Quantum processor architecture

Abstract

There is provided a quantum processor comprising a plurality of qubits, wherein the qubits are arranged such that direct interactions between any two qubits are suppressed, a harmonic oscillator mode coupled transversally to the plurality of qubits, a plurality of adiabatic driving lines associated with said plurality of qubits, wherein each adiabatic driving line is configured for driving a respective qubit with an adiabatic driving signal and a control module configured to drive concurrently at least a first qubit and a second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals and adjust a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal to establish a ZZ gate between the at least first and second qubits.

Description

QUANTUM PROCESSOR ARCHITECTURE TECHNICAL FIELD

[0001] The presently disclosed subject matter relates, in general, to the field of quantum computing. BACKGROUND

[0002] An efficient quantum processor should have a scalable register of qubits that is easy to initialize, read out and manipulate. It should be possible to entangle together a large number of qubits. At the same time, the register should be well-protected from its environment in order to avoid decoherence and loss of information. In particular, the coupling between the qubits of the register needs to be controlled so that two qubit gates can be quickly applied with high fidelity and substantially no crosstalk is present when one wishes to manipulate a specific qubit. The quantum processor architecture plays a critical role in determining the performance and capabilities of a quantum processor.

[0003] Most of the current implementations of superconducting quantum processors rely on transmon qubits (J. Koch et al., “Charge-insensitive qubit design derived from the cooper pair box”, Physical Review A, 76, 042319,2007). These qubits are relatively easy to control (R. Barends et al., “Coherent Josephson qubit suitable for scalable quantum integrated circuits”, Physical Review Letters, 111, 080502, 2013) and can exhibit long coherence time (A. P. M. Place et al., “New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds”, Nature Communications, 12(1), 2021). Yet, transmon qubits have also a large electric dipole moment, which induces unintentional couplings between qubits that need to be cancelled for the proper functioning of the processor.

[0004] A possible way to control the coupling consists of using tunable couplers (Y. Sung et al., “Realization of high-fidelity cz and zz-free iswap gates with a tunable coupler”, Physical Review X, 11, 021058, 2021). In various implementations, the tunable coupler is itself a flux-tunable transmon qubit. Tunable couplers allow turning on and off the coupling between nearest neighbor qubits. This control is establishedby a dedicated flux line which modifies the resonant frequency of the coupler. A specific bias current is chosen to completely cancel the coupling between the two qubits. To implement a two-qubit gate, the bias current is changed from its default value and a pulse of appropriate length and amplitude is applied to the flux line of the tunable coupler.

[0005] In this approach, a quantum processor unit consists of data qubits and coupler qubits arranged in a lattice. Both data and coupler qubits include a SQUID for tuning their frequency by magnetic flux. Each data qubit is in addition coupled to a flux line for manipulation of its state and to a readout resonator. For large qubit register, the number of control lines of both data and coupler qubit increases quickly and becomes an issue.

[0006] Moreover, coupling between qubits is possible only between nearest neighbors. This means that several successive two-qubit gates are needed if one wishes to entangle two distant qubits. A programmer of such a processor should therefore optimize the algorithm to minimize two-qubit gates between distant qubits, which will have systematically a lower fidelity.

[0007] Another approach to control quantum processors consists of tuning the coupling between qubits dynamically by applying well-chosen microwave drives (C. Rigetti, A. Blais, and M. Devoret, “Protocol for universal gates in optimally biased superconducting qubits”, Physical Review Letters, 94, 240502, 2005). One of the main advantages of these techniques is that they can be directly generalized to large registers with minimal extra hardware and control lines and allow potentially an all-to-all connectivity. Along these lines, several protocols have been proposed and realized in recent years. For instance, one can apply a resonant Rabi frequency drive on both qubits at the same time as suggested in Ref. (C. Rigetti and M. Devoret, “Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies”, Physical Review B, 81, 134507, 2010) or one can even drive one qubit at the resonant frequency of the other (S. Kirchhoff et al., “Optimized cross-resonance gate for coupled transmon systems”.,Physical Review A, 97, 042348, 2018)., inducing dynamics in the latter across the connecting resonator. These techniques (FLICFORQ, cross-resonance, ...) have been demonstrated in recent experiments but are generally considered less efficient than controlled direct coupling.Indeed, one of the main limitations of these techniques is the low anharmonicity of transmons, which limits the possible drive strength.

