Linear inductive coupler

Abstract

Systems and methods for coupling quantum modes using a linear inductive coupler are provided. The linear inductive coupler includes an inductor and two Josephson junctions having approximately equal Josephson energies, each coupled in parallel with the inductor. During operation, the linear inductive coupler is biased by a magnetic flux approximately equal to Φ0 / 2 such that a phase drop across each of the two Josephson junctions is approximately equal to π / 2. Driving the linear inductive coupler thereafter using at least one electromagnetic signal enables coupling between two quantum modes.

Description

Attorney Docket No.: Y0087.70175WO00 LINEAR INDUCTIVE COUPLER CROSS-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 / 680,015, filed August 6, 2024, and titled “LINEAR INDUCTIVE COUPLER,” and of U.S. Provisional Patent Application No. 63 / 751,743, filed January 30, 2025, and titled “LINEAR INDUCTIVE COUPLER,” each of which is incorporated by reference herein in its entirety. GOVERNMENT FUNDING

[0002] This invention was made with government support under W911NF-23-1-0051 awarded by the United States Army Research Office. The government has certain rights in the invention. BACKGROUND

[0003] 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, process and retrieve 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 quantum superpositions and thereby potentially exist in both of those states at once.

[0004] To perform quantum computing operations, two or more qubits may be coupled to permit interaction between the two quantum states. Parametric coupling techniques permit tunable coupling between qubits within a quantum information processing system. SUMMARY

[0005] Some embodiments are directed to a linear inductive coupler, comprising: an inductor and two Josephson junctions having approximately equal Josephson energies, each of which is coupled in parallel with the inductor, wherein: during operation, the linear inductive coupler is configured to be biased by a magnetic flux approximately equal to Φ^ / 2 such that a phase drop across each of the two Josephson junctions is approximately equal to π / 2.Attorney Docket No.: Y0087.70175WO00

[0006] In some embodiments, the inductor comprises a high kinetic inductance material. In some embodiments, the inductor comprises granular aluminum or niobium nitride.

[0007] In some embodiments, the inductor comprises a plurality of Josephson junctions arranged in series. In some embodiments, the plurality of Josephson junctions comprises at least 5 Josephson junctions. In some embodiments, the plurality of Josephson junctions comprises between 5 and 100 Josephson junctions.

[0008] In some embodiments, the linear inductive coupler is coupled between a first quantum mode and a second quantum mode.

[0009] In some embodiments, the first quantum mode is stored in a quantum harmonic oscillator, and the second quantum mode is stored in an ancilla qubit. In some embodiments, the ancilla qubit comprises one of a transmon qubit, a fluxonium qubit, or a charge qubit. In some embodiments, the quantum harmonic oscillator comprises a microwave cavity resonator.

[0010] In some embodiments, the first quantum mode is stored in a first quantum harmonic oscillator, and the second quantum mode is stored in a second quantum harmonic oscillator. In some embodiments, at least one of the first quantum harmonic oscillator and / or the second quantum harmonic oscillator comprises a microwave cavity resonator.

[0011] In some embodiments, the first quantum mode and the second quantum mode are configured to support quantum mode photons, and when driven by an electromagnetic drive waveform comprising drive photons, the linear inductive coupler is configured to activate an interaction between only an odd number of drive photons and an even number of the quantum mode photons.

[0012] In some embodiments, when driven by an electromagnetic drive waveform, the linear inductive coupler is configured to cause three-wave mixing between the first quantum mode and the second quantum mode.

[0013] In some embodiments, wherein the linear inductive coupler is further configured to exhibit approximately equal and opposite AC phase drops across the two Josephson junctions when driven by an electromagnetic drive waveform.

[0014] In some embodiments, the linear inductive coupler is configured to cause three- wave mixing between the first quantum mode and the second quantum mode when driven by an electromagnetic drive waveform having a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode.Attorney Docket No.: Y0087.70175WO00

[0015] In some embodiments, when not driven by an electromagnetic drive waveform during operation, the linear inductive coupler is configured to behave approximately like a linear oscillator.

[0016] Some embodiments are directed to a circuit quantum electrodynamic (cQED) system comprising: a first quantum device configured to store a first quantum mode during operation of the cQED system; a second quantum device configured to store a second quantum mode during operation of the cQED system; a linear inductive coupler configured to couple the first quantum mode and the second quantum mode when driven by an electromagnetic drive waveform during operation of cQED system. The linear inductive coupler comprises: an inductor; and two Josephson junctions having approximately equal Josephson energies, each of which is coupled in parallel with the inductor, wherein: during operation of the cQED system, the linear inductive coupler is configured to be biased by a magnetic flux approximately equal to Φ^ / 2 such that a phase drop across each of the two Josephson junctions is approximately equal to π / 2.

[0017] In some embodiments, the inductor comprises a high kinetic inductance material. In some embodiments, the inductor comprises granular aluminum or niobium nitride.

[0018] In some embodiments, the inductor comprises a plurality of Josephson junctions arranged in series. In some embodiments, the plurality of Josephson junctions comprises at least 5 Josephson junctions. In some embodiments, the plurality of Josephson junctions comprises between 5 and 100 Josephson junctions.

[0019] In some embodiments, the first quantum device comprises a quantum harmonic oscillator; and the second quantum device comprises an ancilla qubit. In some embodiments, the ancilla qubit comprises one of a transmon qubit, a fluxonium qubit, or a charge qubit. In some embodiments, the quantum harmonic oscillator comprises a microwave cavity resonator.

[0020] In some embodiments, the first quantum device comprises a first quantum harmonic oscillator, and the second quantum device comprises a second quantum harmonic oscillator. In some embodiments, at least one of the first quantum harmonic oscillator and / or the second quantum harmonic oscillator comprises a microwave cavity resonator.

[0021] In some embodiments, the cQED system further comprises an electromagnetic drive configured to differentially drive the linear inductive coupler, wherein: the first quantum mode and the second quantum mode are configured to support quantum mode photons, the electromagnetic drive is configured to generate the electromagnetic drive waveform comprising drive photons, and when driven by the electromagnetic drive waveform, the linearAttorney Docket No.: Y0087.70175WO00 inductive coupler is configured to activate an interaction between an odd number of the drive photons and an even number of the quantum mode photons.

[0022] In some embodiments, the electromagnetic drive waveform has a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode, and when driven by the electromagnetic drive waveform, the linear inductive coupler is configured to cause three-wave mixing between the first quantum mode and the second quantum mode.

[0023] In some embodiments, the linear inductive coupler is further configured to exhibit approximately equal and opposite AC phase drops across the two Josephson junctions when driven by an electromagnetic drive waveform.

[0024] In some embodiments, when not driven by an electromagnetic drive waveform during operation, the linear inductive coupler is configured to behave approximately like a linear oscillator.

[0025] Some embodiments are directed to a method of operating a linear inductive coupler, comprising biasing the linear inductive coupler by a magnetic flux approximately equal to Φ^ / 2 such that a phase drop across each of the two Josephson junctions is approximately equal to π / 2; and driving, using at least one electromagnetic signal, the linear inductive coupler to cause parametric coupling between a first quantum mode and a second quantum mode.

[0026] In some embodiments, driving the linear inductive coupler using the at least one electromagnetic signal causes three-wave mixing between the first quantum mode and the second quantum mode.

[0027] In some embodiments, the method further comprises before or after driving the linear inductive coupler using the at least one electromagnetic signal, causing the linear inductive coupler to behave approximately like a linear oscillator by not driving the linear inductive coupler using the at least one electromagnetic signal.

[0028] In some embodiments, driving the linear inductive coupler using the at least one electromagnetic signal causes interactions between only an odd number of drive photons generated by the at least one electromagnetic signal and an even number of quantum mode photons associated with the first quantum mode and the second quantum mode.

[0029] In some embodiments, driving the linear inductive coupler comprises driving a superconducting circuit comprising an inductor and two Josephson junctions with approximately equal Josephson energies, each of the two Josephson junctions being coupled in parallel with the inductor.Attorney Docket No.: Y0087.70175WO00

[0030] In some embodiments, driving the linear inductive coupler comprises driving the linear inductive coupler with an electromagnetic signal having a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode.

[0031] In some embodiments, driving the linear inductive coupler with the at least one electromagnetic signal comprises driving the linear inductive coupler with a differential drive electromagnetic signal configured to cause approximately equal and opposite AC phase drops across the two Josephson junctions of the linear inductive coupler.

[0032] The foregoing apparatus and method embodiments may be implemented with any suitable combination of aspects, features, and acts described above or in further detail below. These and other aspects, embodiments, and features of the present teachings can be more fully understood from the following description in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF DRAWINGS

[0033] Various aspects and embodiments will be described with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing.

[0034] FIG.1 is a schematic circuit diagram of an example of a linear inductive coupler, in accordance with some embodiments of the technology described herein.

[0035] FIG. 2 is a schematic circuit diagram of another example of a linear inductive coupler, in which the inductor includes a plurality of Josephson junctions arranged in series, in accordance with some embodiments of the technology described herein.

[0036] FIG. 3 is an illustrative block diagram of a circuit quantum electrodynamics (cQED) system including a linear inductive coupler, in accordance with some embodiments of the technology described herein.

[0037] FIG.4 shows equivalent circuit diagrams of the quantum system of FIG.3 when the linear inductive coupler is driven by an electromagnetic drive waveform and when the linear inductive coupler is undriven by an electromagnetic drive waveform, in accordance with some embodiments of the technology described herein.

[0038] FIG. 5 illustrates static dipole coupling between pairs of the quantum modes in the quantum system of FIG.3 when the linear inductive coupler is driven by an electromagnetic drive waveform, in accordance with some embodiments of the technology described herein.Attorney Docket No.: Y0087.70175WO00

[0039] FIG. 6 illustrates a chip architecture including a linear inductive coupler, two harmonic oscillators, and an ancilla qubit, in accordance with some embodiments of the technology described herein.

[0040] FIG. 7 is a plot of the simulated coupling rate versus the drive amplitude for a photon-swap operation that occurs between two quantum modes when the linear inductive coupler is driven by a frequency approximately equal to a difference between the resonant frequencies of the two quantum modes, in accordance with some embodiments of the technology described herein.

[0041] FIG. 8 is a plot of the simulated driven coupler shift for each of the frequency, Kerr, and Kerr (arrayed) as a function of the drive amplitude applied to the linear inductive coupler, in accordance with some embodiments of the technology described herein.

[0042] FIG. 9 is a plot of the simulated steady state impurity of the linear inductive coupler as a function of driven frequency and drive amplitude, in accordance with some embodiments of the technology described herein.

[0043] FIG.10 is a plot depicting an illustrative pulse scheme for operating the quantum system of FIG.3, in accordance with some embodiments of the technology described herein.

[0044] FIG. 11 is a schematic diagram of an illustrative implementation of a computer system that may be used in connection with some embodiments of the technology described herein.

[0045] FIG. 12A shows a schematic illustration of types of mixer balancing. By taking two identical nonlinear elements and driving them 180∘out of phase (e.g., with a balun), a single-balanced mixer can be created. If the nonlinear element is a single junction, this circuit is the differentially-driven SQUID (DDS). If the nonlinear element is instead a DC flux-biased RF-SQUID, this circuit results in the linear inductive coupler (LINC).

[0046] FIG.12B shows an illustration of the advantage of a balanced mixer. In general, a driven nonlinear element will permit numerous nonlinear processes, of which only a small subset is typically desired. Balancing the nonlinear element will suppress a significant fraction of spurious processes (long dashed lines) but preserve the desired processes (solid lines) at high strength. Any remaining parasitic processes (short dashed lines) still permitted by the symmetry are avoided through other means, like careful selection of drive frequencies.

[0047] FIGs. 13A-13C show static Hamiltonian and flux noise sensitivity. FIG. 13A shows the static LINC frequency and Kerr as a function of DC flux. Numerical values (solidlines) are calculated by exact diagonalization of a LINC Hamiltonian with ^^ / ℎ = 52.8GHz,Attorney Docket No.: Y0087.70175WO00^^ / ℎ = 15.84GHz, ^^ / ℎ = 100MHz, with the center shunt composed of an array of 10junctions, where ℎ is Planck's constant. Overlayed analytic curves follow Eqs.3 and 4. At theoperating point of ^^^ = ^ / 2, the coupler is linear. FIG. 13B shows three-wave mixingstrength (dashed black) and sensitivity to flux-noise (yellow) as a function of DC flux. The former is calculated from a Floquet simulation of a parametric squeezing process. The latter is a simple derivative of the static frequencies calculated in part a ^1 / 2^^^^ / ^Φ^. These quantities are equal, up to a scaling factor of 2 (see Example, Section A). FIG. 13C shows inherited dephasing for a coupled quantum memory, as a function of DC flux. Thermal noise- induced dephasing is calculated for a coupler thermal population of 2% and!of 20 "s, and is minimized at the operating point, where the coupler is linear. The low-frequency dephasingdue to inherited flux noise is calculated through Eq.7, with noise amplitude $% =The latter dominates memory coherence but may be suppressed through dynamical decoupling.

[0048] FIGs. 14A-14D are graphs showing properties of the LINC as a driven mixer. FIG. 14A shows LINC beamsplitting strength^)*+^and Kerr^,^^as a function of drive strength (^-^), from an exact Floquet simulation. While the LINC is Kerr-free when idle, higher-order nonlinearities can induce a driven Kerr. This can be suppressed by arraying multiple LINCs while preserving the (driven) flux through each LINC loop (arrayed 3x, dashed). FIG.14B shows driven LINC frequency shift (solid purple) vs drive strength. Unlike in charge-driven mixers like the SNAIL, this frequency shift is independent of the coupler Kerr and persists even if the coupler is arrayed. It is possible to minimize this shift by biasing to a flux point where the coupler frequency has an inflection point as a function of DC flux. FIG. 14C shows steady-state driven purity of the LINC (solid line) and SNAIL (dotted line) atconstant drive amplitude ^^.^ = 0.2^^ as a function of drive frequency. Both couplers areoperated at a bias point where they are Kerr-free, and their parameters are optimized to match their beamsplitting strength and frequency at this bias point. Each coupler is simulated withequal decay, dephasing, and coupling to an external qubit (^ / = 4.9GHz). The LINC displaysa significantly cleaner driven spectrum due to its parity protection. FIG. 14D shows steady- state driven purity of the LINC (solid line) and SNAIL (dotted line) in the presence of twodrive tones (each with ^.^ = 0.1^), displaying the effect of parasitic intermodulation products.The two drive frequencies are scaled simultaneously maintaining a fixed ratio of 2: 3, to easeFloquet-Markov simulations.

