =^^^ = ^^.

[0037] Continuing with the above equation,may refer to properties of the “bare” resonator coupler (C) 104, prior to coupling with fluxonium (A) 102 and (B) 106, wherein ^^ற^^ may be defined as creationand annihilation (^^) operators for a harmonic oscillator, such as resonator coupler (C) 104. The final three terms of the above Hamiltonian, ^^^,^^^^^^^, ^^^,^^^^^^^, and ^^^,^^^^^^^, refer to two-body couplings of the system described by quantum hardware device 100. As introduced above, ^^^,^, ^^^,^, and ^^^,^refer to capacitive coupling strengths between respective components of quantum hardware device 100, and ^^^, ^^^, and ^^^therefore refer to charge operators for said respective components. As additionally described herein, charge modulation of resonator coupler (C) 104 provides a mechanism for performing a microwave- activated two-qubit entangling gate. By tuning ratios between values of ^^^,^, ^^^,^, and ^^^,^for a given implementation of quantum hardware device 100, two-qubit entangling gates, such as those described herein, may be engineered to be microwave-activated, such that the entangling gate is initiated when a microwave pulse of microwave pulse generator 114 is emitted (see also the following paragraph), and entangling gate stops when said microwave pulse ceases. Furthermore, tuning ratios between values of ^^^,^, ^^^,^, and ^^^,^may additionally be used to suppress parasitic, “always-on” ^^^^ interactions, as additionally described herein.

[0038] Quantum hardware device 100 may additionally be described by using the dressed basis(e.g., Hamiltonianwhich is discussed in additional detail herein with regard to FIG. 4Aand energy state diagram 400 of the system.

[0039] In some embodiments, a microwave pulse generator, such as microwave pulse generator 114, may be housed within quantum hardware device 100 and / or may be locally coupled to at least resonator coupler (C) 104 of quantum hardware device 100. As shown in FIG. 1, a microwave pulse may be emitted from microwave pulse generator 114 in order to perform a microwave-activated entangling gate between fluxonium qubits implemented using physical components arranged as shown in quantum hardware device 100. As additionally described herein with regard to FIG.4A – FIG.6B, generated waveform 116 may refer to a microwave pulse that has been determined and selected using simulation techniques (see also description hereinpertaining to FIG.7) and may then be applied for a microwave-activated two-qubit entangling gate between fluxonium qubits implemented using hardware components of quantum hardware device 100. Generated waveform 116 may additionally refer to properties of said microwave pulse, such as a microwave pulse having a Gaussian-shaped envelope and a given full-width-half-maximum (FWHM).

[0040] Microwave pulse generator 114 may be further configured to receive drive control instructions pertaining to generated waveform 116 from a classical computing device, such as classical computing device 960, classical computing device 1000, etc., wherein the drive control instructions may include information such as duration of the pulse to be emitted, frequency of the pulse to be emitted, timing of when to emit the pulse within an overall execution of a given quantum computation, etc.

[0041] In some embodiments, simulation techniques may be used to determine the various degrees of freedom of components within quantum hardware device 100 (see also description pertaining to FIG. 7 herein). In a first example, a given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that thefollowing circuit parameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 2 GHz,^^^,^⁄ 2^^ = 0.3 GHz, and ^^^^,^⁄ 2^^ = 7.1 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 2 GHz,^^^,^⁄ 2^^ = 0.3 GHz, and ^^^^,^⁄ 2^^ = 7.2 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ =7.08 GHz, with an impedance ^^ of 190 Ω. In such an example, it may be further determinedthrough such simulation techniques that capacitive coupling strengths= 0.33 GHz,^^^,^⁄ 2^^ = 0.33 GHz, and ^^^,^⁄ 2^^ = 0.1 GHz.

[0042] In a second example, a given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that the following circuitparameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.1 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.2 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ = 7.08 GHz, with an impedance^^ of 2000 Ω. As indicated by the impedance value of 2000 Ω, resonator coupler (C) 104 is withina high impedance regime of operation in this second example. In addition, in such an example, itmay be further determined through such simulation techniques that capacitive coupling strengths^^^,^⁄ 2^^ = 1.07 GHz, ^^^,^⁄ 2^^ = 1.07 GHz, and ^^^,^⁄ 2^^ = 0.1 GHz.

[0043] In a third example, another given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that the following circuitparameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.1 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.2 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ = 7 GHz and an impedance ^^ of5000 Ω. As indicated by the impedance value of 5000 Ω, resonator coupler (C) 104 is within a high impedance regime of operation in this third example. In addition, in such an example, it maybe further determined through such simulation techniques that capacitive coupling strengths^^^,^⁄ 2^^ = 2.9 GHz, ^^^,^⁄ 2^^ = 2.9 GHz, and ^^^,^⁄ 2^^ = 0.16 GHz.

[0044] In a fourth example, another given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that the following circuitparameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.1 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.2 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ = 7 GHz and an impedance ^^ of2000 Ω. As indicated by the impedance value of 2000 Ω, resonator coupler (C) 104 is within a high impedance regime of operation in this fourth example. In addition, in such an example, it maybe further determined through such simulation techniques that capacitive coupling strengths^^^,^⁄ 2^^ = 1.88 GHz, ^^^,^⁄ 2^^ = 1.88 GHz, and ^^^,^⁄ 2^^ = 0.16 GHz.

[0045] In a fifth example, another given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that the following circuitparameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.1 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 2 GHz, ^^^,^⁄ 2^^ = 0.3 GHz, and^^^^,^⁄ 2^^ = 7.2 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ = 7.08 GHz and an impedance ^^of 3600 Ω. As indicated by the impedance value of 3600 Ω, resonator coupler (C) 104 is within a high impedance regime of operation in this fifth example. In addition, in such an example, it maybe further determined through such simulation techniques that capacitive coupling strengths^^^,^⁄ 2^^ = 1.43 GHz, ^^^,^⁄ 2^^ = 1.43 GHz, and ^^^,^⁄ 2^^ = 0.1 GHz.

[0046] In a sixth example, another given implementation of quantum hardware device 100 for use in performing two-qubit entangling gates may be optimized such that the following circuitparameters are configured for fluxonium (A) 102: ^^^^^,^⁄ 2^^ = 1 GHz, ^^^,^⁄ 2^^ = 1 GHz, and^^^^,^⁄ 2^^ = 7.0 GHz; for fluxonium (B) 106: ^^^^^,^⁄ 2^^ = 1 GHz, ^^^,^⁄ 2^^ = 1 GHz, and^^^^,^⁄ 2^^ = 6.9 GHz; and for resonator coupler (C) 104: ^^^⁄ 2^^ = 5.18 GHz and an impedance ^^of 2000 Ω. As indicated by the impedance value of 2000 Ω, resonator coupler (C) 104 is within a high impedance regime of operation in this sixth example. In addition, in such an example, it maybe further determined through such simulation techniques that capacitive coupling strengths^^^,^⁄ 2^^ = −1.15 GHz, ^^^,^⁄ 2^^ = −1.15 GHz, and ^^^,^⁄ 2^^ = 0.137 GHz.

[0047] Furthermore, it should be understood that, in some embodiments, hardware components of quantum hardware device 100 may be connected to one or more other sets of fluxonium hardware components and / or resonator coupler elements of a larger-scale quantum hardware device that implements additional fluxonium qubits. For example, fluxoniums (A) 102 and (B) 106 and resonator coupler (C) 104 may be implemented within quantum processing core 920, as shown in FIG.9. Moreover, it should be understood that a grounding of one or more of the hardware components of quantum hardware device 100 may also be required, and is meant to be encompassed within discussion of a given larger architecture design implementation that includes quantum hardware device 100, such as that which is described with regard to implementation of a quantum computer 930.

[0048] FIG.2 illustrates examples of two-qubit entangling gates that may be performed using a quantum hardware device such as that which is shown in FIG. 1, according to some embodiments.

[0049] In some embodiments, two-qubit entangling gates that may be performed between fluxonium qubits implemented using a hardware layout such as quantum hardware device 100 may resemble two-qubit entangling gates 200, as shown in FIG.2. For example, a two-qubit Controlled- Z (CZ) gate 210 may be defined herein as a quantum gate that is performed on two qubits, such as fluxonium qubits 212 and 214. When written in matrix form, an operation defined by a CZ gate on to two qubits may resemble the following:

[0050] A truth table for such a CZ gate is also shown in truth table 220, wherein “output” states represent possible quantum states that may be measured, following performance of CZ gate 210, for respective possible “input” states. It may be understood that notation for said input and outputstates, as shown in truth table 220, are written in a form resembling a notation of|fluxonium A, fluxonium B^, and, specifically for CZ gate 210,|fluxonium qubit 212, fluxonium qubit 214^. In some embodiments, quantum information, asused throughout the description herein, may refer to one or more of said quantum states. For example, quantum information following a performance of CZ gate 210 may refer to one of the four output states shown in truth table 220.

[0051] Furthermore, as pertaining to performance of an entangling gate between two fluxonium qubits as described herein, a control sequence for a CZ gate may include emission of a microwave pulse that drives resonator coupler (C) 104 from a ground state to a first excited state,and back down to the ground state of the resonator coupler, conditional on fluxonium qubits 212 and 214 being in their respective first excited states, and while causing a phase shift of ^^ within a duration of the CZ gate (e.g., with respect to an initial phase prior to commencing the performance of CZ gate 210). It is therefore noted that an outcome of such a control sequence may be labeled herein as a −|11^ state.

[0052] Similarly, the control sequence for the CZ gate may include emission of a microwave pulse that maintains resonator coupler (C) 104 in the ground state of the resonator coupler when one or both of fluxonium qubits 212 and 214 are not in their respective first excited states (e.g., one or both of fluxonium qubits 212 and 214 are in their respective ground states). It is thereforenoted that an outcome of such a control sequence may be labeled herein as a |00^, the |01^, or the|10^ quantum state, depending upon which fluxonium qubit, if either, is in a first excited state.

[0053] Parameters pertaining to such a microwave pulse are additionally detailed with regard to FIGs.4A – 6B herein.