[0008] Superconducting flux qubits are a possible solution to these problems. These circuits are formed by one or two superconducting aluminum loops intersected by three or four Josephson junctions (T. P. Orlando et al., “Superconducting persistent-current qubit”, Physical Review B, 60, 15398, 1999). They behave as two-level systems with both high anharmonicity and long coherence time (T. Chang et al., “Reproducibility and control of superconducting flux qubits”, Physical Review Applied, 18, 064062, 2022). Moreover, it is possible to arrange the areas and orientations of the loops to form gradiometer qubits (F. G. Paauw et al., “Tuning the Gap of a Superconducting Flux Qubit”, Physical Review Letters, 102, 090501,2009) and thus cancel their dipole magnetic moment. This last property may be useful to cancel spurious couplings between qubits, which strongly mitigate the performances of state-of-the-art transmon architectures. GENERAL DESCRIPTION

[0009] In accordance with certain aspects of the present disclosure, there is provided a quantum processor comprising a plurality of qubits, wherein the qubits are arranged such that direct interactions between any two qubits are suppressed, a harmonic oscillator mode coupled transversally to the plurality of qubits, a plurality of adiabatic driving lines associated with said plurality of qubits, wherein each adiabatic driving line is configured for driving a respective qubit with an adiabatic driving signal and a control module configured to drive concurrently at least a first qubit and a second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals and adjust a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal to establish a ZZ gate between the at least first and second qubits.

[0010] In some embodiments, the quantum processor further comprises a plurality of bias lines associated with said plurality of qubits, wherein each bias line is configured for biasing each respective qubit to an operating point thereof and the control module is further configured to provide a DC bias signal to each dedicated bias line.

[0011] In some embodiments, the quantum processor further comprises a plurality of resonant driving lines associated with said plurality of qubits, wherein each resonantdriving line is configured for driving each respective qubit with a resonant driving signal to establish single qubit gates.

[0012] In some embodiments, the quantum processor is configured so that at least one of the bias lines is further configured to be used as a adiabatic driving line or a resonant driving line, and / or at least one of the adiabatic driving lines is further configured to be used as a bias line or a resonant driving line, and / or at least one of the resonant driving lines is further configured to be used as a bias line or a adiabatic driving line.

[0013] In addition, the quantum processor may comprise a readout module configured to measure the state of each qubit.

[0014] In addition, the control module may be configured to adjust the first and second waveforms so that an integral of the first waveform over a waveform time range and an integral of the second waveform over the waveform time range are zero so that single qubit rotation of the first and second qubits are cancelled.

[0015] In addition, the control module may be configured to adjust the first and second waveforms so that an integral of a product of the first and second waveforms over the waveform time range is configured to cause a two-qubit rotation of the first and second qubits.

[0016] In some embodiments, the plurality of qubits are superconducting gradiometer flux qubits.

[0017] In some embodiments, the harmonic oscillator mode is a mode of a microwave superconducting resonator. The microwave superconducting resonator mode may be coupled transversally to the qubits by magnetic coupling.

[0018] In addition, the bias lines may be configured as flux lines and a DC current applied to the flux lines modifies a magnetic flux threading the superconducting gradiometer flux qubit.

[0019] In addition, the resonant driving lines may be configured as microwave transmission lines and a microwave signal drives the superconducting gradiometer flux qubit.

[0020] In addition, the adiabatic driving lines may be configured as radiofrequency transmission lines and a radiofrequency signal drives adiabatically the superconducting gradiometer flux qubit.

[0021] In some embodiments, the radiofrequency signal has a power spectrum limited to components below 200 MHz. In some embodiments, the microwave signal has a power spectrum that is limited to components within 1-26GHz. In some embodiments, the DC current is below 50 mA.

[0022] There is further provided, according to another aspect of the present disclosure, a method for coupling a first and second qubits of a quantum processor comprising a plurality of qubits, wherein the qubits are arranged such that direct interactions between any two qubits are suppressed, a harmonic oscillator mode coupled transversally to the plurality of qubits, a plurality of adiabatic driving lines associated with said plurality of qubits, wherein each adiabatic driving line is configured for driving a respective qubit with an adiabatic driving signal, the method comprising: (a) driving concurrently at least the first qubit and the second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals, and (b) adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal to establish a ZZ gate between the at least first and second qubits.

[0023] Additionally, the method may comprise adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal comprises adjusting the first and second waveforms so that an integral of the first waveform over a waveform time range and an integral of the second waveform over the waveform time range are zero so that single qubit rotation of the first and second qubits are cancelled.

[0024] Additionally, the method may comprise adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal comprises adjusting the first and second waveforms so that an integral of a product of the first and second waveforms over the waveform time range is configured to cause a two-qubit rotation of the first and second qubits.

