Asymmetric kerr parametric oscillator for quantum information processing

Abstract

The inventors have developed techniques to facilitate the operation of a quantum oscillator with increased bit-flip lifetimes by inducing an energetic asymmetry between qubit states. The technology may be used for the storage of quantum information or for providing a system for quantum information processing with longer bit flip lifetimes. The technology includes a qubit along with corresponding drives for controlling the energetic barrier and energetic asymmetry between the energetic potentials for each qubit state.

Description

Attorney Docket No.: Y0087.70177WO00ASYMMETRIC KERR PARAMETRIC OSCILLATOR FOR QUANTUM INFORMATION PROCESSINGCROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit under 35 U. S. C. § 119(e) of U. S. Provisional Patent Application No. 63 / 673,594 filed July 19, 2024, titled “ASYMMETRIC KERR PARAMETRIC OSCILLATOR FOR QUANTUM INFORMATION PROCESSING,” which is incorporated by reference herein in its entirety.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] This invention was made with government support under Grant No. W911NF-23- 1-0051 awarded by the US Army Research Office. The government has certain rights in the invention.BACKGROUND

[0003] This disclosure relates to quantum information processing techniques. Quantum information processing techniques perform computation by manipulating one or more quantum objects. These techniques are sometimes referred to as “quantum computing.” In order to perform computations, a quantum information processor utilizes quantum objects to reliably store, retrieve, and / or manipulate information. According to some quantum information processing approaches, a quantum analogue to the classical computing “bit” (being equal to 1 or 0) has been developed, which is referred to as a quantum bit, or “qubit.” A qubit can be composed of any quantum system that has two distinct states (which may be thought of as 1 and 0 states) but also has the special property that the system can be placed into a quantum superposition and thereby potentially exist in both of those states at once.SUMMARY

[0004] Some embodiments provide for a quantum information system comprising: one or more quantum elements, each quantum element of the plurality of quantum elements being configured with one or more quantum states in a potential energy double well; a first drive configured toAttorney Docket No.: Y0087.70177WO00control an energetic barrier of the potential energy well such that the potential energy well is a potential energy double well; and a second drive configured to control an energetic symmetry of the potential energy double well.

[0005] In some embodiments, the plurality of quantum elements comprises a plurality of superconducting nonlinear asymmetric inductive elements.

[0006] In some embodiments, the plurality of quantum elements comprises a plurality of superconducting nonlinear Josephson elements.

[0007] In some embodiments, the plurality of quantum elements comprises a plurality of ion traps.

[0008] In some embodiments, the plurality of quantum elements comprises a plurality of optical potentials configured with cold atoms.

[0009] In some embodiments, the first drive is a first microwave drive configured to operate at a first amplitude and a first frequency, and the second drive is a second microwave drive configured to operate at a second amplitude and a second frequency.

[0010] In some embodiments, the first amplitude and the first frequency is different than the second amplitude and the second frequency.

[0011] In some embodiments, the first amplitude is between 0.01 to 50 times the Kerr coefficient.

[0012] In some embodiments, the second amplitude is between 0.01 to 50 times the Kerr coefficient.

[0013] Some embodiments provide for a method of storing quantum information in a quantum system that includes one or more quantum elements, each quantum element of the plurality of quantum elements being configured with one or more quantum states in a potential energy well, the method comprising: driving the quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well; initializing an initial quantum state of a plurality of interacting quantum states; and subsequent to initializing the initial quantum state, driving the quantum system using a second drive, configured to control an energetic symmetry of the potential energy double well such that the potential energy double well is asymmetric.Attorney Docket No.: Y0087.70177WO00

[0014] In some embodiments, the method further comprises, subsequent to initializing the initial quantum state, driving the quantum system using the second drive such that the potential energy double well is symmetric; and detecting an electronic output indicative of a final quantum state of the plurality of interacting quantum states.

[0015] In some embodiments, initializing the quantum state comprises transmitting a microwave pulse through the plurality of quantum elements.

[0016] In some embodiments, driving the quantum system using the first drive comprises operating a microwave drive at a first amplitude.

[0017] In some embodiments, the first amplitude is between 0.01 to 50 times the Kerr coefficient.

[0018] In some embodiments, driving the quantum system using the second drive comprises operating the microwave drive at a second amplitude.

[0019] In some embodiments, the second amplitude is between 0.01 to 50 times the Kerr coefficient.

[0020] Some embodiments provide for a method of operating a quantum system for quantum information processing, wherein the quantum system includes a one or more quantum elements, each quantum element of the plurality of quantum elements being configured with one or more quantum states in a potential energy well, the method comprising: driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well; driving the quantum system using a second drive, configured to control an energetic symmetry of the potential energy double well such that the potential energy double well is asymmetric; subsequent to driving the quantum system using the second drive, initializing an initial quantum state of a plurality of interacting quantum states.

[0021] In some embodiments, the method further comprises detecting an electronic output indicative of a final quantum state of the plurality of interacting quantum states.

[0022] In some embodiments, driving the quantum system using first drive comprises operating a microwave drive at a first amplitude.

[0023] In some embodiments, the first amplitude is between 0.01 to 50 times the Kerr coefficient.Attorney Docket No.: Y0087.70177WO00

[0024] In some embodiments, driving the quantum system using the second drive comprises operating the microwave drive at a second amplitude.

[0025] In some embodiments, the second amplitude is between 1 to 2.5 times the Kerr coefficient.BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The accompanying drawings may not be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. In the drawings:

[0027] FIG. 1 illustrates a diagram of a single double well system 100, in accordance with some embodiments of the technology described herein.

[0028] FIG. 2A illustrates a diagram of an optical trap quantum information processing system, in accordance with some embodiments of the technology described herein.

[0029] FIG. 2B illustrates a diagram of a double optical trap for quantum information processing system, in accordance with some embodiments of the technology described herein.

[0030] FIG. 2C illustrates a diagram of a SNAIL quantum information processing system, according to some embodiments of the technology described herein.

[0031] FIG. 3A illustrates a process 300 for storing quantum information in a quantum system, according to some embodiments of the technology described herein.

[0032] FIG. 3B illustrates a process 310 for storing quantum information in a quantum system, according to some embodiments of the technology described herein.

[0033] FIG. 4A shows the Bloch sphere representation of the qubit.

[0034] FIG. 4B illustrates a process 400 for performing a gate operation using an asymmetric potential energy double well system in connection with quantum information processing, in accordance with some embodiments of the technology described herein.

[0035] FIG. 5A illustrates a rendering of a half-aluminum, half-copper sample package containing two Sapphire chips for use in quantum information processing, in accordance with some embodiments of the technology described herein.Attorney Docket No.: Y0087.70177WO00

[0036] FIG. 5B illustrates a magnification of a Sapphire chip, such as the Sapphire chips included in the package shown in FIG. 5 A, in accordance with some embodiments of the technology described herein.

[0037] FIG. 5C illustrates a schematic of the SNAIL-transmon, in accordance with some embodiments of the technology described herein.

[0038] FIG. 5D illustrates a scanning electron micrograph of a two-SNAIL array, in accordance with some embodiments of the technology described herein.

[0039] FIG. 6A illustrates the semiclassical double well potential energy V(x), in accordance with some embodiments of the technology described herein.

[0040] FIG. 6B illustrates the spectrum of the parametric oscillator Hamiltonian as a function of £2 controlling the barrier height, in accordance with some embodiments of the technology described herein.

[0041] FIG. 6C illustrates the semiclassical double well potential energy for the asymmetric case and the Hamiltonian energies, in accordance with some embodiments of the technology described herein.

[0042] FIG. 6D illustrates the spectrum as a function of ei controlling the asymmetry, in accordance with some embodiments of the technology described herein.

[0043] FIG. 7A illustrates the experimental population dynamics when initializing the system in the shallower well, in accordance with some embodiments of the technology described herein.

[0044] FIG. 7B illustrates theory calculation modeling the experiment shown in FIG. 7A.

[0045] FIG. 8A illustrates a semiclassical Hamiltonian prediction for the resonance conditions (parabolic dashed lines) and Bohr’s quantization condition for n and m allowed quantum orbits in the small and large lemniscate’s lobes.

[0046] FIG. 8B illustrates a measurement of activation time T as a function of C2 / K and el / K, in accordance with some embodiments of the technology described herein.

[0047] FIG. 8C illustrates a theory prediction from a Lindbladian model including single photon loss and gain corresponding to the experimental results shown in FIG. 8B.Attorney Docket No.: Y0087.70177WO00

[0048] FIG. 9A illustrates experimental data for the activation time T as a function of e K, effectively controlling the barrier height, in accordance with some embodiments of the technology described herein.

[0049] FIG. 9B illustrates theoretical modeling for the activation time T as a function of e / K, effectively controlling the barrier height, in accordance with the experimental data shown in FIG.9A.

[0050] FIG. 10 illustrates the response to a dispersive readout to varying the probe drive amplitude, in accordance with some embodiments of the technology described herein.

[0051] FIG. 11 A illustrates time-resolved quantum coherent oscillation as a function of relative phase <[) between the squeezing and linear drives, in accordance with some embodiments of the technology described herein.

[0052] FIG. 11B illustrates a linear drive with relative phase of <[) = 90°, in accordance with some embodiments of the technology described herein.

[0053] FIG. 11C illustrates a similar experiment as depicted in Fig. 7A, but with a Rabi drive phase of 180° instead of 0°.

[0054] FIG. 12A illustrates time-resolved quantum coherent Rabi-like oscillations as a function of squeezing amplitude, in accordance with some embodiments of the technology described herein.

[0055] FIG. 12B illustrates the photon number as a function of applied voltage for the digital control of the squeezing drive, in accordance with some embodiments of the technology described herein.

[0056] FIG. 13 A illustrates pulse sequence for the determination of the activation time, in accordance with some embodiments of the technology described herein.