[0054] Similarly, a two-qubit Controlled-X (CX) gate 230 may be defined herein as a quantum gate that is performed on two qubits, such as fluxonium qubits 232 and 234. Such a CX gate 230 may additionally be referred to as a CNOT gate, according to some embodiments. When written in matrix form, an operation defined by a CX gate on to two qubits may resemble the following: 10 0 0൦0 1 0 00 0 0 1൪.0 0 1 0

[0055] As additionally shown in CX gate 230 of FIG.2, a CX gate may be transformed into a series of quantum gates including a CZ gate and two Hadamard gates, wherein a Hadamard gate onto fluxonium qubit 234 is performed both before and after performing the CZ gate onto fluxonium qubits 232 and 234.

[0056] FIG. 3 illustrates a process of performing an entangling gate between two fluxonium qubits, coupled together using a resonator coupler, according to some embodiments.

[0057] In the following description of FIG.3, it is meant to be understood that references to fluxonium qubits refer to fluxonium qubits that have been implemented using fluxonium hardware components, such as those shown in quantum hardware device 100 in FIG.1, and that references to a resonator coupler refers to a resonator coupler that has been implemented using additional hardware components, such as those also shown in quantum hardware device 100.

[0058] In block 300, drive control instructions may be generated and subsequently provided to microwave pulse generator 114, such that generated waveform 116 may be emitted in order to perform a two-qubit entangling gate between two fluxonium qubits, as further described in blocks302 – 310. In some embodiments, drive control instructions may include indications of a frequency to be used for the microwave pulse, a duration of time which the pulse is to be emitted for, and / or any other parameters that may be pertinent to performance of the two-qubit entangling gate.

[0059] At a moment in time just prior to commencing performance of an entangling gate between two fluxonium qubits coupled together using a resonator coupler, the two fluxonium qubits that are to be involved in the entangling gate may be reset (e.g., initialized) into some initial quantum states, such as their ground states, according to some embodiments. Furthermore, a two- qubit entangling gate, such as CZ gate 210 and CX gate 230, may resemble one quantum gate of a larger quantum circuit, quantum algorithm, or other quantum computation that includes at least two or more quantum gates other than the current entangling gate being performed. If, in some embodiments, the given two-qubit entangling gate, discussed with regard to FIG. 3, is a first quantum gate within such a quantum circuit, then the two fluxonium qubits may be reset into their ground states. If, in some other embodiments, the given two-qubit entangling gate, discussed with regard to FIG.3, is a second, third, etc. quantum gate within such a quantum circuit, then a readout step (e.g., a non-destructive measurement step in which a quantum state is read out to a classical measurement device) of one or more of the fluxonium qubits involved in the quantum circuit may have occurred just prior to block 302, in which case the two fluxonium qubits may be reset into their ground states prior to commencing the performance of the entangling gate described in blocks 302, 304, and 306.

[0060] In block 202, a two-qubit entangling gate is performed between two fluxonium qubits, such as those implemented using fluxonium (A) 102 and fluxonium (B) 106, wherein the two fluxoniums are coupled together using a resonator coupler, such as resonator coupler (C) 104. Following a moment in time marking the commencement of the entangling gate and the beginning of the emission of generated waveform 116, an energy state of the resonator coupler is made conditionally dependent upon the respective energy states of the two fluxonium qubits until the end of the entangling gate and the termination of the emission of generated waveform 116.

[0061] As depicted in response to block 304, when both fluxonium qubits are in their first excited states at a moment in time marked by the commencement of the entangling gate, then the resonator coupler is driven from the ground state of the resonator coupler to the first excited state of the resonator coupler, and back down to the ground state within a duration of time of the entangling gate, as described in block 306. Furthermore, a phase shift of ^^ occurs within the duration of time of the entangling gate.

[0062] As additionally depicted in response to block 304, when one or both of the fluxonium qubits are not in their first excited states at a moment in time marked by the commencement of theentangling gate, then the resonator coupler remains unaffected, and is maintained in the ground state of the resonator coupler for the duration of the time of the entangling gate, as described in bock 308.

[0063] In block 310, results of the entangling gate may be provided. In some embodiments, another readout step of the fluxonium qubits involved in the entangling gate may proceed, and the information extracted during the readout step may be provided to a classical measurement device. In other embodiments, in which the entangling gate described in blocks 302 – 308 is not a final quantum gate to be performed within a given quantum circuit, algorithm, computation, etc., then one or both quantum states of the respective fluxonium qubits may be stored and / or provided (e.g., in one or more magic states 910 locally accessible to quantum processing core 920, as depicted in FIG. 9 herein) for later use. In yet other embodiments, in which quantum hardware device 100 may be implemented into a quantum repeater, for example, block 310 may refer to a heralding signal provided to other components of the quantum repeater and / or quantum entanglement network, following completion of the performance of the entangling gate.

[0064] FIG. 4A an energy state diagram for two fluxonium qubits and a resonator coupling element that are configured in a hardware layout such as that which is shown in FIG.1, wherein a corresponding drive frequency for performing an entangling gate between said two fluxonium qubits is additionally depicted in the figure, according to some embodiments.

[0065] In some embodiments, an energy state diagram for a system such as quantum hardware device 100, which comprises two fluxonium qubits, implemented using fluxonium hardware components, that are coupled together using a resonator coupler, may resemble energy state diagram 400, as shown in FIG.4A. As additionally described in the Key of FIG. 4A, respectiveenergy states of energy state diagram 400 are written in a form resembling a notation of|fluxonium A, resonator coupler C, fluxonium B^. For example, the |000^ energy state may bedefined as an energy state of the system wherein both fluxoniums and the resonator coupler are in their ground states. In another example, the|103^energy state may be defined as an energy state of the system wherein fluxonium A is at its first excited state, the resonator coupler is at its ground state, and fluxonium B is at its third excited state. Such notations are meant to carry through into energy state diagrams shown in FIGs.5A and 6A as well.

[0066] Furthermore, it may be understood that, while a ground state (|0^), a first excited state(|1^), a second excited state (|2^), and a third excited state (|3^) are illustrated in FIG. 4A forrespective fluxoniums, additional higher energy states, enabled by the configuration of fluxonium hardware components, may also be encompassed within a discussion of energy state diagram 400 herein. Similarly, while a ground state (|0^), a first excited state (|1^), and a second excited state(|2^) are illustrated in FIG.4A for the resonator coupler, additional higher energy states, enabled by the configuration of hardware components for resonator coupler (C) 104 in FIG.1, may also be encompassed within a discussion of energy state diagram 400 herein.

[0067] As should additionally be understood by energy state diagram 400 in FIG. 4A, two energy states, enabled by a given configuration of fluxonium hardware components, may be logically mapped to computational basis states of a corresponding fluxonium qubit. For example, computational basis states for fluxonium qubit A may be logically mapped to a ground state (|0^) and a first excited state (|1^) of energy states of fluxonium A within energy state diagram 400, and similarly for fluxonium B.

[0068] Furthermore, and as additionally described herein with regard to FIG.5A, a frequency that may be used to excite a particle, such as a photon, into a given energy state from some other energy state of the energy states depicted in energy state diagram 400 may also be referred to as “state transition frequencies” herein. For example, fluxonium qubit frequency may refer to a frequency that would be needed to drive a photon from the ground state to the first excited state, or vice versa. Fluxonium 1-2 state transition frequency may refer to a frequency that would be needed to drive a photon from the first excited state to the second excited state, or vice versa. Using methods and techniques described herein, such fluxonium qubit frequency and / or state transition frequencies may be used within control sequences in order to perform entangling gates between respective fluxonium qubits.

[0069] In some embodiments, driving frequency 410 corresponds to a frequency of generated waveform 116, which may be applied in order to perform an entangling gate (e.g., a CZ gate) between the two fluxoniums described using energy state diagram 400. As shown in FIG. 4A, driving frequency 410 refers to a microwave pulse that, within a duration of time of said pulse, the resonator coupler is excited from its ground state, to its first excited state, and back down to its ground state, all while both fluxoniums A and B remain in their respective first excited states. Driving frequency 410 may be additionally referred to herein as a “resonance frequency” of the resonator coupler, due to this exchange between the ground state and first excited state of the resonator coupler. Selectivity of the microwave pulse may depend, at least in part, on dispersive coupling properties of the dispersive two-body couplings between fluxonium (A) 102, fluxonium (B) 106, and resonator coupler (C) 104, respectively, according to some embodiments. Using CZ gate 210 and the corresponding truth table as an example, when either fluxonium qubit 212 or fluxonium qubit 214, or both fluxonium qubits 212 and 214, are in their respective ground states, the energy state of the resonator coupler remains unaffected for a duration of time corresponding to the performance of the CZ gate. This is additionally described herein with regard to transitionfrequency 510 and detuning 508, transition frequency 512 and detuning 506, and transition frequency 516 and detuning 514, which are depicted in FIG.5A. However, when both fluxonium qubits 212 and 214 are in their first excited states, the energy state of the resonator coupler is driven from its ground state to its first excited state, and back to its ground state, due to the emission of the microwave pulse that corresponds to driving frequency 410. “Selectivity” of the microwave pulse may therefore refer to a dependence of the system having both fluxoniums A and B in their first excited states in order to drive a 0-1 state transition of the resonator coupler.

[0070] FIG.4B illustrates a duration of time and a Gaussian-shaped envelope of a microwave pulse used to perform an entangling gate between two fluxonium qubits, coupled together using a resonator coupler, according to some embodiments.

[0071] In some embodiments, generated waveform 116 may resemble a single microwave pulse which has certain parameters that enable resonator coupler (C) 104 to be driven from its ground state to its first excited state, and then back down to its ground state, during performance of a two-qubit entangling gate. More specifically, a time duration of the pulse that defines a start and an end to the performance of the entangling gate causes resonator coupler (C) 104 to be driven from its ground state to its first excited state, and then back down to its ground state exactly within said duration of the pulse. Furthermore, generated waveform 116 may be defined as having aFWHM time of ^^⁄ √2 , and a gate duration time of ^^^. Yet another parameter of generatedwaveform 116 is that there is a Gaussian-shaped envelope around the microwave pulse.