[0025] There is further provided, in accordance with another aspect of the present disclosure a quantum processor comprising a plurality of superconducting gradiometerflux qubits arranged such that direct interactions between any two superconducting gradiometer flux qubits are suppressed, a microwave superconducting resonator mode coupled magnetically to the plurality of qubits, a plurality of radiofrequency transmission lines associated with said plurality of superconducting gradiometer flux qubits, wherein each radiofrequency transmission line is configured for driving a respective superconducting gradiometer flux qubit with a radiofrequency signal and a control module configured to perform all to all ZZ coupling by driving concurrently any two superconducting gradiometer flux qubits with radiofrequency signals.

[0026] Unless otherwise defined, all technical and / or scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the present disclosure pertains. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and are not intended to be necessarily limiting.

[0027] Implementation of the method and / or system of embodiments of the present disclosure can involve performing or completing selected tasks manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of embodiments of the method and / or system of the present disclosure, several selected tasks could be implemented by hardware, by software or by firmware or by a combination thereof using an operating system.

[0028] For example, hardware for performing selected tasks according to embodiments of the present disclosure could be implemented as a chip or a circuit. As software, selected tasks according to embodiments of the present disclosure could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In an exemplary embodiment of the present disclosure, one or more tasks according to exemplary embodiments of method and / or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes a volatile memory for storing instructions and / or data and / or a non-volatile storage, for example, a magnetic hard-disk and / or removable media, for storing instructions and / or data. Optionally, a network connection is provided as well. A display and / or a user input device such as a keyboard or mouse are optionally provided as well.BRIEF DESCRIPTION OF THE DRAWINGS

[0029] In order to understand the present disclosure and to see how it may be carried out in practice, embodiments will now be described, by way of non-limiting example only, with reference to the accompanying drawings, in which: FIG.1 is a schematic block diagram showing the different components of the quantum processor according to embodiments of the present disclosure; FIG.2A represents a squared waveform that can be applied to two qubits in order to generate a ZZ gate according to embodiments of the present disclosure; FIG. 2B represents an average fidelity of the ZZ gate versus an amplitude of the waveform |ε| according to embodiments of the present disclosure; FIG.2C illustrates an average fidelity of the ZZ gate versus the waveform time range T and the amplitude of the waveform |ε| according to embodiments of the present disclosure; FIG. 3A and 3B are a schematic drawing showing a current flowing in the two degenerate minima of the potential energy of a gradiometer flux qubit at its optimal point according to embodiments of the present disclosure; FIG. 4A shows calculated eigenenergies of the circuits of FIGS. 3A-3B versus a magnetic flux difference threading the loops; FIG.4B shows a calculated qubit spectroscopy when a magnetic flux difference in thecircuits of FIGS. 3A-3B is close to a flux quantumΦ0 = h / 2e, where h is the Planckconstant and e the electron charge. At precisely Φ0, the transition energy of the qubit is minimal and equal to the gap Δ; FIG.4C shows a gradiometer flux qubit minimal frequency versus the ratio α of the junctions of the circuits of FIGS.3A-3B; FIG. 5A represents a magnetic field generated by a current flowing in a microwave resonator. The magnetic field threads the loops of a gradiometer flux qubit situated in the close vicinity of the microwave resonator according to embodiments of the present disclosure;FIG.5B represents a possible implementation of magnetic coupling of a gradiometer flux qubit to a transmission line resonator, where the gradiometer flux qubit and the resonator share a common coupling element according to embodiments of the present disclosure; FIG. 6 represents a circuit diagram of a quantum processor composed by superconducting gradiometer flux qubits magnetically (galvanically) coupled to a single transmission line resonator and controlled by adiabatic driving lines according to embodiments of the present disclosure. Each adiabatic driving line may further be configured to be used as a bias line in order to bias a magnetic field gradient in the qubit loops; DETAILED DESCRIPTION OF EMBODIMENTS

[0030] In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. However, it will be understood by those skilled in the art that the presently disclosed subject matter may be practiced without these specific details. In other instances, well-known methods, procedures, components and circuits have not been described in detail so as not to obscure the presently disclosed subject matter.

[0031] It is appreciated that, unless specifically stated otherwise, certain features of the presently disclosed subject matter, which are described in the context of separate embodiments, can also be provided in combination in a single embodiment. Conversely, various features of the presently disclosed subject matter, which are described in the context of a single embodiment, can also be provided separately or in any suitable sub-combination. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the methods and apparatus.