[0057] FIG. 13B illustrates a decay of a coherent state initiated on the shallower well for different asymmetry values ei / K, in accordance with some embodiments of the technology described herein.Attorney Docket No.: Y0087.70177WO00DETAILED DESCRIPTIONI. Introduction

[0058] The inventors have developed techniques to facilitate the operation of a quantum oscillator with increased bit-flip lifetimes by inducing an energetic asymmetry between qubit states. The technology may be used for the storage of quantum information or for providing a system for quantum information processing with longer bit flip lifetimes. The technology includes a qubit along with corresponding drives for controlling the energetic barrier and energetic asymmetry between the energetic potentials for each qubit state.

[0059] A quantum system that can be manipulated and measured tends to interact with uncontrolled degrees of freedom in its environment leading to decoherence. This presents a challenge to the experimental investigation of quantum effects in particular to the field of quantum computing, where qubits must remain coherent while operations are performed. Most noisy environments are only locally correlated and thus cannot decohere quantum information encoded in a non-local manner.

[0060] One choice for non-locally encoding a qubit into the phase space of an oscillator is in the superpositions of macroscopically distinct coherent states, which are sometimes referred to as Schrodinger cat states, or simply “cat states.” Using the cat states, one bit of quantum information (e.g., the state of a physical qubit) may be represented by a multi-component coherent state of a bosonic oscillator. Bosonic oscillators have useful properties such as being a closed qubit space under the action of the annihilation operator; and deterministically evolving in a predictable manner for the state, when in the absence of the boson losses.

[0061] Implementations of cat states in a high-Q resonator are very promising for hardware efficient universal quantum computing. One way that these states have been prepared, for instance, is in a Kerr-nonlinear resonator. A Kerr-nonlinear resonator is a type of nonlinear resonator. Nonlinear resonators are a type of resonator (i.e., systems with resonant frequencies where the oscillating fields, such as electro-magnetic fields, constructively interfere) whose effective restoring force has a nonlinear dependence on the amplitude of the oscillation. In Kerr-nonlinear resonators, the nonlinearity of the system can be controlled through the Kerr effect.Attorney Docket No.: Y0087.70177WO00The Kerr-effect is a four-wave mixing process that results in a frequency of the system that changes proportionally with the power of the driving field.

[0062] Kerr-nonlinear resonators exhibiting bifurcation can be used to generate squeezed radiation, can be used for quantum-limited amplification, and have been proposed for quantum logic. However, because of their sensitivity to undesirable interactions and photon loss, high-fidelity preparation and manipulation of these states is challenging. In particular, the microwave drives necessary to drive such a nonlinear resonator can have undesirable effects. Sufficiently strong drives, for example, may produce anomalous state transitions and / or may degrade gate operation fidelities.

[0063] The inventors have recognized and appreciated techniques to design and operate a quantum oscillator. Harmonic oscillators have equally spaced energy levels, leading to a high probability of transitions between states and increasingly the likelihood of errors for quantum information systems. Conventionally, to improve the performance of quantum oscillators for quantum information processing, quantum oscillators have been designed with a bare nonlinearity to increase the spacing between adjacent states of the quantum oscillator. The inventors have appreciated that, counter to conventional design, quantum oscillators may be designed with less bare nonlinearity but may produce even higher error tolerance when driven with a drive to break the energetic symmetry of the oscillator.

[0064] The inventors have further recognized and appreciated that quantum computing promises disruptive applications in finance, cryptography, material design, etc. However, current qubits based on quantum oscillators are prone to errors. One approach to making quantum computing robust against such errors is to configure a collection of qubits into a quantum error correcting code and operate them as a single unit with fewer errors. Cat-qubits, of which Kerr-cat qubits are one variant, promise to reduce the number of qubits necessary by a factor of 60 to reach any given error tolerance. The Kerr-cat qubit is stabilized in a nonlinear oscillator driven into parametric resonance. The parametric resonance creates a squeezing term in the Hamiltonian that in competition with a Kerr nonlinearity stabilizes the ground state manifold used to encode a qubit. This Kerr-cat qubit is composed of the superposition of two distinct semiclassical coherent states. When having equal weights, these superpositions are typically called “Schrodinger catAttorney Docket No.: Y0087.70177WO00states.” The bit-flip lifetime, the typical timescale for a logical error connecting one of the coherent states with the other, is dominated by resonant tunneling mediated by leakage into the excited states. Therefore, the bit-flip lifetime, as a function of the parametric drive amplitude, is modulated in a nontrivial way by resonances in the system. The quantum gates connecting the different Schrodinger cat states in this system are made possible by a perturbative linear (e.g., nonparametric) drive. The perturbative drive and the squeezing drive are operated in phase.

[0065] Accordingly, Kerr-cat qubits provide an avenue for the improvement of quantum information processing systems. However, one of the limiting factors in Kerr-cat qubits is the bitflip lifetime between states of the Kerr-cat qubit. The bit-flip lifetime of such systems is limited by resonant quantum tunneling mediated by leakage into the excited states. By breaking the resonance condition with a drive producing a strong asymmetry, the tunneling rate may be decreased; therefore, the bit-flip lifetime may be increased.

[0066] Accordingly, the inventors have developed systems and methods to improve the performance quantum information processing by controlling the energetic asymmetry in the system to reduce errors. The energetic asymmetry in the system may be controlled by using a second drive that drives the system beyond the perturbative regime. When driven beyond the perturbative regime, the drive breaks the Hamiltonian resonance conditions modulating the lifetime. Therefore, the bit-flip lifetime increases substantially. This symmetry breaking depends on the amplitude of the strong linear drive and on its phase relative to the squeezing drive. The strong linear drive and the squeezing drive are operated in phase. The inventors have recognized and appreciated that the broken symmetry reduces the effective energetic cost of errors, yet surprisingly it reduces their rate. In a classical scenario reducing the energetic cost of errors would reduce the lifetime, not increase it, by making errors more likely. Since the available states in this oscillator are quantized, this argument is invalid. As detailed below for a quantum oscillator, this allows an increase of the bit-flip lifetime in a Kerr-cat qubit. Despite the use of a drive beyond the perturbative regime to break the symmetry of the system, spurious effects introduced by the drive do not deteriorate the performance of the system. Accordingly, this technique provides an alternative to strong squeezing drives that are known to cause the breakdown of the model and degradation of the bit-flip lifetime. Therefore, the systems andAttorney Docket No.: Y0087.70177WO00methods described herein reduce the number of qubits needed to suppress errors for quantum information processing applications. Additionally, the systems and methods described herein modify the error model for the Kerr-cat qubit, that changes the error model from a nonpolarizing channel to a slightly polarized channel.

[0067] According to some aspects of the systems and methods developed herein, the quantum information system includes a first drive configured to control the energetic barrier between two potential energy wells of a quantum system and a second drive configured to control an energetic asymmetry between the two potential energy wells.

[0068] The first amplitude is configured to control the energetic barrier of the double well potential. Accordingly, the first amplitude is configured to generate states for the double well potential that decrease the probability of a state change occurring between the wells. For example, the state change may be a bit flip of the transmon. In some embodiments, the first amplitude is between 2 to 8 times the Kerr coefficient, 1.5 to 12 times the Kerr coefficient, 0.5 to 15 times the Kerr coefficient, or 0.01 to 50 times the Kerr coefficient.

[0069] The second amplitude is configured to control the asymmetry of the double well potential. A double well potential includes two minima. In an asymmetric double well, the two minima have different energetic depths. Accordingly, the second amplitude is configured to control the energetic difference between the different energetic depths. In some embodiments, the second amplitude is between 1 to 2 times the Kerr coefficient, 1 to 2.5 times the Kerr coefficient, between 0.5 to 5 times the Kerr coefficient, or 0.01 to 50 times the Kerr coefficient.

[0070] According to some aspects of the system and methods described herein, the quantum information system may be used for storing quantum information as a quantum memory unit and / or may be used for quantum information processing. To store quantum information, a quantum state is initialized (e.g., initializing a qubit to an initial state) and then subsequent to initializing the quantum state, the second drive is used to induce an asymmetry between the minima of the double well potential.

[0071] According to some aspects of the system and methods described herein, the quantum information system may be used to process quantum information, and the second drive is applied to induce the asymmetry between the first energy well and the second energy well. Prior toAttorney Docket No.: Y0087.70177WO00inducing the asymmetry between the first well potential and the second well potential, a quantum state is initialized.

[0072] Initialization of the quantum state may occur naturally once the system is driven to induce the potential energy double well. For example, when the desired initialization state is the ground state, the system may naturally populate the ground state, absent an exciting field.Therefore, initialization of the quantum state may include a waiting time after inducing the potential energy double well and / or inducing the asymmetry, such that the system is given time to decay into the ground state. The time to decay may be determined from the lifetime of the energetic states of the system.

[0073] Initialization of the quantum state may also occur with a driving pulse. Depending on the desired initial state of the system, different driving pulses may be used. For example, if the desired initial state of the system is an excited state of the potential energy double well, then a driving pulse tuned to the energy difference between the excited state and the ground state may be used to initialize the system in the excited state.

[0074] Where the desired initial state of the system is the ground state, initialization of the quantum state may involve a driving pulse to couple the system to a dissipative loss channel for coupling excess energy out of the system. Thus, the driving pulse induces decay of the system from potential excited states into the ground state. Alternatively, or additionally, a driving pulse may be used to induce a transfer into or out of the qubit of the system to cause the system to transition to the ground state.

[0075] In some embodiments, the pulses used to initialize the system are microwave drives with frequencies tuned to induce a transition between states of the potential energy double well. In some embodiments, the pulses used to initialize the system are microwave drives with frequencies tuned to induce a coupling between the system and a readout cavity or a different qubit.

[0076] In some embodiments, optical pulses generated by lasers may be used to initialize the system, such as lasers pulses being tuned to frequencies that correspond to the transition between states of a potential energy well.Attorney Docket No.: Y0087.70177WO00II.A Quantum Information Systems

[0077] FIG. 1 illustrates a diagram of a single double well system 100, in accordance with some embodiments of the technology described herein. Quantum information system 100 includes multiple potential minima 102 and drive system 104 for driving the energetic relationship between the multiple potential minima 102.