[0072] As introduced above, for example, a given parameter of generated waveform 116 is that said generated waveform has a frequency corresponding to driving frequency 410, as shown in FIG.4A, which is detuned slightly from a resonant frequency of the resonator coupler that drivesa transition from|101^to|111^. As additionally shown in FIG. 4B, generated waveform 116refers to a charge modulation of resonator coupler (C) 104 at driving frequency 410.

[0073] FIG.4C illustrates an example of the population of a single energy state of an energy state diagram, such as that which is shown in FIG.3A, during performance of an entangling gate between two fluxonium qubits, according to some embodiments.

[0074] Once again referencing truth table for a CZ gate 220, as shown in FIG.2, time domain for the duration of the entangling gate 430 demonstrates performance of such a CZ gate in a scenario in which both fluxonium qubits 212 and 214 are in their first excited states. For ease of discussion in the following paragraphs, a scenario such as that which is shown in FIG.4C may bedefined as an entangling gate that begins at a time of ^^ = 0 and that ends at a time of ^^ = ^^^.However, this is not meant to be restrictive, as the given CZ gate being performed in FIG.4C mayinstead have a duration of any time from to ^^ଶ, such that ^^ = ^^ଶ − ^^^ = ^^^, and so on.

[0075] In some embodiments, an example of entangling gate dynamics for an initial state of|101^ may resemble that which is shown in FIG. 4C. At a moment in time during performance ofthe entangling gate (e.g., at time 0 < ^^ < ^^^), the resonator coupler has been excited from theground state of the resonator coupler to the first excited state of the resonator coupler, such thatthe state becomes |111^. Then, by the end of the performance of the entangling gate (e.g., at time^^ = ^^^), the state returns to |101^ with a phase shift of ^^. Such a phase shift is additionallyillustrated by a transition from the |11^ input state to the −|11^ output state in truth table for a CZ gate 220, as shown in FIG.2.

[0076] In another example, but in a scenario in which the initial state is |100^ when theentangling gate begins at entangling gate at ^^ = 0, the resonator coupler is not excited from theground state to the first excited state and back down to the ground state, and, instead, remains inthe ground state for the duration of the entangling gate (e.g., for a duration of ^^ = ^^ᇱ − ^^ = ^^^).Moreover, similar examples for the additional definitions within truth table for a CZ table 220 may similarly be constructed based on the material disclosed herein.

[0077] FIG.5A illustrates an energy state diagram, such as that which is also shown in FIG. 4A, wherein various transition frequencies that are not the drive frequency for performing an entangling gate between two fluxonium qubits are determined, such that they are detuned from proximity to the frequency of the drive, according to some embodiments.

[0078] FIG.5A continues the discussion herein of the energy state diagram that is introduced using FIG.4A, and that represents a quantum hardware device, such as quantum hardware device 100 that is depicted in. As illustrated in engineered detunings 500 in FIG.5A, a contrast is made between an intended driving frequency 410 for a two-qubit entangling gate, that is made conditional on energy states of the two fluxonium qubits, and other energy state transition frequencies, which are engineered to be detuned from driving frequency 410.

[0079] As introduced above, a system, such as quantum hardware device 100, may bedescribed using a Hamiltonian equation in the bare basis, ℋୠୟ୰^⁄ ℏ . The interaction Hamiltonianmay then be written in the dressed basis as follows:+^^(|1^^^1^|−|0^^^0^|)(|1^^^1^|−|0^^^0^|)(|0^^^0^|) + ⋯.

[0080] In the above Hamiltonian ℋூ⁄ ℏ , variables ^^ and ^^ refer to energy states |0^ and 1^ offluxoniums A and B, respectively. In addition, energy states with subscripts of ^^ refer to energy states of the resonator coupler.

[0081] As shown in Hamiltonian ℋூ⁄ ℏ , the |^^, 1, ^^^^^^, 1, ^^|term represents a shift in drivingfrequency 410 with respect to respective components of quantum hardware device 100 as uncoupled, bare components, wherein there is a frequency shift of ^^^^that is conditional on the energy states of the two fluxonium qubits. As illustrated in FIG.5A, transition frequency 510 and detuning 508, transition frequency 512 and detuning 506, and transition frequency 516 and detuning 514 each resemble energy state transition frequencies in which the resonator coupler is driven from the ground state (|0^^) to the first excited state (|1^^), but in which only one or neither of the two fluxonium qubits are in their first excited states.

[0082] An addition, and as additionally described in the following paragraphs herein, transition frequencies 510, 512, and 516 are respectively detuned from a frequency defined by driving frequency 410, as shown by detunings 508, 506, and 514 in FIG.5A. Detunings 508, 506, and 514, as described in the following paragraphs, may refer to respective frequency shifts ^^^^and / or combinations of frequency shifts ^^^^. For example, ^^^^represents a frequency shift of a frequency required to drive when the resonator coupler from the ground state (|0^^) to the first excited state (|1^^), while both fluxonium qubits begin at and remain at their respective ground states. In another example, ^^^^represents a frequency shift of a frequency required to drive the resonator coupler from the ground state (|0^^) to the first excited state (|1^^), while both fluxonium qubits begin at and remain at their respective first excited states.

[0083] In summary, transition frequencies 510, 512, and 516, along with their respective detunings 508, 506, and 514, and in addition to driving frequency 410, illustrate respective lines of truth table for a CZ gate 220, as introduced above with regard to description for FIG.2 herein. The following paragraphs provide additional description as pertains to each of the respectivetransition frequencies and how their descriptions within Hamiltonian

[0084] In a first example of transition frequencies illustrated in FIG.5A, transition frequency 510 represents an energy state transition frequency in which the resonator coupler is driven fromthe ground state (|0^^) to the first excited state (|1^^), and in which both fluxonium qubits beginat and remain at their respective ground states (e.g., for a duration of a two-qubit entangling gate). Furthermore, and in order to suppress potential sources of coherent error during performance of a two-qubit entangling gate in which the resonator coupler is engineered to be driven between the ground state and first excited state conditional on both fluxonium qubits being in their first excited states, transition frequency 510 is engineered to be detuned far from driving frequency 410, suchthat detuning 508 may be defined as ^^^^ − ^^^^. Such an engineered detuning of detuning 508 maybe simulated, evaluated, optimized, and / or refined during evaluations 716 and / or 718, as additionally described with regard to FIG.7 herein.

[0085] In a second example of transition frequencies illustrated in FIG. 5A, transition frequency 512 represents an energy state transition frequency in which the resonator coupler is driven from the ground state (|0^^) to the first excited state (|1^^), and in which fluxonium A is in its first excited state (|1^^) while fluxonium B begins at and remains at the ground state (|0^^). Furthermore, and in order to suppress potential sources of coherent error during performance of a two-qubit entangling gate in which the resonator coupler is engineered to be driven between the ground state and first excited state conditional on both fluxonium qubits being in their first excited states, transition frequency 512 is engineered to be detuned far from driving frequency 410, suchthat detuning 506 may be defined as ^^^^ − ^^^^. Such an engineered detuning of detuning 506 maybe simulated, evaluated, optimized, and / or refined during evaluations 716 and / or 718, as additionally described with regard to FIG.7 herein.

[0086] In a third example of transition frequencies illustrated in FIG.5A, transition frequency 516 represents an energy state transition frequency in which the resonator coupler is driven from the ground state (|0^^) to the first excited state (|1^^), and in which fluxonium B is in its first excited state (|1^^) while fluxonium A begins at and remains at the ground state (|0^^). Furthermore, and in order to suppress potential sources of coherent error during performance of a two-qubit entangling gate in which the resonator coupler is engineered to be driven between the ground state and first excited state conditional on both fluxonium qubits being in their first excited states, transition frequency 516 is engineered to be detuned far from driving frequency 410, suchthat detuning 514 may be defined as ^^^^ − ^^^^. Such an engineered detuning of detuning 514 maybe simulated, evaluated, optimized, and / or refined during evaluations 716 and / or 718, as additionally described with regard to FIG.7 herein.

[0087] Moreover, the above examples of transition frequencies 510, 512, and 516 and their respective detunings 508, 506, and 514 illustrate a difference between the intended on-resonance driving frequency 410 and transition frequencies 510, 512, and 516, which are not intended to be driven during performance of a two-qubit entangling gate. For example, detuning 508 is detunedfrom driving frequency 410 by an amount of ^^^^ − ^^^^. In a second example, detuning 506 isdetuned from driving frequency 410 by an amount of ^^^^ − ^^^^. In a third example, detuning 514is detuned from driving frequency 410 by an amount of ^^^^ − ^^^^. Engineering such detunings,prior to the fabrication of quantum hardware device 100, ensures that sources of coherent error, caused by unintended driving of transition frequencies 510, 512, and 516 during performance of a two-qubit entangling gate, are suppressed. Description below, with regard to FIG.7, further details how simulation techniques may be used to simulate, evaluate, optimize, and / or refine such detunings prior to fabrication of quantum hardware device 100.

[0088] Referring once again to Hamiltonian equation ℋூ⁄ ℏ , the ^^|1, 2, 1^^1, 2, 1| termrepresents a frequency shift ^^ in order to drive the resonator coupler from the first excited state(|1^^) to the second excited state (|2^^) when both fluxonium qubits are in their respective firstexcited states (|1^1^^). As additionally illustrated in FIG. 5A, transition frequency 504 and detuning 502 represents an energy state transition frequency in which the resonator coupler isdriven from the first excited state (|1^^) to the second excited state (|2^^) when both fluxoniumqubits are in their respective first excited states (|1^1^^). Furthermore, and in order to suppress potential sources of coherent error, and cause said sources to destructively interfere with one another, during performance of a two-qubit entangling gate in which the resonator coupler is engineered to be driven, conditional on both fluxonium qubits being in their first excited states, between the ground state and first excited state of the resonator coupler, and not between the first and second excited states of the resonator coupler, transition frequency 504 is engineered such that a positive anharmonicity exists between transition frequency 504 and driving frequency 410. In some embodiments, a positive anharmonicity ^^ may be defined herein as the absolute value of transition frequency 504 being greater than the absolute value of driving frequency 410 by an amount of ^^.