[0032] It will also be understood that a system according to the present disclosure may be, at least partly, implemented on a suitably programmed computer. Likewise, the present disclosure contemplates a computer program being readable by a computer for executing the method of the present disclosure. The present disclosure further contemplates a non-transitory computer-readable memory tangibly embodying a program of instructions executable by the computer for executing the methods of thepresent disclosure. The term “each” may not be exclusively understood as referring to each and every, and when technically relevant may also refer to “at least some”.

[0033] FIG. 1 generally illustrates a quantum processor according to embodiments of the present disclosure. The quantum processor may be regarded as a processing unit for a larger quantum computing hardware. The quantum processor may contain a qubit register 1 including a plurality of qubits 11, 12, 13 coupled transversally (i.e. a position or momentum operator of a harmonic oscillator mode is coupled to a ^^^^or ^^^^operators of each qubit) to a harmonic oscillator mode 2. The plurality of qubits 11, 12, 13 may be coupled by coupling elements 31, 32, 33 to the harmonic oscillator mode 2. The processor may further include a plurality of adiabatic driving lines 41, 42, 43. Each adiabatic driving line 41, 42, 43 may be associated with a corresponding qubit 11, 12, 13 and be configured for driving said qubit with an adiabatic driving signal (i.e. a signal having a power spectrum with spectral components at frequencies at least one order of magnitude lower than a transition frequency of the qubit) as explained below. The quantum processor may further include a control module 5 configured to drive concurrently (i.e. simultaneously) at least a first qubit and a second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals. The first and second qubits may be any two qubits of the processor e.g. the first and second qubits may be remote from each other i.e. they may not be nearest neighbors. The control module 5 may further be configured to adjust a first waveform of the first adiabatic driving signal (in other words, to control parameters of the waveform within a bandwidth of the control module) and a second waveform of the second adiabatic driving signal to establish a ZZ gate (i.e. controlled phase shift) between the at least first and second qubits. The control module 5 may further be configured to provide a DC bias signal for biasing each respective qubit to an operating point by a plurality of bias lines 61, 62, 63. Additionally, the control module 5 may further be configured to control resonant driving lines 71, 72, 73 for driving each respective qubit with a resonant driving signal to establish single qubit gates. The register 1 including the plurality of qubits 11, 12, 13 may be connected to a readout module 8 configured to measure the state of each qubit. It may be advantageous to reduce the number of lines per qubit and thus to use at least one of the adiabatic driving lines as bias line or as resonant driving line or use at least one of the bias lines as adiabatic driving line or asresonant driving line or use at least one of the resonant driving lines as adiabatic driving line or bias line.

[0034] In order to illustrate the coupling mechanism, we write the Hamiltonian of a system which includes a first qubit of transition frequency Δ1and a second qubit of transition frequency Δ2, both coupled transversally to a harmonic oscillator mode of frequency ^^^^by coupling constant ^^1and ^^2respectively. The two qubits are respectively driven by a first and second adiabatic driving signals ^^^^1and ^^^^2, eachhaving a first and second waveforms ^^1(^^), ^^2(^^). The Hamiltonian of the system canthus be written as ^^ = ^^0 + ^^^^ + ^^^^1 + ^^^^2where ^^

[0035] The adiabatic drives ^^^^1and ^^^^2may be non-resonant for both qubits but can entangle them when their waveforms are controlled adequately. To demonstrate this, we apply to the total Hamiltonian a unitary transformation ^^^^where ^^ is a well-chosen anti-Hermitian operator. Using Baker Campbell Hausdorff formula, the transformed Hamiltonian can be approximated as ^̃^ = ^^^^^^^^−S ≃ ^^ + [S, ^^] +1 [S, [S, ^ ]1 []2^ ] +6[S, [S, S, ^^ ]]Specific forms of ^^ can be chosen for the expansion to be diagonal up to a given order.

[0036] In a first transformation we apply ^^^^1for obtaining a cavity mediated interaction, with }This unitary transformation is such that [^^1, ^^0] = −^^^^ and thus using Baker CampbellHausdorff formula we cancel the transversal coupling terms, while the other terms in the expansion take the form: ^}2Higher terms in the expansion are) and thus will not be considered in thisfirst approach.