[0078] The multiple potential minima 102 are configured as one or more qubits with quantum elements that interact with one another to create states with quantum behavior (e.g., states which may be in a superposition with one another). The multiple potential minima 102 include a first potential minimum 106 and a second potential minimum 108. The first potential minimum 106 and the second potential minimum 108 are separated by an energetic barrier 118. The first potential minimum 106 is described by a first energetic depth 120 and the second potential minimum 108 is described by a second energetic depth 122. Although depicted as separate potential energy wells, the potential minimum may have any suitable form. In some embodiments, the potential minimum may be described by a quantum oscillator system.

[0079] According to some embodiments, the quantum oscillator may include a plurality of superconducting nonlinear asymmetric inductive elements, also known as “SNAILs.” SNAILs are described in International Application No. PCT / US2018 / 064922 filed on 11 December 2018, titled “Superconducting Nonlinear Asymmetric Inductive Element and Related Systems and Methods,” which is hereby incorporated by reference in its entirety. As discussed above, in contrast to a conventional Kerr resonator - which may contain a single nonlinear element - the quantum oscillator described herein may include multiple nonlinear elements (such as SNAILs). Forming a resonator from multiple nonlinear elements may compromise or otherwise reduce the bare nonlinearity of the resonator, which runs contrary to conventional approaches to quantum resonators based quantum information processing. In some embodiments, the quantum oscillator may include 2, 3, 4, 5, or more SNAILs, which may be coupled together in series.

[0080] According to some embodiments, the quantum oscillator may include other quantum elements, as aspects of the technology described herein is not limited in this respect. For example, the multiple potential minima may be achieved in superconducting qubits, trapped ionAttorney Docket No.: Y0087.70177WO00qubits, neutral atom qubits, quantum dot qubits, topological qubits, diamond nitrogen- vacancy center qubits, nuclear magnetic resonance qubits, photonic qubits, or other qubits.

[0081] The drive system 104 includes multiple drives for controlling the energetic relationship between the multiple potential minima. Drive system 104 includes a first drive 110 that is configured to generate a first drive amplitude 112 and a second drive 114 that is configured to generate a second drive amplitude 116. In some embodiments, the first drive 110 and the second drive 114 may be configured to generate sinusoidal microwave signals with adjustable amplitude and phase. In some embodiments, the first drive 110 and the second drive 114 are microwave signal generators (e.g., Agilent N5183A MXG Microwave Analog Signal Generators or other suitable commercial microwave generators).

[0082] The first drive is used to control the energetic spacing 118 between potential minimum 106 and 108. The second drive 114 is used to control the energetic symmetry between the first potential minimum 106 and the second potential minimum 108 such that the first energetic depth 120 is different than the second energetic depth 122. The drive system may also include a measurement device such as a read-out cavity and cavity state detector or other suitable measurement device.

[0083] In some embodiments, the system may include external ports coupled to waveguides. The waveguides carry signals to a SNAIL oscillator system, as described herein. In some embodiments, the waveguides are directly coupled to one or more SNAILs. In some embodiments, an antenna pad may be configured to transmit the microwave signals to the SNAIL.

[0084] According to some embodiments, a Kerr resonator may be driven at a frequency ωpthat is twice the resonator frequency ωq(i.e., ωp= 2ωq). Arranging the Kerr resonator drive to have this frequency may require careful calibration, since a deviation of the drive frequency away from this frequency may inhibit the three-way mixing process that the resonator relies upon to produce coherent states. The Kerr resonator is described in International Application No.PCT / US2019 / 039945, filed on 28 June 2019, titled “Quantum Information Processing with an Asymmetric Error Channel,” which is hereby incorporated by reference in its entirety.Attorney Docket No.: Y0087.70177WO00

[0085] FIG. 2A illustrates a diagram of an optical trap quantum information processing system 200, in accordance with some embodiments of the technology described herein. Optical trap quantum information processing 200 includes an optical trap 202 for confining an ion or cold atom 206. The trapped ion or cold atom 206 is modulated by drive 210 at twice its oscillation frequency to create a bifurcation with two states that are described by a double well energetic barrier. Accordingly, the energetic asymmetry between the bifurcated states may involve the application of one or more additional drives to change the electronic states of one or more of the ions or cold atoms 206 (e.g., by causing electronic excitations or perturbing the electronic states of the ion or cold atom).

[0086] FIG. 2B illustrates a diagram of a double optical trap for quantum information processing system 212, in accordance with some embodiments of the technology described herein. The double optical trap for quantum information processing system 212 includes a first optical drive 216 and a second optical drive 218 for trapping a cold atom 214 or ion. The first and second optical drives are configured to generate potential energy wells within which ions or cold atoms may accumulate. Additionally, or alternatively, radiofrequency (RF) traps may be used to trap ions. RF ion traps may include electrodes driven by RF signals to confine ions within a volume between the electrodes. The spacing between the first optical drive 216 and the second optical drive 218 may control the energetic barrier between the generated potential energy wells. The optical power of the respective lasers may control the energetic depth of the respective wells. For example, as shown in FIG. 2B, optical drive 216 is configured with a larger optical power than optical drive 218. Accordingly, the potential energy well generated by optical drive 216 is deeper than the potential energy well generated by optical drive 218.

[0087] FIG. 2C illustrates a diagram of a SNAIL quantum information processing system 220, according to some embodiments of the technology described herein. SNAIL quantum information processing system 220 includes first SNAIL 226, second SNAIL 228, first node 222, and second node 224. The first node 222 and second node 224 couple the SNAIL array to a readout resonator. Additionally, microwave driving fields can be transmitted to the SNAIL array through the first node 222 and second node 224. A first driving field may be transmitted to the SNAIL array to control an energy barrier in the well potential, generating a dual well potential. AAttorney Docket No.: Y0087.70177WO00second driving field may be transmitted to the SNAIL array to control an energetic asymmetry between the wells. The first and second driving fields are described in further detail below in section III.

[0088] FIG. 3A illustrates a process 300 for storing quantum information in a quantum system, according to some embodiments of the technology described herein. The quantum information system may be any quantum system that includes one or more quantum elements with one or more quantum states in a potential energy well, such as the quantum elements described herein in connection with FIGs. 2A-2C.

[0089] Process 300 begins at act 302 by driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well. In the absence of the first drive (e.g., an amplitude of zero), the potential energy well is a single potential energy well. When driven with a sufficient amplitude and frequency, the first drive modifies the potential energy of the system such that the potential energy is described by a potential energy double well. The two wells of the potential energy wells are separated by an energetic barrier, as shown in FIG. 6A. At larger amplitudes, the energetic barrier between the two wells is larger.

[0090] In embodiments using SNAILs, such as the embodiment described in 2C above and 5A-5D below, the first drive is a microwave drive having an amplitude, frequency, and phase. The amplitude and frequency may be quantified relative to other parameters of the system. For example, the amplitude may be quantified relative to the Kerr coefficient of the one or more quantum elements of the quantum information system. As another example, the frequency may be quantified relative to the frequency of the Kerr parametric oscillator.

[0091] Accordingly, in some embodiments, the amplitude of the first drive may be between 0.01 to 50 times the Kerr coefficient, between 1.5 to 20 times the Kerr coefficient, or between 4 to 12 times the Kerr coefficient.

[0092] Similarly, in some embodiments, the frequency of the first drive is selected such as to facilitate a three- wave mixing process. For example, the frequency of the first drive may be twice the frequency of the mode of the Kerr parametric oscillator.

[0093] In some embodiments, the drive phase of the first drive may be configured to be zero.Attorney Docket No.: Y0087.70177WO00

[0094] Process 300 continues at act 304 by initializing an initial quantum state of a plurality of interacting quantum states. Initializing the quantum state puts the system in a state of the double well potential. As shown in FIG. 6A, double well potential systems have a manifold of states of which the higher energy states are above the energetic barrier between the two wells. The initialized state may be any of the states lower in energy than the top of the energetic barrier. The number of available states will depend on the type of system and the drives used.

[0095] In some embodiments, initializing the quantum state is achieved by transmitting a microwave pulse through the plurality of quantum elements.

[0096] In some embodiments, additional operations will be performed on the initialized state. For example, the initialized state may interact with other qubits and / or may be driven into a different state or superposition of states.

[0097] Process 300 continues at act 306 by, subsequent to initializing the initial quantum state, driving the quantum system using a second drive to control an energetic asymmetry of the potential energy double well such that the potential energy double well is asymmetric. The second drive adjusts the asymmetry of the potential energy double well. Accordingly, larger amplitudes may result in a larger asymmetry. In some embodiments, the second drive is a second microwave drive. Once appropriately tuned, the asymmetric system is protected from errors, as explained further below in connection with FIGs. 7A-8C. Accordingly, the state of the asymmetric system is considered stored, as it is protected from errors.

[0098] In some embodiments, the amplitude of the second drive may be between 0.01 to 50 times the Kerr coefficient, between 1.5 to 20 times the Kerr coefficient, or between 4 to 12 times the Kerr coefficient. The specific amplitude of the second drive should be selected based in part of the amplitude of the first drive, as explained further below in connection with FIGs. 8A-8C.

[0099] In some embodiments, the frequency of the second drive is set to the frequency of the mode of the Kerr parametric oscillator.

[0100] In some embodiments, the phase of the second drive is set to zero. In some embodiments, the phase of the second drive is set to n.

[0101] After act 306, process 300 concludes. Following the conclusion of process 300, the second drive may be modified such that the system symmetrizes. For example, the amplitudeAttorney Docket No.: Y0087.70177WO00of the second drive may be reduced such that the potential energy double well is symmetric. In this way, the stored state may be retrieved and may once again interact with states of the symmetric potential energy double well that were suppressed from interacting in the asymmetric configuration. Additionally, following the conclusion of process 300, an electronic output may be detected that is indicative of a final quantum state of the plurality of interacting quantum states.