[0089] Additional explanation of a positive anharmonicity ^^ is discussed with regard to FIG. 5C herein.

[0090] Referring once again to Hamiltonian equation ℋூ⁄ ℏ , theterm represents quantities of ^^^^-typeinteractions between the two fluxonium qubits. As should be understood by the(|1^^^1^|−|0^^^0^|)|0^^^0^| term, such ^^^^ interactions may occurbetween the two fluxonium qubits, such as when the resonator coupler is in its ground state.

[0091] A ^^^^-type interaction may, for example, include a dependency of driving fluxonium Abetween the ground state (|0^^) and the first excited state (|1^^) while fluxonium A is capacitivelycoupled to fluxonium B with a capacitive coupling strength of ^^^,^, as well as each fluxonium qubit with the resonator coupler with strengths ^^^,^and ^^^,^, as defined above with regard toHamiltonian equation ℋୠୟ୰^⁄ ℏ . In a second example, a ^^^^-type interaction may additionallyinclude a dependency of driving fluxonium B between the ground state (|0^^) and the first excited state (|1^^) while fluxonium A is capacitively coupled to fluxonium B with the capacitive coupling strength of ^^^,^.

[0092] Such ^^^^ interactions that are based, at least in part, on the capacitive coupling of fluxoniums A and B and resonator coupler C, represent some amount of parasitic entanglementbetween fluxoniums A and B that exists outside of said performance time of the gate. Therefore, in order to ensure that two-qubit entangling gates, such as those described herein, remain“microwave-activated,” such “always-on,” parasitic entanglements are suppressed by engineering^^ within theterm to be small.

[0093] Furthermore, by simulating, evaluating, optimizing, and / or refining respective ones of the values of the circuit parameters of quantum hardware device 100 with respect to one another, sources of potential coherent error, such as those defined above with regard to detunings 508, 506, and 514 and positive anharmonicity ^^, may be engineered to destructively interfere with one another. By causing such destructive interference, sources of potential coherent error are further suppressed, prior to fabrication of quantum hardware device 100.

[0094] A suppression of ^^^^ interactions between two fluxonium qubits is additionally discussed with regard to FIG.5C herein, and a “maximal gate fidelity” is additionally discussed with regard to FIGs.6A – 6B.

[0095] FIG. 5B illustrates relationships between respective ones of the detuned transition frequencies shown in FIG.5A, according to some embodiments.

[0096] As shown in plot 520 in FIG.5B,values within the ∑^,^∈^^,^^ ൫^^^^|^^, 1, ^^^^^^, 1, ^^|൯ termof Hamiltonian equationmay be engineered according to a particular frequency of drivingfrequency 410. For example, the line depicted by line 528 in FIG. 5B is made to represent ^^^^, wherein line 522 (e.g., ^^^^), line 524 (e.g., ^^^^), and line 526 (e.g., ^^^^) are engineered, using simulation techniques described herein, to be far from a frequency defined by line 528 (e.g., ^^^^) in order to suppress potential sources of coherent error during performance of a two-qubit entangling gate.

[0097] FIG.5C illustrates parasitic, always-on ^^^^ interactions and an inherited anharmonicityof the resonator coupler, both with respect to a hybridization h =|^2, 0, 1|1, 1, 1^|ଶ / 2 +|^1, 0, 2|1, 1, 1^|ଶ / 2 between the two fluxoniums’ plasmon transition and the resonator coupler,according to some embodiments.

[0098] In some embodiments, an amount of ^^^^ interactions, defined by ^^ within the(|1^^^1^|−|0^^^0^|)|0^^^0^| term of Hamiltonian equation ℋூ⁄ ℏ , maybe engineered to be small for various values of hybridization between the resonator coupler and the two fluxonium qubits, respectively. As shown using the left-hand y-axis of plot 530, line 532 represents how ^^^^ interactions may be engineered to be suppressed (e.g., suppressed to a point of being within a range of 0.1 – 10 kHz) according to given circuit parameters of quantum hardware device 100 that result in a corresponding value of resonator-coupler-fluxonium hybridization on the x-axis of plot 530.

[0099] Furthermore, an amount of positive anharmonicity, ^^, between driving the resonatorcoupler between the ground state (|0^^) and the first excited state (|1^^), and between the firstexcited state (|1^^) and the second excited state (|2^^) may be illustrated using line 534 and theright-hand y-axis of plot 530, wherein positive anharmonicity may be engineered to be large according to given circuit parameters of quantum hardware device 100 that result in a corresponding value of resonator-coupler-fluxonium hybridization on the x-axis of plot 530. In some embodiments, a large positive anharmonicity, ^^, may be defined as having a frequency of 30 – 150 MHz.

[0100] FIG.6A illustrates an energy state diagram, such as that which is also shown in FIGs. 4A and 5A, wherein various other transition frequencies are evaluated in order to further optimize performance of a two-qubit entangling gate between fluxonium qubits, according to some embodiments.

[0101] FIG.6A continues the discussion herein of the energy state diagram that is introduced using FIGs.4A and 5A, and that represents a quantum hardware device, such as quantum hardware device 100 that is depicted in FIG. 1. In some embodiments, relationships between additional energy state transition frequencies may be used to further engineer and tune circuit parameters of quantum hardware device 100, prior to fabrication. Such relationships are illustrated as transition frequency relationships 600 in FIG.6A.

[0102] In some embodiments, in order to further refine amounts of detunings of transitionfrequencies from driving frequency 410 of a two-qubit entangling gate, ^^^^ values within the∑^,^∈^^,^^ ൫^^^^|^^, 1, ^^^^^^, 1, ^^|൯ term of Hamiltonian equationmay be simulated, andsubsequently and / or iteratively refined, through an analysis using perturbation theory.

[0103] As shown in FIG. 6A, transition frequency 602 represents energy state transitions in which the resonator coupler is driven between the first excited state (|1^^) and the second excited state (|2^^), fluxonium A is driven between the first excited state (|1^^) and the second excited state (|2^^), and fluxonium B begins at and remains at the first excited state (|1^^).

[0104] Transition frequency 604 represents energy state transitions in which the resonator coupler is driven between the first excited state (|1^^) and the second excited state (|2^^), fluxonium B is driven between the first excited state (|1^^) and the second excited state (|2^^), and fluxonium A begins at and remains at the first excited state (|1^^).

[0105] Transition frequency 606 represents energy state transitions in which the resonator coupler begins at and remains at the ground state (|0^^), fluxonium A is driven between the first excited state (|1^^) and the second excited state (|2^^), and fluxonium B is driven between the first excited state (|1^^) and the second excited state (|2^^).

[0106] Transition frequency 608 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), fluxonium A isdriven between the first excited state (|1^^) and the second excited state (|2^^), and fluxonium Bbegins at and remains at the first excited state (|1^^).

[0107] Transition frequency 610 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), fluxonium A isdriven between the first excited state (|1^^) and the second excited state (|2^^), and fluxonium Bbegins at and remains at the ground state (|0^^).

[0108] Transition frequency 612 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), fluxonium A begins at and remains at the first excited state (|1^^), and fluxonium B is driven between the groundstate (|0^^) and the third excited state (|3^^).

[0109] Transition frequency 614 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), fluxonium A begins at and remains at the first excited state (|1^^), and fluxonium B is driven between the firstexcited state (|1^^) and the second excited state (|2^^).

[0110] Transition frequency 616 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), the secondexcited state (|2^^), fluxonium A begins at and remains at the ground state (|0^^), and fluxoniumB is driven between the first excited state (|1^^) and the second excited state (|2^^).

[0111] Transition frequency 618 represents energy state transitions in which the resonator coupler is driven between the ground state (|0^^) and the first excited state (|1^^), fluxonium A isdriven between the ground state (|0^^) and the third excited state (|3^^), and fluxonium B beginsat and remains at the first excited state (|1^^).

[0112] In some embodiments, particular combinations of perturbative pathways, which may be defined as combinations of particular transition frequencies such as those illustrated by transition frequencies 602, 604, 606, 608, 610, 612, 614, 616, and 618 in FIG.6A, may be pertinentto the refining of ^^^^ values within the ∑^,^∈^^,^^ ൫^^^^|^^, 1,1, ^^|൯ term of Hamiltonian equationℋூ⁄ ℏ and pertinent to the refining of ^^. The following paragraphs illustrate examples of thoseparticular combinations that may be used to refine respective values of ^^^^and ^^. However, the following description is not meant to be restrictive, as additional perturbative pathways may be simulated and refined in order to further suppress sources of coherent error during performance of two-qubit entangling gates.

[0113] In a first example, a given perturbative pathway with a large charge matrix element for a system described by energy state diagram 400 may be defined using combinations of transition frequencies 608, 610, 614, and 616. In a second example, another given perturbative pathway with large charge matrix elements for a system described by energy state diagram 400 may be defined using combinations of transition frequencies 602 and 604. In a third example, yet another given perturbative pathway with large charge matrix elements for a system described by energy state diagram 400 may be defined using combinations of transition frequencies 608, 606, and 614. In a fourth example, another given perturbative pathway with large charge matrix elements for a system described by energy state diagram 400 may be defined using combinations of transition frequencies 612 and 618.

[0114] Such perturbative pathways may be further defined by the following, wherein notationfollows from notations introduced above with regard to ℋୠୟ୰^⁄ ℏ , and ^^(^)^,^ represents an ^^th ordercorrection to ^^^^:

[0115] In the above definitions of a perturbation theory analysis of respective values of ^^^^,^^ ∈ ^^^, ^^^ and ^^ ∈ ^^^, ^^^ refer to fluxonium qubits A and B. Furthermore, ^^^ refers to a bareresonator coupler frequency, rather than to driving frequency 410, and ^^(^)^,ଶrefers to a bare frequency of an energy state transition between the first and second excited states for the respectivetwo fluxonium qubits, and such thatwherein

[0116] FIG. 6B illustrates usage of simulation techniques to further optimize a two-qubit entangling gate towards unit gate fidelity, according to some embodiments.