[0049] FIGs. 15A-15C show analysis of a LINC circuit. FIG. 15A shows modes of operation of a circuit. The circuit, with an arbitrary inductive element as its center shunt, isAttorney Docket No.: Y0087.70175WO00defined by the three variables 34!, 45, 467. These can be re-written in terms of the moreoperationally relevant charge-dipole symmetric flux (^6̂<=), and anti-symmetric flux (^>̂6<=) modes. FIG. 15B shows LINC three-wave mixing strength as a function of the operating point, comparing analytic formula (Eq. A10) to exact Floquet results from a squeezing operation within the LINC ()+^) and a beamsplitter operation between two external resonators (details in Example, Section B). The discrepancy in the beamsplitting prediction may be due to driven changes in the effective resonator-LINC participations, which are captured in the Floquet simulation but not in the analytics. FIG.15C shows a comparison of the analytic formulae (Eq. A12) to exact Floquet simulation for driven coupler Zeeman shift asa function of DC flux, for fixed drive amplitude ^.^ = 0.1^.

[0050] FIGs.16A-16C show characterization of parametric beamsplitting using a LINC. FIG. 16A shows driven avoided crossing due to the beamsplitter interaction between two storage modes, Alice (4.9 GHz) and Bob (6.0 GHz), as a function of drive detuning frombeamsplitter resonance, at a fixed drive amplitude of ^.^ = 0.2^. FIG. 16B shows drivenfrequency shift of each mode for an unarrayed (solid lines) and arrayed (dashed lines) LINC. Arraying does not make a noticeable difference. FIG.16C shows driven Kerr of each mode for an unarrayed (solid lines) and arrayed (dashed lines) LINC. Arraying three LINCs reduces theinherited Kerr by a factor of ∼ 9.

[0051] FIG. 17A shows dispersion in beamsplitting strength as a function of coupler(undriven) state |0^, … , |3^. The LINC (dotted) has significantly less dispersion than the SNAIL(hatched), which is important since the SNAIL is always driven to a displaced state and has coherent shot noise, but the LINC remains in its undriven ground state. FIG. 17B shows dispersion in the beamsplitting resonance condition as a function of coupler state. The LINC practically has no dispersion at low drive amplitudes, while the SNAIL has an idle dispersive shift to the resonators even when its self-Kerr is set to zero. At larger drives strengths these dispersions become similar. FIG.17C shows steady-state impurity of the driven LINC and the (charge) driven SNAIL, in the presence of decay and flux-noise dephasing, as a function of drive frequency and amplitude. This represents the available drive space for either three-wave mixer without being driven to a mixed state. Any steady state impurity can spoil the parametric process fidelity due to the above state-dependent shifts.

[0052] FIG. 18A shows Kerr of the LINC mode at ^^^ = ^ / 2, computed by directdiagonalization. Black contour lines correspond to ‖DLiNC ‖ = 250, 500, 750, and 1000 kHzrespectively. FIG.18B shows operating point ^Fwhere the LINC is Kerr-free, as a function ofAttorney Docket No.: Y0087.70175WO00 asymmetry. With a minor change in the symmetric DC flux, the Kerr induced by smallasymmetry can be nulled. Black contour lines correspond to flux values from 98% × ^ / 2 to102% × ^ / 2 in 0.5% increments around the asymmetryfree DC operating point. FIG. 18Cshows third-order nonlinearity of the LINC dipole mode, )H^. With finite junction asymmetry,the self-Kerr and cross-Kerr-free points will differ. Contour lines correspond to ‖)H^‖ = 3,6,9,and 12 MHz respectively.

[0053] FIGs. 19A-19C show driven process strength in the presence of asymmetries. FIG.19A shows fractional strength of the desired )5!process in the presence of asymmetries. The strength is primarily dependent upon the asymmetric flux ^J, and is second order with respect to both asymmetries. FIG.19B shows strength of )!!, the linear coupling of the LINC mode to a symmetric flux drive, as compared to the LINC's linear coupling to a charge drive)charge!! ≈ ^^4LMN. FIG. 19C shows parity protection within the third order nonlinearity.Strength of )!5^-5^ , that would permit subharmonic-driving-like processes, as compared to thedesired )5!^-^, is shown as a function of asymmetry for the drive value |^-^| = 0.2^. Theabove simulations are performed by numerically computing appropriate derivatives of the potential and 4LOPas detailed in Eqs. C10 to C12. The behavior of these driven process strengths with respect to asymmetry has been separately verified by Floquet and / or time-domain simulations.

[0054] FIG. 20A shows a schematic circuit diagram for the LINC with parasitic loop inductance (Qloop) and series inductance^QR^. Their effects are analyzed by quantifying theirfractional branch participations ST = Qloop / QU and SM = QR / QV. FIG. 20B shows Kerr at theoperating point ^^^ = ^ / 2 as a function of inductance fractions ST and SM. The seriesinductance has a minimal effect, but a large loop inductance can introduce significant Kerr. FIG. 20C show frequency and Kerr as a function of operating point ^^^, in the presence of parasitic inductances. A series inductor changes the participation of the loop and therefore the frequency tuning range but does not change the Kerr-free point. A loop inductance has minimalchanges to the frequency but can change the LINC's Kerr. For reasonable loop sizes (≤ 100"m,SM ∼ 0.01), the Kerr can be nulled by only a small shift in the operating point.DETAILED DESCRIPTION

[0055] The present application relates to an improved technique for coupling quantum modes stored in quantum multi-level systems to longer-lived bosonic modes stored in quantum harmonic oscillator systems. Quantum multi-level systems (e.g., qubits) exhibit quantum statesAttorney Docket No.: Y0087.70175WO00 that can decohere too quickly for practical quantum computation. It is therefore beneficial to couple a quantum multi-level system to another system that exhibits much longer decoherence times to improve fidelity of the quantum information processing system as a whole. A system configured with bosonic modes may be particularly desirable for coupling to a quantum multi- level system. Through this coupling, the multi-level system’s state may be represented by the bosonic mode(s) instead, thereby maintaining the same information yet in a longer-lived state than would otherwise exist in the multi-level system alone. When used in this manner, the bosonic system is often referred to as a “logical qubit” because it stores information like a qubit, though indirectly and potentially across two or more quantum modes.

[0056] Bosonic systems may additionally be desirable because they are linear quantum systems. Controlling and reading out information from such linear quantum systems may conventionally be performed by coupling non-linear ancillary circuits and / or qubits to the linear quantum systems. As one example, coupling ancilla qubits (e.g., transmon, fluxonium, and / or charge qubits) to one or more resonator cavities (e.g., superconducting and / or microwave resonator cavities) is an excellent architecture for instantiating and manipulating quantum information stored in such desirable bosonic systems. However, one limitation of such an architecture is that the ancilla qubits are generally shorter-lived and can introduce additional errors, such as dephasing, into the quantum states stored in the coupled resonator cavities. This introduction of errors by the ancilla qubits reduces the fidelity of idle storage by the coupled resonator cavities and requires the addition of error correction schema (e.g., error correction codes and / or additional qubit hardware). Mitigating the effects of such ancilla qubits would have a significant impact on practically every aspect of quantum computing with bosonic systems–from enabling more complex continuous variable simulations to improved quantum error correction gain.

[0057] Parametric couplers are devices that may be used to couple ancilla qubits to longer-lived quantum systems such as bosonic systems. Parametric couplers are typically superconducting, non-linear elements that activate a desired non-linear process when driven by electromagnetic drive signals (e.g., microwave and / or radiofrequency signals) with appropriate parameters (e.g., frequency, length, etc.). Specifically, parametric couplers may activate a photon swap process between two quantum modes, where a quantum mode is any quantum information carrier. Examples of systems capable of supporting quantum modes include quantum harmonic oscillators (e.g., cavity resonators), qubits, and couplers. However, parametric couplers typically generate parasitic coherent and incoherent processes at strongAttorney Docket No.: Y0087.70175WO00 drive amplitudes, thereby limiting fidelity of a quantum system by propagating error channels (e.g., coupler decay and dephasing) to the desired photon swap process.

[0058] The inventors have recognized and appreciated that high-fidelity parametric control may be realized by couplers where: (i) parasitic non-linear processes are minimized, and (ii) the coupler acts like a linear quantum system. Accordingly, the inventors have developed a linear inductive coupler configured to couple pairs of quantum modes (e.g., quantum modes stored in quantum harmonic oscillators, ancilla qubits, and / or combinations thereof) when driven by a driving waveform (e.g., an electromagnetic drive waveform such as a microwave and / or RF signal). Additionally, the linear inductive coupler is configured to act like a linear harmonic oscillator when not coupling the pair of quantum modes (e.g., when not being driven by an electromagnetic drive waveform). In this manner, the linear inductive coupler, for the majority of the operation of the quantum system, is in an idle state (e.g., such that the coupler acts as a linear component) that does not inject non-linear errors into the quantum information stored in the quantum modes.

[0059] In some embodiments, the linear inductive coupler includes an inductor coupled in parallel with each of two Josephson junctions such that the inductor is arranged as a shunt. In some embodiments, the two Josephson junctions may have approximately equal Josephson energies.

[0060] In some embodiments, the inductor may be formed of a plurality of Josephson junctions arranged in series (e.g., 5 or more Josephson junctions, 10 or more Josephson junctions, in a range from 5 to 100 Josephson junctions, in a range from 10 to 100 Josephson junctions, in a range from 5 to 80 Josephson junctions, in a range from 5 to 50 Josephson junctions, in a range from 5 to 25 Josephson junctions, in a range from 5 to 10 Josephson junctions, or any suitable range within those ranges).

[0061] While reasonable approximations for the required shunting inductance can be made from an array of Josephson junctions in series, the inventors have appreciated that the linear inductive coupler benefits from having an inductor that is as linear as possible. Therefore, in some embodiments, the inductor may be formed of a high kinetic inductance material. As non-limiting examples, the high kinetic inductance material may be, for example, granular aluminum or niobium nitride.

[0062] In some embodiments, during operation, the linear inductive coupler is configured to be biased by a direct current (DC) magnetic flux approximately equal to one half of the magnetic flux quantum (Φ^ / 2) such that the magnitude of phase drops over each of the two Josephson junctions are approximately equal to ^ / 2. In this operating regime, the linearAttorney Docket No.: Y0087.70175WO00 inductive coupler acts like a linear circuitry component when not driven by an additional applied electromagnetic drive waveform (e.g., a microwave or radiofrequency drive signal). In contrast, in this operating regime and when driven by an applied electromagnetic drive waveform, the linear inductive coupler acts as a nonlinear coupler between quantum modes. The applied electromagnetic drive waveform may be, for example, a microwave or radiofrequency signal provided to a circuitry component disposed adjacent the linear inductive coupler. The applied electromagnetic drive waveform may couple to the linear inductive coupler by mutual inductance such that an effective AC magnetic flux is applied to the linear inductive coupler in addition to the DC magnetic flux.

[0063] In some embodiments, the linear inductive coupler may be coupled between two quantum modes. One of the quantum modes may be, for example, stored in an ancilla qubit (e.g., a transmon, fluxonium, and / or charge qubit), while the other quantum mode may be a bosonic mode stored in a quantum harmonic oscillator (e.g., a microwave resonator cavity). In such embodiments, the linear inductive coupler may be configured to enable interactions (e.g., entanglement, encoding of quantum information in the bosonic mode, reading out from the bosonic mode, etc.) between the bosonic mode and the quantum mode stored in the ancilla qubit. Alternatively or additionally, each of the quantum modes may be, for example, bosonic modes stored in quantum harmonic oscillators (e.g., superconducting and / or microwave resonator cavities). In such embodiments, the linear inductive coupler may be configured to enable interactions (e.g., quantum gates between the two bosonic modes) between the bosonic modes stored in the quantum harmonic oscillators.

[0064] In some embodiments, the linear inductive coupler may be included in a circuit quantum electrodynamic (cQED) system including the quantum devices storing the quantum modes between which the linear inductive coupler may be coupled. The cQED system may optionally include an electromagnetic drive (e.g., a microwave source, a radiofrequency source) configured to differentially drive the linear inductive coupler during operation of the cQED system. As used herein, “differential” driving of the linear inductive coupler is driving in which the linear inductive coupler, in response to the application of an alternating current (AC) electromagnetic drive signal, exhibits approximately equal and opposite AC phase drops across each of the two Josephson junctions disposed in the outer loop of the linear inductive coupler. Additionally, when differentially driven, a phase drop across the central inductor of the linear inductive coupler is approximately zero.

[0065] In some embodiments, when driven by an electromagnetic drive waveform (e.g., as generated by an electromagnetic drive such as a microwave generator), the linear inductiveAttorney Docket No.: Y0087.70175WO00 coupler may be configured to only activate interactions between an odd number of drive photons (e.g., from the driving electromagnetic drive waveform) and an even number of quantum mode photons (e.g., as supported by the quantum modes coupled across the linear inductive coupler). Such interactions suppress parasitic processes typically observed in other parametric couplers and also suppress non-resonant effects like driven frequency shifts.

[0066] In some embodiments, the driving electromagnetic drive waveform has a frequency equal to the difference between the resonant frequencies of the two quantum modes which are being coupled by the linear inductive coupler. In such embodiments, when driven by an electromagnetic drive waveform during operation, the linear inductive coupler is configured to cause three-wave mixing between quantum states stored in the two quantum modes between which the linear inductive coupler is coupled (e.g., between states stored in two quantum harmonic oscillators, or between states stored in at least one quantum harmonic oscillator and an ancilla qubit). In some embodiments, when not driven by a microwave field during operation, the linear inductive coupler is configured to act as a linear oscillator. Thus, the linear inductive coupler is highly linear when idle while suppressing parasitic processes when driven.

[0067] Following below are more detailed descriptions of various concepts related to, and embodiments of, techniques for coupling quantum modes using a linear inductive coupler. It should be appreciated that various aspects described herein may be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only. In addition, the various aspects described in the embodiments below may be used alone or in any combinations and are not limited to the combinations explicitly described herein.

[0068] FIG. 1 is a schematic circuit diagram of a first example of a linear inductive coupler 100, in accordance with some embodiments of the technology described herein. As shown in the example of FIG.1, the linear inductive coupler 100 is a circuit including an outer loop 110 including two Josephson junctions 122 and 124. The Josephson junctions 122 and 124 are each arranged in parallel with an inductor 120. The inductor 120 may act like a shunt.

[0069] In some embodiments, it may be desirable for the inductor 120 to act as linearly as possible (e.g., to approximate a linear harmonic oscillator). Therefore, in some embodiments, the inductor 120 may be formed of a high kinetic inductance material. As non- limiting examples, the high kinetic inductance material may be, for example, granular aluminum (grAl) or niobium nitride. The high kinetic inductance material may be grown in a multi-step fabrication process compatible with the fabrication of the other circuit elements. In embodiments in which the inductor 120 is composed of a high kinetic inductance material, theAttorney Docket No.: Y0087.70175WO00 inductor 120 may have an inductive energy greater than the sum of the Josephson energies of the two Josephson junctions. This may prevent multi-stability and hysteresis within the linear inductive coupler 100.