[0102] FIG. 3B illustrates a process 310 for storing quantum information in a quantum system, according to some embodiments of the technology described herein. The quantum information system may the same quantum system as described above in connection with FIG.3 A. Relative to process 300, process 310 initiates the quantum state after the second drive is initiated and the system is an asymmetric potential energy double well.

[0103] Process 310 begins at act 312 by driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well. Act 312 may be implemented in the same way as act 302 described above.

[0104] Process 310 continues at act 314 by, subsequent to initializing the initial quantum state, driving the quantum system using a second drive to control an energetic asymmetry of the potential energy double well such that the potential energy double well is asymmetric. Act 314 may be implemented using the same parameters as act 306 described above.

[0105] Process 310 continues at act 316 by initializing an initial quantum state of the asymmetric potential energy double well. As described below, the asymmetry of the potential energy double well protects the initialized state from errors. Accordingly, the initialized state may be kept as a stored state or may be acted upon through the application of additional drive signals.

[0106] After act 316, process 310 concludes. Following the conclusion of process 310 the stored state may be retrieved through a readout process and subsequently used in other operations.II. B Quantum GatesAttorney Docket No.: Y0087.70177WO00 o o'' O O r rHH

[0107] In quantum information processing, gates are typically described in terms of their \ \NNNN ( ( ( ( jjjjjj < < < effects on basis states. A single qubit has two basis states, which are customarily d + T T +enoted as |0) and |1). Having two qubits creates four different combinations: |00), |01), 110) T T T T, 111 ).o o''

[0108] Like its symmetric counterpart, an asymmetric Kerr param O O r rHHetric oscillator encodes a single qubit. Its |0) state corresponds to when the system is in the right well of the energy double well. Its |1) state corresponds to when the system is in the left well of the energy double well. FIG. 4A shows the Bloch sphere representation of the qubit. The expressions for the labeled cardinal states are approximate, and a schematic representation of the Wigner function is associated with each labeled state.

[0109] Asymmetry may be utilized to enhance the fidelity of quantum gates, as described herein. In some embodiments, the operation of rotational Z gates, X gates, X(π / 2) gates, rotational ZZ gates, and CNOT gates may be improved through the control of the asymmetry of the quantum system. General descriptions of these gates are included in Table 1 below.Table 1Gate Operates on Description Truth table applies a different phase factor |0)e+^ / 2|0) Z(0) single qubitdepending on which state it is |1> e~ie / 2\l)|o> |i>X single qubit implements a bit flip|i> |o>|0> (|0> - i|l» / V2 X(π / 2) single qubit creates superposition states_ |l) ^ (|0) + i|l)) / V2 _ applies a different phase factorZZ(0) two qubits depending on whether the states ofthe two qubits agree|00) ^ |00) applies a bit-flip to the second qubit |01) ^ |01) CNOT two qubitsonly if the first qubit is in |1) |10) ^ |ll)|ll) ^ |10)

[0110] Prior to implementing the above gate operations, the control drives are configured to store quantum information, as described above in connection with FIGs. 3A-3B.Attorney Docket No.: Y0087.70177WO00

[0111] FIG. 4B illustrates a process 400 for performing a gate operation using an asymmetric potential energy double well system for quantum information processing, in accordance with some embodiments of the technology described herein. Acts 402-406 may be implemented in the same way as acts 312-316 of FIG. 3B.

[0112] Process 400 begins at act 402 by driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well. In some embodiments, the quantum information processing system includes a Kerr parametric oscillator. Accordingly, the first drive is driven with a frequency equal to twice that of the Kerr parametric oscillator. The amplitude of first drive is e2, which may have any value described herein. The phase of the first drive is set to zero.

[0113] Process 400 continues at act 404 by driving the quantum system using a second drive to control an energetic asymmetry of the potential energy double well such that the potential energy double well is asymmetric. In some embodiments, the quantum information processing system includes a Kerr parametric oscillator. Accordingly, the second drive is driven with a frequency equal to that of the Kerr parametric oscillator. The amplitude of the second drive is approximately 1.2 K where K is the Kerr coefficient of the system. The phase of the second drive is 0 or 7t.

[0114] Process 400 continues at act 406 by initializing the initial quantum state of the asymmetric potential energy double well.

[0115] After initializing the initial quantum state, FIG. 4B continues at act 408 by varying the first drive and / or the second drive to perform a gate operation on the initial quantum state. For the ZZ rotation gate, which is a two-qubit gate, an additional linear coupling drive is applied. For the CNOT gate, which is also a two-qubit gate, additional linear coupling and nonlinear coupling drives are also applied.

[0116] The gate operation may be a rotational Z gate operation, in accordance with some embodiments. A rotational Z gate operation may be performed by varying the amplitude of the second drive to cause a rotation around a Z-axis of a Bloch sphere, such as the Bloch sphere shown in FIG. 4A. For example, the amplitude of the second drive is set to non-zero value at act 404 to control an asymmetry of the system. During implementation of the rotational Z gateAttorney Docket No.: Y0087.70177WO00operation, the amplitude of the second drive is temporarily switched to a different value. During the Z gate operation the frequency and phase of the first and second drive may be unchanged relative to the initialization parameters.

[0117] During the rotational Z gate operation, the second drive serves two purposes: it both enacts the gate operation and maintains the required asymmetry for bit-flip error suppression. The amplitude of the second drive, e1;should be chosen on a case-by-case basis based on the value of e2■ The inventors have recognized and appreciated that there is a maximum amount of asymmetry that a double well with an energy barrier of ^2)2 / ^ will tolerate before the higher well becomes unstable. Accordingly, the maximum value of the amplitude of the second drive is er / K < (62 / / f)3^2. Additionally, to prevent leakage outside of the qubitsubspace, the drive amplitude e1should be much smaller than the energy gap 4c2between the qubit states and other excited states.

[0118] Of the optimal values for the amplitude of the second drive e1, the amplitude is selected to reduce the rate of bit-flips during the gate by optimizing the product e Tx. For example, if c2 / / = 7.7. Then the amplitude of the second drive is limited to e^ / K < 8.Accordingly, the gate operation may be performed with e^ / K=6.5, so that the resonance condition is broken and the lifetime is maximal.

[0119] Although described in connection with performing a rotational Z gate operation, these amplitude parameters may be used in connection with other gate operations, as aspects of the technology described herein are not limited in this regard.

[0120] The gate may be an X gate to implement a bit flip, in accordance with some embodiments. An X gate operation may be performed by varying the phase of the first drive and the phase of the second drive such that the bit value is flipped. For example, to perform an X gate operation, the phase of the second drive is varied from <[) to <[) — TT, and the phase of the first drive is varied from 0 to -2TT. The phase changes are implemented adiabatically or approximately adiabatically. During the X gate operation the frequency and amplitude of the first and second drive may be unchanged relative to the initialization.Attorney Docket No.: Y0087.70177WO00

[0121] The gate may be a X(π / 2) gate to create a superposition of states, in accordance with some embodiments described herein. A X(π / 2) gate operation may be performed by varying the amplitude of the second drive and varying the frequencies of both the first drive and the second drive such that a superposition of states is formed. For example, a frequency of the second drive is varied from a first initial frequency by a value of Aco and subsequently varying the frequency of the second drive back to the first initial frequency; and a frequency of the first drive is varied from a second initial frequency by a value of 2Aco and subsequently varying the frequency of the first drive back to the second initial frequency. To reduce the probability of errors, the frequencies are varied smoothly. A summary of parameters for performing the X(π / 2) gate operation is summarized in Table 2 below.Table 2Drive Frequency Amplitude Phaseω₁ → ω₁ + Δmax→ ω₁, temporarily switched to aDrive 2 unchanged smoothly different valueω₂ → ω₂ + 2Δmax→ ω₂,Drive 1 unchanged unchangedsmoothly

[0122] The gate may be a ZZ(0) gate, in accordance with some embodiments described herein. A ZZ(0) gate implements a rotation to a two qubit system that depends on the respective states of each qubit. Accordingly, where the first drive and the second drive control a first qubit, a third drive and a fourth drive may be used to control a second qubit. The second qubit may be identical to the first qubit. The third drive may control an energetic barrier such that a potential energy well of the second element is a potential energy double well, and the fourth drive may control an energetic asymmetry of the potential energy double well of the second element such that the potential energy double well is asymmetric. In some embodiments, both the first qubit and the second qubit are Kerr parametric oscillators.

[0123] To perform a ZZ(0) gate operation, a fifth drive is used to induce a linear coupling between the two qubits. In some embodiments, the fifth drive may be applied to a coupling element (e.g., a SNAIL) coupling between each qubit. In some embodiments, the fifth drive mayAttorney Docket No.: Y0087.70177WO00be applied to one of the two SNAIL transmon qubits to generate the desired linear coupling. The fifth drive is operated with an amplitude larger than zero to induce a phase rotation that depends on respective states of the first element and the second element. For example, the frequency of the fifth drive is tuned to the frequency difference between the first element and the second element. A summary of parameters for performing the ZZ(0) gate operation is summarized in Table 3 below.Table 3Drive Frequency Amplitude Phase Drive 1 unchanged unchanged unchanged Drive 2 unchanged unchanged unchanged Drive 3 unchanged unchanged unchanged Drive 4 unchanged unchanged unchanged temporarily switched to aLinear coupling ωlin= ωKPO1− ωKPO2φlin= 0nonzero value