[0117] In some embodiments, the description above pertaining to FIG.6A has focused largely on frequency shift considerations under two-body charge-charge couplings ^^^,^, ^^^,^, and ^^^,^tomaximize a resulting ^^^,^ − ^^^^ and ^^. However, in order to trend towards maximally entanglinggate fidelity during a two-qubit entangling gate between said fluxonium qubits, circuit parameters for phase shifts of the computational states under the gate being performed are also tuned and / orrefined using simulation techniques described herein. In reference once again to truth table for a CZ gate 220, there should be a phase shift of close to ^^ when both fluxonium qubits are in their first excited states (e.g.,and a phase shift of close to zero when one or more of thefluxonium qubits are not in their first excited states (e.g.,|0^0^^,|1^0^^, and|0^1^^).

[0118] As illustrated in plot 650 in FIG. 6B, line 658 depicts an expected phase shift (^^^^) from ^^ when both fluxonium qubits are in their first excited states (e.g.,|1^1^^), and for various durations of the gate (^^). Line 656 depicts an expected phase shift (^^^^) when fluxonium A is in the first excited state and fluxonium B is in the ground state (e.g., |1^0^^), and for various durations of the gate (^^). Line 654 depicts an expected phase shift (^^^^) when fluxonium A is in the ground state and fluxonium B is in the first excited state (e.g., |0^1^^), and for various durations of the gate (^^). An expected phase shift ^^^^may similarly be defined as when fluxoniumqubits A and B are both in their ground states (e.g.,|0^0^^).

[0119] Furthermore, such expected phase shifts may be simulated and further refined, using simulation techniques described herein, in order to trend closely towards the following relationshipfor a maximally entangling gate:^^ ≈ ^^^^ + ^^^^ − ^^^^ − ^^^^ ≈ 0.

[0120] In some embodiments, under a condition of ^^, the entangling gate may correspond to truth table for a CZ gate 220 when combined with single-qubit ^^ rotations on both fluxoniums,which may be applied as “virtual” gates. Line 652 represents ^^ ≈ ^^^^ + ^^^^ − ^^^^ − ^^^^ and, asshown in FIG.6B, line 652 trends close to zero.

[0121] Moreover, for a condition of a non-zero ^^, an expected coherent error due to such adependency may be represented as follows:

[0122] In order to further ensure that such sources of coherent error destructively interfere with one another, the following relationship may be further simulated, evaluated, refined, and / or tuned, prior to fabrication of quantum hardware device 100: 3−1.58 ଶ√2^^ 2^^ 2^^^^ = √ √ఝ 20^^ଶ ^^^ −2(^^ − ^^ ) −2(^^ − ^^ ) +2( )^ .^^ ^^ ^^ ^^ ^^^^ − ^^^^

[0123] FIG. 7 illustrates a visual representation of simulation techniques used to determine and refine circuit parameters for a quantum hardware device that is to be configured to perform two-qubit entangling gates between fluxonium qubits that are coupled together using a resonator coupler, according to some embodiments.

[0124] In some embodiments, one or more classical computing devices, such as classical computing device 1000, may be configured to implement quantum hardware device simulator 714, as shown in FIG. 7, which may be used to generate refined circuit parameters for quantum hardware device 100, prior to fabrication (e.g., pre-fabrication stage 700). Using methods and techniques described above with regard to FIGs.1 – 6B, such circuit parameters may be tuned in order to provide high gate fidelity and fast gate performance upon fabricating quantum hardware device 100. In some embodiments, certain additional parameters, such as a finalized driving frequency for a two-qubit entangling gate, may be additionally refined post-fabrication of quantum hardware device 100. For example, upon executing a calibration protocol using a post-fabricated version of quantum hardware device 100, driving frequency 410 may be further refined in order to compensate for slight imperfections incurred during one or more steps in a fabrication process for quantum hardware device 100, or for a non-zero ^^.

[0125] In some embodiments, constraints on time duration 712 may refer to initial parameters of generated waveform 116, such as a duration of the entangling gate ^^^. In addition, constraints on time duration 712 may also refer to maximum time limits within which the entangling gate should be expected to be performed. For example, constraints on time duration 712 may indicate expected coherence and / or lifetime expectancies of fluxonium qubits that are implemented using the given circuit parameters defined by 704, 706, 708, and 710.

[0126] In some embodiments, circuit parameters for fluxonium A 704 may refer to ^^^^^,^,^^^,^, and ^^^^,^, as defined above with regard to Hamiltonian equation ℋୠୟ୰^⁄ ℏ . Similarly, circuitparameters for fluxonium B 706 may refer to ^^^^^,^, ^^^,^, and ^^^^,^; circuit parameters for resonator coupler C 708 may refer to ^^^and ^^; and respective capacitive coupling strengths 710 may refer to ^^^,^, ^^^,^, and ^^^,^.

[0127] Evaluation 716 may then refer to a simulation of the Hamiltonian equation ℋூ⁄ ℏ , suchthat initial parameters of ^^^^, ^^, and ^^ may be determined. As further illustrated by evaluation 716, one or more of the initial circuit parameters 702 may then be refined (see refinement 720) in order to ensure that detunings 502, 506, 508, and 514 are large, given the refined values of the circuit parameters (see also descriptions pertaining to FIGs.4A – 5C herein). Evaluation 718 may then refer to tuning dependencies between values of ^^^^, ^^, and ^^ that have been determined during evaluation 716, such as additionally described herein with regard to frequency and phase shifts in FIGs.5A – 6B.

[0128] One or more iterations of evaluations 716 and / or 718 may be completed in order to further suppress sources of coherent error, increase gate fidelity, and increase speed of gate performance, etc.

[0129] Furthermore, additional parameters may be included within initial quantum hardware device parameters 702, such as material properties of hardware components of quantum hardware device 100, according to some embodiments. For example, a high-quality material, such as Tantalum, may be used to fabricate resonator coupler (C) 104 in order to suppress potential sources of incoherent error. Such additional system dynamics may be considered to be relevant input values to provide to quantum hardware device simulator 714.

[0130] As additionally shown with regard to FIG.1, classical computing device(s) 1000 may be additionally configured to provide drive control instructions to microwave pulse generator 114 such that selected driving frequency 410 may be applied, for performance of a given two-qubit entangling gate, and according to additional drive control instructions used to construct a Gaussian shape, duration, etc. of generated waveform 116. Such drive control instructions may be generated based, at least in part, on a predicted frequency of driving frequency 410 that is determined by quantum hardware device simulator 714, following a refinement of the initial circuit parameters for quantum hardware device 100.

[0131] FIG.8 illustrates a process of determining and refining circuit parameters for a quantum hardware device that is to be configured to perform two-qubit entangling gates between fluxonium qubits that are coupled together using a resonator coupler, according to some embodiments.

[0132] In block 800, initial values of circuit parameters that are to be used to fabricate a quantum hardware device, such as quantum hardware device 100, are determined and provided to classical computing devices that are to perform simulations in order to further tune, refine, and / or optimize said initial values.

[0133] In block 802, a series of simulation techniques may be performed in order to evaluate performance metrics and / or relationships between the initial values of the circuit parameters. Forexample, the initial values of circuit parameters may be applied to a Hamiltonian, such as ℋୠୟ୰^⁄ ℏdefined above, in order to determine driving frequency 410, and detunings 502, 506, 508, and 514.

[0134] In some embodiments, one or more of the initial values of the circuit parameters may then be refined, as described in block 804. For example, once driving frequency 410, and transition frequencies 504, 510, 512, and 516 have been determined, then detunings 502, 508, 506, and 514 may also be refined such that performance metrics pertaining to suppression of coherent error, maximization of gate fidelity, and enhancement of an effect of positive anharmonicity betweenrespective energy state transition frequencies of the resonator coupler, etc. may be optimized with respect to providing faster entangling gates between fluxonium qubits and with higher fidelity.

[0135] As indicated by the arrow between blocks 804 and 802 in FIG.8, blocks 802 and 804 may be performed iteratively and / or multiple times, such that refined values of the circuit parameters are then re-simulated, then additionally refined, etc. until one or more performance metrics pertaining to circuit parameters for fabrication of a quantum hardware device are met, according to some embodiments.

[0136] In block 806, the refined values of the circuit parameters are then provided for use in fabricating the quantum hardware device. In some embodiments, classical computing devices that implement a quantum hardware device simulator, such as that which is described in FIGs.7 and 8, may be devices that are within a service provider network. Said classical computing devices may be configured to provide the refined values of the circuit parameters for fabrication using fabrication methods and techniques available within the service provider network, or may be configured to provide the refined values for fabrication by a third party entity outside of the service provider network, according to some embodiments.

[0137] FIG.9 is a block diagram illustrating an example quantum hardware device that may be configured to execute quantum gates (e.g., entangling gates) and perform readout measurements following the execution of those quantum gates, according to some embodiments.

[0138] As shown in FIG.9, a quantum hardware device 900 may comprise one or more central quantum processing units (QPUs) and / or quantum processing cores 920 that, collectively, implement a quantum computer 930. Various configurations of physical qubits may be included in implementation of quantum computer 930 wherein a given subset of a total number of qubits may represent quantum processing core 920 and another given subset of qubits may be used to implement magic state factories, additional routing space, and / or additional quantum processing cores that are accessible via lattice surgery, as shown in block 910. Portions of quantum computations and / or operations may be performed in quantum processing core 930, wherein computationally intensive logical computations may use magic state factories within block 910 in order to produce magic states that may be used to store intermediate computations such that they are held in memory during such quantum computations. In some embodiments, a given magic state factory of block 910 may be merged with quantum processing core 920 during a procedure such as lattice surgery in order for information to pass between such components of the quantum computer.