[0037] The terms on ^^ ^^ ^^^^^^^^is the flip-flop interaction term and can be reduced to ZZ coupling using a second unitary transformation ^^^^2where−)This passive ZZ coupling is proportional to ^^4 / (^^^^ − Δ)3. Including this lattertransformation, the Hamiltonian expansion up to the second order in ^^ / (^^^^ − Δ) takesthe form^̃̃^ = ^^^^2^̃^^^−^^2^^

[0038] The terms in (^^2 + (^^+)2)^^^^^^can be removed in making a third unitary transformation ^^^^3with ^^ ^^The only terms that remain in the expansion up to second order are the one coming from the adiabatic drives ^^ ^^After these three transformations we are left with ^̃ ̃̃^= ^^^^3 ^̃̃^^^−^^3^^ 2

[0039] A ZZ interaction comes out of this expression when we apply a last unitary time-dependent transformation ^^^^4(^^)where ^^Leading to a term^^ ^^1(^^)^^2(^^)^^1^^2^^^^=( 2 2 2^^ ^2^^^^^^1^^2^^^^^ − Δ1)(^^^^ − Δ2)The terms coming )− ^^ arenegligeable if the frequency of the adiabatic drive is small enough in front of the transition frequencies of the qubits (adiabatic driving signal).

[0040] One can choose a pulseand ε2(^^)(i.e. adjust a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal) such that the ^^1^^and ^^^^2 single qubit rotations (related to ^^^^1, ^^^^2 termsin ^̃ ̃̃^ ) are cancelled while the ^^^^^^^^interaction term, which is proportional to ^^1(^^)^^2(^^), is non zero.

[0041] In some embodiments, the first wave form and the second waveform of the first and second adiabatic driving signals are equal on the waveform time range i.e.^^1(^^) = ^^2(^^) = ^^(^^). Then, the qubit-qubit coupling arises from the square of thepulse drive, so the amplitude can be adjusted to achieve the desired 2-qubit rotation. Coupling distant qubits is thus achieved in a selective and controllable way.

[0042] To test the validity of the approximations described hereinabove, we calculated numerically the evolution operator ^^(^^)of the total Hamiltonian andchecked under which conditions it can reproduce a ZZ entangling gate, ^^ =exp [−^^^^^^ ^^4 ^^1^^2].

[0043] Every two-qubit operation can be decomposed using the Cartan decomposition as ^^ = ^^1^^^^2Where ^^1,^^2 ∈ ^^^^2^^^^^2 are products of single qubit unitary operators on the qubitsand ^^ is a two qubit operation that cannot be decomposed in a product of single-qubit gates and can be written as ^^]

[0044] The computation was done in two steps where first the local operations ^^1,^^2are found by optimizing the operational fidelity in the subspace of zero photons in the cavitywhere ^^ = ^^1^^^2^|0^^0|. In the second step, we calculated the average fidelity byusing the optimized ^^1,^^2and by sampling uniformly 1000 random points in ^^^^4Where ^^(^^) is the reduced density matrix of the two-qubit subspace obtained by evolving a product state composed of a random initial state in ^^^^4and the cavity at its ground state and tracing out the cavity after the evolution under the propagator ^^(^^).

[0045] FIG. 2A represents the square of a waveform according to embodiments of the present disclosure. Results of a simulation to test numerically the fidelity of the ZZ gate using the waveform of FIG.2A are presented in FIG.2B. The waveform may be defined by the following relation ^^(^^) = |^^|^^(^^) cos(^^^^^^ + ^^)With ^^(^^) an envelope function with a cosine ramp of length ^^, defined as follows ^^^^^^And ^^(^^) = 0 otherwise.

[0046] In FIG. 2A, a waveform time range ^^ is chosen to be 150 ns, the ramping time ^^ is taken to be 20 ns, the adiabatic drive^^2^^ is 10 MHz and the phase of the adiabatic drive ^^ is chosen to be zero. The control pulse length is 150 ns with driving frequency around 10 MHz. The waveform may be chosen to be symmetric with respect to half of the pulse time, so that its integral over the waveform time range is zero. Thus, any single qubit rotation is effectively removed from the dynamics.

[0047] FIG. 2B represents the results of the ZZ gate fidelity versus|^^|. In this Δ simulation, the qubit register contains two qubits with12^^= 7 GHz andΔ22^^= 7.5 GHz while the harmonic oscillator mode=3 GHz and the^^ transversal couplings of the qubits to the harmonic oscillator mode are12^^ = ^^22^^ =200 MHz. The qubit-qubit coupling arises from the square of the pulse drive, so that ^^(^^) can be adjusted to the desired 2-qubit rotation. At the maximum, the average fidelity is 99.91% and only single qubit Z-rotations contribute^^1and ^^2.

