Zz reduction via third-order dispersive interactions

Abstract

Described is a method for reducing parasitic ZZ interactions using third-order dispersive interactions and its implementation in superconducting qubits. In one example embodiment, a system comprises a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, wherein interactions between the resonant circuit and the first and second qubits alters at least one energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

Description

ZZ REDUCTION VIA THIRD-ORDER DISPERSIVE INTERACTIONSCROSS-REFERENCE TO RELATED APPLICATIONSThis application claims the benefit of U.S. Provisional Application No. 63 / 488,219 filed on March 3, 2023 of which is hereby incorporated herein by reference in its entireties.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0001] The invention was made with government support under W911 NF-18-1- 0116 awarded by the U.S. Army Research Office and FA8702-15-D-0001 awarded by the U.S. Air Force. The government has certain rights in the invention.PARTIES TO A JOINT RESEARCH AGREEMENT

[0002] Not applicable.BACKGROUND

[0003] One obstacle in scaling-up quantum processors is the parasitic always- on ZZ interaction that arises when qubits are coupled. The ZZ interaction between two qubits is defined as the energy difference between when both qubits are in their excited state and when they are individually in their excited state. This interaction, which shifts the frequency of one qubit depending on the state of another qubit, is an entangling interaction that occurs during the normal idling state of a quantum system. When large, the ZZ interaction prevents straightforward implementation of high-fidelity one- and two-qubit gates.

[0004] The parasitic, always-on ZZ interaction arises naturally between coupled superconducting qubits. The underlying cause of non-zero ZZ interaction is the many extra higher energy levels of real-life qubits, which end up pushing on some of the important computational states of the system. Mathematically, ifdenotes a transition frequency of a two-qubit state corresponding to qubit 1 being in its ithstate and qubit 2 being in its jthstate, then the ZZ interaction rate is given mathematically by = 6) - a>01- co10+ t0o- A common scenario in superconducting qubits consists of two capacitively coupled transmon qubits. Here, the |02> and |20> states push on the 111> state through capacitive couplingsuch that the frequency of the 111> state is no longer simply the sum of the frequencies of the individual |01> and 110> states, resulting in = 0. It is possible to compensate for this ZZ interaction in gates which commute with the ZZ operator, such as isolated single qubit gates or cPHASE type gates. In general, however, the ZZ interaction will manifest as a gate error and is at the forefront of infidelity processes in large scale quantum processors.

[0005] There exists a number of different methods for canceling out ZZ interaction in superconducting qubits. The two broad categories for these methods are cancelation of ZZ interaction by applying a microwave cancellation pulse, or by choosing the processor architecture carefully so that the native interactions between the qubits suppresses ZZ interaction.

[0006] One drawback of methods in the first category is that the extra microwave tone needed to cancel ZZ cancelation increases resource requirements for the control of the quantum processor and greatly complicates calibrating the device. While methods in the second category attempt to natively solve the problem of cancelation of ZZ interaction, methods in the second category rely upon particular qubit parameters which puts demands on the precision of the design and fabrication.SUMMARY OF DISCLOSED EMBODIMENTS

[0007] In accordance with the concepts, systems and techniques disclosed herein, described is a method of reducing parasitic ZZ interactions using third- order dispersive interactions and its implementation in superconducting qubits.

[0008] In accordance with one example embodiment illustrating the concepts described herein, ZZ interaction between two computational, fluxonium qubits is reduced by inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions. These 3rdorder interactions are in many cases oppositely signed than the 2ndand 4thorder terms, giving a mechanism for reducing the ZZ interaction between the computational qubits.

[0009] The concepts, systems and techniques described herein may, in a sense, be considered by some to belong to the second category of ZZcancelation, i.e., choosing the processor architecture carefully so that the native interactions between the qubits suppresses the ZZ.

[0010] However, unlike traditional approaches of using oppositely signed anharmonicity qubits or the 4thorder terms from a tunable coupler to cancel ZZ interaction, the concepts, systems and techniques described herein work with low frequency qubits (e.g., less than about 1 GHz) without a lower frequency coupler . An additional advantage over existing methods is that the reduction of ZZ interaction is stable for a wide range of circuit parameters and coupler frequencies.

[0011] In contrast, using oppositely signed anharmonicity qubits as in the prior art, results in large ZZ variations with device parameters which relies heavily on fabrication accuracy (i.e. , prior art approaches require tolerances which are very difficult, and thus very expensive, to meet and consistently maintain during a fabrication process). Furthermore, traditional tunable couplers have a ZZ interaction characteristic that depends strongly on the coupler’s frequency (e.g., depends strongly on the transition frequency between the ground and first excited state of the coupler). Moreover, reducing the ZZ interaction during idle simplifies quantum logic gate design.

[0012] In accordance with one aspect of concepts described herein a two-qubit gate can be performed by driving to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by a resonant circuit. In embodiments, an operation frequency can be adjusted by changing a characteristic of the resonant circuit. In embodiments, an operation frequency can be adjusted by adjusting a frequency of the resonant circuit.

[0013] In accordance with a further aspect of concepts described herein, described is a method for reducing parasitic ZZ interactions using third-order dispersive interactions and its implementation in superconducting qubits. In one example embodiment, a system comprises a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, wherein interactions between the resonant circuit and the first and second qubitsalters at least one energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

[0014] In accordance with a further aspect of concepts described herein, a system comprises first and second fluxonium qubits and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, wherein interactions between the resonant circuit and the first and second qubits alters energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

[0015] In embodiments, the resonant circuit may comprise a transmon qubit.

[0016] In embodiments the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

[0017] In embodiments, the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

[0018] In embodiments, the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

[0019] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

[0020] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0021] In accordance with a still further aspect of concepts described herein a system comprises first and second qubits a resonant circuit coupled to both the first and second qubits according to respective coupling strengths wherein interactions between the resonant circuit and the first and second qubits alter at least one energy level of computational states of the system to thereby reduce an always-on ZZ interaction.

[0022] In embodiments, the resonant circuit may comprise a transmon qubit.

[0023] In embodiments the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

[0024] In embodiments, the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

[0025] In embodiments, the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

[0026] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

[0027] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0028] In embodiments, the first qubit is provided as a first fluxonium qubit and the second qubit is provided as a second fluxonium qubit.

[0029] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of the resonant circuit.

[0030] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0031] In accordance yet a further aspect of concepts described herein, a method for reducing ZZ interaction between two qubits includes inserting a coupler transmon qubit between the between the two qubits, wherein the coupler transmon qubit relies on third-order (3rdorder) dispersive interactions oppositely signed with respect to second (2nd) and (4th) order dispersive interactions such that the third-order (3rdorder) dispersive interactions cancel at least some of thesecond (2nd) and (4th) order dispersive interactions to thereby reduce ZZ interaction between the two qubits.