[0124] The gate may be a CNOT gate, in accordance with some embodiments. The CNOT gate applies a bit-flip to the second qubit only if the first qubit is in 11). The CNOT gate may be implemented by driving the third drive from an amplitude of s to an amplitude of - s and decreasing a phase of the third drive by 71, driving the fourth drive from a non-zero amplitude to an amplitude of zero and decreasing a phase of the fourth drive by TT / 2, driving the fifth drive at a frequency tuned to the frequency difference between the first element and the second element, from a zero amplitude to a gate amplitude, and decreasing a driving phase by of the fifth drive by 7t / 2, and driving a sixth drive at a frequency, to induce a nonlinear coupling between the two qubits, tuned to the difference of twice a frequency of the second element minus the frequency of the first element, from a zero amplitude to a gate amplitude and subsequently back to zero amplitude, and decreasing a phase of the sixth drive by it. A summary of parameters for performing the CNOT gate operation is summarized in Table 4 below.Attorney Docket No.: Y0087.70177WO00Table 4Drive Frequency Amplitude Phase Drive 1 unchanged unchanged unchanged Drive 2 unchanged unchanged unchanged φ₁ → φ₁ − π / 2 adiabatically Drive 4 unchanged ε1→ 0, adiabaticallyadiabatically 0 → −π Drive 3 unchanged ε2→ −ε2, adiabaticallyadiabatically 0Drive 5 ωlin= ωKPO1− ωKPO20 → max, adiabaticallyadiabatically 0 -» — 7T Drive 6 ωnl= 2ωKPO2− ωKPO10 → max → 0, adiabaticallyadiabaticallyIII. Appendix

[0125] The inventors have experimentally realized a strongly asymmetric double well Kerr parametric oscillator operated in the quantum regime, which is reported herein. The experiments use a low-noise, all-microwave control, and continuously tunable driven Hamiltonian realized in a superconducting quantum circuit that is well described by a static effective theory. Measurement of the activation rate across the double well barrier demonstrates two counterintuitive effects: i) a weak asymmetry can significantly decrease the activation rate even when the system is initialized in the shallower well, and ii) the width of the resonant activation lines alternates between narrow and broad as a function of the well depth and the wellAttorney Docket No.: Y0087.70177WO00asymmetry. Based on the experiments and modeling described herein, the inventors have recognized and appreciated that both these effects will manifest in ordinary chemical double well systems operating in the quantum regime.III. A. Introduction

[0126] The Kerr parametric oscillator, a nonlinear oscillator made bi-stable by a parametric drive, is an elementary quantum optical model that has continuously received attention for decades because, despite its simplicity, it is surprisingly rich. In the early 1990s it was predicted that this system should exhibit quantum tunnelling between the two stable points, opening a pathway to engineer a highly controllable simulator for the physics involved in quantum chemistry, nuclear structure, two- level systems and other problems involving a double well energy surface. The physical implementation of such a promising model was, however, challenging and experiments in the quantum regime have been realized only recently. The refinement of experiments, enabled by control of experimental parameters over a large range, not only verified a battery of longstanding theoretical predictions but also revealed new physics, like the modulation of the activation rate by the quantization of quasienergies and their resonances, which has direct implications for quantum chemistry and quantum computation.

[0127] The exploration of the activation dynamics in the case of a continuously tunable asymmetric double well parametric oscillator is discussed herein. The inventors have recognized and appreciated two unexpected effects. First, that the asymmetric double well can experience a significantly longer activation time from one well to the other (well-switching) relative to the symmetric case, even when the system is initialized in the shallower well. This is counterintuitive because one would think that by reducing the barrier height, the activation time should decrease. The inventors have recognized and appreciated that this is not the case in the system described herein due to a fine-tuned quantum effect that can be heuristically explained by the breaking of resonances inside the double well that occurs when the asymmetry of the double well is increased. As a result of the breaking of resonances inside the double well tunneling between the two wells is hindered. Yet, excessive asymmetry revives the resonances by aligning energy levels again. Thus, the inventors have recognized and appreciated that a sweet spot existsAttorney Docket No.: Y0087.70177WO00between these extremes, maximizing the activation time. This is in contrast to the resonant quantum tunnelling that can be observed in the symmetric case.

[0128] The second unexpected effect is observable thanks to the simultaneous and continuous tunability of barrier height and asymmetry in system described herein. The activation of the system exhibits pronounced quantum resonances whose width alternates with both the depth and the asymmetry of the wells. The incoherent activation rate of the system reflects the width of the Hamiltonian anti-crossing of the energy levels close to the separatrix of the double well energy surface. The location and width of these resonances clearly reveal Hamiltonian properties and are remarkably well explained by a time-independent quantum model obtained from the parametrically driven system under a rotating wave approximation (RWA).Semiclassical predictions are also in excellent agreement with the data and provide a deeper insight. A basic Lindbladian model fails, however, to capture quantitatively the purely dissipative activation rates. This deviation from the model may be explained by the breakdown of the RWA in under the experimental conditions described herein, as further discussed below in connection with the control measurements discussed in section III. E.III. B Setup and Model System

[0129] An example setup for demonstrating aspects of the technologies described herein is shown in FIG. 5A. The example setup consists of two chips with superconducting circuits that are addressable by microwave drives via charge-coupling. In the following examples, only one of the two chips is used, however the concepts described herein are not limited to a single chip and may be applied to multichip systems. The relevant chip contains an array of two superconducting nonlinear asymmetric inductive elements (SNAILs) shunted by a large capacitor, as depicted in FIGs. 5B, 5C, and 5D.

[0130] FIG. 5A illustrates a rendering of the half-aluminum, half-copper sample package 500 containing two Sapphire chips 502a and 502b, in accordance with some embodiments of the technology described herein. The package 500 includes copper block 506 and aluminum block 504. The copper block 506 includes Sapphire chips 502a and 502b mounted to the copper block and further includes a solenoid for generating a magnetic field. The aluminum block 504 includesAttorney Docket No.: Y0087.70177WO00cavities for accommodating the Sapphire chips when the aluminum block 504 is affixed to the copper block, such that the Sapphire chips are enclosed between the aluminum and copper blocks. The aluminum block 504 includes waveguides 508 and 510 for receiving and transmitting microwave pulses 512 and 514 to one or more of the Sapphire chips. The microwave pulses 512 and 514 are generated by respective microwave drives.

[0131] FIG. 5B illustrates a magnification of a Sapphire chip 502a, such as the Sapphire chips included in the package shown in FIG. 5 A, in accordance with some embodiments of the technology described herein. Each chip has a SNAIL-transmon 822, readout resonator, and Purcell filter. Only one chip is used in the examples discussed below. Applying a strong microwave drive at ω₂ ~ 2ωatransforms the SNAIL-transmon Hamiltonian into the parametric oscillator Hamiltonian.

[0132] FIG. 5C illustrates a schematic 530 of the SNAIL-transmon 532, in accordance with some embodiments of the technology described herein. As shown in FIG. 5C, the SNAIL-trasmon includes a two-SNAIL array that serves as the nonlinear element. The two-SNAIL array includes first SNAIL 534 and second SNAIL 536. The capacitor pads are shifted with respect to the axis of the array to couple it to the readout resonator.

[0133] FIG. 5D illustrates a scanning electron micrograph of the two-SNAIL array 532, in accordance with some embodiments of the technology described herein. The SNAIL loops are biased with an external magnetic flux / o = 0.31, where o is the magnetic flux quantum. Each of the SNAILs include a branch with multiple Josephson junctions and a branch with a single Josephson junction. As shown in FIG. 5D, the first SNAIL includes multiple Josephson junctions 556 on a first branch and a single Josephson junction 552 on a second branch. The second SNAIL includes multiple Josephson junctions 558 on a first branch and a single Josephson junction 554 on a second branch.

[0134] The Hamiltonian of a SNAIL transmon with charge drives can be approximated by Equation 1, shown below.= ωoâ†â + g₃ / 3(â + â†)³ + g₄ / 4(â + â†)⁴ − iΩ₁sin(ω₁t + φ)(â − â†) − iΩ₂sin(ω₂t)(â − â†) Equation 1Attorney Docket No.: Y0087.70177WO00

[0135] In Equation 1, ω0is the bare resonance frequency of the SNAIL transmon, g₃, g₄ are the third- and fourth-order nonlinearities of the circuit and d is the bosonic annihilation operator. Here, Ω₁ is the amplitude and ω₁ is the frequency of the drive that will henceforth be referred to as the linear (additive) drive, while Ω2and ω₂ are the amplitude and frequency of what we refer to as the squeezing, or two-photon, parametric drive. This Hamiltonian is the so-called (asymmetric) parametric oscillator Hamiltonian when ω₂ ~ 2ω0and ω₁ ~ ω0. The phase <[) is the relative phase between the two drives. By applying displaced frame transformations, transforming into the rotating frame ω₂ / 2 and keeping some terms beyond the RWA, the effective Hamiltonian describing the asymmetric parametric oscillator is given by Equation 2 below.= −Kâ†2â2+ ε2(â2+ â†2) + |ε1|(eiφâ + Equation 2

[0136] In Equation 2, the leading order Kerr non-linearity, K, is described by K =with ωathe renormalized SNAIL transmon resonance frequency(which is Lamb and Stark-shifted from ω0). The drive coefficients are given by |ε1| = andε₂ = g₃Ω₁ and Ω₂ may be derived based on the values of ε1and ε2described above in 3ωasection II. B. The relation to a double well becomes apparent in the classical limit as shown in Equation 3.= V|φ=0= k₄x⁴ − k₂x² + k₁x₁ Equation 3

[0137] In Equation 3, d i->â = (x + ip) / √2 is mapped together with kk1= √2 |ε1| cos φ, k2= −ε2, and k4= −K / 4. Two instances of V(x) are shown in FIG. 6A and 6C, while in FIG. 6B and D the associated energy spectra are shown as a function of the control parameters si and S2 for 0 = 0.

[0138] FIG. 6A illustrates the semiclassical double well potential energy V(x), in accordance with some embodiments of the technology described herein. The semiclassical double well potential energy is shown with the allowed quantum energies E / K obtained from the diagonalization of the parametric oscillator Hamiltonian.Attorney Docket No.: Y0087.70177WO00

[0139] FIG. 6B illustrates the spectrum of the parametric oscillator Hamiltonian as a function of controlling the barrier height, in accordance with some embodiments of the technology described herein.

[0140] FIG. 6C illustrates the semiclassical double well potential energy for the asymmetric case and the Hamiltonian energies, in accordance with some embodiments of the technology described herein.

[0141] FIG. 6D illustrates the spectrum as a function of ci controlling the asymmetry, in accordance with some embodiments of the technology described herein.