[0070] In some embodiments, and as shown in the additional example of FIG.2, a linear inductive coupler 200 may include an inductor 220 formed from a plurality of Josephson junctions arranged in series. It should be appreciated that while FIG.2 shows inductor 220 as including four Josephson junctions, the inductor 220 may include more Josephson junctions or fewer Josephson junctions than shown in FIG.2, as aspects of the technology described herein are not limited in this respect. In some embodiments, the inductor 220 may be formed of, as non-limiting examples, 5 or more Josephson junctions, 10 or more Josephson junctions, in a range from 5 to 100 Josephson junctions, in a range from 10 to 100 Josephson junctions, in a range from 5 to 80 Josephson junctions, in a range from 5 to 50 Josephson junctions, in a range from 5 to 25 Josephson junctions, in a range from 5 to 10 Josephson junctions, or any suitable number of Josephson junctions within those ranges. In some embodiments, the inductor 220 may be formed of, as another non-limiting examples, 8, 9, or 10 Josephson junctions.

[0071] In some embodiments, during operation of the linear inductive coupler 100, 200, the linear inductive coupler 100, 200 may be biased by a direct (DC) magnetic flux 130. Themagnetic flux 130 may be approximately equal to Φ^ / 2, where Φ^ = ℎ / ^2Y^ is thesuperconducting magnetic flux quantum, where ℎ is the Planck constant and Y is the electron charge. The magnetic flux 130 may be generated, for example, by the application of a DC electrical signal to a circuit component (e.g., a wire, electronic trace, coaxial or coplanar waveguide, or other means of transmitting a DC electric current) disposed adjacent the linear inductive coupler 100, 200. For example, the circuit component may be disposed on a same chip as the linear inductive coupler 100, 200.

[0072] In some embodiments, the magnetic flux 130 may be configured to cause a phase drop across the Josephson junctions 122 and 124. The phase drops across both of the Josephson junctions 122 and 124 may be approximately equal in magnitude (e.g., equal to ^ / 2). Additionally, the phase drops across both of the Josephson junctions 122 and 124 may be opposite in sign relative to one another. For example, the phase drop across Josephson junction 122 may be approximately equal to ^ / 2 while the phase drop across Josephson junction 124 may be approximately equal to −^ / 2, or vice versa.Attorney Docket No.: Y0087.70175WO00

[0073] In some embodiments, when the linear inductive coupler 100, 200 is biased by magnetic flux 130, the linear inductive couplers 100, 200 may be described by the following Hamiltonian: 55 4̂' = 4^^[\ + ^^ 2 − 2^^ sin ^>: cos 4̂where ^^ is the charging energy provided, for example, by the capacitive pads of the coupler,^^ is the inductive energy of the inductors 120, 220, ^^ is the Josephson energy of eachJosephson junction 122, 124, and ^ is the driven alternating current (AC) flux threading the>:loop. The conjugate charge and phase variables of the circuit are given by [ and 4, respectively.

[0074] The Hamiltonian of the linear inductive coupler can be separated into two deportions: a first undriven portion, 4^ [\5 + ^ c^ ^ 5 , arising from the center inductor 120, 220,and a second driven portion, 2^ sin ^ cos 4̂, arising from the outer loop 110 and Josephson^ >:junctions 122, 124. When the linear inductive coupler is driven by an AC electromagnetic drive waveform, ^ is non-zero such that both the first undriven portion and the second driven>:portion of the Hamiltonian are non-zero. During operation of the linear inductive coupler 100, 200, this is equivalent to an activation of the outer loop 110.

[0075] In some embodiments, the linear inductive coupler 100, 200 is driven by an effective AC magnetic flux, ^ , generated by the application of an electromagnetic drive>:waveform. The applied electromagnetic drive waveform may be, for example, a microwave or radiofrequency signal provided to a circuit component (e.g., a wire, electronic trace, coaxial or coplanar waveguide, or other means of transmitting an AC electromagnetic drive waveform) disposed adjacent the linear inductive coupler. The applied electromagnetic drive waveform may couple to the linear inductive coupler 100, 200 by mutual inductance such that an effective AC magnetic flux is applied to the linear inductive coupler 100, 200 in addition to the DC magnetic flux. In some embodiments, the circuit component used to apply the AC magnetic flux may be the same as the circuit component used to apply the DC magnetic flux. Alternatively, the AC and DC magnetic fluxes may be applied by separate circuit components disposed adjacent to the linear inductive coupler 100, 200.

[0076] However, when the circuit is not driven by an AC electromagnetic drive waveform, ^ is zero, causing the second driven portion of the Hamiltonian to also be equal>:to zero. In practice, this corresponds to a deactivation of the outer loop 110, causing the linear inductive coupler 100, 200 to act like a linear circuit element because the inductor 120, 220 acts as a linear central shunt. This deactivation of the outer loop 110 is different from otherAttorney Docket No.: Y0087.70175WO00 coupler devices because, when undriven, the linear inductive coupler 100, 200 operates with all orders of residual non-linearity simultaneously turned off. In this manner, when undriven and in an idle state, the linear inductive coupler 100, 200 does not propagate non-linear errors into quantum modes within a larger quantum system.

[0077] In some embodiments, the inductor 120, 220 may be disposed in a central manner within the outer loop 110 to create two symmetric subloops (e.g., a first subloop including the inductor 120, 220 and Josephson junction 122 and a second subloop including the inductor 120, 220 and Josephson junction 124). The symmetric arrangement of the circuit components of the linear inductive coupler 100, 200 causes, when magnetic flux 130 is applied, an approximately equal magnetic flux to thread the two subloops. This results in there being no net driven current through the inductor 120, 220, allowing for a separation between the driven portions (e.g., Josephson junctions 122 and 124) and undriven portions (e.g., inductor 120, 220) of the linear inductive coupler 100, 200. Additionally, the symmetric placement of the inductor 120, 220 prevents instabilities during operation that could occur in response to strong drive signals passing through multi-junction branches within the linear inductive coupler 100, 200, as can occur in other coupler architectures. The exact location of the inductor may be calibrated using linear electromagnetic simulations.

[0078] In some embodiments, when the linear inductive coupler 100, 200 is biased by magnetic flux 130 and driven by an AC electromagnetic drive waveform, the non-linearity of the linear inductive coupler 100, 200 is activated in a selective manner. In particular, only processes in which an interaction occurs between an odd number of drive photons (e.g., supplied by the driving AC electromagnetic drive waveform) and an even number of quantum mode photons (e.g., supported by any quantum systems coupled to the linear inductive coupler 100, 200) are allowed to occur. This can be seen in the last term of the Hamiltonian, whichincludes the product sin ^>: cos 4̂, where sine is an odd function and cosine is an even function.This limitation on the kinds of interactions that can be generated strongly suppresses parasitic processes that are typically permitted by other coupler architectures. Additionally, this limitation suppresses any average non-resonant effects (e.g., driven frequency shifts). Any remaining non-linear processes introduced by the linear inductive coupler 100, 200 can be shifted to be off resonance by choosing an appropriate frequency of the electromagnetic drive waveform such that the effects of these non-linear processes are suppressed.

[0079] In some embodiments, the linear inductive coupler 100, 200 may be integrated into a circuit quantum electrodynamic (cQED) system. An example of a schematic of a cQEDAttorney Docket No.: Y0087.70175WO00 system 300 is shown in FIG.3. The cQED system 300 includes a linear inductive coupler 305, which may be either of linear inductive couplers 100 or 200 as described in connection with FIGs.1 and 2 herein. The linear inductive coupler 305 is coupled to a first harmonic oscillator 310, a second harmonic oscillator 320, and an ancilla qubit 330. The first and / or second harmonic oscillator 310 and / or 320 may be, for example, resonator cavities (e.g., superconducting and / or microwave resonator cavities), and the ancilla qubit 330 may be, for example, a transmon qubit, a fluxonium qubit, a charge qubit, and / or any suitable multi-state quantum system.

[0080] In some embodiments, components of the cQED system 300 may be coupled to an energy source 340 configured to generate one or more electromagnetic signals in response to received control signals from controller 350. The one or more electromagnetic signals generated by the energy source 340 may be configured to drive the linear inductive coupler 305 and / or the ancilla qubit 330. For example, the energy source 340 may be configured to generate one or more electromagnetic signals (e.g., AC electromagnetic signals) configured to drive the linear inductive coupler 305, as described herein, to cause coupling between various components of the cQED system 300. The energy source energy source 340 may alternatively or additionally be configured to generate one or more electromagnetic signals configured to drive and / or cause readout from the ancilla qubit 330. In some embodiments, the one or more electromagnetic signals may be microwave or radio frequency signals.

[0081] In the arrangement of FIG. 3, the linear inductive coupler 305 may be used to implement bosonic control and readout from the first and second harmonic oscillators 310, 320. In particular, the linear inductive coupler 305 may be arranged to, when biased by a suitable DC magnetic flux and driven by an AC electromagnetic drive waveform, cause a dipole coupling between pairs of the first and second harmonic oscillators 310, 320 and / or ancilla qubit 330 (e.g., coupling between the first and second harmonic oscillators 310, 320; coupling between the first harmonic oscillator 310 and the ancilla qubit 330; or coupling between the second harmonic oscillator 320 and the ancilla qubit 330). The top panel of FIG.4 shows the mechanism by which this coupling occurs, in accordance with some embodiments described herein. When driven, the outer loop 110 of the linear inductive coupler 305 is activated, generating non-linear inductive coupling between the linear inductive coupler 305 and pairs of quantum modes in the cQED system 300.

[0082] In some embodiments, when the linear inductive coupler 305 is in a driven state, the linear inductive coupler 305 may couple any two of the first harmonic oscillator 310, the second harmonic oscillator 320, and / or the ancilla qubit 330 to one another. As one example,Attorney Docket No.: Y0087.70175WO00 when in a driven state, the linear inductive coupler 305 may couple the first harmonic oscillator 310 to the second harmonic oscillator 320. Alternatively or additionally, when in a driven state, the linear inductive coupler 305 may couple the first harmonic oscillator 310 to the ancilla qubit 330 and / or the second harmonic oscillator 320 to the ancilla qubit 330. In this manner, the linear inductive coupler 305 may be used to implement bosonic control of the first and second harmonic oscillators 310, 320 by strategically coupling either: (i) a quantum mode stored in the ancilla qubit 330 to quantum modes stored in either of the first and second harmonic oscillators 310 and 320, and / or (ii) quantum modes stored in the first and second harmonic oscillators 310 and 320 (e.g., to perform a quantum gate between the quantum modes stored in the first and second harmonic oscillators 310 and 320).

[0083] In some embodiments, when not driven by a suitable AC electromagnetic drive waveform, the linear inductive coupler 305 may act as a linear oscillator such that the quantum modes of the first harmonic oscillator 310, the second harmonic oscillator 320, and / or the ancilla qubit 330 remain uncoupled while the linear inductive coupler 305 is in an idle state. The bottom panel of FIG. 4 illustrates the effective circuit diagram of the cQED system 300 when the linear inductive coupler 305 is undriven. In this case, the driven AC flux threading the loop is zero such that the outer loop 110 of the linear inductive coupler 305 is deactivated, causing the linear inductive coupler 305 to act approximately like a linear circuit component due to the shunting caused by the central inductor (e.g., inductor 120 and / or 220). There is approximately zero coupling between any of the linear inductive coupler 305, the first harmonic oscillator 310, the second harmonic oscillator 320, and / or ancilla qubit 330.

[0084] In some embodiments, when the linear inductive coupler 305 is undriven, the static coupling between either of the first harmonic oscillator 310 or the second harmonic oscillator 320 and the ancilla qubit 330 is additionally strongly suppressed by geometric separation. This suppression of static coupling protects the harmonic oscillators 310, 320 from injection of ancilla errors that may occur in the ancilla qubit 330. Furthermore, ancilla operations that are independent of the harmonic oscillators 310, 320 (e.g., unselective gates, readout, etc.) can be performed using the ancilla qubit 330 without affecting the quantum modes stored in the harmonic oscillators 310, 320.

[0085] Multi-oscillator gates (e.g., a parametric SWAP operation between the harmonic oscillators 310, 320 causing the photon(s) stored in each of the harmonic oscillators 310, 320 to swap positions between the harmonic oscillators 310, 320) can also be activated by driving the linear inductive coupler 305 at approximately the frequency difference of the harmonicAttorney Docket No.: Y0087.70175WO00 oscillators 310, 320, in some embodiments. Such quantum gates do not suffer from ancilla errors that are conventionally observed in cQED systems using other coupling circuitry.

[0086] In some embodiments, when driven with a suitable AC electromagnetic drive waveform, the linear inductive coupler 305 is configured to cause a three-wave mixing process resulting in parametric coupling between two quantum modes. To cause three-wave mixing, the drive frequency of the AC electromagnetic drive waveform may be approximately equal to a difference between the resonant frequencies of the two quantum modes being coupled. This may be derived from the Hamiltonian of the linear inductive coupler 305 when it is driven by an electromagnetic drive waveform:where )>,nare the static dipole coupling strengths between participating quantum modes (labelled o and q), Δ>,nare the frequency detunings between the linear inductive coupler 305 and each of the participating quantum modes, andand o (or qpand q) are the raising and lowering operators, respectively, for each quantum mode. This driven process is a photon-swap operation (e.g., causing the swapping of photons between each of the participating quantum modes). The photon-swap operation can be configured to be implemented on-resonant or off- resonant as desired (e.g., by selecting a frequency of the electromagnetic drive waveform to be resonant or not resonant with the photon swap process), with the strength of the photon-swap operation scaling linearly with the drive amplitude, ^>:.

[0087] In some embodiments, the AC electromagnetic drive waveform, when applied, may drive the linear inductive coupler 305 differentially. The linear inductive coupler, in response to the application of the AC electromagnetic drive signal, exhibits approximately equal and opposite AC phase drops across each of the two Josephson junctions disposed in the outer loop of the linear inductive coupler. Additionally, when differentially driven, a phase drop across the central inductor of the linear inductive coupler is approximately zero.

[0088] FIG.5 is an illustrative diagram showing dipole coupling within the cQED system 300 of FIG. 3, in accordance with some embodiments of the technology described herein. In the example of FIG.5, )>n^s^ is the static dipole coupling strength between the quantum modes stored in harmonic oscillator 310 and harmonic oscillator 320, t>^s^ is the static dipole coupling strength between the quantum modes stored in harmonic oscillator 310 and ancilla qubit 330, and tn^s^ is the static dipole coupling strength between the quantum modes stored in harmonic oscillator 320 and ancilla qubit 330. A microwave readout strip 510 is coupled to ancilla qubit 330 to enable readout from the cQED system 300. As shown in FIG.5, couplingAttorney Docket No.: Y0087.70175WO00 a non-linear mode (e.g., the ancilla qubit 330) and a linear mode (e.g., either of the harmonic oscillators 310, 320) creates an effective dispersive interaction between the coupled quantum modes. This dispersive interaction allows the control of the linear mode using cQED techniques, and the linear inductive coupler permits this dispersive interaction only when driven appropriately.