[0032] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational qubits.

[0033] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational fluxonium qubits.

[0034] In embodiments, inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions between two fluxonium qubits reduces ZZ interaction between the two fluxonium qubits.

[0035] In embodiments, inserting a coupler transmon qubit between two fluxonium qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the fluxonium qubits, but only affects ZZ interaction at higher orders.

[0036] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the two qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the two qubits, but only affects ZZ interaction between the two qubits at higher orders.DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0037] The manner and process of making and using the disclosed embodiments may be appreciated by reference to the figures of the accompanying drawings. It should be appreciated that the components and structures illustrated in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principals of the concepts described herein. Likereference numerals designate corresponding parts throughout the different views. Furthermore, embodiments are illustrated by way of example and not limitation in the figures, in which:

[0038] FIGs. 1A, 1 B, 1C are energy level diagrams representing 2ndorder (FIG. 1A), 3rdorder (FIG. 1 B), and 4thorder (FIG.. 10) perturbation theory assuming a dispersive interaction;

[0039] FIG. 2 is a block diagram of an example two-qubit system in accordance with the concepts, techniques, and structures disclosed herein;

[0040] FIG. 2A is a mask design of an example two-qubit system in accordance with the concepts, techniques, and structures disclosed herein;

[0041] FIG. 3 is a plot of fluxonium qubit-to-qubit coupling vs. qubit-to- transmon coupling (both in MHz) for the system of FIG. 2A; and

[0042] FIG. 4 shows a plot of the ZZ interaction between qubits (in kHz) as a function of magnetic flux applied to the coupler (in units of the flux quantum $0).

[0043] FIG. 5 is a schematic diagram of a device comprising a pair of qubits coupled through an intermediate circuit in accordance with the concepts, techniques, and structures disclosed herein.

[0044] FIG. 6 is an energy level diagram of a device comprising a pair of qubits coupled by a coupling circuit in accordance with the concepts, techniques, and structures disclosed herein.DETAILED DESCRIPTION

[0045] Quantum bits (or more simply qubits) may be described as quantum mechanical systems having quantized energy levels and a transition frequency at which they transition between those energy levels. Thus, in a sense, qubits can be viewed as oscillators. The transition frequency for a qubit depends upon the state of excitation of the qubit. This transition frequency can be tuned in a variety of ways, including by applying external, biasing magnetic fluxes to the qubit.

[0046] A multi-qubit system behaves similarly, except that coherent quantum mechanical coupling terms play a role. A system with multiple qubits may have several total energy levels that are relatively close (i.e., the energy level difference is not large compared to the coupling energy) up to a 1storder approximation. Such situations occur, for example in a system with two substantially identical qubits that are capacitively coupled. The coupling between the two systems pushes the energy levels of the two quantum states in opposite directions. The magnitude of this pushing is proportional to the coupling strength.

[0047] Typical prior art quantum logic systems are constructed from transmon qubits coupled by transmon couplers. These systems have excited states with energy levels that are designed to push a third energy level in opposite directions, thereby stabilizing that third level. However, repulsions between the levels are designed to be deliberately strong to avoid accidental transitions between the states, while simultaneously there is a need to fine-tune the energy of the third level, and this balance is difficult to achieve in practice. Also, manufacturing flaws can make it difficult to tune some components (e.g., the coupler) after fabrication.

[0048] For concreteness, described herein are example embodiments of the concepts, techniques, and structures disclosed herein when the two qubits are superconducting fluxonium qubits, which have a very low excited state frequency (e.g., less than about 1 GHz).

[0049] Fluxonium qubits have a large (e.g., greater than about 1 GHz), positive anharmonicity. As a result, the higher excited states of fluxonium qubits (i.e. the |20> state, the |21> state, the |02> state, etc.) push down on the energy levels of the computational states (i.e. |00>, |01 >, |10>, and |11 >).

[0050] Normally, a fluxonium-fluxonium (“F-F”) system doesn’t have 3rdorder terms in its Hamiltonian; however introducing a third qubit would add these terms. Intuitively, introducing a transmon coupler in the system should be disadvantageous since its higher energy levels should push down on the levels of the computational basis.

[0051] However, the inventors have recognized that introducing a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits does not affect the 2ndorder ZZ interaction between the qubits, but only affects it at higher orders.

[0052] Further, in accordance with the concepts, systems, devices and techniques described herein, the inventors have discovered that introducing a transmon coupler introduces dispersive terms that can be used to reduce or ideally eliminate parasitic ZZ interaction between the fluxonium qubits. These 3rdorder interactions are oppositely signed to the 2ndand 4thorder terms, and thus provide a mechanism for reducing the overall ZZ interaction.

[0053] Accordingly, and more generally, described herein is a method for reducing parasitic ZZ interactions using third-order dispersive interactions and its implementation in superconducting qubits.

[0054] In this connection, FIG. 1A is a level diagram representing 2ndorder perturbation theory in such embodiments, assuming a dispersive interaction; FIG. 1 B is a level diagram representing 3rdorder perturbation theory assuming a dispersive interaction; and FIG. 1C is a level diagram representing 4thorder perturbation theory assuming a dispersive interaction.

[0055] In FIGs. 1A-1C, E® represents the energy of the computational state |n) accurate to jthorder perturbation theory, and Vjkrepresents the matrix element of the coupling Hamiltonian <j|V|k>. Here, the state |n) is labeled generically but represents the |11> or 1101> computational state in the described scheme (i.e., both qubits are in their first excited state). When all intermediate states (labeled by kj) lie above |n>, only the 3rdorder interaction provides a positive energy shift on state |n).

[0056] As may be seen from FIG. 1A, the computational state |n>, written as general state here, is meant to represent the 111) state of a two-fluxonium system. Second-order interactions from higher energy states, although far away, push down on this state causing a negative ZZ interaction. In this situation,conventional methods of using 2ndorder interactions to push up on the computational state from below are unfeasible.

[0057] By contrast, in accordance with embodiments of the concepts, systems, devices, circuits, techniques, and structures disclosed herein and as illustrated in FIG. 1 B, inserting an intermediate circuit such as a resonant circuit coupled between the two fluxoniums creates 3rdorder perturbation terms to pull up the 1101) state (where we introduce the middle index in the notation to represent the coupler state). In embodiments, a resonant circuit (or other intermediate circuit) acts as a coupler between the two fluxoniums to create 3rdorder perturbation terms to pull up the 1101) state. In embodiments, inserting a resonant circuit such as a transmon between the two fluxoniums creates 3rdorder perturbation terms to pull up the 1101) state. In embodiments, a coupler may be inserted between the two fluxoniums to create 3rdorder perturbation terms to pull up the 1101) state.