[0142] As shown by Equation 2 and Equation 3, ei controls the asymmetry of the wells and that £2 controls their depth. Note, however, that the parametric oscillator Hamiltonian cannot be written as a sum of kinetic [T (p)] and potential [V (x)] energies since cross terms like x2p2are present. These terms can lead to interesting effects but they do not play a critical role in the present analysis.

[0143] For ci = 0, this is the conventional (symmetric) parametric oscillator Hamiltonian that creates a double well along the position axis. If ei = 0, then the phase <[) becomes relevant. For <[) = 90°, the linear drive is momentum-like, thus not breaking the symmetry of the double well. For <[) = 0, this drive is position-like and lifts the degeneracy between the two wells (see FIG. 6C and 6D). The case <[) = 0 is discussed in section III. C and the experimental study of the phase dependence is discussed in section III. E. An ordinary Lindbladian model containing only single photon gain and single photon loss with phenomenological rates and temperature is used to model the activation rate (see section III. E).III. C Experiment and Analysis

[0144] To measure the activation rate in this system, the states localized at the bottom of the wells need to be prepared and measured as a function of time. A number of steps are required for this. To ready the setup for measurements, the SNAIL loops are biased with an external magnetic field sourced by a solenoid lying below the copper part of the enclosure (block 506 in FIG. 5A). This flux allows the Hamiltonian parameters coaand K to be set. In this work, a flux point is selected at which ωa / 2π and K / 2π are measured to be ωa / 2π = 6.086 GHz and K / 2π =Attorney Docket No.: Y0087.70177WO00(528+8) kHz (see section III. E.). From the flux dependence of both ωaand K a model for the SNAIL is fit to extract g₃ / 3 = 2π and g₄ = 2π. The values of Qi and Q2 are directly proportional to the microwave amplitude applied to the sample and, therefore, the microwave amplitudes provide precise control over ε₁ and ε₂ (see section III. E.). In the example setup, dielectric loss sets the single-photon lifetime to 20 μs.

[0145] By turning on the squeezing drive and waiting five times the single-photon lifetime to prepare the oscillator in its steady state. A bifurcation of the SNAIL oscillator is observable by homodyning the emitted radiation (e.g., fluorescent readout). The fluorescence is microwave- activated by a tone parametrically coupling the parametric oscillator with the on-chip readout resonator coupled to a quantum-limited amplifier detection line. The homodyne signal clearly shows the typical pair of stable oscillations out-of-phase by 180°. The photons emitted by the oscillator during readout are continuously replenished by the squeezing drive. Accordingly, the driven oscillator in presence of dissipation remains in one of its two(quasi-)steady states. Taking a single data point for the discrimination between parametric oscillator states, takes 4 μs. This is typically much shorter than the activation time across the double well parametric oscillator barrier. By performing a second measurement after a variable waiting time (see section III. E.), one can detect activation events changing the state of the parametric oscillator from one stable state to the other. By repeating these measurements, one can determine the probability per unit time of having an activation event and therefore directly measure the activation rate for a given set of Hamiltonian parameters ei / K and C2 / K.

[0146] To measure the activation rate in the parametric oscillator, the instances when the system is found initially in the shallower well are measured and posted. Measurements of the population dynamics in between the two steady states for different values of the asymmetry are shown in FIG. 7A. The population dynamics are controlled by ε₁ / K. For these measurements the squeezing amplitude is set to ε₂ / K = 7.7. The probability as a function of time of remaining in the initial state is well approximated by an exponential decay (see section III. E.). The timescale of these exponential T is a direct measurement of the activation rate (1 / T).

[0147] FIG. 7A illustrates the experimental population dynamics when initializing the system in the shallower well, in accordance with some embodiments of the technology describedAttorney Docket No.: Y0087.70177WO00herein. The probability of survival Pupdrops exponentially. The dashed line represents the halflifetime of the state. Resonances with characteristic widths are apparent. Note that the halflifetime at finite asymmetry (ε₁ / K ~ 1) is larger than for the symmetric case. Data is collected up to 900 μs.

[0148] FIG. 7B illustrates theory calculation modeling the experiment shown in FIG. 7A. The theory plot is obtained from independent calibration of all Hamiltonian parameters (coa, e, ci, and K). Without free parameters, the Hamiltonian theory and experiment show agreement. The shift between the resonances is consistent with a 5% uncertainty in the calibration of ci / K and E2 / K.

[0149] In FIG. 7B a theory prediction is shown that is computed from a Lindbladian model including single-photon loss at rate κ / K = 0.025 and gain corresponding to a finite temperature determined by a mean number of thermal photons nth = 0.05. Observable resonances are shown in FIG. 7B for certain values of ci / K, where the activation rate is markedly increased. These are resonances between states in different wells. The resonance for levels deep within the wells behave effectively as level crossings since the coupling is exponentially small due to the suppressing of tunneling under the barrier. Therefore, these activation events are mediated by thermal and quantum heating from the ground state into the tunneling levels at the barrier top. This is observed from the data by noting that the alternating width of the different resonance in FIG. 6D correspond to the strength of the anticrossings of the spectrum at the barrier top (energy levels above the energy barrier in FIG. 6C). This interpretation of the data allows us to predict the location in parameter space where resonant tunneling takes place. This happens when the uncoupled levels in the right and left well align. By realizing that the energy spacing of the states in the wells can be estimated by δ ~ 4ε₂ and that the asymmetry can be estimated fromEquation. 3 as where the resonance condition is written as (A = nS), resulting inEquation 4.ε₁ / K ≈ (ε₂ / K) Equation 4K \nKJAttorney Docket No.: Y0087.70177WO00

[0150] Equation 4 predicts the location of the resonances with remarkable precision (see FIG. 8A-8C).

[0151] FIG. 8A illustrates a semiclassical Hamiltonian prediction for the resonance conditions (parabolic dashed lines) and Bohr’s quantization condition for n and m allowed quantum orbits in the small and large lemniscate’s lobes. The circles mark triple intersections labeled (n, m).

[0152] FIG. 8B illustrates a measurement of activation time T as a function of C2 / K (controlling the barrier height) and ci / K (controlling the asymmetry), in accordance with some embodiments of the technology described herein.

[0153] FIG. 8C illustrates a theory prediction from a Lindbladian model including single photon loss and gain corresponding to the experimental results shown in FIG. 8B. The color scale is logarithmic. The serpentining dashed lines shows the maxima of T at each value of e K.

[0154] The dashed parabolas in FIG. 8A-8C are obtained from Equation 4. The theoretical analysis is complemented with a semiclassical action quantization taken here as a proxy for the quantum levels. The number of allowed quantum orbits is given by the number of action quanta enclosed by the asymmetric “figure eight” lemniscate delineating the phase space separatrix in between the wells. The solid lined curves in FIG. 8A show the pairs ci / K and C2 / K where the Einstein-Brillouin-Keller (EBK) action quantization condition is met, as reflected by Equation 5.1 / 2π ∮ p dx = ℏ(ñ + 1 / 2) Equation 5

[0155] Here, ñ = n, m are the quantum numbers of the shallow and deep wells, respectively. The triple intersection points of the parabolas from Equation 4 with the equi-action curves meeting the EBK quantization conditions mark the point in parameter space where a new level enters the wells in the tunneling resonances condition. The intersection points are marked with circles labeled by (n, m), the quantum number of each well. At these points, where each well contains exactly n and m semiclassical orbits, the inventors have recognized and appreciated that the resonance will broaden due to a new orbit contributing to the activation rate. This is seenAttorney Docket No.: Y0087.70177WO00as sharpening structures in FIG. 8B and FIG. 8C (see also FIG. 7A and FIG. 7B for the broadened resonances).

[0156] In FIG. 8B the measured activation time is shown for a fine scan of both ci / K and e / K. The resonances are clearly shown to have width that changes as the broad anti-crossings are narrowed by the suppression of tunnel splitting up to the point that the tunneling through a given excited pair of states is relayed by the tunneling via the next pair of excited states, now at the top of the barrier and the new limiting processes setting the activation time. See this by following the energies and their tunnel splitting in FIG. 6D. The agreement of the Hamiltonian theory with the experiment is remarkable (see III. E for additional details on the quantum treatment). The Hamiltonian theory predicts, quantitatively, the resonance condition and, qualitatively, their widths.

[0157] The Lindbladian prediction of the experiment is shown in FIG. 8C. The location and behavior of the resonance in parameter space are in qualitative agreement but falls short in quantitatively predicting the activation rates measured. However, many features of the data are correctly captured, like for example, the fact that an asymmetric system can have a longer activation lifetime than the symmetric system, even if one of the wells is markedly shallower. This is shown directly from FIG. 7 at ci / K ~ 1. The serpentining dashed line in FIG. 8B shows the experimentally determined maxima of T as a function of the control parameters. The theoretical maxima are shown in FIG. 8D by the serpentining line. The location of the maxima is nontrivial, and it is relevant because it can readily be used to extend the lifetime of a Kerr-cat qubit. The effect was unknown before the analysis of data described in the present disclosure.

[0158] FIG. 9A illustrates experimental data for the activation time T as a function of e / K, effectively controlling the barrier height, in accordance with some embodiments of the technology described herein. The symmetric case (ei = 0) is modulated by resonances in the quantized energies. Exploiting asymmetry, it is possible to avoid these resonances and substantially increase the activation time. The asymmetric optimum corresponds to the serpentining dashed line in FIG. 8.