[0089] In some embodiments, the external magnetic flux (e.g., magnetic flux 130 and an applied AC electromagnetic drive waveform) may be delivered in a differential manner (e.g., the voltage across the two Josephson junctions 122, 124 is of an equal magnitude and opposite phase) to preserve the selection rules of the linear inductive coupler. This external magnetic flux delivery may contain a DC component, statically biasing the linear inductive coupler circuit at a phase of ^ / 2. In general, “fast flux delivery” in a superconducting, high-Q environment is a difficult task. Accordingly, the inventors have developed a coaxial on-chip filter design, shown in FIG.6, that enables the differential delivery of external magnetic flux. FIG. 6 shows an example of a chip 600 including a linear inductive coupler 305 (which may be either of linear inductive couplers 100 or 200), harmonic oscillators 310 and 320, and ancilla qubit 330, in accordance with some embodiments of the technology described herein.

[0090] In some embodiments, the chip 600 further includes an on-chip filter 610 configured to deliver the AC and DC components of the differential magnetic flux. The on- chip filter 610 also is configured to perform band-stop filtering at or around the resonant frequencies of the linear inductive coupler 305, the ancilla qubit 330, and the harmonic oscillators 310 and 320. The chip 600 supports lithographically defined frequencies and couplings.

[0091] In some embodiments, the linear inductive coupler 305 is directly adjacent to the on-chip filter 610 and couples to the on-chip filter 610 through mutual inductance. The linear inductive coupler 305 also includes two coupling arms configured to allow for the dipole coupling to pairs of quantum modes.

[0092] In some embodiments, the ancilla qubit 330 and microwave readout strips 510 are also patterned onto chip 600, with the ancilla qubit 330 being couplable to the linear inductive coupler 305 but not to the harmonic oscillators 310 and 320. Additionally, the microwave readout strips 510 are coupled to ancilla qubit 330 but not to the linear inductive coupler 305. The chip architecture of chip 600 allows for the condensing of important control and readout elements onto a single chip. Additionally, because the ancilla qubit 330 and the microwave readout strips 510 lie on a single chip in a coaxial geometry that may be inserted between twoAttorney Docket No.: Y0087.70175WO00 bosonic modes, the chip 600 may be tiled in two dimensions to enable coupling to multiple pairs of harmonic oscillators.

[0093] FIG.7 is a plot of a simulated coupling rate as a function of the drive amplitude for a photon-swap operation between two quantum modes (e.g., stored in two harmonic oscillators). The simulation was performed for a driving frequency matching a difference between the resonant frequencies of the two quantum modes. Additionally, the numerical values depicted in FIGs. 7-9 were derived using Floquet simulations with the following parameters: a linear inductive coupler resonant frequency of 6.5 GHz; a static dipole coupling strength between the linear inductive coupler and a first harmonic oscillator of 0.12; a frequency detuning of the linear inductive coupler and the first harmonic oscillators of -1.6 GHz; a static dipole coupling strength of the linear inductive coupler and a second harmonic oscillator of 0.05; a frequency detuning of the linear inductive coupler and the second harmonic oscillator of -0.5 GHz; a charging energy of approximately 100 MHz; and a Josephson energy associated with the Josephson junctions that is approximately 0.3 times the inductive energy of the inductor.

[0094] Curve 700 depicts how the strength of the photon-swap operation scales linearly with the drive amplitude. In contrast, most parametric couplers become unpredictable at very strong drive amplitudes, where the coupler can get excited to highly energetic or unconfined states. The inductive shunt in the linear inductive coupler described herein prevents such ionization to “out-of-well” states, and the linear inductive coupler is simulated to remain in its ground state at drive amplitudes greater than those allowed in the ancilla qubit.

[0095] FIG. 8 is a simulated plot of the driven coupler shift as a function of the drive amplitude applied to the linear inductive coupler, in accordance with some embodiments of the technology described herein. As depicted in FIG. 8, curve 802 illustrates how the simulated driven coupler shift frequency changes as a function of the drive amplitude, with a linearly decreasing coupler shift for larger drive amplitudes. Curve 804 illustrates how the simulated driven coupler shift Kerr frequency increases as a function of the drive amplitude. Curve 806 describes how the simulated driven coupler shift Kerr (arrayed) frequency remains almost constant as a function of the drive amplitude. The comparison of curves 802, 804, and 806 shows how, when driven, the linear inductive coupler suppresses driven frequency shifts. In particular, as described above, the linear inductive coupler, when driven, allows only processes that consist of an odd number of drive photons interacting with an even number of quantum mode photons. This strongly suppresses a large number of parasitic processes conventionally permitted by other coupler architectures.Attorney Docket No.: Y0087.70175WO00

[0096] FIG. 9 is a plot of the simulated steady state impurity of the linear inductive coupler as a function of driven frequency and drive amplitude, in accordance with some embodiments of the technology described herein. The simulations to generate FIG. 9 were performed assuming the presence of realistic coupler decay and dephasing in the cQED system. The steady state impurity of the linear inductive coupler is a proxy for a measure of the linear inductive coupler’s state (e.g., whether it has remained in its ground state), even when driven. Thus, the plot in FIG.9 illustrates how the linear inductive coupler remains in its ground state at large drive amplitudes and when driven at frequencies below the linear inductive coupler’s mode frequency.

[0097] FIG.10 is a plot depicting an illustrative pulse scheme 1000 as might be used to perform a quantum operation using cQED system 300, in accordance with some embodiments of the technology described herein. In some embodiments, the first harmonic oscillator 310 has a resonant frequency, ^>, the second harmonic oscillator 320 has a different resonant frequency, ^n, and the ancilla qubit 330 has a resonant frequency, ^ / .

[0098] The example of FIG. 10 shows two lines of pulses (e.g., electromagnetic drive signals) applied at different times. The top line indicates pulses applied to drive the linear inductive coupler 305, and the bottom line indicates pulses applied to drive the ancilla qubit 330. The example of FIG. 10 shows three pulses 1002, 1004, and 1006 where the linear inductive coupler is driven. In the first pulse 1002, the linear inductive coupler may be drivenat a frequency ^> − ^n, causing coupling between the first harmonic oscillator 310 and thesecond harmonic oscillator 320. This coupling between the first harmonic oscillator 310 and the second harmonic oscillator 320 implements a SWAP gate between the first harmonic oscillator 310 and the second harmonic oscillator 320, causing an exchange of photons between the first harmonic oscillator 310 and the second harmonic oscillator 320.

[0099] In the example of FIG. 10, the second pulse 1004 drives the linear inductivecoupler at a frequency of ^> − ^ / , causing coupling between the first harmonic oscillator 310and the ancilla qubit 330. This coupling between the first harmonic oscillator 310 and the ancilla qubit 330 causes an interaction between the photons stored in the first harmonic oscillator 310 and the photons stored in the ancilla qubit. Additionally, a drive signal 1012 may be applied to the ancilla qubit 330 to enable readout from and / or the application of a quantum gate to the first harmonic oscillator 310.

[0100] In the example of FIG.10, the third pulse 1006 drives the linear inductive couplerat a frequency of ^n − ^ / , causing coupling between the second harmonic oscillator 320 andAttorney Docket No.: Y0087.70175WO00 the ancilla qubit 330. This coupling between the second harmonic oscillator 320 and the ancilla qubit 330 causes an interaction between the photons stored in the second harmonic oscillator 320 and the photons stored in the ancilla qubit 330 (e.g., photons that may have been acquired from the first harmonic oscillator 310 in the previous interaction caused by pulse 1002). Additionally, a drive signal 1014 may be applied to the ancilla qubit 330 to enable readout from and / or the application of a quantum gate to the second harmonic oscillator 320. In this manner, the linear inductive coupler enables the control of a cQED system including bosonic modes and ancilla modes.

[0101] An illustrative implementation of a computer system 1100 that may be used in connection with any of the embodiments of the technology described herein (e.g., such as the system of FIG. 3) is shown in FIG. 11. The computer system 1100 includes one or more processors 1110 and one or more articles of manufacture that comprise non-transitory computer-readable storage media (e.g., memory 1120 and one or more non-volatile storage media 1130). The processor 1110 may control writing data to and reading data from the memory 1120 and the non-volatile storage device 1130 in any suitable manner, as the aspects of the technology described herein are not limited to any particular techniques for writing or reading data. To perform any of the functionality described herein, the processor 1110 may execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., the memory 1120), which may serve as non-transitory computer-readable storage media storing processor-executable instructions for execution by the processor 1110.

[0102] Computing system 1100 may also include a network input / output (I / O) interface 1140 via which the computing device may communicate with other computing devices (e.g., over a network), and may also include one or more user I / O interfaces 1150, via which the computing device may provide output to and receive input from a user. The user I / O interfaces may include devices such as a keyboard, a mouse, a microphone, a display device (e.g., a monitor or touch screen), speakers, a camera, and / or various other types of I / O devices.

[0103] The above-described embodiments can 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 can be executed on any suitable processor (e.g., a microprocessor) or collection of processors, whether provided in a single computing device or distributed among multiple computing devices. It should be appreciated that any component or collection of components that perform the functions described above can be generically considered as one or more controllers that control theAttorney Docket No.: Y0087.70175WO00 above-discussed functions. The one or more controllers can be implemented in numerous ways, such as with dedicated hardware, or with general purpose hardware (e.g., one or more processors) that is programmed using microcode or software to perform the functions recited above.

[0104] In this respect, it should be appreciated that one implementation of the embodiments described herein comprises at least one computer-readable storage medium (e.g., RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or other tangible, non-transitory computer- readable storage medium) encoded with a computer program (i.e., a plurality of executable instructions) that, when executed on one or more processors, performs the above-discussed functions of one or more embodiments. The computer-readable medium may be transportable such that the program stored thereon can be loaded onto any computing device to implement aspects of the techniques discussed herein. In addition, it should be appreciated that the reference to a computer program which, when executed, performs any of the above-discussed functions, is not limited to an application program running on a host computer. Rather, the terms computer program and software are used herein in a generic sense to reference any type of computer code (e.g., application software, firmware, microcode, or any other form of computer instruction) that can be employed to program one or more processors to implement aspects of the techniques discussed herein.

[0105] The foregoing description of implementations provides illustration and description but is not intended to be exhaustive or to limit the implementations to the precise form disclosed. Modifications and variations are possible in light of the above teachings or may be acquired from practice of the implementations. In other implementations the methods depicted in these figures may include fewer operations, different operations, differently ordered operations, and / or additional operations. Further, non-dependent blocks may be performed in parallel.

[0106] It will be apparent that example aspects, as described above, may be implemented in many different forms of software, firmware, and hardware in the implementations illustrated in the figures. Further, certain portions of the implementations may be implemented as a “module” that performs one or more functions. This module may include hardware, such as a processor, an application-specific integrated circuit (ASIC), or a field-programmable gate array (FPGA), or a combination of hardware and software.Attorney Docket No.: Y0087.70175WO00

[0107] Having thus described several aspects and embodiments of the technology set forth in the disclosure, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the embodiments described herein. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation many equivalents to the specific embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described. In addition, any combination of two or more features, systems, articles, materials, kits, and / or methods described herein, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the scope of the present disclosure.

[0108] The above-described embodiments can be implemented in any of numerous ways. One or more aspects and embodiments of the present disclosure involving the performance of processes or methods may utilize program instructions executable by a device (e.g., a computer, a processor, or other device) to perform, or control performance of, the processes or methods. In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, 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 one or more of the various embodiments described above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computers or other processors to implement various ones of the aspects described above. In some embodiments, computer readable media may be non-transitory media.

[0109] The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects as described above. Additionally, it should be appreciated that according to one aspect, one or more computerAttorney Docket No.: Y0087.70175WO00 programs that when executed perform methods of the present disclosure need not reside on a single computer or processor but may be distributed in a modular fashion among a number of different computers or processors to implement various aspects of the present disclosure.

[0110] Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0111] Also, data structures may be stored in computer-readable 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 computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements.

[0112] When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.

[0113] Further, it should be appreciated that 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, as non-limiting examples. 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 smartphone, a tablet, or any other suitable portable or fixed electronic device.

[0114] Also, a computer may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can 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 can 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 formats.

[0115] Such computers may be interconnected by one or more networks in any suitable form, including a local area network or a wide area network, such as an enterprise network, and intelligent network (IN) or the Internet. Such networks may be based on any suitableAttorney Docket No.: Y0087.70175WO00 technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.

[0116] Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. EXAMPLE

[0117] Driven superconducting circuits provide a platform for studying complex nonlinear quantum optics and manipulating quantum information in the microwave domain. The intrinsic nonlinearity of Josephson junctions has been utilized to design numerous circuit elements, ranging from quantum-limited amplifiers, novel nonreciprocal devices, ultra-low- noise detectors, and, most notably, quantum information processing through the field of circuit Quantum Electrodynamics (cQED). A significant portion of these applications are enabled by nonlinear quantum mixers that activate specific desired processes when parametrically driven with a microwave drive. Such parametric mixers have found use as couplers, amplifiers, and even nonlinear oscillators in their own right.

[0118] Ideally, one desires fast high-fidelity operations that manipulate sensitive quantum information without introducing errors. This requires the driven quantum mixer to turn on desired interactions at high strength, without activating parasitic processes such as drive-induced frequency shifts and leakage to uncontrolled states. A further complication is the finite coherence of the quantum mixer, where any unintended excitation of the mixer out of its ground state can lead to mixer decoherence dominating the infidelity of the parametric process. In particular, when the mixer is used as a coupler between high-Q modes, it ideally activates a parametric coupling between the modes on demand, while not reducing both their idle and driven coherence. One thus aims to engineer the mixer such that it is minimally invasive when idling and that it also remains in its ground state and prevents parasitic processes when driven.

[0119] A common approach to mitigate some parasitic effects is to use a “Kerr-free three wave-mixer,” where one utilizes a third-order nonlinearity to generate the desired process with a single drive tone, while nulling the fourth-order (Kerr) nonlinearity. Suppressing the Kerr nonlinearity suppresses spurious processes like the AC Stark shift, which can lead to reduced saturation power in the amplifier context, or frequency collisions with other sensitive modes in the coupler context. However, for high-fidelity applications, this may still be insufficient inAttorney Docket No.: Y0087.70175WO00 such fragile systems, as the loss of even a single photon to some uncontrolled degree of freedom can ruin sensitive quantum information. In particular, at strong drive powers, even Kerr-free mixers are inevitably spoiled by the onset of numerous transitions induced by nonlinearities beyond the fourth order. The effect of these transitions is especially evident at drive powers where the mixer ionizes into higher-lying states or enters a chaotic regime, irreversibly spoiling the system's information.

[0120] To suppress these higher-order nonlinear processes, certain symmetries can be leveraged in a mixer design that further improves their performance by imposing an explicit selection rule on the types of allowed processes that the mixer can activate. These design principles mimic mixer balancing in the classical electrical engineering context, where for example, a single-balanced mixer utilizes symmetries of both the circuit and the drive delivery to coherently suppress approximately half the undesired mixing products (see FIGs.12A-12B). This significantly increases the amplitude and frequencies at which the mixer can be driven without spoiling fidelity, especially in the presence of drive-induced shifts. Such symmetries have been put to use before in circuit QED, both in the amplifier context (for e.g., with the Josephson parametric coupler) as well as in high-Q control. Particularly in the parametric coupler context, employing this symmetry in four-wave mixing with a Differentially Driven SQUID (DDS) has shown significant improvement over regular four-wave mixing with a single junction.