[0048] FIG. 2C represents an average fidelity of the ZZ gate versus the waveform | ^^ time range ^^ and|^^ with an adiabatic drive frequency chosen to be^^2^^=50 MHz in presence of three qubits in the register. In this embodiment of the disclosure, the first Δ12^^= 7 GHz, the secondΔ22^^= 7.25 GHz and the third qubit2^^harmonic oscillator mode frequency=3 GHz and^ the transversal couplings of the qubits to the harmonic oscillator mode^22^^ = ^^22^^= 200 MHz. The first and third qubit are driven by the adiabatic drive while thesecond qubit is not. The white regions on the figure indicate high fidelity. Within this range of parameters, the maximum average fidelity is 99.985% for |^^| / 2^^ = 3.54GHz and ^^ = 52 ns.

[0049] FIG. 3A and 3B represent an exemplary schematic drawing of a superconducting gradiometer flux, showing the current flowing in the two degenerate minima (3A or 3B) of its potential energy at its optimal operating point. The gradiometer flux qubit is a superconducting circuit which consists of two micron- sized aluminum loops intersected by four or more Josephson junctions. Three junctions may be identical (i.e. have the same critical current) while the fourth one is smaller than the others by a factor ^^. A DC magnetic flux Φ^^(resp. Φ^^) is threading the top (resp. bottom) loop. We denote as Φ^^ the flux difference Φ^^ = Φ^^ − Φ^^. The phasedifference over each junction are denoted^^1, ^^2, ^^3, ^^^^.Using Faraday law along each loop, we have ^ℎ where Φ0=2^^ is the flux quantum. The potential energy of the circuit can be written as a sum of the potential energies of each Josephson junction intersecting the loops.]Φ^^Φ0 = 1 and for ^^ ≥1 3, the system exhibits two degenerate minima. The position of the minima is given by solving the partial derivative equations ^^^^^^^^ = 0. The twosolutions verify the simple equation sin ^^∗ = ^^ sin 3^^∗and correspond to opposite persistent currents ^^^^ = ^^0 sin ^^∗flowing in the loops and represented in FIG 3A and 3B respectively. An analysis of the capacitive energy (i.e. kinetic energy term) of the circuit is needed to write the full Hamiltonian of the system.

[0050] FIG. 4A shows calculated eigen-frequencies of the circuit versus the magnetic flux difference threading the loops that is obtained by diagonalizing the Hamiltonian.

[0051] FIG. 4B shows a calculated qubit spectroscopy obtained by subtracting the frequency of the first excited state |1^ from the frequency of the ground state |0^ versus ℎ the magnetic flux.Φ^^is close to a flux quantum Φ0= 2^^ (h the Planck constant and e the electron charge), the system behaves as a qubit. Higher levels are far (typically ~50 GHz above the first excited state) and the spectrum can be fully described by two parameters : • The value of the persistent current, . • The so-called flux qubit gap, denoted as ^^ / 2^^ which corresponds to the tunneling term between the two potential minima. Φ At precisely^^Φ0 = 1, the transition frequency of the qubit is minimal and equal to thegap Δ / 2^^. This point is known as the optimal operating point of the qubit due to its immunity at first order to flux noise.

[0052] FIG. 4C represents the gradiometer flux qubit minimal frequency (Δ / 2^^) versus the ratio α of the junctions. The gap is known to be strongly dependent on α., the Hamiltonian ofthe qubit can be written perturbatively asWhere ^̂^ = ^^^^0 sin^^^^ is the operator of the current flowing in the ^^ junction. Whenthis operator is projected on the eigenstates |0^ and |1^ of ^^0, we get ^0|^̂^|0^ = 0, ^0|^̂^|1^ = ^^^^^1|^̂^|0^ = ^^^^, ^1|^̂^|1^ = 0Therefore, the Hamiltonian of the gradiometer flux qubit can be written in this basis as ^^ =ℏ [Δ^^^^ + ^^^^^^]2−Φ0). One of the main advantages of the gradiometer design compared to standard flux qubit designs is that the transition energy is immune to a magnetic field applied uniformly on the two qubit loops together. In particular, the superconducting gradiometer flux qubit is intrinsically protected from external field fluctuations, which act uniformly on the two loops. For instance, the superconducting gradiometer flux qubit is almost immune to the magnetic field generated by another distant superconducting gradiometer flux qubit. The cancellation of the dipolar moment strongly reduces the possible spurious interactions between distant qubits. Indeed, the remaining quadrupolar interactions are known to decay quickly with distance. This property may be useful for isolating each qubit from others and thus for building a quantum processor according to embodiments of the present disclosure.