[0058] Furthermore, since all dipole allowed interactions with the 1101) state have higher frequency, every existing 3rdorder interaction will pull up on the 1101) state. By tuning the relative coupling strengths between the qubits and the intermediate circuit (e.g., the resonant circuit or coupler), the effects of these third- order terms can counteract the 2ndand 4thorder ZZ contributions (as illustrated in Figs. 1A and 1C, respectively) to within about 10 kHz, with typical circuit parameters. An added benefit is that this ZZ interaction varies slowly with the coupler frequency, allowing for the ZZ interaction rate to remain less than about 10 kHz across the entire spectrum of the coupler.

[0059] FIG. 2 is a block diagram of an example two-qubit system 10 in accordance with the concepts, systems, circuits, devices, techniques, and structures disclosed herein. The system has a pair qubits 12, 14 (i.e. , first and second qubits) coupled through an intermediate circuit 16. This circuit could be generalized to a larger N-qubit system by constructing a lattice of fluxonium qubits (such as qubits 12 and 14) such that there exists an intermediate coupling circuit (such as circuit 16) between each pair of adjacent qubits.

[0060] The system may be driven by one or more signal sources 18a, 18b, 18c. Although three signal sources are shown in FIG. 2, only a single signal source is needed. Further, signal sources 18a - 18c are here shown in phantom since a signal source is not properly a part of system 10. In embodiments a drive signal provided by signal sources 18a - 18c may be oscillating current or voltage signals. In embodiments, a signal source 18a may, for example, provide an oscillating current signal to qubit 12. In embodiments, a signal source 18c may, for example, provide an oscillating current signal to qubit 14. In embodiments, a signal source 18b may, for example, provide an oscillating current signal to coupler 16. Thus, each signal source 18a -18c provides a signal to a different element of system 10.

[0061] In general, multiple signal sources may each couple to different, respective ones of multiple elements in a system such as system 10. In embodiments, it is possible that there may be a desire or a need to have a single signal source provide one or more signals to different elements. For example, in some embodiments it may be desired or required that source 18a provide a signal (e.g., a drive signal such as an oscillating current or voltage signal) to two or more of qubits 12, 14 and coupler 16. In embodiments, a single signal source may provide signals to all elements. In embodiments, a first signal source (e.g., signal source 18a) may provide signals to some elements (e.g., qubits 12, 14) and a second signal source (e.g., signal source 18b) may provide signals to other elements (e.g., coupler 16).

[0062] In accordance with the concepts, systems, circuits, devices, techniques, and structures disclosed herein, FIG. 2A shows a mask design for a two-qubit system 10’ which may be the same as or similar to system 10 described above in conjunction with FIG. 2. The system 10’ has a pair of qubits 12, 14 (i.e., first and second qubits) coupled through an intermediate circuit 16. In this example embodiment, qubits 12, 14 are illustrated as fluxonium qubits 12, 14, and intermediate circuit 16 is illustrated as a transmon coupler 16. Thus, in the example system of FIG. 2A, a pair of fluxonium qubits 12, 14, are coupled through a transmon coupler 16 and system 10’ may be referred to as a fluxonium- transmon-fluxonium system.

[0063] To facilitate the coupling between the two fluxonium qubits, the coupler may be or may comprise any circuit with a resonant mode including but not limited to: a resonator; cooper-pair-box; and / or a flux qubit.

[0064] Fluxonium qubit 12 has an associated resonator 22, charge line 32, and flux line 42. Likewise, Fluxonium qubit 14 has an associated resonator 24, charge line 34, and flux line 44. Finally, the transmon coupler 16 has an associated resonator 26 and flux line 46.

[0065] The fluxonium qubits 12, 14 may be used to perform quantum logic gate operations. Each such qubit 12, 14 is constructed of rectangular plates, coupled by wires having various electrical and magnetic properties (e.g., inductances, Josephson junctions, etc.) as known in the art to produce a fluxonium qubit having desired design parameters such as transition frequencies within a given range. In this example embodiment, each qubit 12, 14 is capacitively coupled to the transmon coupler 16 via a parallel plate capacitance. The capacitance is formed by the metal pads of each qubit being in close proximity. This is roughly illustrated as the two edges of the boxes of each qubit being close to each other (e.g., as illustrated in FIG. 2).

[0066] The resonators 22, 24, 26 may be capacitively coupled to each qubit and lead to an external device to measure the quantum state of the corresponding qubit or coupler at desired times. The charge lines 32, 34 may be used to drive qubit state transitions (i.e. between a ground state |0> and a first excited state |1 >). The flux lines 42, 44, 46 may be used to provide static or varying magnetic flux to the qubits and the coupler, and in particular applied to their respective magnetic flux loops, thereby modulating these components to implement a quantum logic gate. Thus, in the example embodiment of FIG. 2A, signals may be coupled to the system 10' via one or more signal paths 32, 34, 42 and 44.

[0067] FIG. 3 shows a plot of fluxonium qubit-to-qubit coupling vs. qubit-to- transmon coupling (both in MHz) for the system of FIG. 2A. In FIG. 3, curve 54 (and designated “Traditional F-F”) represents the direct fluxonium qubit-to-qubit coupling strength, and curve 50 (and designated “F-T-F”) represents the qubit-to- transmon coupler strength.

[0068] A traditional, two qubit fluxonium-fluxonium (F-F) system has no coupler and is therefore represented by dashed line 54 along the horizontal axis. As may be appreciated, as the coupling strength increases between the qubits in this system (i.e. as curve 54 is followed to the right), the ZZ interaction also increases (as represented by the darkening shade of the plot). This 2ndorder interaction undesirably pushes down on the energy of the useful computational state of the system by upwards of 400 kHz.

[0069] Noticeably, however, there exists a ’’trough region” in ZZ interaction strength through which F-T-F curve 50 substantially passes, which represents the fluxonium-transmon-fluxonium system of an embodiment, such as example system 10’ shown in FIG. 2A. In a system according to an embodiment, the coupling between each fluxonium qubit and the transmon coupler (i.e. the “F-T Coupling” strength on the vertical axis) also increases with “F-F Coupling” strength on the horizontal axis. Advantageously, and unlike in the prior art, the presence of the transmon coupler between the two fluxonium qubits introduces 3rdorder effects that decrease the ZZ interaction strength to nearly zero (as indicated by the lack of shading in the trough region). Thus, it is possible to obtain low ZZ interaction between the two qubits (e.g., on the order of single-digit kHz), even when the direct coupling strength is as large as in conventional, F-F systems that lack a transmon coupler (e.g., between 200 and 400 MHz). The F-T characteristics of one example embodiment (which may be the same as or similar to the example embodiment of FIG. 2A) are labelled with reference numeral 52 in FIG. 3.