[0121] Utilizing similar design principles to the DDS, a mixer is introduced herein that has an identical Hamiltonian to a balanced flux-biased RF-SQUID pair (FIG. 12A) but can manifest as a simple dipole element (FIG.4). When operated at a particular DC bias point, idle effects due to the Josephson junctions disappear, and the mixer is exceptionally linear. When driven, the outer SQUID loop activates a three-wave mixing process that obeys a strict selection rule called parity protection, forbidding a significant fraction of the remaining parasitic processes. This means that the mixer is not just Kerr-free when driven, but the Kerr-free bias point also coincides with the extinguishment of all even-order nonlinearity simultaneously. I. THE LINEAR INDUCTIVE COUPLER

[0122] A new mixer is presented herein, highlighting the advantages that emerge from its inherent symmetry. To start, recall the Differentially Driven SQUID (DDS), a symmetric superconducting loop that contains two junctions, such that the drive always causes an equal and opposite phase drop across the two junctions. Imposing this symmetry on the driven circuit results in the coherent cancellation of half of the nonlinear resonances allowed in a single junction circuit, significantly cleaning up the parasitic processes that this mixer can activate,Attorney Docket No.: Y0087.70175WO00 even when strongly driven. However, this coupler still retains its idle Kerr nonlinearity and has significant driven frequency shifts.

[0123] The improved circuit proposed herein shunts a DDS with a linear inductor, and biases the circuit at a constant external DC magnetic field of half a flux quantum threading the SQUID loop. The inductive shunt is placed in a precise manner such that the drive across it exactly cancels when the outer SQUID loop is differentially driven (see FIG.4). Biasing this circuit at half flux would generally set its frequency to zero in the absence of the linear inductor, since the junctions in its outer loop are biased to effectively infinite Josephson inductance. However, the presence of the shunting inductor stabilizes the circuit and sets the mixer's frequency when undriven.

[0124] More precisely, the Hamiltonian of the mixer can be written in terms of its two available degrees of freedom under the above symmetry constraints, the flux-induceddifferential phase across each junction ^ = and the common modephase across the circuit 4̂ that is conjugate to the charge [̂ on its capacitive island:Following standard notation in circuit quantum electrodynamics, ^^denotes the circuit's charging energy, ^^the Josephson energy of each junction in the outer loop, ^^the inductive energy of the shunting inductor, and ℏ has been set to 1. Note that any static charge offsets are redistributed by the shunting inductor, and hence absent in the Hamiltonian. For the circuit's potential energy to have a single-valued solution at all values of DC flux, the shunt's inductiveenergy must dominate the circuit ^ > 2^ , which is equivalent to its frequency never crossing^ ^zero (FIG.13A).

[0125] An analysis of the circuit's behavior, both when idle and when driven, followsbelow. At the special flux point of ^ = u^^ 5, the nonlinearity of the circuit completelydisappears when idle, resulting in a completely linear static Hamiltonian:This is beyond a simple 'Kerr-free' mixer—at this flux point, the circuit's static nonlinearity at all orders completely disappears. It will be shown later that this property holds even in the presence of experimental imperfections like a parasitic linear inductance, which is often present in realistic mixers due to geometric constraints. The mixer hence provides a completely linear environment to any modes it mediates a coupling between in the coupler context, which isAttorney Docket No.: Y0087.70175WO00 particularly useful for bosonic control. This mixer is thus named the “Linear INductive Coupler” (LINC).

[0126] When the LINC is driven, the drive exactly cancels across the shunting inductor by design and does not displace the common-mode phase. The LINC's Hamiltonian is thus separable into two non-interacting parts, the undriven and the driven, as shown in Eq.1. The driven portion contains beneficial symmetries due to the balancing of the drive, enforcing a selection rule on the allowed nonlinear processes in a manner similar to the DDS. Only processes where the number of coupler and resonator photons are even can be activated. Even within these, processes that are even-order in the drive, like the drive-induced frequency shift, are suppressed. This is in stark contrast to prevalent mixers like the SNAIL, where nominally every process can occur, but individual nonlinearities can be manually tuned to zero at specific flux bias points. It will be shown that this selection rule significantly expands the frequencies and strengths at which the LINC can be driven without causing spurious transitions and reduces inter-modulation products in the presence of multiple drive tones. At leading order, the LINC functions as a three-wave mixer, activating either a beamsplitting (red sideband / conversion) or squeezing (blue sideband / gain) process when driven with a single tone of appropriate frequency. II. IDLE NONLINEARITY AND DECOHERENCE

[0127] While the well-controlled mixing properties of the LINC make it advantageous for a number of applications, the LINC particularly shines as a coupler for high-Q bosonic quantum control. In this context, the LINC activates a parametric coupling to one or more resonators (bosonic modes) when driven, and minimally spoil (e.g., minimally introduce errors into) their quantum information when undriven. Since the resonators in isolation are linear and only incur slow single-photon decay errors (at rate ^res), an ideal bosonic coupler must neither limit this decay rate, nor introduce any additional nonlinearity or dephasing to the resonators. Suppressing the decay inherited from the coupler limits the allowable energy participation ofthe resonator in the LINC mode, which is assumed herein to be ^res = 0.01. Minimizing theresonator nonlinearity and dephasing bounds the acceptable static nonlinearity of the coupler. It is important to note that, while a four-wave mixer like the transmon or DDS induces resonator dephasing through the cross-Kerr (dispersive) interaction, a three-wave mixer like the SNAIL or LINC suppresses such dephasing but can introduce low-frequency fluctuations from flux noise. The effect of both the LINC's linearity and propagated flux noise is analyzed below.Attorney Docket No.: Y0087.70175WO00

[0128] The ideal LINC's nonlinearity goes to zero at the operating point, as shown in FIG.13A. The curious shape of this curve can be explained entirely analytically by considering the flux dependence of the fourth-order nonlinearity (see Section A):Note that the un-driven LINC has no third-order nonlinearity at any bias point ^^^^^, and thus has no perturbative corrections to the coupler's idle Kerr. This means that the LINC's self-Kerr and cross-Kerr to coupled modes always simultaneously go to zero at the same operating point. The LINC's frequency can also be simply calculated through:These analytical results are derived in Section A and are shown overlayed with exact numerical diagonalization in FIG. 13A. These analytics describe static physics near the operating pointexceptionally well and only deviate from numerical solutions around Φ = Φ , where the^^ ^truncated Taylor expansion does not accurately represent the coupler's nonlinearity.

[0129] If instead of an ideal inductor, the LINC's shunt is composed of an array of ^ Josephson Junctions, its static Hamiltonian at the operating point is given by:where each junction in the shunting array has been scaled appropriately to preserve the sametotal inductance 8^^,^ = ^^^;. To keep the effect of any inherited resonator nonlinearitysmaller than the resonator's linewidth, the number of junctions in the LINC shunt should begreater than ^^ 5 [‾^ / ^^^res ^ e, for an intended resonator population of [‾ . However, note that theLINC behaves as a protected three-wave mixer even if it is not this linear, including when its shunting inductor is just a single Josephson Junction.

[0130] While nulling the LINC's nonlinearity suppresses shot-noise-induced dephasing, a major alternative source of dephasing in any LINC architecture will be low-frequency flux noise. In fact, since the LINC is a flux-driven three-wave mixer, there is an exact trade-off in the sensitivity of the coupler to flux noise and the effective three-wave mixing strength, )H^^(see FIG. 13B). This means that at the operating point of ^ = ^ / 2, both the ) and flux-^^ H^^noise sensitivity are close to maximum (i.e., at the anti-sweet spot). However, since the LINC itself remains in the ground state, this dephasing is primarily harmful if it propagates errorsAttorney Docket No.: Y0087.70175WO00 into the desired parametric process, or to the modes that are statically hybridized with the LINC. The effect of flux noise on the strength of the parametric process can be estimated by the simplified infidelity limit:Here is the spectral density of flux noise, )¨©§ª«is a characteristic function defined by the pulse sequence, and the flux-noise sensitivity is analytically derived (see Section A). For an upper bound on the infidelity, )¨is approximated as a rectangular window between therelevant timescales of a single gate 8ª« ∼ 1 ns) and the total experiment (ªwx² ∼ 1 s). Note thatthe quadratic dependence on flux-noise sensitivity assumes that the variance in the strength of )H^^is slow and small and therefore appears as a coherent offset to the intended pulseevolution. For 1 / § type noise^§^ ∼ $5% / §, typical values of the noise amplitude $% ∼at 1 Hz^41 − 43^ lead to an estimate of 1 − ℱ ∼ 10³!^M , which means thismechanism should not be a limiting factor in the mixer's performance.

[0131] A more relevant effect might be the inherited flux noise in coupled information- storing modes, which is estimated analytically in FIG.13C. This inherited dephasing scales ∝ ^µw^:where º = ∼ 3 − 5 is a slow time dependence on whenis^ª^, and the total length of the experiment 8ªwx²;. The inherited dephasing can therefore be significant, but it is low frequency, which means it could be mitigated with techniques like dynamical decoupling or stabilized bosonic codes. This makes it potentially still beneficial to operate near the anti-sweet spot, where the coupled resonator is sensitive to inherited flux noise but not to thermal noise-induced dephasing, since the latter requires non-trivial strategies for suppression. III. THE LINC AS A DRIVEN MIXERAttorney Docket No.: Y0087.70175WO00

[0132] Next, the driven behavior of the LINC and its performance as a balanced quantum mixer was investigated. With ideal symmetry, the LINC's driven behavior is independent of its center shunt (up to a normalization of its impedance), and is given by:There are two important points to note about this driven Hamiltonian. The first is that the drive, ^-^, acts on an entirely orthogonal degree of freedom to the LINC mode. This means that the drive does not displace the mode, and the LINC in general remains in its undriven ground state, similar to the DDS. In stark contrast to charge-driven mixers like the transmon or the SNAIL, this decouples the effective drive strength from the frequency of the LINC mode, allowing for independent optimization of the drive delivery and LINC frequency.

[0133] Second, the order of the allowed parametric processes obeys a strict selection rule—the only processes allowed are of the type ^-=^ 4̂j, where [ is strictly even. This selection rule is called “parity protection,” analogous to the selection rule in the DDS. Interestingly, the LINC has this parity protection at arbitrary operating points, even though it is only linear at^^^ = ^ / 2. Note that despite the drive amplitude appearing as an odd function in theHamiltonian, there is only a weak parity protection in the number of drive photons ¾. This isbecause modulating the drive parameter ^.^ can also modulate the spread of the wave-function4LMN, causing non-trivial corrections to the strength of parametric processes. Additionally, odd-order processes can combine at higher orders in perturbation theory to form even-order processes. The simplest non-trivial effect where this is observable is in the driven frequency shift of the LINC, which is suppressed but non-zero at the operating point (see FIG. 14B). However, since the LINC is a “true” parametric coupler, all such effects are well-predicted by measuring static properties of the LINC as a function of the parameter (^^^) and computing appropriate derivatives. This is explored in more detail in Section A.

[0134] The dominant mixing process in the driven LINC is an effective 3-wave mixing process, which for small drive strengths is given by:where ^^ = Á8^^^^ is the frequency of the LINC mode, À and Àpare its ladder operators,¿!^Â^ is the first-order Bessel function, and ^-^^s^ = |^-^|costhe drive. For twoAttorney Docket No.: Y0087.70175WO00modes Alice ^ô, ^>^ and Bob8q̂, ^n; that are coupled to the LINC (with energy participations^> and ^n), driving at select frequencies can activate various desired processes, such as:with ^> = ^n for the single-mode squeezing process. The strength of the first of these(beamsplitting) has been shown in FIG.14A for illustrative purposes.

[0135] While the LINC is linear when un-driven, turning on a drive can induce some nonlinearity in the coupler due to higher-order effects (FIG.14A). This finite coupler Kerr is not directly harmful, as the LINC remains in its ground state when driven, but it can induce undesired inherited Kerr in coupled linear quantum modes. Since the Kerr arises from higher- order nonlinear effects, it can be suppressed simply by arraying multiple LINCs while keeping the strength of the desired process constant. This results in the following Hamiltonian, assuming each LINC loop is still threaded by the same flux (see Section A):^cos Éwhere the inductance of each LINC in the array has been scaled down to preserve the same total inductance. This suppresses the driven Kerr of the LINC by a factor of 1 / Ê5, as shown in FIG.14A, and all remaining undesired processes are similarly suppressed. In fact, with a large array of LINCs, this circuit element approaches the ideal three-wave mixing of a modulated linear inductor:The beamsplitter performance of both the arrayed and un-arrayed LINC is analyzed in more detail in Section B.

[0136] Even an ideal three wave mixer incurs drive-induced frequency shifts, as can beseen from the average frequency of the above element ≠ 0.fact, in contrast to a charge-driven element like the SNAIL, the driven frequency shift of the LINC is unaffected by arraying (see FIG.14B) and is not related to the coupler's anharmonicity.It is instead given by the susceptibility of the LINC frequency to change in the flux, Δ^^ ∝^5^ / ^^5|¢e. One can minimize this driven shift (for example, when using the LINC in aAttorney Docket No.: Y0087.70175WO00parametric amplifier) by finding the inflection point ^5^ / ^^5^^^^^ = 0, which for theperfectly symmetric LINC (with ^^ / ^^ = 0.3 ) occurs at ^^^ ∼ 0.594^ (dotted line in FIG.14B). However, it is noted that this operating point is not parity-protected and therefore might incur other parasitic processes that may be less desirable than the coupler frequency shift.

[0137] To demonstrate how parity protection helps suppress parasitic processes, one can examine the driven behavior of the LINC in the presence of one or more drive tones and compare it to a Kerr-free SNAIL. Ideally, both couplers remain in their ground state even at strong drive amplitudes, which is a challenge similar to mitigating measurement-induced state transitions during qubit readout. For a realistic simulation, an environment that induces both coupler decay and dephasing can be incorporated, which in the presence of the drive can be transformed into nonlinear coupler heating. A coupled, information-storing qubit mode, which must ideally remain unaffected by the coupler during any parametric process, is also included. Assuming the coupler is not periodically reset, this coupler-qubit system will reach a driven steady state after a sufficient number of operations. This steady state will be pure if the coupler remains in its ground state, i.e. is neither affected by parasitic transition, dressed decoherence, nor stray interactions with the qubit. Any impurity in the coupler state will translate into an infidelity in the desired parametric process, due to state-dependent shifts of the process' strength and resonance condition (see Section B). Thus, the purity of the driven coupler steady state was evaluated as a measure for evaluating coupler performance.