[0159] FIG. 9B illustrates theoretical modeling for the activation time T as a function of e / K, effectively controlling the barrier height, in accordance with the experimental data shownAttorney Docket No.: Y0087.70177WO00in FIG. 9 A. In FIG. 9A, we show that the increase of T as a function of the asymmetry can be large. To provide physical insight into this effect we note that for the symmetric case (ei = 0), T is modulated in a step-like fashion by the orbits falling under the barrier. The linear drive can be exploited to break the (parity) symmetry of these orbits as shown in FIG. 6 and therefore avoid altogether the resonant saturation seen for ci = 0 in FIG. 9A and 9B. This observation explains the trajectory of the serpentining curve in FIG. 8B, 8C which snakes around “avoiding” the resonance conditions omened by the circles. In other words, quantum tunneling produces a hybridization of the classically decoupled orbits under the barrier, so the asymmetry degree of freedom can be used to minimize that hybridization, reducing the tunneling rate via the excited states, and avoid the plateaus in FIG. 9A almost completely.III. D Conclusion

[0160] The measurement of the activation rate in a continuously tunable asymmetric Kerr parametric oscillator and the observed a fine structure that was described herein. The quantized energy level structure is manifest in the activation dependence on the control parameters. The features in the activation times caused by this parameter dependence are captured by a simple model and a semiclassical analysis, which allowed extension of the concepts described herein to be applied to other systems beyond parametric oscillators.

[0161] The inventors have recognized and appreciated that the system described herein may be used for quantum computation having qubits encoded in the well state. In this regard, the inventors have recognized and appreciated aspects contributions of the present disclosure that should be considered with regard to quantum computation. First, o the activation timescale T in between the wells may be increased by a fine control of the asymmetry. This may directly map to a reduction of bit-flip errors with no extra hardware requirements. Second, operation of a highly asymmetric parametric oscillator in the quantum regime is demonstrated by the examples described herein. As described herein, the static effective description is not compromised under strong linear drives, which are required for fast gates and new implementations of hardware efficient readout schemes.Attorney Docket No.: Y0087.70177WOOO

[0162] To conclude, aspects of the technology described herein improve the functioning of high-fidelity quantum computing gates and measurements and in particular, may be further applied to other parametric oscillator experiments operating in the quantum regime. The example system described above, together with the additional configuration details in section III. E, demonstrate some advantages provided by the technologies described herein for advancing quantum computation operations.III. E Additional Configuration DetailsIII. E.l. Measurement of Kerr coefficient

[0163] The Kerr coefficient K, as used in the parametric oscillator Hamiltonian Equation 2, is extracted by spectroscopy. A saturating probe drive ωpris applied to the SNAIL transmon operated with 62 = 0. Varying the probe drive frequency and measuring the response via dispersive readout, results in the data shown in FIG. 10. The spectrum shows two clear dips, which correspond, from lower frequency to higher frequency, to the two-photon gf / 2 transition and the ge transition (the SNAIL levels are labeled following the atomic physics convention in increasing energy order g, e, f,...). The Kerr coefficient is extracted by using the relation ωge- ωgf / 2= K. Fitting two Gaussian peaks to the spectrum in FIG. 10, to find K / 2it = (528+8) kHz.III. E.2. Calibration of relative phase between squeezing drive and linear drive

[0164] Quantum coherent Rabi-like oscillations as a function of this phase <[) are shown in FIG. HA. For <[) = 0, this drive is position-like and lifts the degeneracy between the two wells (see FIG. 6C). This energy difference induces the Rabi-like oscillation. For <[) = 90° (FIG. 11 A), the linear drive is momentum-like (the Hamiltonian term is oc p), thus not breaking the symmetry of the double well.

[0165] FIG. HA illustrates time-resolved quantum coherent oscillation as a function of relative phase <[) between the squeezing and linear drives, in accordance with some embodiments of the technology described herein. The time-resolved quantum coherent oscillation is described as Y = i|cr)(— a\— i| — a){a | as a function of relative phase <[).Attorney Docket No.: Y0087.70177WO00

[0166] FIG. 11B illustrates a linear drive with relative phase of <[) = 90°, in accordance with some embodiments of the technology described herein. Symmetry between wells is preserved and no resonances are visible (compare to FIG. 7A).

[0167] FIG. 11C illustrates a similar experiment as depicted in Fig. 7A, but with a Rabi drive phase of 180° instead of 0°. The flipped phase results in an exchange of the left and right well, meaning that now the left well is the deeper well. In the experiment, the shallower (now right) well is initialized, but the time-resolved population of the deeper (now left) well is measured. As a result, the population of the left well is initially zero and then increases over time. Trends and features, such as the resonant-tunneling also seen in Fig. 7A, are shown. This confirms that the 180° phase shift only exchanges the roles of the wells, but otherwise exhibits the same physical phenomena.III. E.3. Calibration of squeezing (parametric) drive amplitude;:2 and linear drive amplitude si

[0168] In the experimental setup, the drive amplitudes are directly controlled by a digital to analog voltage converter. To calibrate the strength of the drive in MHz time-resolved Rabi oscillations are measured as a function of the digital control of the squeezing drive e. The experimental data is shown in FIG. 12A. The oscillations of the observable X =(|a)(a| — | — a) — a|) / |a|2occur at a rate Ωcat(ε2) ≈ J₁(4αε1α*) where a = and theapproximation is valid for |a| > 1. Using, also, that for S2=0 the oscillation has a frequency of 2si, just like for an ordinary transmon, a is obtained in MHz completing the calibration of the drive amplitudes. To be clear, this relation can be rewritten as <a'ta'> = cz / K = QCat(< T2)2 / 16c2, where <a'ta'> is the average photon number of the coherent states. By extracting the Rabi rate QCat for each voltage of the digital control of Q and using the previously determined value of K, ⟨a'†a'⟩ is found as a function of the digital control of 62. The data is shown in Fig. 12B and shows a clear linear relationship between the applied voltage for the drive and the average photon number. The slope of the linear fit (together with the previously extracted value of Kerr) determines theAttorney Docket No.: Y0087.70177WO00proportionality constant between the digital to analog converter in volts and the drive amplitude 62 in MHz as required by the Hamiltonian description.

[0169] FIG. 12A illustrates time-resolved quantum coherent Rabi-like oscillations as a function of squeezing amplitude, in accordance with some embodiments of the technology described herein. The squeezing amplitude is measured as the voltage of the digital controller.

[0170] FIG. 12B illustrates the photon number as a function of applied voltage for the digital control of the squeezing drive, in accordance with some embodiments of the technology described herein. As shown in FIG. 12B, the photon number (a't a") is plotted as a function of applied voltage for the digital control of squeezing drive C2- The experimental data points are obtained from FIG. 12A using (a't a") = E2 / K = flcat2 / 16ei2. A linear fit is used to convert the voltage set by the digital control of £2 to the squeezing drive e? in MHz.

[0171] FIG. 13A illustrates pulse sequence for the determination of the activation time, in accordance with some embodiments of the technology described herein. The squeezing drive is turned on adiabatically. This is followed by a measurement of the which- well information, projecting the parametric oscillator into either of the wells. Then the linear drive is turned on adiabatically. Next, the state evolves for a variable time t, during which both the squeezing and linear drive remain on. After that, the linear drive is turned off adiabatically. Finally, the which-well information is measured again to find the remaining population.

[0172] FIG. 13B illustrates a decay of a coherent state initiated on the shallower well for different asymmetry values ei / K, in accordance with some embodiments of the technology described herein. Experimental data are dots, solid lines are exponential fits. For small well asymmetry ei / K, the probability to be in the shallower well decays to 0.5. For increasingly large asymmetry, the steady-state population is no longer equally distributed between both wells, becoming increasingly biased towards the deeper well.

[0173] Having thus described several aspects of at least one embodiment of the technology described herein, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art.Attorney Docket No.: Y0087.70177WO00

[0174] Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of disclosure. Further, though advantages of the technology described herein are indicated, not every embodiment of the technology described herein will include every described advantage. Some embodiments may not implement any features described as advantageous herein and in some instances one or more of the described features may be implemented to achieve further embodiments. Accordingly, the foregoing description and drawings are by way of example only.

[0175] The above-described embodiments of the technology described herein may be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software, or a combination thereof. When implemented in software, the software code may be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art by names such as CPU chips, GPU chips, microprocessor, microcontroller, or co-processor.Alternatively, a processor may be implemented in custom circuitry, such as an ASIC, or semicustom circuitry resulting from configuring a programmable logic device. As yet a further alternative, a processor may be a portion of a larger circuit or semiconductor device, whether commercially available, semi-custom or custom. As a specific example, some commercially available microprocessors have multiple cores such that one or a subset of those cores may constitute a processor. However, a processor may be implemented using circuitry in any suitable format.

[0176] Further, a computer may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer.Additionally, a computer may be embedded in a device not generally regarded as a computer but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smart phone, tablet, or any other suitable portable or fixed electronic device.

[0177] Also, a computer may have one or more input and output devices. These devices may be used, among other things, to present a user interface. Examples of output devices thatAttorney Docket No.: Y0087.70177WO00may be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that may be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computer may receive input information through speech recognition or in other audible format.

[0178] Such computers may be interconnected by one or more networks in any suitable form, including as a local area network or a wide area network, such as an enterprise network or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks, fiber optic networks, or any suitable combination thereof.

[0179] Also, the various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and / or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine.

[0180] In this respect, aspects of the technology described herein may be embodied as a computer readable storage medium (or multiple computer readable media) (e.g., a computer memory, one or more floppy discs, compact discs (CD), optical discs, digital video disks (DVD), magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments described above. As is apparent from the foregoing examples, a computer readable storage medium may retain information for a sufficient time to provide computer-executable instructions in a non-transitory form. Such a computer readable storage medium or media may be transportable, such that the program or programs stored thereon may be loaded onto one or more different computers or other processors to implement various aspects of the technology as described above. As used herein, the term "computer-readable storage medium" encompasses only a non-transitory computer readableAttorney Docket No.: Y0087.70177WO00medium that may be considered to be a manufacture (i.e., article of manufacture) or a machine. Alternatively or additionally, aspects of the technology described herein may be embodied as a computer readable medium other than a computer-readable storage medium, such as a propagating signal.

[0181] The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of processor-executable instructions that may be employed to program a computer or other processor to implement various aspects of the technology as described above. Additionally, one or more computer programs that when executed perform methods of the technology described herein need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the technology described herein.

[0182] Processor-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically, the functionality of the program modules may be combined or distributed.

[0183] Also, data structures may be stored in one or more non-transitory computer-readable storage media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a non-transitory computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish relationships among information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationships among data elements.