[0070] FIG. 4 is a plot of ZZ interaction between the qubits (in kHz) as a function of magnetic flux applied to the coupler (in units of the flux quantum (,). This plot shows a solid, model curve 58 of the predicted ZZ interaction, and measured data with error bars. Note that the vertical axis, showing ZZ interaction strength, is in the low single-digit, negative kHz. As predicted, the strength of the ZZ interaction does not depend much on the coupler frequency, but more so on the static coupling strengths (i.e., the F-T coupling strength and the F-F coupling strength).

[0071] FIG. 5 is a schematic diagram of a device comprising a pair of qubits 62, 66 coupled via coupler 64 (e.g., a schematic representation of an example two-qubit system such as the example two-qubit system of FIG. 2A).

[0072] Referring now to FIG. 6, an energy level diagram 70 with state labels |qubit 1 , coupler, qubit 2> denoted with reference numerals 72a - 72g. The indices of each state label are ordered qubit 1 , coupler, qubit 2. For example, looking at the top, there are states |201>, |111>, and 1102> respectively denoted 72a , 72b 72c. State label 72a |201 > represents qubit 1 in state 2, coupler in state 0, and qubit 2 in state 1. Likewise for |111 > and all other labels.

[0073] Driving any of the transitions illustrated by arrows 74a, 74b, 74c for a single-period Rabi oscillation results in a two-qubit gate. This drive may be performed by a signal source (e.g., one or more of signal sources 18 in FIG. 1) which provides an oscillating voltage or current signal which is coupled to the system. Up to single-qubit corrections, this two-qubit gate will be the CZ (CPhase with 180 degrees) gate when driven with a calibrated drive amplitude and drive frequency.

[0074] The frequency of the arrows may be tuned by adjusting the coupler flux allowing for a tunable gate operation frequency.

[0075] Prior art implementing such a gate does not involve coupler qubits, and thus cannot have the gate operation frequency tuned in situ. That is, the gate operation frequencies are fixed after fabrication of the device, which can be inconvenient when using more than two qubits.

[0076] In contrast, in accordance with the techniques described herein, a gate comprising coupler qubits can have the gate operation frequency tuned in situ. Tuning the gate operation frequency allows for flexibility when coordinating many different qubit pairs in a larger device

[0077] The above-described embodiments of the technology described herein can be implemented in any of numerous ways. Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements notspecifically described in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0078] Also, the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0079] Further, some actions are described as taken by a “user.” It should be appreciated that a “user” need not be a single individual, and that in some embodiments, actions attributable to a “user” may be performed by a team of individuals and / or an individual in combination with computer-assisted tools or other mechanisms.

[0080] Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

[0081] The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value. The term “substantially equal” may be used to refer to values that are within ±20% of one another in some embodiments, within ±10% of one another in some embodiments, within ±5% of one another in some embodiments, and yet within ±2% of one another in some embodiments.

[0082] The term “substantially” may be used to refer to values that are within ±20% of a comparative measure in some embodiments, within ±10% in some embodiments, within ±5% in some embodiments, and yet within ±2% in some embodiments. For example, a first direction that is “substantially” perpendicular to a second direction may refer to a first direction that is within ±20% of making a 90° angle with the second direction in some embodiments, within ±10% of making a 90° angle with the second direction in some embodiments, within ±5% of making a 90° angle with the second direction in some embodiments, and yet within ±2% of making a 90° angle with the second direction in some embodiments.

[0083] Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

[0084] Although the disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, the present disclosure has been made only by way of example. Thus, numerous changes in the details of implementation of the disclosed subject matter may be made without departing from the spirit and scope of the disclosed subject matter.

[0085] Accordingly, the scope of this patent should not be limited to the described implementations but rather should be limited only by the spirit and scope of the following claims.

[0086] All publications and references cited in this patent are expressly incorporated herein by reference in their entirety.I claim:1 . A system comprising: a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, such that interactions between the resonant circuit and the first and second qubits alters energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.2. The system according to claim 1 , wherein the resonant circuit comprises a transmon qubit.3. The system according to claim 1 , wherein the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.4. The system according to claim 1 , wherein the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.5. The system according to claim 1 , wherein the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.6. The system according to claim 1 , configured to perform a two-qubit gate in response to being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.7. The system according to claim 6, wherein the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.ZZ REDUCTION VIA THIRD-ORDER DISPERSIVE INTERACTIONSCROSS-REFERENCE TO RELATED APPLICATIONSThis application claims the benefit of U.S. Provisional Application No. 63 / 488,219 filed on March 3, 2023 of which is hereby incorporated herein by reference in its entireties.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0001] The invention was made with government support under W911 NF-18-1- 0116 awarded by the U.S. Army Research Office and FA8702-15-D-0001 awarded by the U.S. Air Force. The government has certain rights in the invention.PARTIES TO A JOINT RESEARCH AGREEMENT

[0002] Not applicable.BACKGROUND

[0003] One obstacle in scaling-up quantum processors is the parasitic always- on ZZ interaction that arises when qubits are coupled. The ZZ interaction between two qubits is defined as the energy difference between when both qubits are in their excited state and when they are individually in their excited state. This interaction, which shifts the frequency of one qubit depending on the state of another qubit, is an entangling interaction that occurs during the normal idling state of a quantum system. When large, the ZZ interaction prevents straightforward implementation of high-fidelity one- and two-qubit gates.

[0004] The parasitic, always-on ZZ interaction arises naturally between coupled superconducting qubits. The underlying cause of non-zero ZZ interaction is the many extra higher energy levels of real-life qubits, which end up pushing on some of the important computational states of the system. Mathematically, ifdenotes a transition frequency of a two-qubit state corresponding to qubit 1 being in its ithstate and qubit 2 being in its jthstate, then the ZZ interaction rate is given mathematically by = 6) - a>01- co10+ t0o- A common scenario in superconducting qubits consists of two capacitively coupled transmon qubits. Here, the |02> and |20> states push on the 111> state through capacitive couplingsuch that the frequency of the 111> state is no longer simply the sum of the frequencies of the individual |01> and 110> states, resulting in = 0. It is possible to compensate for this ZZ interaction in gates which commute with the ZZ operator, such as isolated single qubit gates or cPHASE type gates. In general, however, the ZZ interaction will manifest as a gate error and is at the forefront of infidelity processes in large scale quantum processors.

[0005] There exists a number of different methods for canceling out ZZ interaction in superconducting qubits. The two broad categories for these methods are cancelation of ZZ interaction by applying a microwave cancellation pulse, or by choosing the processor architecture carefully so that the native interactions between the qubits suppresses ZZ interaction.