[0054] FIG.5A illustrates a possible way to couple transversally a superconducting gradiometer flux qubit to a microwave superconducting resonator. For this purpose, the superconducting gradiometer flux qubit is placed at short distance ^^ from a coupling element (i.e. inductor) of the microwave superconducting resonator. The superconducting gradiometer flux qubit is coupled to magnetic field generated by the quantum current fluctuations in the microwave superconducting resonator. Assuming the current is flowing in an infinitely thin wire, it is possible to calculate analytically the magnetic field in the vicinity of the qubit using Biot-Savart law. Namely,^^^^ (^^^^^00 ^) = ^^02^^^^ Where ^^^^0are the quantum current fluctuations in the resonator and ^^ is the distance from the wire. The transversal coupling between the microwave superconducting resonator and the superconducting gradiometer flux qubit may be obtained by integrating the magnetic field threading the loops of the qubit2Where ^^] is the mutual inductance. This simple back-of-the-envelope can be applied for instance to the system shown in FIG. 5A where ^^ = 0.5µm, ^^ = ℓ = 4 µm, ^^^^ = 300 nA and ^^^^0 = 60 nA. We obtain ^^ = 0.4 pH and^^⁄ 2^^ = 10 MHz. Interestingly, when the distance ^^ ≫ ^^, the coupling decreases2.

[0055] FIG. 5B illustrates another possible way to couple transversally a superconducting gradiometer flux qubit to a microwave superconducting resonator. The mutual inductance may be increased by connecting galvanically the superconducting gradiometer flux qubit to the superconducting microwave resonator. In this configuration, the coupling is given by: ℏ^^ = ^^(ℓ + ^^)^^^^^^^^ 20where the mutual inductance per unit length ^^ reaches approximately 3 pH / µm forwires of cross section 200 × 40 nm2 . Thus, the transversal coupling can easily reach^^⁄ 2^^ = 200 − 300 MHz.

[0056] FIG. 6 illustrates a possible realization of the disclosure using a series of superconducting gradiometer flux qubits 101, 102, 103, 104 galvanically connected to a superconducting microwave resonator 20. The transversal coupling between the superconducting gradiometer flux qubits and the superconducting microwave resonator may be obtained like in FIG.5B. In this embodiment of the disclosure, each qubit may be adiabatically driven by a dedicated adiabatic driving line 401, 402, 403, 404 which may be also used as bias line to bias the superconducting gradiometer fluxqubit to its optimal operation point. In this specific case, the resonant driving signals to operate single qubit gates on each qubit are introduced directly via one of the ports 21,21 of the superconducting microwave resonator 20. The control module and the readout module are not represented in this figure. In these conditions, the Hamiltonian of the system can be written as: ^^ = ^^0 + ^^^^ + ^^^^i + ^^^^jwhere ^^^^Where ^^^^is the frequency of a resonator, Δ^^are the gaps of the superconducting gradiometer flux qubits. Theε^^depend linearly on the magnetic flux difference threading the loops Φ^^.(as demonstrated previously) and may be tuned to be zero by using an appropriate DC bias for operating each superconducting gradiometer flux qubit at its optimal operating point. The couplings between each superconducting gradiometer flux qubit and the superconducting microwave resonator are denoted as ^^^^. As already mentioned, the^^direct coupling between qubits may be almost completely cancelled by the special topology of superconducting gradiometer flux qubits.

[0057] It is to be understood that the present disclosure is not limited in its application to the details set forth in the description contained herein or illustrated in the drawings. The present disclosure is capable of other embodiments and of being practiced and carried out in various ways. Hence, it is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting. As such, those skilled in the art will appreciate that the conception upon which this disclosure is based may readily be utilized as a basis for designing other structures, methods, and systems for carrying out the several purposes of the presently disclosed subject matter.

[0058] Those skilled in the art will readily appreciate that various modifications and changes can be applied to the embodiments of the present disclosure as hereinbefore described without departing from its scope, defined in and by the appended claims.

Claims

CLAIMS 1. A quantum processor comprising: − a plurality of qubits, wherein the qubits are arranged such that direct interactions between any two qubits are suppressed, − a harmonic oscillator mode coupled transversally to the plurality of qubits, − a plurality of adiabatic driving lines associated with said plurality of qubits, wherein each adiabatic driving line is configured for driving a respective qubit with an adiabatic driving signal, − a control module configured to: o drive concurrently at least a first qubit and a second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals, and o adjust a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal to establish a ZZ gate between the at least first and second qubits.