[0139] As related to the description herein, one or more fluxonium qubits within implementation of quantum computer 930 may additionally be coupled to a quantum readoutdevice for measurements of quantum states following performance of one or more quantum gates, such as two-qubit entangling gates described herein. The given quantum readout device may be locally connected to various qubits of quantum processing core 920, as shown by interaction arrows to / from block 940.

[0140] Depending upon factors such as type(s) of qubit technologies used (e.g., superconducting architectures), type(s) of gates performed between said qubits (e.g., entangling gates, readout measurements), etc., quantum hardware device 910 may also comprise various control devices (e.g., microwave pulse generators, lasers, devices for temperature, magnetic, and / or other environmental controls pertaining to local environments of the grid of qubits within implementation of quantum computer 930, etc.) that may be used to maintain and / or transform various properties of the qubits and / or other physical components of a given quantum computer, as shown via local environmental control devices within block 940. For example, microwave pulse generator 114 may be locally coupled to one or more quantum hardware components within quantum processing core 920, such that a microwave pulse emitted from microwave pulse generator 114 may be used to initiate, perform, and terminate a two-qubit entangling gate between various qubits of quantum processing core 920.

[0141] In some embodiments in which local environmental control devices 940 include a processor such as processors 1010, local environmental control devices 940 may additionally be configured to interact with other devices 960 via network 950. In some embodiments, other devices 960 may include classical computing devices such as classical computing device 1000, which may be configured to interact with quantum hardware device 900 either locally or remotely.