[0006] One drawback of methods in the first category is that the extra microwave tone needed to cancel ZZ cancelation increases resource requirements for the control of the quantum processor and greatly complicates calibrating the device. While methods in the second category attempt to natively solve the problem of cancelation of ZZ interaction, methods in the second category rely upon particular qubit parameters which puts demands on the precision of the design and fabrication.SUMMARY OF DISCLOSED EMBODIMENTS

[0007] In accordance with the concepts, systems and techniques disclosed herein, described is a method of reducing parasitic ZZ interactions using third- order dispersive interactions and its implementation in superconducting qubits.

[0008] In accordance with one example embodiment illustrating the concepts described herein, ZZ interaction between two computational, fluxonium qubits is reduced by inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions. These 3rdorder interactions are in many cases oppositely signed than the 2ndand 4thorder terms, giving a mechanism for reducing the ZZ interaction between the computational qubits.

[0009] The concepts, systems and techniques described herein may, in a sense, be considered by some to belong to the second category of ZZ2cancelation, i.e., choosing the processor architecture carefully so that the native interactions between the qubits suppresses the ZZ.

[0010] However, unlike traditional approaches of using oppositely signed anharmonicity qubits or the 4thorder terms from a tunable coupler to cancel ZZ interaction, the concepts, systems and techniques described herein work with low frequency qubits (e.g., less than about 1 GHz) without a lower frequency coupler . An additional advantage over existing methods is that the reduction of ZZ interaction is stable for a wide range of circuit parameters and coupler frequencies.

[0011] In contrast, using oppositely signed anharmonicity qubits as in the prior art, results in large ZZ variations with device parameters which relies heavily on fabrication accuracy (i.e. , prior art approaches require tolerances which are very difficult, and thus very expensive, to meet and consistently maintain during a fabrication process). Furthermore, traditional tunable couplers have a ZZ interaction characteristic that depends strongly on the coupler’s frequency (e.g., depends strongly on the transition frequency between the ground and first excited state of the coupler). Moreover, reducing the ZZ interaction during idle simplifies quantum logic gate design.

[0012] In accordance with one aspect of concepts described herein a two-qubit gate can be performed by driving to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by a resonant circuit. In embodiments, an operation frequency can be adjusted by changing a characteristic of the resonant circuit. In embodiments, an operation frequency can be adjusted by adjusting a frequency of the resonant circuit.

[0013] In accordance with a further aspect of concepts described herein, described is a method for reducing parasitic ZZ interactions using third-order dispersive interactions and its implementation in superconducting qubits. In one example embodiment, a system comprises a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, wherein interactions between the resonant circuit and the first and second qubitsalters at least one energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

[0014] In accordance with a further aspect of concepts described herein, a system comprises first and second fluxonium qubits and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, wherein interactions between the resonant circuit and the first and second qubits alters energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

[0015] In embodiments, the resonant circuit may comprise a transmon qubit.

[0016] In embodiments the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

[0017] In embodiments, the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

[0018] In embodiments, the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

[0019] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

[0020] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0021] In accordance with a still further aspect of concepts described herein a system comprises first and second qubits a resonant circuit coupled to both the first and second qubits according to respective coupling strengths wherein interactions between the resonant circuit and the first and second qubits alter at least one energy level of computational states of the system to thereby reduce an always-on ZZ interaction.

[0022] In embodiments, the resonant circuit may comprise a transmon qubit.

[0023] In embodiments the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

[0024] In embodiments, the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

[0025] In embodiments, the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

[0026] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

[0027] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0028] In embodiments, the first qubit is provided as a first fluxonium qubit and the second qubit is provided as a second fluxonium qubit.

[0029] In embodiments, the system is configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of the resonant circuit.

[0030] In embodiments, the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

[0031] In accordance yet a further aspect of concepts described herein, a method for reducing ZZ interaction between two qubits includes inserting a coupler transmon qubit between the between the two qubits, wherein the coupler transmon qubit relies on third-order (3rdorder) dispersive interactions oppositely signed with respect to second (2nd) and (4th) order dispersive interactions such that the third-order (3rdorder) dispersive interactions cancel at least some of thesecond (2nd) and (4th) order dispersive interactions to thereby reduce ZZ interaction between the two qubits.

[0032] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational qubits.

[0033] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational fluxonium qubits.

[0034] In embodiments, inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions between two fluxonium qubits reduces ZZ interaction between the two fluxonium qubits.

[0035] In embodiments, inserting a coupler transmon qubit between two fluxonium qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the fluxonium qubits, but only affects ZZ interaction at higher orders.

[0036] In embodiments, inserting a coupler transmon qubit between the between the two qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the two qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the two qubits, but only affects ZZ interaction between the two qubits at higher orders.DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0037] The manner and process of making and using the disclosed embodiments may be appreciated by reference to the figures of the accompanying drawings. It should be appreciated that the components and structures illustrated in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principals of the concepts described herein. Like6reference numerals designate corresponding parts throughout the different views. Furthermore, embodiments are illustrated by way of example and not limitation in the figures, in which:

[0038] FIGs. 1A, 1 B, 1C are energy level diagrams representing 2ndorder (FIG. 1A), 3rdorder (FIG. 1 B), and 4thorder (FIG.. 10) perturbation theory assuming a dispersive interaction;

[0039] FIG. 2 is a block diagram of an example two-qubit system in accordance with the concepts, techniques, and structures disclosed herein;

[0040] FIG. 2A is a mask design of an example two-qubit system in accordance with the concepts, techniques, and structures disclosed herein;

[0041] FIG. 3 is a plot of fluxonium qubit-to-qubit coupling vs. qubit-to- transmon coupling (both in MHz) for the system of FIG. 2A; and

[0042] FIG. 4 shows a plot of the ZZ interaction between qubits (in kHz) as a function of magnetic flux applied to the coupler (in units of the flux quantum $0).

[0043] FIG. 5 is a schematic diagram of a device comprising a pair of qubits coupled through an intermediate circuit in accordance with the concepts, techniques, and structures disclosed herein.

[0044] FIG. 6 is an energy level diagram of a device comprising a pair of qubits coupled by a coupling circuit in accordance with the concepts, techniques, and structures disclosed herein.DETAILED DESCRIPTION

[0045] Quantum bits (or more simply qubits) may be described as quantum mechanical systems having quantized energy levels and a transition frequency at which they transition between those energy levels. Thus, in a sense, qubits can be viewed as oscillators. The transition frequency for a qubit depends upon the state of excitation of the qubit. This transition frequency can be tuned in a variety of ways, including by applying external, biasing magnetic fluxes to the qubit.7

[0046] A multi-qubit system behaves similarly, except that coherent quantum mechanical coupling terms play a role. A system with multiple qubits may have several total energy levels that are relatively close (i.e., the energy level difference is not large compared to the coupling energy) up to a 1storder approximation. Such situations occur, for example in a system with two substantially identical qubits that are capacitively coupled. The coupling between the two systems pushes the energy levels of the two quantum states in opposite directions. The magnitude of this pushing is proportional to the coupling strength.