2. The quantum processor of claim 1, further comprising a plurality of bias lines associated with said plurality of qubits, wherein each bias line is configured for biasing each respective qubit to an operating point thereof and the control module is further configured to provide a DC bias signal to each dedicated bias line.

3. The quantum processor of any of the preceding claims, further comprising a plurality of resonant driving lines associated with said plurality of qubits, wherein each resonant driving line is configured for driving each respective qubit with a resonant driving signal to establish single qubit gates.

4. The quantum processor of any of the preceding claims, wherein at least one of the bias lines is further configured to be used as a adiabatic driving line or a resonant driving line, and / or at least one of the adiabatic driving lines is further configured to be used as a bias line or a resonant driving line, and / or at least one of the resonant driving lines is further configured to be used as a bias line or a adiabatic driving line.

5. The quantum processor of any of the preceding claims, further comprising a readout module configured to measure the state of each qubit.

6. The quantum processor of any of the preceding claims, wherein the control module is configured to adjust the first and second waveforms so that an integral of the first waveform over a waveform time range and an integral of the second waveform over the waveform time range are zero so that single qubit rotation of the first and second qubits are cancelled.

7. The quantum processor of any of the preceding claims, wherein the control module is configured to adjust the first and second waveforms so that an integral of a product of the first and second waveforms over the waveform time range is configured to cause a two-qubit rotation of the first and second qubits.

8. The quantum processor of any of the preceding claims, wherein the plurality of qubits are superconducting gradiometer flux qubits.

9. The quantum processor of claim 8, wherein harmonic oscillator mode is a mode of a microwave superconducting resonator.

10. The quantum processor of any of claims 8 to 9, wherein the microwave superconducting resonator mode is coupled transversally to the superconducting gradiometer flux qubits by magnetic coupling.

11. The quantum processor of any of claims 8 to 10 as dependent on claim 2, wherein the bias lines are configured as flux lines and a DC current applied to the flux lines modifies a magnetic flux threading the superconducting gradiometer flux qubit.

12. The quantum processor of any of claims 8 to 11 as dependent on claim 3, wherein the resonant driving lines are configured as microwave transmission lines and a microwave signal drives the superconducting gradiometer flux qubit.

13. The quantum processor of any of claims 8 to 12, wherein the adiabatic driving lines are configured as radiofrequency transmission lines and a radiofrequency signal drives adiabatically the superconducting gradiometer flux qubit.

14. The quantum processor of claim 13, wherein the radiofrequency signal has a power spectrum limited to components below 200 MHz.

15. The quantum processor of claim 12, wherein the microwave signal has a power spectrum that is limited to components within 1-26 GHz.

16. The quantum processor of claim 11, wherein the DC current is below 50 mA.

17. A method for coupling a first and second qubits of a quantum processor comprising: − a plurality of qubits, wherein the qubits are arranged such that direct interactions between any two qubits are suppressed, − a harmonic oscillator mode coupled transversally to the plurality of qubits, − a plurality of adiabatic driving lines associated with said plurality of qubits, wherein each adiabatic driving line is configured for driving a respective qubit with an adiabatic driving signal, the method comprising: o driving concurrently at least the first qubit and the second qubit by their respective adiabatic driving lines with a first and second adiabatic driving signals, and o adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal to establish a ZZ gate between the at least first and second qubits.

18. The method of claim 17, wherein adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal comprises adjusting the first and second waveforms so that an integral of the first waveform over a waveform time range and an integral of the second waveform over the waveform time range are zero so that single qubit rotation of the first and second qubits are cancelled.

19. The method of any of claims 17 to 18, wherein adjusting a first waveform of the first adiabatic driving signal and a second waveform of the second adiabatic driving signal comprises adjusting the first and second waveforms so that an integral of a product of the first and second waveforms over the waveform time range is configured to cause a two-qubit rotation of the first and second qubits.

20. A quantum processor comprising: − a plurality of superconducting gradiometer flux qubits arranged such that direct interactions between any two superconducting gradiometer flux qubits are suppressed, − a microwave superconducting resonator mode coupled magnetically to the plurality of qubits, − a plurality of radiofrequency transmission lines associated with said plurality of superconducting gradiometer flux qubits, wherein each radiofrequency transmission line is configured for driving a respective superconducting gradiometer flux qubit with a radiofrequency signal, − a control module configured to perform all to all ZZ coupling by driving concurrently any two superconducting gradiometer flux qubits with radiofrequency signals.

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