[0142] Embodiments of the present disclosure may be described in view of the following clauses: Clause 1. A method, comprising: determining initial values of circuit parameters to be used to fabricate a quantum hardware device, wherein: the quantum hardware device, when fabricated, will comprise: a first set of fluxonium hardware components, used to implement a first fluxonium qubit; a second set of fluxonium hardware components, used to implement a second fluxonium qubit; and a third set of hardware components, used to implement a resonator coupler; and the quantum hardware device, when fabricated, will be configured to:capacitively couple the first and second sets of fluxonium hardware components to one another; capacitively couple the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components and to the second sets of fluxonium hardware components; and perform entangling gates between the first and second fluxonium qubits based, at least in part, on charge modulation of the resonator coupler; simulating an expected resonance frequency of the resonator coupler based, at least in part, on the initial values of the circuit parameters; simulating other expected energy state transition frequencies of the resonator coupler and of the first and second fluxonium hardware components based, at least in part, on the initial values of the circuit parameters; refining one or more of the initial values of the circuit parameters such that the other expected energy state transition frequencies are further detuned from the expected resonance frequency of the resonator coupler; and fabricating the quantum hardware device based, at least in part, on the refined values of the circuit parameters. Clause 2. The method of clause 1, wherein the circuit parameters to be used to fabricate a quantum hardware device comprise: a first energy corresponding to an inductance of the first set of fluxonium hardware components; a second energy corresponding to a capacitance of the first set of fluxonium hardware components; a third energy corresponding to a Josephson junction of the first set of fluxonium hardware components; a fourth energy corresponding to an inductance of the second set of fluxonium hardware components; a fifth energy corresponding to a capacitance of the second set of fluxonium hardware components; a sixth energy corresponding to a Josephson junction of the second set of fluxonium hardware components; a seventh energy corresponding to an inductance of the third set of hardware components;an eighth energy corresponding to a capacitance of the third set of hardware components; a first capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; a second capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; and a third capacitive coupling strength of coupling the first set of fluxonium hardware components to the second set of fluxonium hardware components. Clause 3. The method of clause 1 or clause 2, further comprising:additionally refining one or more of the initial values of the circuit parameters such that:^^^^ interactions between the first and the second fluxonium qubits are suppressed;and the entangling gates are microwave-activated. Clause 4. The method of any of clauses 1 through 3, further comprising: subsequent to said fabricating the quantum hardware device, determining a frequency of a microwave pulse to be emitted for performance of entangling gates between the first and second fluxonium qubits. Clause 5. The method of clause 4, further comprising: performing, using the determined frequency of the microwave pulse, an entangling gate between the first and second fluxonium qubits, wherein said performing comprises: conditional on the first and second fluxonium qubits being in respective first excited states, driving the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; driving the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler; and driving, within a duration of the entangling gate, a phase shift of ^^; and conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintaining the resonator coupler in a ground state of the resonator coupler for the duration of the entangling gate. Clause 6. A system, comprising:one or more classical computing devices configured to implement a quantum hardware device simulator, wherein the quantum hardware device simulator is configured to: receive an initial set of circuit parameters to be used to fabricate a quantum hardware device, wherein: the quantum hardware device, when fabricated, will comprise: a first set of fluxonium hardware components, used to implement a first fluxonium qubit; a second set of fluxonium hardware components, used to implement a second fluxonium qubit; and a third set of hardware components, used to implement a resonator coupler; and the quantum hardware device, when fabricated, will be configured to: capacitively couple the first and second sets of fluxonium hardware components to one another; capacitively couple the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components and to the second sets of fluxonium hardware components; and perform entangling gates between the first and second fluxonium qubits based, at least in part, on charge modulation of the resonator coupler; simulate, using simulation techniques, an expected resonance frequency of the resonator coupler based, at least in part, on the initial values of the circuit parameters; simulate, using the simulation techniques, other expected energy state transition frequencies of the resonator coupler and of the first and second fluxonium hardware components based, at least in part, on the initial values of the circuit parameters; refine one or more of the initial values of the circuit parameters such that the other expected energy state transition frequencies are further detuned from the expected resonance frequency of the resonator coupler; and provide the refined values of the circuit parameters to be used to fabricate the quantum hardware device. Clause 7. The system of clause 6, wherein:the circuit parameters to be used to fabricate a quantum hardware device comprise: a first capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; a second capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; and a third capacitive coupling strength of coupling the first set of fluxonium hardware components to the second set of fluxonium hardware components; and the quantum hardware device simulator is further configured to: additionally refine one or more of the first, second, and third capacitive coupling strengths such that the resonator coupler is defined as having a positive anharmonicity. Clause 8. The system of clause 7, wherein to refine the one or more of the first, second, and third capacitive coupling strengths such that the resonator coupler is defined as having a positive anharmonicity, the quantum hardware device simulator is further configured to: determine a transition frequency corresponding to a transition frequency between a first excited state and a second excited state of energy states enabled by the configuration of the resonator coupler such that the determined frequency is greater than the expected resonance frequency of the resonator coupler. Clause 9. The system of clause 8, wherein a difference between the determined transition frequency and the expected resonance frequency is between 30 MHz and 150 MHz. Clause 10. The system of clause 6, wherein, to refine one or more of the initial values of the circuit parameters such that the other expected energy state transition frequencies are further detuned from the expected resonance frequency of the resonator coupler, the quantum hardware device simulator is further configured to: determine relationships between the detuned expected energy state transition frequencies and the expected resonance frequency such that the entangling gates are further optimized towards being maximally entangling gates. Clause 11. The system of clause 6, wherein the circuit parameters to be used to fabricate a quantum hardware device comprise: a first energy corresponding to an inductance of the first set of fluxonium hardware components;a second energy corresponding to a capacitance of the first set of fluxonium hardware components; a third energy corresponding to a Josephson junction of the first set of fluxonium hardware components; a fourth energy corresponding to an inductance of the second set of fluxonium hardware components; a fifth energy corresponding to a capacitance of the second set of fluxonium hardware components; a sixth energy corresponding to a Josephson junction of the second set of fluxonium hardware components; a seventh energy corresponding to an inductance of the third set of hardware components; and an eighth energy corresponding to a capacitance of the third set of hardware components. Clause 12. The system of clause 11, wherein, subsequent to the simulation, using the simulation techniques, of the other expected energy state transition frequencies of the resonator coupler and of the first and second fluxonium hardware components, the quantum hardware device simulator is further configured to: determine respective amounts of expected sources of coherent errors, wherein: a first source of coherent error comprises excitations to frequencies of the other expected energy state transition frequencies of the first and second fluxonium hardware components; and a second source of coherent error comprises excitations to frequencies of the other expected energy state transition frequencies of the resonator coupler. Clause 13. The system of clause 12, wherein the quantum hardware device simulator is further configured to: additionally refine one or more of the first, second, third, fourth, fifth, sixth, seventh, and eighth energies such that the first source of coherent error and the second source of coherent error destructively interfere with one another. Clause 14. The system of clause 6, wherein: the circuit parameters to be used to fabricate a quantum hardware device comprise: a first energy corresponding to an inductance of the first set of fluxonium hardware components; a second energy corresponding to a capacitance of the first set of fluxonium hardware components;a third energy corresponding to a Josephson junction of the first set of fluxonium hardware components; a fourth energy corresponding to an inductance of the second set of fluxonium hardware components; a fifth energy corresponding to a capacitance of the second set of fluxonium hardware components; and a sixth energy corresponding to a Josephson junction of the second set of fluxonium hardware components; and the quantum hardware device simulator is further configured to: additionally refine one or more of the first, second, third, fourth, fifth, and sixths energies such that:^^^^ interactions between the first and the second fluxonium qubits aresuppressed; and the entangling gates are microwave-activated. Clause 15. The system of clause 6, wherein the simulation techniques comprise an analysis, using perturbation theory, of a Hamiltonian used to define respective energies and capacitive couplings of the first and second sets of fluxonium hardware components and of the third set of hardware components. Clause 16. One or more non-transitory, computer-readable, media storing program instructions that, when executed on or across one or more processors, cause the one or more processors to: receive an initial set of circuit parameters to be used to fabricate a quantum hardware device, wherein: the quantum hardware device, when fabricated, will comprise: a first set of fluxonium hardware components, used to implement a first fluxonium qubit; a second set of fluxonium hardware components, used to implement a second fluxonium qubit; and a third set of hardware components, used to implement a resonator coupler; and the quantum hardware device, when fabricated, will be configured to: capacitively couple the first and second sets of fluxonium hardware components to one another;capacitively couple the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components and to the second sets of fluxonium hardware components; and perform entangling gates between the first and second fluxonium qubits based, at least in part, on charge modulation of the resonator coupler; simulate, using simulation techniques, an expected resonance frequency of the resonator coupler based, at least in part, on the initial values of the circuit parameters; simulate, using the simulation techniques, other expected energy state transition frequencies of the resonator coupler and of the first and second fluxonium hardware components based, at least in part, on the initial values of the circuit parameters; refine one or more of the initial values of the circuit parameters such that the other expected energy state transition frequencies are further detuned from the expected resonance frequency of the resonator coupler; and provide the refined values of the circuit parameters to be used to fabricate the quantum hardware device. Clause 17. The one or more non-transitory, computer-readable media of clause 16, wherein: the circuit parameters to be used to fabricate a quantum hardware device comprise: a first capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; a second capacitive coupling strength of coupling the third set of hardware components, used to implement the resonator coupler, to the first set of fluxonium hardware components; and a third capacitive coupling strength of coupling the first set of fluxonium hardware components to the second set of fluxonium hardware components; and the program instructions further cause the one or more processors to: additionally refine one or more of the first, second, and third capacitive coupling strengths such that the resonator coupler is defined as having a positive anharmonicity. Clause 18. The one or more non-transitory, computer-readable media of clause 16, wherein the circuit parameters to be used to fabricate a quantum hardware device comprise:a first energy corresponding to an inductance of the first set of fluxonium hardware components; a second energy corresponding to a capacitance of the first set of fluxonium hardware components; a third energy corresponding to a Josephson junction of the first set of fluxonium hardware components; a fourth energy corresponding to an inductance of the second set of fluxonium hardware components; a fifth energy corresponding to a capacitance of the second set of fluxonium hardware components; a sixth energy corresponding to a Josephson junction of the second set of fluxonium hardware components; a seventh energy corresponding to an inductance of the third set of hardware components; and an eighth energy corresponding to a capacitance of the third set of hardware components. Clause 19. The one or more non-transitory, computer-readable media of clause 18, wherein, subsequent to the simulation, using the simulation techniques, of the other expected energy state transition frequencies of the resonator coupler and of the first and second fluxonium hardware components, the program instructions further cause the one or more processors to: determine respective amounts of expected sources of coherent errors, wherein: a first source of coherent error comprises excitations at frequencies of the other expected energy state transition frequencies of the first and second fluxonium hardware components; and a second source of coherent error comprises excitations at frequencies of the other expected energy state transition frequencies of the resonator coupler; and additionally refine one or more of the first, second, third, fourth, fifth, sixth, seventh, and eighth energies such that the first source of coherent error and the second source of coherent error destructively interfere with one another. Clause 20. The one or more non-transitory, computer-readable media of clause 18, wherein the program instructions further cause the one or more processors to: additionally refine one or more of the first, second, third, fourth, fifth, and sixths energies such that:^^^^ interactions between the first and the second fluxonium qubits are suppressed;andthe entangling gates are microwave-activated. Clause 21. A system, comprising: one or more classical computing devices, configured to provide drive control instructions to one or more quantum hardware devices for performance of an entangling gate; and the one or more quantum hardware devices comprising: a first set of fluxonium hardware components, configured to implement a first fluxonium qubit; a second set of fluxonium hardware components, configured to: implement a second fluxonium qubit; and be capacitively coupled to the first set of fluxonium hardware components; and a resonator coupler, wherein resonator coupler is configured to be capacitively coupled to the first set of fluxonium hardware components and to the second set of fluxonium hardware components, wherein the one or more quantum hardware devices are further configured to: receive the drive control instructions; and perform the entangling gate between the first and second fluxonium qubits based, at least in part, on charge modulation of the resonator coupler. Clause 22. The system of clause 21, wherein the entangling gate between the first and second fluxonium qubits is a Controlled-^^ gate. Clause 23. The system of clause 21 or clause 22, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to: conditional on the first and second fluxonium qubits being in respective first excited states, drive the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; and drive the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler. Clause 24. The system of clause 23, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to: conditional on the first and second fluxonium qubits being in respective first excited states, cause, within a duration of the entangling gate, a phase shift of ^^. Clause 25. The system of clause 21, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to:conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintain the resonator coupler in a ground state of the resonator coupler for a duration of the entangling gate. Clause 26. The system of any of clauses 21 through 25, wherein the resonator coupler is fabricated using Tantalum. Clause 27. The system of any of clauses 21 through 25, wherein the resonator coupler is a harmonic oscillator. Clause 28. The system of clause 27, wherein the capacitive coupling to the first and second sets of fluxonium hardware components causes the harmonic oscillator to inherit a positive anharmonicity, such that: a frequency corresponding to a transition frequency between a first excited state and a second excited state of energy states enabled by the configuration of the harmonic oscillator is greater than another frequency corresponding to a transition frequency between a ground state and the first excited state of the energy states enabled by the configuration of the harmonic oscillator. Clause 29. The system of any of clauses 21 through 28, wherein the first set of fluxonium hardware components and the second set of fluxonium hardware components respectively comprise: an inductor; a capacitor; and a Josephson junction. Clause 30. The system of any of clauses 21 through 29, wherein: the one or more classical computing devices are further configured to: provide additional drive control instructions to the one or more quantum hardware devices for performance of a calibration protocol; and the one or more quantum hardware devices are further configured to: perform the calibration protocol; and provide results of the calibration protocol to the one or more classical computing devices. Clause 31. The system of clause 30, wherein the one or more classical computing devices are further configured to: determine an updated performance metric, with respect to a performance metric prior to the performance of the calibration protocol, of the one or more quantum hardwaredevices, based at least in part, on the results of the calibration protocol, wherein the performance metric and the updated performance metric comprise a rate of expected coherent errors; adjust one or more parameters of the drive control instructions for the entangling gate based, at least in part, on a change in the updated performance metric with respect to the performance metric prior to the performance of the calibration protocol, wherein the one or more parameters comprise one or more of: a frequency of the microwave pulse; the duration of the entangling gate; or an envelope shape of the microwave pulse; and provide the adjusted drive control instructions to the one or more quantum hardware devices for performance of another entangling gate. Clause 32. A method, comprising: performing an entangling gate between a first fluxonium qubit and a second fluxonium qubit, coupled together using a resonator coupler, wherein said performing comprises: modulating charge of the resonator coupler, using a microwave pulse, for a duration of the entangling gate, wherein said modulating the charge comprises: conditional on the first and second fluxonium qubits being in respective first excited states, driving the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; and driving the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler; and conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintaining the resonator coupler in a ground state of the resonator coupler for the duration of the entangling gate. Clause 33. The method of clause 32, further comprising: subsequent to said performing the entangling gate, providing quantum information pertaining to updated quantum states of the first and the second fluxonium qubits, with respect to initial quantum states ofthe first and the second fluxonium qubits, prior to said performing the entangling gate. Clause 34. The method of clause 32 or clause 33, wherein said modulating the charge of the resonator coupler, using the microwave pulse, for the duration of the entangling gate further comprises: conditional on the first and second fluxonium qubits being in respective first excited states, driving, within the duration of the entangling gate, a phase shift of ^^. Clause 35. The method of any of clauses 32 through 34, further comprising: performing a calibration protocol on quantum hardware components that implement the first and second fluxonium qubits and on hardware components that implement the resonator coupler; determining an updated performance metric, with respect to a performance metric prior to the performance of the calibration protocol, of the first and second fluxonium qubits and of the resonator coupler, based at least in part, on results of the calibration protocol; and adjusting one or more parameters of the entangling gate based, at least in part, on determining that the updated performance metric has changed with respect to the performance metric prior to the performance of the calibration protocol, wherein the one or more parameters comprise one or more of the following: a frequency of the microwave pulse; the duration of the entangling gate; and an envelope shape of the microwave pulse. Clause 36. The method of clause 35, wherein the performance metric and the updated performance metric comprise a rate of expected coherent errors. Clause 37. A quantum hardware device, comprising: a first set of fluxonium hardware components configured to implement a first fluxonium qubit; a second set of fluxonium hardware components, configured to: implement a second fluxonium qubit; and be capacitively coupled to the first set of fluxonium hardware components; a resonator coupler, configured to be respectively coupled to the first set of fluxonium hardware components and to the second set of fluxonium hardware components; anda drive, configured to emit a microwave pulse to cause an entangling gate between the first and second fluxonium qubits to be performed based, at least in part, on charge modulation of the resonator coupler. Clause 38. The quantum hardware device of clause 37, wherein, to perform the entangling gate, the quantum hardware device is further configured to: conditional on the first and second fluxonium qubits being in respective first excited states, drive, using the emitted microwave pulse, the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; drive, using the emitted microwave pulse, the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler; and drive, within a duration of the entangling gate, a phase shift of ^^. Clause 39. The quantum hardware device of clause 37, wherein, to perform the entangling gate, the quantum hardware device is further configured to: conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintain, using the emitted microwave pulse, the resonator coupler in a ground state of the resonator coupler for a duration of the entangling gate. Clause 40. The quantum hardware device of any of clauses 37 through 39, wherein the resonator coupler is a linear oscillator and is fabricated using Tantalum. Illustrative computer system

[0143] FIG.10 is a block diagram illustrating an example classical computing device that may be used in at least some embodiments.

[0144] FIG.10 illustrates such a general-purpose classical computing device 1000 as may be used in any of the embodiments described herein. In the illustrated embodiment, classical computing device 1000 includes one or more processors 1010 coupled to a system memory 1020 (which may comprise both non-volatile and volatile memory modules) via an input / output (I / O) interface 1030. Classical computing device 1000 further includes a network interface 1040 coupled to I / O interface 1030.

[0145] In various embodiments, classical computing device 1000 may be a uniprocessor system including one processor 1010, or a multiprocessor system including several processors 1010 (e.g., two, four, eight, or another suitable number). Processors 1010 may be any suitable processors capable of executing instructions. For example, in various embodiments, processors 1010 may be general-purpose or embedded processors implementing any of a variety of instructionset architectures (ISAs), such as the x86, PowerPC, SPARC, or MIPS ISAs, or any other suitable ISA. In multiprocessor systems, each of processors 1010 may commonly, but not necessarily, implement the same ISA. In some implementations, graphics processing units (GPUs) may be used instead of, or in addition to, conventional processors.