[0047] Typical prior art quantum logic systems are constructed from transmon qubits coupled by transmon couplers. These systems have excited states with energy levels that are designed to push a third energy level in opposite directions, thereby stabilizing that third level. However, repulsions between the levels are designed to be deliberately strong to avoid accidental transitions between the states, while simultaneously there is a need to fine-tune the energy of the third level, and this balance is difficult to achieve in practice. Also, manufacturing flaws can make it difficult to tune some components (e.g., the coupler) after fabrication.

[0048] For concreteness, described herein are example embodiments of the concepts, techniques, and structures disclosed herein when the two qubits are superconducting fluxonium qubits, which have a very low excited state frequency (e.g., less than about 1 GHz).

[0049] Fluxonium qubits have a large (e.g., greater than about 1 GHz), positive anharmonicity. As a result, the higher excited states of fluxonium qubits (i.e. the |20> state, the |21> state, the |02> state, etc.) push down on the energy levels of the computational states (i.e. |00>, |01 >, |10>, and |11 >).

[0050] Normally, a fluxonium-fluxonium (“F-F”) system doesn’t have 3rdorder terms in its Hamiltonian; however introducing a third qubit would add these terms. Intuitively, introducing a transmon coupler in the system should be disadvantageous since its higher energy levels should push down on the levels of the computational basis.8

[0051] However, the inventors have recognized that introducing a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits does not affect the 2ndorder ZZ interaction between the qubits, but only affects it at higher orders.

[0052] Further, in accordance with the concepts, systems, devices and techniques described herein, the inventors have discovered that introducing a transmon coupler introduces dispersive terms that can be used to reduce or ideally eliminate parasitic ZZ interaction between the fluxonium qubits. These 3rdorder interactions are oppositely signed to the 2ndand 4thorder terms, and thus provide a mechanism for reducing the overall ZZ interaction.

[0053] Accordingly, and more generally, described herein is a method for reducing parasitic ZZ interactions using third-order dispersive interactions and its implementation in superconducting qubits.

[0054] In this connection, FIG. 1A is a level diagram representing 2ndorder perturbation theory in such embodiments, assuming a dispersive interaction; FIG. 1 B is a level diagram representing 3rdorder perturbation theory assuming a dispersive interaction; and FIG. 1C is a level diagram representing 4thorder perturbation theory assuming a dispersive interaction.

[0055] In FIGs. 1A-1C, E® represents the energy of the computational state |n) accurate to jthorder perturbation theory, and Vjkrepresents the matrix element of the coupling Hamiltonian <j|V|k>. Here, the state |n) is labeled generically but represents the |11> or 1101> computational state in the described scheme (i.e., both qubits are in their first excited state). When all intermediate states (labeled by kj) lie above |n>, only the 3rdorder interaction provides a positive energy shift on state |n).

[0056] As may be seen from FIG. 1A, the computational state |n>, written as general state here, is meant to represent the 111) state of a two-fluxonium system. Second-order interactions from higher energy states, although far away, push down on this state causing a negative ZZ interaction. In this situation,9conventional methods of using 2ndorder interactions to push up on the computational state from below are unfeasible.

[0057] By contrast, in accordance with embodiments of the concepts, systems, devices, circuits, techniques, and structures disclosed herein and as illustrated in FIG. 1 B, inserting an intermediate circuit such as a resonant circuit coupled between the two fluxoniums creates 3rdorder perturbation terms to pull up the 1101) state (where we introduce the middle index in the notation to represent the coupler state). In embodiments, a resonant circuit (or other intermediate circuit) acts as a coupler between the two fluxoniums to create 3rdorder perturbation terms to pull up the 1101) state. In embodiments, inserting a resonant circuit such as a transmon between the two fluxoniums creates 3rdorder perturbation terms to pull up the 1101) state. In embodiments, a coupler may be inserted between the two fluxoniums to create 3rdorder perturbation terms to pull up the 1101) state.

[0058] Furthermore, since all dipole allowed interactions with the 1101) state have higher frequency, every existing 3rdorder interaction will pull up on the 1101) state. By tuning the relative coupling strengths between the qubits and the intermediate circuit (e.g., the resonant circuit or coupler), the effects of these third- order terms can counteract the 2ndand 4thorder ZZ contributions (as illustrated in Figs. 1A and 1C, respectively) to within about 10 kHz, with typical circuit parameters. An added benefit is that this ZZ interaction varies slowly with the coupler frequency, allowing for the ZZ interaction rate to remain less than about 10 kHz across the entire spectrum of the coupler.

[0059] FIG. 2 is a block diagram of an example two-qubit system 10 in accordance with the concepts, systems, circuits, devices, techniques, and structures disclosed herein. The system has a pair qubits 12, 14 (i.e. , first and second qubits) coupled through an intermediate circuit 16. This circuit could be generalized to a larger N-qubit system by constructing a lattice of fluxonium qubits (such as qubits 12 and 14) such that there exists an intermediate coupling circuit (such as circuit 16) between each pair of adjacent qubits.10

[0060] The system may be driven by one or more signal sources 18a, 18b, 18c. Although three signal sources are shown in FIG. 2, only a single signal source is needed. Further, signal sources 18a - 18c are here shown in phantom since a signal source is not properly a part of system 10. In embodiments a drive signal provided by signal sources 18a - 18c may be oscillating current or voltage signals. In embodiments, a signal source 18a may, for example, provide an oscillating current signal to qubit 12. In embodiments, a signal source 18c may, for example, provide an oscillating current signal to qubit 14. In embodiments, a signal source 18b may, for example, provide an oscillating current signal to coupler 16. Thus, each signal source 18a -18c provides a signal to a different element of system 10.

[0061] In general, multiple signal sources may each couple to different, respective ones of multiple elements in a system such as system 10. In embodiments, it is possible that there may be a desire or a need to have a single signal source provide one or more signals to different elements. For example, in some embodiments it may be desired or required that source 18a provide a signal (e.g., a drive signal such as an oscillating current or voltage signal) to two or more of qubits 12, 14 and coupler 16. In embodiments, a single signal source may provide signals to all elements. In embodiments, a first signal source (e.g., signal source 18a) may provide signals to some elements (e.g., qubits 12, 14) and a second signal source (e.g., signal source 18b) may provide signals to other elements (e.g., coupler 16).