[0138] In FIG. 14C, this driven purity was computed for a 6.5 GHz LINC throughFloquet-Markov simulations, with decay= ^26.7" s^³!, flux-noise dephasing ¦%%^^^ =©1"Φ^ / √'(¬5 / ^, and coupling to a qubit ^4.9GHz^ in FIG. 14C. The comparison to an equivalent SNAIL, with identical frequency, beamsplitting strength and decoherence at the Kerr-free point, makes the advantage of parity protection clear—the LINC remains significantly purer at all drive frequencies. This advantage is even more apparent in the presence of multiple drive tones, where the SNAIL sees substantial spurious transitions over most of the frequency range, but a large fraction of these intermodulation products are suppressed by parity protection in the LINC (FIG.14D). Overall, these simulations show that in realistic settings, the LINC should offer important advantages over the SNAIL in high-Q and multi-tone applications. IV. EFFECT OF ASYMMETRY AND PARASITIC INDUCTANCE

[0139] The analysis of the LINC so far has been restricted to a perfectly symmetric circuit under a purely differential DC flux and drive. However, practical implementations of theAttorney Docket No.: Y0087.70175WO00 coupler will come with experimental imperfections, and it is important to consider the effect of finite asymmetry on the LINC's static and driven performance. Specifically, whether the LINC remains quasi-linear when idle, and whether it retains the parity protection in its driven processes, is evaluated below.

[0140] To quantify the asymmetry of the outer junctions, the ratio 8^^! − ^^5; / ^^: =is used, which from fabrication imperfections is expected to be ≲ 2%. The second imperfectionis a finite difference in the DC flux in the two LINC sub-loops, arising from a gradient in the residual magnetic field in which the experimental package cooled down. This imperfection is in principle possible to cancel in-situ with two dedicated flux lines per coupler, such that the relative currents in the flux lines can be tuned to achieve arbitrary flux biases in the two sub- loops. However, the LINC ideally only requires a single fast flux line for its drive and DC flux, which always applies symmetric flux to both sub-loops. In this scenario, a residual DC flux difference will change both the LINC's driven and undriven behavior, which is analyzed below. In the fully general case, one may also have an asymmetry in the applied drive due to imperfect drive line engineering, which might also affect performance. However, this asymmetry can be nearly entirely nulled through careful microwave design, so its effect will not be explored here.

[0141] First the effect of asymmetries on the linearity of the idle LINC was evaluated. Since the drive is off, the asymmetries to consider are the junction and DC flux asymmetries, ^ÒÓ both of which result in an effective shift in the potential minima of the LINC to 4: ≠ 0.Specifically,to lowest order in SJand ^J. This new minima results in a change in the undriven potential, meaning that Taylor expansions to compute parameters of interest should be expanded about this new point. In light of this shift, the static Kerr at the operating point becomes (see Section C):for small asymmetries. Thus any static nonlinearity gained by the LINC in the presence of these 37 asymmetries is only second-order in SJ, ^J . Notably, the sign of the individual asymmetriesis important, as the two effects can either constructively or destructively interfere. Additionally, in the absence of junction asymmetry, the static Kerr is nulled for any value of ^ . As aJreference, for a S = 2% and a ^ = − uJ J 5 × 1%, the static Kerr of the LINC is roughly -130Attorney Docket No.: Y0087.70175WO00KHz for an ^: = 100MHz. Additionally, the shifted Kerr-free point can be found by changingthe symmetric flux bias by less than 0.005^ (see Section C).

[0142] When a LINC with non-zero asymmetry is driven, the LINC may turn on processes that would have otherwise been parity-protected. As an example, consider the third- order mixing process involving two drive photons and a single LINC photon, corresponding to a parasitic sub-harmonic displacement of the coupler. This process is always allowed in any charge-driven mixer (like the SNAIL) yet is forbidden in the ideal LINC. The strength of thisprocess, which is represented as 'sub =is given by (see Section C):and is thus linearly sensitive to both junction and flux asymmetry, with their relative sensitivitydepending on. For modest asymmetries (again= − u5 × 1% ), the relativesuppression of such parasitic resonances is )! ^55 -^ / )5!^-^ ≈ 5% for ^-^ = 0.2^, stillachieving over an order of magnitude reduction compared to an ordinary three-wave mixer.

[0143] Finally, for reasonable coupler geometries, one may expect the LINC to have a non-negligible parasitic linear inductance in series with the coupler. Such a parasitic inductor often has an impact beyond diluting the driven nonlinearity of a general coupler—for example, in the SNAIL, this can cause the Kerr-free operating point to shift significantly. In the LINC, this linear inductance is far less harmful, since it does not directly interact with the flux (drive) degree of freedom. In fact, any parasitic series inductance fully preserves the linearity of theidle LINC at the same operating point of ^^^ = ^ / 2, where every even order nonlinearity(including Kerr) is simultaneously suppressed. On the other hand, parasitic inductances within the LINC loop, while relatively small in magnitude, may shift the linear operating point slightly. A detailed analysis is included in Sections D and E. Overall, these simulations predict that the LINC remains a robust and protected quantum mixer in the presence of realistic experimental imperfections. V. CONCLUSION

[0144] Presented herein is a protected quantum mixer that combines the benefits of Kerr- free three-wave mixing and mixer balancing. This results in a nonlinear element that is nearly linear when idling, and only activates its nonlinearity when it is driven to turn on gates. Even when driven, the mixer balancing enforces selection rules that prevent a large fraction of the parasitic processes allowed by a general Josephson nonlinearity. The benefits of such a mixerAttorney Docket No.: Y0087.70175WO00 are significant, offering possible advantages in bosonic and qubit control, frequency conversion, and amplification.

[0145] In the bosonic context, the LINC breaks the trade-off between fast nonlinear control and the idle errors introduced by a nonlinear ancillary mode. This is particularly important for multi-photon encodings, where inherited Kerr and thermal-noise-induced dephasing irreversibly spoil logical information. Even in single photon encodings like the dual- rail bosonic qubit, the LINC should reduce static dispersive interactions between neighboring oscillators, while enabling fast and clean universal control through parametric beamsplitting. This advantage also extends to non-Gaussian control, where the LINC serves as an interface between the bosonic mode and its ancillary qubit, effectively shielding the bosonic mode from the ancilla qubit’s nonlinearity when idling.

[0146] The parity-protection in the LINC also provides important advantages in its power handling and multi-tone operation. This is useful in multiple contexts, with the simplest being the activation of a simultaneous parametric coupling between multiple neighboring elements. When the LINC is used as an amplifier, an array of LINCs should perform equivalently to an array of balanced RF-SQUIDs, potentially outperforming both SNAIL and regular RF-SQUID based amplifiers. Such an implementation allows simultaneous gain at multiple frequencies, easing the constraints on multiplexed readout. Finally, the LINC can also simultaneously activate multiple types of parametric processes, giving rise to new bosonic control techniques. For example, by activating a resonant beamsplitting and two-mode squeezing between two oscillators in the high-Q regime, direct parametric quadrature-quadrature coupling can be realized, enabling two-qubit gates for the Gottesman-Kitaev-Preskill (GKP) code. Section A: Analyzing the LINC circuit

[0147] Derived here is the Hamiltonian of the LINC circuit, along with analytical formulae that describe its dominant characteristics, as a function of its operating DC flux point. The circuit in FIGs. 15A-15C contains three inductive branches, and two galvanic loops that can be independently threaded by DC magnetic flux, setting its operating point. In the presence of non-zero field, each of these three branches can incur a different voltage drop across them,with corresponding superconducting phase drops across the three inductive elements, 4!̂, 46̂and 4̂5. To rewrite these phase drops in terms of more convenient independent variables,Attorney Docket No.: Y0087.70175WO00 4:̂ corresponds to the common mode of the circuit and is the phase variable conjugate to the charge on the capacitive pads^[̂:^. In the absence of a magnetic field, this phase fully describes the circuit, which behaves similar to an inductively shunted transmon (IST). A symmetric flux in both loops displaces ^ŝym, and forms the bias and the differential drive for the LINC. An antisymmetric flux on the other hand, displaces ^âsymand forms the bias for the asymmetrically threaded SQUID (ATS) mode of operating the circuit.

[0148] The full Hamiltonian, with no assumptions on symmetry, is given by:For the rest of this section, assume that all symmetry constraints in the circuit are satisfied, specifically that the outer junctions have equal Josephson energy ^^, and any DC flux and drive enter both loops symmetrically. The effects of deviating from this ideal are explored in SectionC. This constraint lets ^âsym = 0, thus giving:It is additionally assumed the symmetric flux degree of freedom ^ŝymis stiff and can bereplaced by a classical variable ^F =where Φloop is the total flux threadingthe outer loop and Φ^is the flux quantum. Note that this assumption is only valid for a weakly coupled flux-drive port and is similar to the usual treatment of the differential mode in a SQUID circuit. Under these assumptions,reduces to the LINC Hamiltonian in the main text:Attorney Docket No.: Y0087.70175WO00where at the operating point, ^F = ^ / 2 + ^.^ .

[0149] The LINC's frequency and nonlinearity underwent analytical analysis using a Taylor expansion. Since the two variables 4:̂ and ^Fare independent, the approach focuses onexamining the bi-variate Taylor expansion of the LINC potential ØLINC 84:̂, ^F; about itsminima at 4:̂^ÒÓ = 0 and ^F = ^^^. However, this expansion comes with a subtlety—studyingthe LINC's physics benefits from a bosonic basis defined by ladder operators3À̂, À̂p7, where:The bosonic basis is thus itself a function of the LINC flux point, and modulating this flux modulates not just the LINC potential, but also the effective spread of the un-driven wavefunction 4LMN^^F^. This latter modulation results in an important re-normalization of theLINC physics, for example, resulting in a non-zero driven coupler shift at ^^^ = ^ / 2. Thephysics can be captured by expanding the LINC potential as follows:with generalized parametric strengths )=jdefined as:The expansion in Eq. A6 explicitly only contains the even terms in ¾, highlighting that theinherent parity protection rule holds at arbitrary operating points. Note that at ^^^ = ^ / 2, anadditional protection is enforced which sets part of the LINC potential to zero, which makes the un-driven Hamiltonian linear:

[0150] The LINC's frequency, effective three-wave mixing strength, and anharmonicity are then derived explicitly. First, Eq. A6 was used to expand the undriven LINC Hamiltonian to fourth order, deriving its frequency^^^^and Kerr nonlinearity^,^^:Attorney Docket No.: Y0087.70175WO00At the operating point, ^^^^ / 2^ = Á8^^^^ and ,^^^ / 2^ = 0. This analytical prediction isoverlaid on results from an exact diagonalization of the Hamiltonian of a LINC with a realistic center shunt comprised of an array of 10 junctions in FIG.13A.

[0151] Similar to the static derivation, one can also find the effective three-wave mixing strength when driven:Note that this is almost exactly half the flux noise sensitivity ^^^ / ^^^^ = 4)5,! +^,^ / ^^^^, due to conventions for the definition of )H^= chosen in Eq. 10. This analyticformula matches the numerically derived three-wave mixing strength reasonably well (FIG. 15B). The latter is calculated from a Floquet simulation of either a self-squeezing of the LINC(with an added Kerr to induce |0^ ↔ |2^ oscillations), or a full beamsplitting process (seeSection B). It is also simple to derive the relative sensitivity of this process to flux noise, for Eq.6 in the main text:where ^^^ = u%v Φ^^.

[0152] Finally, the driven Zeeman shift, which is independent of the coupler's Kerr nonlinearity, is given by:Attorney Docket No.: Y0087.70175WO00This matches realistic Floquet simulations of the Zeeman shift with ^.^ = 0.1^sin ^^Fs^ well,as seen in FIG.15C. Note that this Zeeman shift can also be used to analytically account for changes in participations during driven operations.

[0153] In pursuit of additional linearity, one could also choose to array the LINC with Ê LINC loops, shunted by a single capacitor. To preserve the same frequency and parametric process strength as the single-loop LINC, the inductive energy ^^and Josephson energy ^^of each loop are scaled by Ê, and the drive and bias in each loop must be nominally identical to the single-loop device. This simple extension of the circuit can be analyzed by thetransformation 4:̂ → 4:̂ / ^,resulting in Eq. 11 in the main text. Ineffect, this preserves the inductance and three-wave mixing of the LINC but suppresses allhigher processes ∝ À̂5j by a factor ofAs Ê → ∞, this approaches the ideal three-wave mixing properties of a linear resonator with a modulated inductance (Eq.12). Section B: Driven operations with the LINC

[0154] The LINC was then analyzed as a parametric coupler and evaluate its performance in a realistic frequency stack. Consider the LINC coupled to two resonators (Alice and Bob)with linear coupling strengths and frequencies )>, ^> and )n , ^n. One can analyze the resultingdressed Hamiltonian through a simple transformation:where Δ^>,n = ^^ − ^>,n. In a frame rotating at ^>, the Hamiltonian of the system up to thethird order is then:It is then simple to derive the strength of each parametric process when the drive frequencyis set to the appropriate resonance condition, as in Eq.10.Attorney Docket No.: Y0087.70175WO00

[0155] While it is convenient to follow analytic derivations from 'H^^, from an experimental perspective, it is difficult to directly measure )H^^and ^>, ^n. Since these processes are truly a result of modulating the parameter ^, one can instead just measure the frequency dependence of the Alice and Bob modes as a function of operating flux ^^^, and then take appropriate derivatives of these flux curves as in Eq. A7. For the beamsplitting process, this results in a simplified relation (ignoring driven frequency shifts):which is similar to other flux-driven couplers.

[0156] To numerically demonstrate such a parametric beamsplitting process, and highlight the difference between the LINC and the SNAIL coupler devices, the specificfrequency stack of ^> = 2^ × 4.9GHz, ^n = 2^ × 6.0GHz, and ^: = 2^ × 6.5GHz, isconsidered with Rabi coupling strengths of )>: = 2^ × 120MHz and )n: = 2^ × 50MHz.The LINC circuit parameters are chosen as ^^ = 2^ × 100MHz, ^^ = 2^ × 15.84GHz, and^^ = 2^ × 52.8GHz. With M LINC loops in an array, the system Hamiltonian is given byassuming that each LINC loop is modulated by ^ext = ^ / 2 + ^-^.

[0157] Using standard Floquet theory, the quasi-energies of the Floquet modes can be numerically computed, and drive-induced frequency shifts and Kerr shifts can be extracted as a function of the drive frequency and amplitude, which are shown in FIGs.16A-16C. As in the case of the LINC itself, it is clear that the resonator driven frequency shifts do not change upon arraying, but their driven Kerr can be significantly suppressed.

[0158] For comparison on driven performance, a SNAIL circuit was also simulated withparameters ^^ = 2^ × 100MHz, ^^ = 2^ × 276GHz, and , = 0.193. The driven SNAILHamiltonian in the displaced frame is modeled as:Attorney Docket No.: Y0087.70175WO00 where Ê is the number of SNAIL loops in an array, ^ is the number of Josephson junctions in the array of each SNAIL loop, and 4w^,Òis the frustration phase across the i-th branch of the SNAIL loop, as a result of the DC flux penetrating the loop. For consistency, the phase displacement of the SNAIL was denoted as ^.^, even though it has a different physical origination than flux modulation in the LINC circuit. Under these parameters, the SNAILcircuit yields the same frequency of ^: = 2^ × 6.5GHz as the LINC, as well as the same )H^=,at its Kerr-free point of Φext = 0.442Φ^.