[0146] System memory 1020 may be configured to store instructions and data accessible by processor(s) 1010. In at least some embodiments, the system memory 1020 may comprise both volatile and non-volatile portions; in other embodiments, only volatile memory may be used. In various embodiments, the volatile portion of system memory 1020 may be implemented using any suitable memory technology, such as static random-access memory (SRAM), synchronous dynamic RAM or any other type of memory. For the non-volatile portion of system memory (which may comprise one or more NVDIMMs, for example), in some embodiments flash-based memory devices, including NAND-flash devices, may be used. In at least some embodiments, the non-volatile portion of the system memory may include a power source, such as a supercapacitor or other power storage device (e.g., a battery). In various embodiments, memristor based resistive random access memory (ReRAM), three-dimensional NAND technologies, Ferroelectric RAM, magnetoresistive RAM (MRAM), or any of various types of phase change memory (PCM) may be used at least for the non-volatile portion of system memory. In the illustrated embodiment, program instructions and data implementing one or more desired functions, such as those methods, techniques, and data described above, are shown stored within system memory 1020 as code 1025 and data 1026.

[0147] In some embodiments, I / O interface 1030 may be configured to coordinate I / O traffic between processor 1010, system memory 1020, and any peripheral devices in the device, including network interface 1040 or other peripheral interfaces such as various types of persistent and / or volatile storage devices. In some embodiments, I / O interface 1030 may perform any necessary protocol, timing or other data transformations to convert data signals from one component (e.g., system memory 1020) into a format suitable for use by another component (e.g., processor 1010). In some embodiments, I / O interface 1030 may include support for devices attached through various types of peripheral buses, such as a variant of the Peripheral Component Interconnect (PCI) bus standard or the Universal Serial Bus (USB) standard, for example. In some embodiments, the function of I / O interface 1030 may be split into two or more separate components, such as a north bridge and a south bridge, for example. Also, in some embodiments some or all of the functionality of I / O interface 1030, such as an interface to system memory 1020, may be incorporated directly into processor 1010.

[0148] Network interface 1040 may be configured to allow data to be exchanged between classical computing device 1000 and other devices 1060 attached to a network or networks 1050, such as other computer systems or devices as illustrated in FIG.1 through FIG.9, for example. In various embodiments, network interface 1040 may support communication via any suitable wired or wireless general data networks, such as types of Ethernet network, for example. Additionally, network interface 1040 may support communication via telecommunications / telephony networks such as analog voice networks or digital fiber communications networks, via storage area networks such as Fibre Channel SANs, or via any other suitable type of network and / or protocol.

[0149] In some embodiments, system memory 1020 may represent one embodiment of a computer-accessible medium configured to store at least a subset of program instructions and data used for implementing the methods and apparatus discussed in the context of FIG.1 through FIG. 9. However, in other embodiments, program instructions and / or data may be received, sent or stored upon different types of computer-accessible media. Generally speaking, a computer- accessible medium may include non-transitory storage media or memory media such as magnetic or optical media, e.g., disk or DVD / CD coupled to classical computing device 1000 via I / O interface 1030. A non-transitory computer-accessible storage medium may also include any volatile or non-volatile media such as RAM (e.g., SDRAM, DDR SDRAM, RDRAM, SRAM, etc.), ROM, etc., that may be included in some embodiments of classical computing device 1000 as system memory 1020 or another type of memory. In some embodiments, a plurality of non- transitory computer-readable storage media may collectively store program instructions that when executed on or across one or more processors implement at least a subset of the methods and techniques described above. A computer-accessible medium may further include transmission media or signals such as electrical, electromagnetic, or digital signals, conveyed via a communication medium such as a network and / or a wireless link, such as may be implemented via network interface 1040. Portions or all of multiple classical computing devices such as that illustrated in FIG. 10 may be used to implement the described functionality in various embodiments; for example, software components running on a variety of different devices and servers may collaborate to provide the functionality. In some embodiments, portions of the described functionality may be implemented using storage devices, network devices, or special- purpose computer systems, in addition to or instead of being implemented using general-purpose computer systems. The term “classical computing device”, as used herein, refers to at least all these types of devices, and is not limited to these types of devices.Conclusion

[0150] Various embodiments may further include receiving, sending or storing instructions and / or data implemented in accordance with the foregoing description upon a computer-accessible medium. Generally speaking, a computer-accessible medium may include storage media or memory media such as magnetic or optical media, e.g., disk or DVD / CD-ROM, volatile or non- volatile media such as RAM (e.g. SDRAM, DDR, RDRAM, SRAM, etc.), ROM, etc., as well as transmission media or signals such as electrical, electromagnetic, or digital signals, conveyed via a communication medium such as network and / or a wireless link.

[0151] The various methods as illustrated in the Figures and described herein represent exemplary embodiments of methods. The methods may be implemented in software, hardware, or a combination thereof. The order of method may be changed, and various elements may be added, reordered, combined, omitted, modified, etc.

[0152] Various modifications and changes may be made as would be obvious to a person skilled in the art having the benefit of this disclosure. It is intended to embrace all such modifications and changes and, accordingly, the above description is to be regarded in an illustrative rather than a restrictive sense.

Claims

CLAIMS WHAT IS CLAIMED IS:

1. A system, comprising: one or more classical computing devices, configured to provide drive control instructions to one or more quantum hardware devices for performance of an entangling gate; and the one or more quantum hardware devices comprising: a first set of fluxonium hardware components, configured to implement a first fluxonium qubit; a second set of fluxonium hardware components, configured to: implement a second fluxonium qubit; and be capacitively coupled to the first set of fluxonium hardware components; and a resonator coupler, wherein resonator coupler is configured to be capacitively coupled to the first set of fluxonium hardware components and to the second set of fluxonium hardware components, wherein the one or more quantum hardware devices are further configured to: receive the drive control instructions; and perform the entangling gate between the first and second fluxonium qubits based, at least in part, on charge modulation of the resonator coupler.

2. The system of claim 1, wherein the entangling gate between the first and second fluxonium qubits is a Controlled-^^ gate.

3. The system of claim 1, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to: conditional on the first and second fluxonium qubits being in respective first excited states, drive the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; and drive the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler.

4. The system of claim 3, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to: conditional on the first and second fluxonium qubits being in respective first excited states, cause, within a duration of the entangling gate, a phase shift of ^^.

5. The system of claim 1, wherein, to perform the entangling gate, the one or more quantum hardware devices are configured to: conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintain the resonator coupler in a ground state of the resonator coupler for a duration of the entangling gate.

6. The system of any one of claims 1 through 5, wherein the resonator coupler is fabricated using Tantalum.

7. The system of any one of claims 1 through 5, wherein the resonator coupler is a harmonic oscillator.

8. The system of claim 7, wherein the capacitive coupling to the first and second sets of fluxonium hardware components causes the harmonic oscillator to inherit a positive anharmonicity, such that: a frequency corresponding to a transition frequency between a first excited state and a second excited state of energy states enabled by the configuration of the harmonic oscillator is greater than another frequency corresponding to a transition frequency between a ground state and the first excited state of the energy states enabled by the configuration of the harmonic oscillator.

9. The system of any one of claims 1 through 5, wherein the first set of fluxonium hardware components and the second set of fluxonium hardware components respectively comprise: an inductor; a capacitor; and a Josephson junction.

10. The system of any one of claims 1 through 5, wherein: the one or more classical computing devices are further configured to: provide additional drive control instructions to the one or more quantum hardware devices for performance of a calibration protocol; and the one or more quantum hardware devices are further configured to: perform the calibration protocol; and provide results of the calibration protocol to the one or more classical computing devices.

11. The system of claim 10, wherein the one or more classical computing devices are further configured to: determine an updated performance metric, with respect to a performance metric prior to the performance of the calibration protocol, of the one or more quantum hardware devices, based at least in part, on the results of the calibration protocol, wherein the performance metric and the updated performance metric comprise a rate of expected coherent errors; adjust one or more parameters of the drive control instructions for the entangling gate based, at least in part, on a change in the updated performance metric with respect to the performance metric prior to the performance of the calibration protocol, wherein the one or more parameters comprise one or more of: a frequency of the microwave pulse; the duration of the entangling gate; or an envelope shape of the microwave pulse; and provide the adjusted drive control instructions to the one or more quantum hardware devices for performance of another entangling gate.

12. A method, comprising: performing an entangling gate between a first fluxonium qubit and a second fluxonium qubit, coupled together using a resonator coupler, wherein the first fluxonium qubit and the second fluxonium qubit are implemented using a first set of fluxonium hardware components that are capacitively coupled to a second set of fluxonium hardware components.

13. The method of claim 12, further comprising: subsequent to said performing the entangling gate, providing quantum information pertaining to updated quantum states of the first and the second fluxonium qubits, with respect to initial quantum states of the first and the second fluxonium qubits, prior to said performing the entangling gate.

14. The method of claim 12 or claim 13, wherein said performing the entangling gate comprises: modulating charge of the resonator coupler, using a microwave pulse, for a duration of the entangling gate, wherein said modulating the charge comprises: conditional on the first and second fluxonium qubits being in respective first excited states, driving the resonator coupler from a ground state of the resonator coupler to a first excited state of the resonator coupler; and driving the resonator coupler back from the first excited state of the resonator coupler to the ground state of the resonator coupler; and conditional on one or both of the first and second fluxonium qubits being in respective ground states, maintaining the resonator coupler in a ground state of the resonator coupler for the duration of the entangling gate.

15. The method of claim 14, wherein said modulating the charge of the resonator coupler, using the microwave pulse, for the duration of the entangling gate further comprises: conditional on the first and second fluxonium qubits being in respective first excited states, driving, within the duration of the entangling gate, a phase shift of ^^.

Images