[0062] In accordance with the concepts, systems, circuits, devices, techniques, and structures disclosed herein, FIG. 2A shows a mask design for a two-qubit system 10’ which may be the same as or similar to system 10 described above in conjunction with FIG. 2. The system 10’ has a pair of qubits 12, 14 (i.e., first and second qubits) coupled through an intermediate circuit 16. In this example embodiment, qubits 12, 14 are illustrated as fluxonium qubits 12, 14, and intermediate circuit 16 is illustrated as a transmon coupler 16. Thus, in the example system of FIG. 2A, a pair of fluxonium qubits 12, 14, are coupled through a transmon coupler 16 and system 10’ may be referred to as a fluxonium- transmon-fluxonium system.11

[0063] To facilitate the coupling between the two fluxonium qubits, the coupler may be or may comprise any circuit with a resonant mode including but not limited to: a resonator; cooper-pair-box; and / or a flux qubit.

[0064] Fluxonium qubit 12 has an associated resonator 22, charge line 32, and flux line 42. Likewise, Fluxonium qubit 14 has an associated resonator 24, charge line 34, and flux line 44. Finally, the transmon coupler 16 has an associated resonator 26 and flux line 46.

[0065] The fluxonium qubits 12, 14 may be used to perform quantum logic gate operations. Each such qubit 12, 14 is constructed of rectangular plates, coupled by wires having various electrical and magnetic properties (e.g., inductances, Josephson junctions, etc.) as known in the art to produce a fluxonium qubit having desired design parameters such as transition frequencies within a given range. In this example embodiment, each qubit 12, 14 is capacitively coupled to the transmon coupler 16 via a parallel plate capacitance. The capacitance is formed by the metal pads of each qubit being in close proximity. This is roughly illustrated as the two edges of the boxes of each qubit being close to each other (e.g., as illustrated in FIG. 2).

[0066] The resonators 22, 24, 26 may be capacitively coupled to each qubit and lead to an external device to measure the quantum state of the corresponding qubit or coupler at desired times. The charge lines 32, 34 may be used to drive qubit state transitions (i.e. between a ground state |0> and a first excited state |1 >). The flux lines 42, 44, 46 may be used to provide static or varying magnetic flux to the qubits and the coupler, and in particular applied to their respective magnetic flux loops, thereby modulating these components to implement a quantum logic gate. Thus, in the example embodiment of FIG. 2A, signals may be coupled to the system 10' via one or more signal paths 32, 34, 42 and 44.

[0067] FIG. 3 shows a plot of fluxonium qubit-to-qubit coupling vs. qubit-to- transmon coupling (both in MHz) for the system of FIG. 2A. In FIG. 3, curve 54 (and designated “Traditional F-F”) represents the direct fluxonium qubit-to-qubit coupling strength, and curve 50 (and designated “F-T-F”) represents the qubit-to- transmon coupler strength.12

[0068] A traditional, two qubit fluxonium-fluxonium (F-F) system has no coupler and is therefore represented by dashed line 54 along the horizontal axis. As may be appreciated, as the coupling strength increases between the qubits in this system (i.e. as curve 54 is followed to the right), the ZZ interaction also increases (as represented by the darkening shade of the plot). This 2ndorder interaction undesirably pushes down on the energy of the useful computational state of the system by upwards of 400 kHz.

[0069] Noticeably, however, there exists a ’’trough region” in ZZ interaction strength through which F-T-F curve 50 substantially passes, which represents the fluxonium-transmon-fluxonium system of an embodiment, such as example system 10’ shown in FIG. 2A. In a system according to an embodiment, the coupling between each fluxonium qubit and the transmon coupler (i.e. the “F-T Coupling” strength on the vertical axis) also increases with “F-F Coupling” strength on the horizontal axis. Advantageously, and unlike in the prior art, the presence of the transmon coupler between the two fluxonium qubits introduces 3rdorder effects that decrease the ZZ interaction strength to nearly zero (as indicated by the lack of shading in the trough region). Thus, it is possible to obtain low ZZ interaction between the two qubits (e.g., on the order of single-digit kHz), even when the direct coupling strength is as large as in conventional, F-F systems that lack a transmon coupler (e.g., between 200 and 400 MHz). The F-T characteristics of one example embodiment (which may be the same as or similar to the example embodiment of FIG. 2A) are labelled with reference numeral 52 in FIG. 3.

[0070] FIG. 4 is a plot of ZZ interaction between the qubits (in kHz) as a function of magnetic flux applied to the coupler (in units of the flux quantum (,). This plot shows a solid, model curve 58 of the predicted ZZ interaction, and measured data with error bars. Note that the vertical axis, showing ZZ interaction strength, is in the low single-digit, negative kHz. As predicted, the strength of the ZZ interaction does not depend much on the coupler frequency, but more so on the static coupling strengths (i.e., the F-T coupling strength and the F-F coupling strength).13

[0071] FIG. 5 is a schematic diagram of a device comprising a pair of qubits 62, 66 coupled via coupler 64 (e.g., a schematic representation of an example two-qubit system such as the example two-qubit system of FIG. 2A).

[0072] Referring now to FIG. 6, an energy level diagram 70 with state labels |qubit 1 , coupler, qubit 2> denoted with reference numerals 72a - 72g. The indices of each state label are ordered qubit 1 , coupler, qubit 2. For example, looking at the top, there are states |201>, |111>, and 1102> respectively denoted 72a , 72b 72c. State label 72a |201 > represents qubit 1 in state 2, coupler in state 0, and qubit 2 in state 1. Likewise for |111 > and all other labels.

[0073] Driving any of the transitions illustrated by arrows 74a, 74b, 74c for a single-period Rabi oscillation results in a two-qubit gate. This drive may be performed by a signal source (e.g., one or more of signal sources 18 in FIG. 1) which provides an oscillating voltage or current signal which is coupled to the system. Up to single-qubit corrections, this two-qubit gate will be the CZ (CPhase with 180 degrees) gate when driven with a calibrated drive amplitude and drive frequency.

[0074] The frequency of the arrows may be tuned by adjusting the coupler flux allowing for a tunable gate operation frequency.

[0075] Prior art implementing such a gate does not involve coupler qubits, and thus cannot have the gate operation frequency tuned in situ. That is, the gate operation frequencies are fixed after fabrication of the device, which can be inconvenient when using more than two qubits.