[0159] Time-dependent perturbation theory was employed for both couplers in the Floquet mode basis, to calculate the Alice-Bob beamsplitter rate given a particular coupler state, |¾^. This amounts to finding the coefficient of the effective Hamiltonian,When the coupler is in its ground state (¾ = 0), this calculation provides the beamsplitter ratereported in the main text. When the coupler is in an excited state, we study the state-dependent shifts of the beamsplitter rate and resonance condition (FIGs.17A- 17B), which can dephase the beamsplitting process if the coupler is spuriously excited by the parametric drive. Specifically, the amount of state-dependent dispersion in )ÆVand ΔÆVsets the magnitude of process infidelity, given a driven incoherent excitation to that coupler state.

[0160] Finally, using the Floquet-Markov method, the transition rates among different Floquet modes, their time-domain evolution, as well as the steady state of the system underdrive can be calculated. A thermal relaxation spectrum of the coupler was assumed to bewhere Ð ³! ℏ^ / ö!^^^ = ^26.7" s^ , and [*^^^ = 1 / 8Y ^^− 1; isthe Bose occupation factor. It was also assumed that 1 / § was the dephasing spectrum for thecoupler, ¦%%^^^^, with a cut-off frequency of 1 Hz. The impurity of the steady statewas examined over different drive frequencies and amplitudes, as a proxy for process infidelity induced by spurious coupler excitation, as well as other parasitic coupler-resonator interactions that may result in a mixed state. The contrast in the available drive space for the driven LINC and driven SNAIL (see FIG. 17C) illustrates the clear advantage of the LINC as a cleaner parametric coupler. Section C: Effect of junction and DC flux asymmetries

[0161] It was shown above how the LINC can leverage symmetries of the circuit, of the bias point, and of the drive to realize a parity-protected three-wave mixer. Here, the LINC isAttorney Docket No.: Y0087.70175WO00 examined in the presence of some asymmetries inherent to any experimental realization, and the effect of these asymmetries on device performance is predicted.

[0162] To begin, similar to the treatment of ^ŝymin Section A, it is assumed that the asymmetric flux degree of freedom ^âsymis stiff and can be replaced by the classical variableHere Φ^,^ is the flux threading the left and right loops respectively,with ^Jrepresents an asymmetry in the flux threading the two loops, which may arise due to residual local gradients in the ambient magnetic field. Further, to model the effect of imperfectjunction fabrication of the outer arms, S =introduced, with ^^setting the energy scale of the system.

[0163] With these definitions, the static inductive potential can be expressed in the presence of asymmetries as Ø!full / ^^ = 5 ^4: − 2^J / 3^5−SÖcos ^^F^^cos ^^J / 3^cos ^4:^ − sin ^^J / 3^sin ^4:^^(C1) +SJsin ^^F^^cos ^^J / 3^sin ^4:^ + sin ^^J / 3^cos ^4:^^

[0164] To study the effects that these asymmetries will have on static properties of theLINC (such as the static Kerr), the LINC's potential at the ^F = ^ / 2 operating point can beexamined, where the potential simplifies toNotably, the presence of asymmetry shifts the potential minimum with respect to 4:̂ from4̂^ÒÓ: = 0 toto lowest order in SJand ^J̇ Due to this shift, the parameters of interest must be recomputed by evaluating derivatives at this new 4:^ÒÓ, following the general framework given in Section A. From this, it is found that the LINC frequency and Kerr nonlinearity are modified to beAttorney Docket No.: Y0087.70175WO00

[0165] When focusing on the LINC's behavior at the operating point, this framework provides an approximate expression of the induced Kerr, given asto lowest order in asymmetry. However, when comparing with exact diagonalization of the LINC Hamiltonian in the presence of asymmetries, it is found that an additional factor must be added to agree with numerics. These nonidealities, to leading order, therefore induce a static LINC Kerr nonlinearity ofWhile it is not entirely clear where this additional factor comes from (possibly higher-order corrections to Eq. (C6)), it is clear that the LINC's Kerr is only second-order sensitive to asymmetry. To get a sense of the Kerr landscape, FIGs. 18A-18C shows the LINC Kerr computed by direct diagonalization of the LINC Hamiltonian as a function of asymmetry. Notably, in the absence of any junction asymmetry, the LINC is Kerr-free for arbitraryasymmetric DC flux threading the loops, so long as ^F = ^ / 2, interpolating the devicebetween the LINC ^^F = 0^ and the ATS^^F = ^ / 2^. Intuitively, when the two outerjunctions are identical, their potentials can entirely destructively interfere, leaving only thepotential of the linear shunt as seen in Eq. C2 for= 0.

[0166] Importantly, for small asymmetries, the LINC Kerr always has a zero-crossing. Thus for applications requiring truly zero residual Kerr, the symmetric DC flux can be tunedslightly away from ^F = 0.5^ to arrive at a Kerr- free point, as shown in FIG. 18B. However,straying from this bias point may compromise some of the protected driven qualities of the system.

[0167] In addition to an ideal LINC's Kerr going to zero at the operating point, any cross- Kerr to other modes are also zero in the absence of asymmetry due to the simultaneous extinguishment of all )=ý5,^. However, in the presence of asymmetry, these other nonlinearities re-appear and can cause the self-Kerr- and cross-Kerr-free points to no longer coincide. Specifically, the nonlinear termAttorney Docket No.: Y0087.70175WO00will enter into the induced cross-Kerr as tres = 24^ 5res− ^but enter into the self-Kerr as D = 12^) − 5)5^ H^ / ^^^. In effect, these two quantities now relyon an interplay between ) and ) , meaning that they can no longer be eliminatedH^ ^^simultaneously.

[0168] Now that the implications of asymmetry on the static behavior of the LINC has been studied, the driven behavior is examined. To do this, the driven Hamiltonian at the operating point can be examined in the presence of asymmetries(C9)^ ^ ^ ^ ^ ^ ^ ^ ^ ^+S sin ^ ©cos ^ / 3 cos 4 − sin ^ / 3 sin 4 ;¬Ö .^ J : J :^ ^ ^ ^ ^ ^ ^ ^ ^ ^+S cos ^ ©cos ^ / 3 sin 4 + sin ^ / 3 cos 4 ;¬J .^ J : J :^ÒÓ Utilizing Eqs. A6 and A7 but evaluating at 4 = 4 , the strength of terms of interest can be: :computed, including those typically forbidden by the LINC's parity protection. In general one may also have an asymmetry in the applied drive, but this asymmetry can be nearly entirely nulled through careful microwave design, so only on junction and DC flux asymmetry are focused on here.

[0169] The strength of the desired three-wave mixing process, ) À̂ +examined to see how the beamsplitting strength degrades. It was found that the strength of the desired process suffers only quadratically with asymmetry,To show the fractional change in beamsplitting strength, the relative change in ) was plotted5!as a function of asymmetry in FIG.19A and only a moderate reduction for reasonable asymmetries was found.

[0170] Next, the leading-order undesired process was examined, the linear drive The emergence of this linear coupling term scalesfor small asymmetries. As a benchmark to compare against, the alternative case of chargedriving a LINC can be considered, which would result in a linear coupling of ≈ ^ 4 .^FIG.19B shows that the linear drive strength from the flux drive is still only a fraction of the linear drive from a charge displacement even for moderate asymmetry.Attorney Docket No.: Y0087.70175WO00

[0171] From the simulations in FIGs.19A-19B, it is found that both )5!and )!!have a rather strong dependence on ^J. This can be well understood through the relationship between the LINC and the ATS. In the absence of junction and DC flux asymmetry, the circuit operates as a LINC and generates a pure )5!nonlinearity with no )!!from the parity protection. In theother extreme, with ^ = uJ 5, the behavior of an ATS is recovered and has precisely the oppositeparity protection, meaning that )5!becomes forbidden while )!!is amplified. So, as ^Jischanged between the two regimes, the corresponding reduction in )5!and enhancement of

[0172] Next, attention shifts to specifically checking parity protection within the three-wave mixing processes, by looking at the term )! ^À̂ + À̂p5 ^^5-^ . For context, note that for anunprotected mixer like the SNAIL, this driven term would always only be off by a factor of 4LMNfrom the desired mixing process^)5!^. It is found that the protected term, in the presence of asymmetry, becomeswhich is again linearly sensitive to asymmetry. This linear dependence on SJcan be seen from the form of Eq. C9 as the term explicitly appears in the Hamiltonian. Interestingly, the linear dependence in ^Joriginates from a modulation of 4LMN, similar to the LINC's AC Zeeman shift. To get a better understanding of how much these forbidden processes are suppressed as compared to permitted processes of the same order, FIG. 19C shows the ratio of processstrengthsfor a chosen drive strength of ^- = 0.2^. As seen below,LINC still offers substantial suppression for reasonable asymmetries. Notably, for a charge-driven mixer, the )5! process would be a factor of 1 / 4LMN ≈ 4 times stronger than )!5.

[0173] Overall, while asymmetries can spoil the perfect cancellation of many undesired quantities, the LINC is still rather robust. The LINC's static properties are preserved quite well in the presence of small asymmetry, with the LINC Kerr depending only quadratically on asymmetry and always having a zero crossing at a slightly offset symmetric flux. Additionally, while the strength of some undesired driven processes are first-order sensitive to asymmetry, a slightly unbalanced LINC will still suppress these processes significantly more than an otherwise unbalanced mixer like the SNAIL. Section D: Effects of parasitic loop inductanceAttorney Docket No.: Y0087.70175WO00

[0174] Next, the effect that a loop parasitic inductance has on the static nonlinearity of the LINC potential energy is examined. In particular, it is assumed that each Josephson junction in the outer SQUID loop has a stray series inductance (QRin FIG. 20A) and apply nonlinear current conservation methods. The equations for the analysis are established below, followed by a numerical evaluation for actual comparison. Note that because the loop inductance always shares current with an outer junction, it is difficult to directly give analytic solutions that are non-parametric. The case of the series inductance in the coupler arms^QRin FIG.20A) does not face this issue, and it is given full analytical treatment in Section E.

[0175] In the absence of the central linear inductance of the LINC (e.g. in the simple DC- SQUID case) a small loop inductance would already bring the circuit's potential energy to a multi-stable configuration. Thankfully, the linear shunt protects the LINC from becoming multi-stable in realistic geometries, as long as the ratio of loop inductance to shunt inductance is small. To compute the potential energy expansion coefficients in the presence of these inductors, the starting point is a generic expression for the potential energy of a two-loop superconducting dipole with a symmetric external flux ^Fwhere Ø^,^are the potential energies of the left and right branch, respectively, while Ø^is the potential energy of the center branch, which is nominally just the shunting inductor's energy.The m-th order derivative of the LINC with respect to 4: is thenAssuming perfect symmetry, these derivatives are evaluated 4:,^ÒÓ = 0. These expansioncoefficients can be rewritten aswhere '=, )= and (= are the m-th order derivatives of the potential energy functions of the left,shunting and right LINC branches, respectively, evaluated at ± ^5. Finally, for driven behavior,the [-th order derivative can be computed with respect to external flux, giving the general expansion coefficientAttorney Docket No.: Y0087.70175WO00

[0176] For a symmetric LINC with identical junctions and loop inductances, the potential energies Ø^and Ø&have the same functional expression, expressed parametrically as: ^+Ø = loop8S sin , ;5^,& 2 T ^^,- − ^^cos ,^^,-^^,& = ,^^,- + STsin ,^^,-where ^loopis the energy of the stray loop inductance, STis the ratio between the stray loop inductance and the Josephson inductance 8Qloop / Q^;, and ,^is the phase across the Josephsonjunction. The effect of the symmetric flux bias is captured in the junction phases, forcing ,^^ =−,^- = ,^. Higher order derivatives of Ø^,& can be expressed in the same form.

[0177] Combining these ingredients, the potential energy expansion coefficients Ø=jcan be related to external flux via a parametric relation, noticing that the external flux ^Fcan be expressed asGraphical evaluation of the exact expressions of the LINC Hamiltonian expansion coefficients then becomes possible. Numerical methods are used to evaluate effects on the static LINC in FIGs. 20B-20C. The loop inductance of the LINC is expected to be geometrically limited to∼ 1 pH / "m, which for a 100"m loop side implies Qloop = 0.2nH ^ ST ∼ 0.02 (with Q+ ∼3nH, Q^ ∼ 10nH). It is observed that the static frequency of the LINC as a function of flux isbarely affected for in this range of loop inductance. This also implies that the parametric mixingstrength )H^= ∼ 0.5^^^ / ^^F remains similar. The static Kerr of the LINC is moresignificantly affected, with ,^ = 1 − 10 MHz at ^.^ = ^ / 2, but the Kerr still has a zero-crossing very close to half flux and can therefore be nulled out for sensitive operations. Section E: Effect of a series parasitic inductance

[0178] The effect of a parasitic inductance was considered in series with the LINC loop, which may arise from the geometric inductance of the metallic arms of the coupler that form the electric dipole necessary for self-capacitance and coupling. As in Section A, the phase across the LINC is labeled as 4:(with the flux threading it as ^F), with potential energy ØLINC. The phase across the whole effective dipole (LINC + parasitic inductor) is then considered as 4, with corresponding potential energy Øtot. Each nonlinear process activated by the coupler is then described through derivatives of this total potential energy and can be analytically derived. It is seen that for a symmetric LINC, this parasitic inductance will primarily deform the potential and dilute the nonlinearity but not shift the potential minima—thus preserving theAttorney Docket No.: Y0087.70175WO00LINC's linearity at ^^^ = ^ / 2. The derivation is outlined in a pedagogical manner below, withthe actual analytic results for the LINC Hamiltonian detailed in Section E3.

[0179] Since much of the LINC's advantageous properties, including its static linearity, come from its symmetry enforced selection rules, the objective is to quantify how well its parity protection is preserved in the presence of a parasitic series inductance, i.e. whetherTo evaluate the correspondence in Eq. E1, the current conservation condition / = / LINC is usedto relate first-order partial derivatives of Øthe following relation always holds:Higher order partial derivatives can then be computed from this identity, making it possible to relate partial derivatives of Øtotas linear combinations of partial derivatives of ØLINC, weightedby partial derivatives of 4:^4, ^F^. This requires a change of variables from ^4:, ^F^ to theeffective dipole variables ^4, ^F^, outlined below.1. Representing the total potential energy and phase

[0180] The change of variables from the LINC variables ^4:, ^F^ to the effective dipolevariables ^4:, ^F^ is analyzed and its invertibility is studied here. The bivariate functionimplementing the change of variables is defined as 0, such that ^4, ^F^ = 0^4:, ^F^. Thesuperconducting phase drop across the parasitic inductor is defined to be 4M, satisfying 4 =4: + 4M, and its current to be / M. The derivation primarily uses the current conservation relation / LINC = / M.