[0184] Techniques described herein may be embodied as computer-executable instructions, these computer-executable instructions may be implemented in any suitable manner, including as a number of functional facilities, each providing one or more operations to complete execution of algorithms operating according to these techniques. A “functional facility,”Attorney Docket No.: Y0087.70177WO00however instantiated, is a structural component of a computer system that, when integrated with and executed by one or more computers, causes the one or more computers to perform a specific operational role. A functional facility may be a portion of or an entire software element. For example, a functional facility may be implemented as a function of a process, or as a discrete process, or as any other suitable unit of processing. If techniques described herein are implemented as multiple functional facilities, each functional facility may be implemented in its own way; all need not be implemented the same way. Additionally, these functional facilities may be executed in parallel and / or serially, as appropriate, and may pass information between one another using a shared memory on the computer(s) on which they are executing, using a message passing protocol, or in any other suitable way.

[0185] Generally, functional facilities include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically, the functionality of the functional facilities may be combined or distributed as desired in the systems in which they operate. In some implementations, one or more functional facilities carrying out techniques herein may together form a complete software package. These functional facilities may, in alternative embodiments, be adapted to interact with other, unrelated functional facilities and / or processes, to implement a software program application.

[0186] Various aspects of the technology described herein may be used alone, in combination, or in a variety of arrangements not specifically described in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0187] Also, the technology described herein may be embodied as a method, of which examples are provided herein. The acts performed as part of any of the methods may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.Attorney Docket No.: Y0087.70177WO00

[0188] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, for example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and / or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

[0189] The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and / or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and / or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and / or B,” when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0190] Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed. Such terms are used merely as labels to distinguish one claim element having a certain name from anotherAttorney Docket No.: Y0087.70177WO00element having a same name (but for use of the ordinal term). The phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” “having,” “containing,” “involving,” and variations thereof, is meant to encompass the items listed thereafter and additional items.

[0191] Unless otherwise specified, the terms “approximately,” “substantially,” and “about” may be used to mean within ±10% of a target value in some embodiments. The terms “approximately,” “substantially” and “about” may include the target value.

[0192] Having described several embodiments of the techniques described herein in detail, various modifications, and improvements will readily occur to those skilled in the art. Such modifications and improvements are intended to be within the spirit and scope of the disclosure. Accordingly, the foregoing description is by way of example only, and is not intended as limiting. The techniques are limited only as defined by the following claims and the equivalents thereto.

Claims

Attorney Docket No.: Y0087.70177WO00CLAIMS1. A quantum information system comprising:one or more quantum elements, each quantum element of the one or more quantum elements being configured with one or more quantum states in a potential energy well;a first drive configured to control an energetic barrier of the potential energy well such that the potential energy well is a potential energy double well; anda second drive configured to control an energetic symmetry of the potential energy double well.

2. The system of claim 1, wherein the one or more quantum elements comprise a plurality of superconducting nonlinear elements.

3. The system of claim 1, wherein the one or more quantum elements comprise a plurality of ion traps.

4. The system of claim 1, wherein the one or more quantum elements comprise a plurality of optical potentials configured with a lattice of cold atoms.

5. The system of claim 1, wherein the first drive is a first microwave drive configured to operate at a first amplitude and a first frequency and the second drive is a second microwave drive configured to operate at a second amplitude and a second frequency.

6. The system of claim 5, wherein the first amplitude and the first frequency are different than the second amplitude and the second frequency.

7. The system of claim 6, wherein the first amplitude is between 0.01 to 50 times a Kerr coefficient of the one or more quantum elements of the quantum information system.Attorney Docket No.: Y0087.70177WO008. The system of claim 7, wherein the second amplitude is between 0.01 to 50 times the Kerr coefficient of the one or more quantum elements of the quantum information system.

9. A method of storing quantum information in a quantum system that includes one or more quantum elements, each quantum element of the one or more quantum elements being configured with one or more quantum states in a potential energy well, the method comprising:driving the quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well;initializing an initial quantum state of a plurality of interacting quantum states; andsubsequent to initializing the initial quantum state, driving the quantum system using a second drive, configured to control an energetic symmetry of the potential energy double well such that the potential energy double well is asymmetric.

10. The method of claim 9, further comprising:subsequent to initializing the initial quantum state, driving the quantum system using the second drive such that the potential energy double well is symmetric; and detecting an electronic output indicative of a final quantum state of the plurality of interacting quantum states.

11. The method of claim 9, wherein initializing the initial quantum state comprises transmitting a microwave pulse through the one or more quantum elements.

12. The method of claim 9, wherein driving the quantum system using the first drive comprises operating a microwave drive at a first amplitude and a first frequency.Attorney Docket No.: Y0087.70177WO0013. The method of claim 12, wherein the first amplitude is between 0.01 to 50 times a Kerr coefficient of the one or more quantum elements of the quantum system.

14. The method of claim 9, wherein driving the quantum system using the second drive comprises operating a microwave drive at a second amplitude and a second frequency.

15. The method of claim 14, wherein the second amplitude is between 0.01 to 50 times a Kerr coefficient of the one or more quantum elements of the quantum system.

16. The method of claim 9, further comprising detecting a lifetime of the quantum system and adjusting an amplitude of the second drive to yield an increase to the lifetime of the quantum system relative to the lifetime of the quantum system when the potential energy double well is symmetric.

17. The method of claim 9, further comprising detecting a lifetime of the quantum system and adjusting an amplitude of the second drive to optimize the lifetime of the quantum system.

18. A method of operating a quantum system for quantum information processing, wherein the quantum system includes one or more quantum elements, each quantum element of the one or more quantum elements being configured with one or more quantum states in a potential energy well, the method comprising:driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well;driving the quantum system using a second drive, configured to control an energetic symmetry of the potential energy double well such that the potential energy double well is asymmetric;subsequent to driving the quantum system using the second drive, initializing an initial quantum state of a plurality of interacting quantum states.Attorney Docket No.: Y0087.70177WO0019. The method of claim 18, further comprising detecting an electronic output indicative of a final quantum state of the plurality of interacting quantum states.

20. The method of claim 18, wherein driving the quantum system using first drive comprises operating a microwave drive at a first amplitude and a first frequency.

21. The method of claim 20, wherein the first amplitude is between 0.01 to 50 times the Kerr coefficient of the one or more quantum elements of the quantum system.

22. The method of claim 18, wherein driving the quantum system using the second drive comprises operating the microwave drive at a second amplitude and a second frequency.

23. The method of claim 22, wherein the second amplitude is between 0.01 to 50 times the Kerr coefficient of the one or more quantum elements of the quantum system.

24. A method of operating a quantum system for quantum information processing, wherein the quantum system includes one or more quantum elements, each quantum element of the one or more quantum elements being configured with one or more quantum states in a potential energy well, the method comprising:driving a quantum system using a first drive configured to control an energetic barrier such that the potential energy well is a potential energy double well;driving the quantum system using a second drive, configured to control an energetic asymmetry of the potential energy double well such that the potential energy double well is asymmetric;subsequent to driving the quantum system using the second drive, initializing an initial quantum state of a plurality of interacting quantum states; andAttorney Docket No.: Y0087.70177WO00subsequent to initializing the initial quantum state, varying the first drive and / or the second drive to perform a gate operation on the initial quantum state.

25. The method of claim 24, wherein the gate operation is a rotational Z gate operation and performing the gate operation comprises varying an amplitude of the second drive to perform a rotation around a Z-axis of a Bloch sphere.

26. The method of claim 24, wherein the gate operation is a X gate operation and performing the gate operation comprises varying a phase of the first drive and the phase of the second drive to implement a bit flip.

27. The method of claim 26, wherein the phase of the second drive is varied adiabatically from <|) to <|) — TT, and the phase of the first drive is varied from 0 to -2TT.

28. The method of claim 24, wherein the gate operation is a X(n / 2) gate operation and performing the gate operation comprises varying an amplitude of the second drive and varying frequencies of both the first drive and the second drive to create a superposition of states.

29. The method of claim 28, wherein a frequency of the second drive is varied from a first initial frequency by a value of Aco and subsequently varying the frequency of the second drive back to the first initial frequency; and a frequency of the first drive is varied from a second initial frequency by a value of 2Aco and subsequently varying the frequency of the first drive back to the second initial frequency.

30. The method of claim 24, wherein the first drive and the second drive are used to drive a first element of the quantum system, and the method further comprises:Attorney Docket No.: Y0087.70177WO00driving a second element of the quantum system using a third drive configured to control an energetic barrier such that a potential energy well of the second element is a potential energy double well;driving the second element of the quantum system using a fourth drive, configured to control an energetic asymmetry of the potential energy double well of the second element such that the potential energy double well is asymmetric;driving the first element and the second element of the quantum system with a fifth drive to induce a linear coupling between the first element and the second element, wherein a frequency of the fifth drive is tuned to a frequency difference between the first element and the second element.

31. The method of claim 30, wherein the gate operation is a rotational ZZ gate operation and performing the gate operation comprises driving the fifth drive with an amplitude larger than zero to induce a phase rotation that depends on respective states of the first element and the second element.

32. The method of claim 30, further comprising driving the first element and the second element of the quantum system with a sixth drive to induce a nonlinear coupling between the first element and the second element.

33. The method of claim 32, wherein the gate operation is a CNOT gate operation and performing the gate operation comprises:driving the third drive from an amplitude of s to an amplitude of - s and decreasing a phase of the third drive by 7t;driving the fourth drive from a non-zero amplitude to an amplitude of zero and decreasing a phase of the fourth drive by TT / 2;driving the fifth drive at a frequency tuned to the frequency difference between the first element and the second element, from a zero amplitude to a gate amplitude, and decreasing a driving phase by of the fifth drive by TT / 2;Attorney Docket No.: Y0087.70177WO00driving the sixth drive at a frequency tuned to a difference of twice a frequency of the second element minus the frequency of the first element, from a zero amplitude to a gate amplitude and subsequently back to zero amplitude, and decreasing a phase of the sixth drive by it.

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