[0076] In contrast, in accordance with the techniques described herein, a gate comprising coupler qubits can have the gate operation frequency tuned in situ. Tuning the gate operation frequency allows for flexibility when coordinating many different qubit pairs in a larger device

[0077] The above-described embodiments of the technology described herein can be implemented in any of numerous ways. Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements notspecifically described in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0078] Also, the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0079] Further, some actions are described as taken by a “user.” It should be appreciated that a “user” need not be a single individual, and that in some embodiments, actions attributable to a “user” may be performed by a team of individuals and / or an individual in combination with computer-assisted tools or other mechanisms.

[0080] Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

[0081] The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value. The term “substantially equal” may be used to refer to values that are within ±20% of one another in some embodiments, within ±10% of one another in some embodiments, within ±5% of one another in some embodiments, and yet within ±2% of one another in some embodiments.15

[0082] The term “substantially” may be used to refer to values that are within ±20% of a comparative measure in some embodiments, within ±10% in some embodiments, within ±5% in some embodiments, and yet within ±2% in some embodiments. For example, a first direction that is “substantially” perpendicular to a second direction may refer to a first direction that is within ±20% of making a 90° angle with the second direction in some embodiments, within ±10% of making a 90° angle with the second direction in some embodiments, within ±5% of making a 90° angle with the second direction in some embodiments, and yet within ±2% of making a 90° angle with the second direction in some embodiments.

[0083] Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

[0084] Although the disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, the present disclosure has been made only by way of example. Thus, numerous changes in the details of implementation of the disclosed subject matter may be made without departing from the spirit and scope of the disclosed subject matter.

[0085] Accordingly, the scope of this patent should not be limited to the described implementations but rather should be limited only by the spirit and scope of the following claims.

[0086] All publications and references cited in this patent are expressly incorporated herein by reference in their entirety.16

Claims

8. A system comprising: a first qubit; a second qubit; and a resonant circuit coupled to both the first and second qubits according to respective coupling strengths such that interactions between the resonant circuit and the first and second qubits alter at least one energy level of computational states of the system to thereby reduce an always-on ZZ interaction.

9. The system according to claim 8, wherein the resonant circuit comprises a transmon qubit.

10. The system according to claim 8, wherein the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

11. The system according to claim 8, wherein the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

12. The system according to claim 8, wherein the first qubit is provided as a first fluxonium qubit and the second qubit is provided as a second fluxonium qubit.

13. The system according to claim 8, configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of the resonant circuit.

14. The system according to claim 13, wherein the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

15. A method for reducing ZZ interaction between two qubits, the method comprise inserting a coupler transmon qubit between the between the two qubits, wherein the coupler transmon qubit relies on third-order (3rdorder) dispersive interactions oppositely signed with respect to second (2nd) and (4th) orderdispersive interactions such that the third-order (3rdorder) dispersive interactions cancel at least some of the second (2nd) and (4th) order dispersive interactions to thereby reduce ZZ interaction between the two qubits.

16. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational qubits.

17. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational fluxonium qubits.

18. The method of claim 17 wherein inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions between two fluxonium qubits reduces ZZ interaction between the two fluxonium qubits.

19. The method of claim 17 wherein inserting a coupler transmon qubit between two fluxonium qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the fluxonium qubits, but only affects ZZ interaction at higher orders.

20. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the two qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the two qubits, but only affects ZZ interaction between the two qubits at higher orders.I claim:1 . A system comprising: a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, such that interactions between the resonant circuit and the first and second qubits alters energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

2. The system according to claim 1 , wherein the resonant circuit comprises a transmon qubit.

3. The system according to claim 1 , wherein the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

4. The system according to claim 1 , wherein the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

5. The system according to claim 1 , wherein the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

6. The system according to claim 1 , configured to perform a two-qubit gate in response to being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

7. The system according to claim 6, wherein the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

8. A system comprising: a first qubit; a second qubit; and a resonant circuit coupled to both the first and second qubits according to respective coupling strengths such that interactions between the resonant circuit and the first and second qubits alter at least one energy level of computational states of the system to thereby reduce an always-on ZZ interaction.

9. The system according to claim 8, wherein the resonant circuit comprises a transmon qubit.

10. The system according to claim 8, wherein the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

11. The system according to claim 8, wherein the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

12. The system according to claim 8, wherein the first qubit is provided as a first fluxonium qubit and the second qubit is provided as a second fluxonium qubit.

13. The system according to claim 8, configured to perform a two-qubit gate by being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of the resonant circuit.

14. The system according to claim 13, wherein the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

15. A method for reducing ZZ interaction between two qubits, the method comprise inserting a coupler transmon qubit between the between the two qubits, wherein the coupler transmon qubit relies on third-order (3rdorder) dispersive interactions oppositely signed with respect to second (2nd) and (4th) order18dispersive interactions such that the third-order (3rdorder) dispersive interactions cancel at least some of the second (2nd) and (4th) order dispersive interactions to thereby reduce ZZ interaction between the two qubits.

16. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational qubits.

17. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a coupler transmon qubit between two computational fluxonium qubits.

18. The method of claim 17 wherein inserting a coupler transmon qubit which relies on third-order (3rdorder) dispersive interactions between two fluxonium qubits reduces ZZ interaction between the two fluxonium qubits.

19. The method of claim 17 wherein inserting a coupler transmon qubit between two fluxonium qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the fluxonium qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the fluxonium qubits, but only affects ZZ interaction at higher orders.

20. The method of claim 15 wherein inserting a coupler transmon qubit between the between the two qubits comprises inserting a transmon coupler having an operating frequency which is different from an operating frequency of the two qubits such that the coupler transmon qubit does not affect a second (2nd) order ZZ interaction between the two qubits, but only affects ZZ interaction between the two qubits at higher orders.19I claim:1 . A system comprising: a first fluxonium qubit; a second fluxonium qubit; and a resonant circuit coupled to both the first fluxonium qubit and the second fluxonium qubit according to respective coupling strengths, such that interactions between the resonant circuit and the first and second qubits alters energy levels of computational states of the system to thereby reduce an always-on ZZ interaction.

2. The system according to claim 1 , wherein the resonant circuit comprises a transmon qubit.

3. The system according to claim 1 , wherein the first qubit is coupled to the second qubit according to a coupling strength of less than 400 MHz.

4. The system according to claim 1 , wherein the resonant circuit is coupled to the first qubit, or to the second qubit, or to each such qubit, according to a coupling strength of less than 800 MHz.

5. The system according to claim 1 , wherein the first qubit and the second qubit have a ZZ interaction of between -200 kHz and 0 kHz.

6. The system according to claim 1 , configured to perform a two-qubit gate in response to being driven to an energy level outside of a computational subspace, with an operation frequency that can be adjusted by changing a characteristic of a resonant circuit.

7. The system according to claim 6, wherein the operation frequency of the two-qubit gate can be adjusted by a frequency of the resonant circuit.

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