[0181] From the individual potential energies of the LINC and the parasitic inductor, the LINC current is known to beand the current through the parasitic inductor is known to be Φ / M = ^2^Q 4M ^^4^MApplying current conservation and phase addition to Eqs. E3 and E4, the effective dipole phase reads,Attorney Docket No.: Y0087.70175WO00where SM = QM / Q and S^ = 2^QM1^ / Φ^ = QM / Q^ have been defined. Eq. E5 thus defines thechange of variable function 0. To compute the desired derivatives, the partial derivatives of theinverse of this function are instead required, implementing the change of variable ^4:, ^F^ =0³!^4, ^F^.

[0182] Next, the conditions under which 2 is invertible were computed. A necessary andsufficient condition is that the determinant of 3 = ∇2 (the Jacobian matrix) does not changesign for any value of the independent variables. By computing partial derivatives of 2 andcombining them into the determinant, the following is obtainedAs det3^0,0^ > 0, and by noticing that it is minimized when cos ^Fcos 4: = −1, thecondition for invertibility of 2 iswhich is automatically satisfied if the LINC potential ØLINC is single-valued, i.e.det^∇2^ ≠ 0 ^^8^2. Computing derivatives of the total potential

[0183] The partial derivatives of 2³!may now be computed by composing it with 2 to form the identity 4= 4^4:^4, ^F^, ^F^4, ^F^^ ^^9^Defining the generalized participation of the LINC asand computing first-order partial derivatives with respect to 4 and ^Fon both sides of (E9), the following set of equations is obtained ìÜ41 =Ü4 ^!^:íÜ4 Ü4î0 =Ü4 ^^! +: Ü^FNote that the implicit dependence of 4 on ^Fvia 4:is included in the ^^!term. As a consequence, the partial derivative of 4 with respect to ^Fhas to only account for the explicit dependence on ^F(consistent with the definition of the partial derivative).

[0184] It is useful to define the dimensionless Josephson part of the LINC potential, and its partial derivatives, asAttorney Docket No.: Y0087.70175WO00The first-order participation ratios of the LINC are then obtained from (E11) asAdditionally, the following algebraic rules hold for Eq. (E2):which ease the computation of higher-order coefficients. One can now apply these rules to compute arbitrary partial derivatives of the effective dipole potential energy function, notingthat the right hand side term of Eq. E2 can be written asDefining the partial derivatives of Øtotasand incorporating Eq. E2 givesImportantly, Ø=jcan be expressed as a function of ^!^and =÷g>?g¹!…=only. For instance, Ø5^ö¹!…jcan be computed applying the algebraic rules to (E15), obtainingØ5^ = ^^^!^ + ^^^!^÷5^ ^^18^and, deriving Ø5^ with respect to 4 obtainsThe expression for the second order participation ratio ^5^can also be obtained by applying the algebraic rules to the first line of (E13), resulting in ^H5^ = −S^^!^ ÷H^ ^^20^Attorney Docket No.: Y0087.70175WO00 Using this, the following is obtainedwhich can be further simplified to ØHH^ = ^^^!^ ÷H^ ^^22^by performing the substitutionwhich can be demonstrated from the definitions of S^, S^ and ^!^. For the LINC, ÷H^ = 0,which implies )H^ ∝ ØH^ = 0. For higher order nonlinearities, performing an additionalderivative with respect to 4 will generate a ^5^term, which can be again substituted with the previously obtained expression. It is thus clear that Ø=^can be expressed as a multinomialfunction of ^!^ and 3÷5^, ÷H^, … ÷=^7.

[0185] The same approach may also be taken to compute the arbitrary derivative Ø=jstarting from Ø!!, which readsHere, ^^! can be substituted by the expression obtained in (E13), givingØ!! = ©^^ − S^^!^8^^ + ^^÷5^;¬÷!! ^^25^This expression can be further simplified by performing the substitution in Eq. (E23), resultinginØ!! = ^^^!^÷!! ^^26^A further derivative with respect to ^F will generate Ø!5, which will contain the participation^!!. However, ^!! can be obtained by applying the algebraic rules to (E13), asIn combination with the expression for ^5^previously obtained, this gives an expression of Ø!5that again only containes ^!^and ÷=jterms. Thus by recursively substituting the expressionsfor ^^!, ^!!and ^5^, any partial derivative of Ø can be expressed as a multinomial function of^!^ and the set of nonlinear coefficients ÷=j.

[0186] Finally, after the desired order of derivative is obtained, all the quantities have tobe evaluated at the generic flux operating point ^4^ÒÓ, ^^ÒÓ^ = ^0, ^F^. Importantly, theinvertibility of 2, along with Eq. E5, imply that this expansion point corresponds to^4: = 0, ^F^. This means that the potential minima is unchanged, and one can then evaluatearbitrary Ø=jat this minima. 3. The LINC Hamiltonian with parasitic inductanceAttorney Docket No.: Y0087.70175WO00

[0187] Given the derivatives of the total potential Øy^ywith respect to 4̂ and ^F, the relevant coupler Hamiltonian can be found in the presence of linear inductance, which is derived here. Similar to Eq. A6, the derived Hamiltonian describes the equations of motioncorresponding to the bosonic operator 4̂ = 4LMN^^^^^^À̂ +which can be expanded interms of Ø=jas:Note that the parasitic inductance also affects Ø5,^ and hence the spread of the wavefunction4LMN, whose intrinsic flux dependence affects every flux driven process.

[0188] First, consider the static Hamiltonian, given by:It is known that the frequency and impedance of the LINC shifts due to the parasitic inductance, specificallyThis change is just a renormalization of the LINC's inductive energy by its linear participation in the parasitic inductor.

[0189] Higher-order nonlinearities are also similarly renormalized by the parasitic inductor, which means they can be recursively proven to be zero at the operating point, for the ideal LINC. As an example, the LINC Kerr in the presence of this parasitic inductance is given byThus, the undriven LINC remains linear at the operating point of ^^^ = ^ / 2, and its staticbehavior is computed in FIGs. 20B-20C. The driven LINC terms are also similarly renormalized, for example with the three-wave mixing strength changing to: H / 5^!^ )H^^^0, ^^^^ ^^30^Attorney Docket No.: Y0087.70175WO00 Overall, the LINC with parasitic series inductance retains its static linearity at the same operating point and has analytically predictable changes in its driven properties, which make its behavior significantly simpler than charge-driven mixers like the transmon or the SNAIL. EQUIVALENTS

[0190] Having thus described several aspects and embodiments of the technology set forth in the disclosure, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the embodiments described herein. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation many equivalents to the specific embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described. In addition, any combination of two or more features, systems, articles, materials, kits, and / or methods described herein, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the scope of the present disclosure.

[0191] Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0192] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0193] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0194] 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 elementsAttorney Docket No.: Y0087.70175WO00 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.

[0195] 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, as a non-limiting 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.

[0196] In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively.

[0197] The use of “coupled” or “connected” is meant to refer to circuit elements, or signals, which are either directly linked to one another or through intermediate components. Elements that are not “coupled” or “connected” are “decoupled” or “disconnected.”

[0198] The term “approximately” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, within ±2% of a target value in some embodiments, and / orAttorney Docket No.: Y0087.70175WO00 within ±1% of a target value in some embodiments. The term “approximately” may include the target value.

Claims

Attorney Docket No.: Y0087.70175WO00 What is claimed is: CLAIMS 1. A linear inductive coupler, comprising: an inductor; and two Josephson junctions having approximately equal Josephson energies, each of which is coupled in parallel with the inductor, wherein: during operation, the linear inductive coupler is configured to be biased by a magnetic flux approximately equal tosuch that a phase drop across each of the two Josephson junctions is approximately equal to π / 2.

2. The linear inductive coupler of claim 1, wherein the inductor comprises a high kinetic inductance material.

3. The linear inductive coupler of claim 2, wherein the inductor comprises granular aluminum or niobium nitride.

4. The linear inductive coupler of claim 1, wherein the inductor comprises a plurality of Josephson junctions arranged in series.

5. The linear inductive coupler of claim 4, wherein the plurality of Josephson junctions comprises at least 5 Josephson junctions.

6. The linear inductive coupler of claim 4, wherein the plurality of Josephson junctions comprises between 5 and 100 Josephson junctions.

7. The linear inductive coupler of any one of claims 1-6, wherein the linear inductive coupler is coupled between a first quantum mode and a second quantum mode.

8. The linear inductive coupler of claim 7, wherein: the first quantum mode is stored in a quantum harmonic oscillator; and the second quantum mode is stored in an ancilla qubit.Attorney Docket No.: Y0087.70175WO00 9. The linear inductive coupler of claim 8, wherein the ancilla qubit comprises one of a transmon qubit, a fluxonium qubit, or a charge qubit.

10. The linear inductive coupler of claim 8, wherein the quantum harmonic oscillator comprises a microwave cavity resonator.

11. The linear inductive coupler of claim 7, wherein: the first quantum mode is stored in a first quantum harmonic oscillator; and the second quantum mode is stored in a second quantum harmonic oscillator.

12. The linear inductive coupler of claim 11, wherein at least one of the first quantum harmonic oscillator and / or the second quantum harmonic oscillator comprises a microwave cavity resonator.

13. The linear inductive coupler of claim 7, wherein: the first quantum mode and the second quantum mode are configured to support quantum mode photons, and when driven by an electromagnetic drive waveform comprising drive photons, the linear inductive coupler is configured to activate an interaction between only an odd number of drive photons and an even number of the quantum mode photons.

14. The linear inductive coupler of claim 7, wherein, when driven by an electromagnetic drive waveform, the linear inductive coupler is configured to cause three-wave mixing between the first quantum mode and the second quantum mode.

15. The linear inductive coupler of claim 7, wherein the linear inductive coupler is configured to cause three-wave mixing between the first quantum mode and the second quantum mode when driven by an electromagnetic drive waveform having a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode.

16. The linear inductive coupler of claim 7, wherein the linear inductive coupler is further configured to exhibit approximately equal and opposite AC phase drops across the two Josephson junctions when driven by an electromagnetic drive waveform.Attorney Docket No.: Y0087.70175WO00 17. The linear inductive coupler of claim 7, wherein, when not driven by an electromagnetic drive waveform during operation, the linear inductive coupler is configured to behave approximately like a linear oscillator.

18. A circuit quantum electrodynamic (cQED) system comprising: a first quantum device configured to store a first quantum mode during operation of the cQED system; a second quantum device configured to store a second quantum mode during operation of the cQED system; and a linear inductive coupler configured to couple the first quantum mode and the second quantum mode when driven by an electromagnetic drive waveform during operation of cQED system, the linear inductive coupler comprising: an inductor; and two Josephson junctions having approximately equal Josephson energies, each of which is coupled in parallel with the inductor, wherein: during operation of the cQED system, the linear inductive coupler is configured to be biased by a magnetic flux approximately equal tosuch that a phase drop across each of the two Josephson junctions is approximately equal to π / 2.

19. The cQED system of claim 18, wherein the inductor comprises a high kinetic inductance material.

20. The cQED system of claim 19, wherein the inductor comprises granular aluminum or niobium nitride.

21. The cQED system of claim 18, wherein the inductor comprises a plurality of Josephson junctions arranged in series.

22. The cQED system of claim 21, wherein the plurality of Josephson junctions comprises at least 5 Josephson junctions.Attorney Docket No.: Y0087.70175WO00 23. The cQED system of claim 21, wherein the plurality of Josephson junctions comprises between 5 and 100 Josephson junctions.

24. The cQED system of any one of claims 18-23, wherein: the first quantum device comprises a quantum harmonic oscillator; and the second quantum device comprises an ancilla qubit.

25. The cQED system of claim 24, wherein the ancilla qubit comprises one of a transmon qubit, a fluxonium qubit, or a charge qubit.

26. The cQED system of claim 24, wherein the quantum harmonic oscillator comprises a microwave cavity resonator.

27. The cQED system of any one of claims 18-23, wherein: the first quantum device comprises a first quantum harmonic oscillator; and the second quantum device comprises a second quantum harmonic oscillator.

28. The cQED system of claim 27, wherein at least one of the first quantum harmonic oscillator and / or the second quantum harmonic oscillator comprises a microwave cavity resonator.

29. The cQED system of any one of claims 18-23, further comprising an electromagnetic drive configured to differentially drive the linear inductive coupler, wherein: the first quantum mode and the second quantum mode are configured to support quantum mode photons, the electromagnetic drive is configured to generate the electromagnetic drive waveform comprising drive photons, and when driven by the electromagnetic drive waveform, the linear inductive coupler is configured to activate an interaction between an odd number of the drive photons and an even number of the quantum mode photons.

30. The cQED system of any one of claims 18-23, wherein:Attorney Docket No.: Y0087.70175WO00 the electromagnetic drive waveform has a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode, and when driven by the electromagnetic drive waveform, the linear inductive coupler is configured to cause three-wave mixing between the first quantum mode and the second quantum mode.

31. The cQED system of any one of claims 18-23, wherein the linear inductive coupler is further configured to exhibit approximately equal and opposite AC phase drops across the two Josephson junctions when driven by an electromagnetic drive waveform.

32. The cQED system of any one of claims 18-23, wherein, when not driven by an electromagnetic drive waveform during operation, the linear inductive coupler is configured to behave approximately like a linear oscillator.

33. A method of operating a linear inductive coupler, comprising: biasing the linear inductive coupler by a magnetic flux approximately equal to Φ^ / 2 such that a phase drop across each of two Josephson junctions of the linear inductive coupler is approximately equal to π / 2; and driving, using at least one electromagnetic signal, the linear inductive coupler to cause parametric coupling between a first quantum mode and a second quantum mode coupled to the linear inductive coupler.

34. The method of claim 33, wherein driving the linear inductive coupler using the at least one electromagnetic signal causes three-wave mixing between the first quantum mode and the second quantum mode.

35. The method of claim 33, further comprising, before or after driving the linear inductive coupler using the at least one electromagnetic signal, causing the linear inductive coupler to behave approximately like a linear oscillator by not driving the linear inductive coupler using the at least one electromagnetic signal.

36. The method of any of claims 33-35, wherein:Attorney Docket No.: Y0087.70175WO00 driving the linear inductive coupler using the at least one electromagnetic signal causes interactions between only an odd number of drive photons generated by the at least one electromagnetic signal and an even number of quantum mode photons associated with the first quantum mode and the second quantum mode.

37. The method of any of claims 33-35, wherein driving the linear inductive coupler comprises driving a superconducting circuit comprising an inductor and two Josephson junctions with approximately equal Josephson energies, each of the two Josephson junctions being coupled in parallel with the inductor.

38. The method of any one of claims 33-35, wherein driving the linear inductive coupler comprises driving the linear inductive coupler with an electromagnetic signal having a frequency approximately equal to a difference between a first resonant frequency of the first quantum mode and a second resonant frequency of the second quantum mode.

39. The method of any one of claims 33-35, wherein driving the linear inductive coupler with the at least one electromagnetic signal comprises driving the linear inductive coupler with a differential drive electromagnetic signal configured to cause approximately equal and opposite AC phase drops across the two Josephson junctions of the linear inductive coupler.

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