Self-correcting GKP qubit in a superconducting circuit with an oscillating voltage bias

Abstract

The present disclosure relates to methods and systems for generating and stabilising Gottesman–Kitaev-Preskill (GKP) encoded quantum states in a superconducting resonator. A first method enables autonomous correction of phase-flip and bit-flip errors by driving a superconducting resonator with a voltage tone and coupling the resonator to a thermodynamic environment for providing error correction via dissipation of noise-induced entropy. A second method implements a read-out protocol with Josephson junction coupled to an intermediate node of the resonator and driven with an auxiliary voltage tone to distinguish logical GKP states via a measured time-averaged supercurrent. A system comprises a superconducting resonator including a Josephson junction, a voltage source for applying the drive tone, and a coupling to an thermodynamic environment arranged to realise dissipative error correction.

Description

[0001] P7918PC00

[0002] 1

[0003] SELF-CORRECTING GKP QUBIT IN A SUPERCONDUCTING CIRCUIT WITH AN OSCILLATING VOLTAGE BIAS

[0004] The disclosure relates to methods and systems for generating and stabilising Gottesman-Kitaev-Preskill (GKP) encoded states in a superconducting resonator. A method for autonomous correction of phase- and bit-flip errors is provided by driving the resonator with a voltage tone and coupling the resonator to a thermodynamic environment that dissipates noise-induced entropy. A read-out method is also provided using a Josephson junction coupled to an intermediate node and driven with an auxiliary tone to distinguish logical states via a time-averaged supercurrent. A system comprises the resonator, a drive-tone source, and an thermodynamic environment dissipative error correction.

[0005] Statement regarding Federally Sponsored Research and Development

[0006] This invention was made with government support under FA9550-22-1-0166 awarded by the Air Force Office of Scientific Research. The government has certain rights in the invention.

[0007] Background

[0008] Dissipative quantum error correction (DEC) offers an alternative route to quantum information processing that may alleviate some of the scalability challenges associated with conventional, measurement-based approaches. By using a thermodynamic reservoir to remove noise-induced entropy, DEC frameworks aim to stabilize encoded quantum information without requiring large numbers of ancillary qubits, continuous readout, or real-time feedback operations.

[0009] Proposals for implementing genuine DEC, capable of protecting against both phase-flip and bit-flip errors, have recently begun to emerge in circuit-QED architectures. These approaches make use of Gottesman-Kitaev-Preskill (GKP) codes, which encode a qubit in grid-like structures in the phase space of an electromagnetic resonator.

[0010] Previous attempts have explored periodic pulse-comb driving as well as stepwise activation protocols for generating effective dynamics that approximate GKP stabilizer potentials. Numerical and analytic results from these works indicate a potential for exponential improvements in qubit stability; at the same time, the requirement for high P7918PC00

[0011] 2

[0012] control resolution and fidelity constitutes an important challenge. Representative examples of such approaches include the works of [Sellem2023] and [Natham2024],

[0013] In view of this, it is an objective of the present disclosure to provide a GKP architecture and method that can achieve stabilisation and error correction of encoded states under experimentally attainable conditions and with reduced control complexity.

[0014] References

[0015] [Gottesman2001] Gottesman, Kitaev, and Preskill, “Encoding a qubit in an oscillator”, Phys. Rev. A 64, 012310 (2001)

[0016] [Sellem2023] Sellem et al., “A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb”, arXiv:2304.01425 (2023)

[0017] [Nathan2024] Nathan et al., “Self-correcting GKP qubit and gates in a driven-dissipative circuit”, arXiv:2405.05671 (2024)

[0018] [Sivak2023] Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50-55 (2023)

[0019] [Campagne-lbarcq2020] Campagne-lbarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368-372 (2020)

[0020] [Lachance-Quirion2024] Lachance-Quirion et al., “Autonomous Quantum Error Correction of Gottesman-Kitaev-Preskill States”, Phys. Rev. Lett. 132, 150607 (2024)

[0021] Summary

[0022] The invention provides a novel design for an operational superconducting qubit where an error correction mechanism is achieved with minimal technical overhead. This principle increases qubit coherence time by several orders of magnitude at potentially minimal cost.

[0023] The design approaches some of the most fundamental challenges in quantum computation:

[0024] 1. Increasing qubit coherence time

[0025] Current implementations of superconducting qubits have insufficient coherence time, prohibiting scaling of existing qubit architectures, such as the transmon-based quantum computers of Google and IBM. Our proposal provides a technologically P7918PC00

[0026] 3

[0027] simple way to achieve error correction within a single circuit element, or physical qubit, increasing its lifetime by orders of magnitude.

[0028] 2. Error-corrected gates

[0029] Besides long coherence, qubits require high-fidelity gates. We show that our proposal admits protected single-qubit Clifford gates, where small deviations from the expected outcome are corrected by the autonomous error correcting mechanism.

[0030] 3. Technologically simple way to generate GKP states.

[0031] Our design is technologically simpler to achieve than other current designs to generate GKP states and qubits based thereon. Besides application as qubits, GKP states have applications in quantum metrology and quantum communication.

[0032] This summary introduces a selection of concepts in simplified form that are described further below in the detailed description. This summary neither identifies key or essential features, nor limits the scope, of the claimed subject matter.

[0033] In a first aspect of the disclosure herein is a method comprising:

[0034] a. driving a superconducting resonator with a voltage tone;

[0035] b. coupling the superconducting resonator to a thermodynamic environment;

[0036] c. generating wave function representations of Gottesman-Kitaev-Preskill error correcting codes (GKP states) that that can encode a qubit or qudit in the wavefunction of the resonator quadratures; and

[0037] d. autonomously error correcting phase and bit flip errors of the encoded qubit by dissipation.

[0038] Harnessing dissipation for quantum error correction is seen as a high-value goal in large parts of the quantum computing community, since the current paradigm of active quantum error correction is costly and its scalability potential remains unclear. Even with the last years’ advances by Google and other companies, only a suppression of error rates for a single qubit by a factor of <10 has been achieved. Further suppression by a factor of order 107is still needed to reach the quality anticipated as necessary for large-scale applications. Even this would not resolve the challenge of combining and controlling hundreds of logical qubits. P7918PC00

[0039] 4

[0040] Our proposed approach aims to address this enormous challenge via a qubit that is autonomously stabilized, or self-correcting. By avoiding, or significantly reducing, the need for costly active error correction, such a qubit could be a game-changer by drastically reducing the cost and complexity of utility-scale quantum information processing. The self-stabilization principle is analogous to classically self-stabilizing bits, such as magnetic bits in hard disks, where dissipation stabilizes encoded information even in the presence of external noise. Despite intensive searches, realizations of an analogous mechanism for dissipative quantum error correction have remained elusive until very recently. Here, we aim to change this state of affairs by proposing a novel scheme for dissipative quantum error correction that brings the industry closer to achieving practical self-correcting qubits or qudits.

[0041] It is worth noting that partial dissipative stabilization is already being adopted by industry, including companies such as Alice & Bob (France) and AWS (US). These companies base their technology on “cat” qubits that self-correct via dissipation but only against bit flips and not against dephasing (phase flips). Hence their approach does not solve the fundamental challenge of quantum computing: protecting qubits against decoherence (wavefunction collapse). As a result, they still rely on costly active error correction to operate.

[0042] Our approach aims to achieve genuine dissipative error correction, which protects against both bit flips and phase flips (also known as decoherence or wavefunction collapse). Thus, our approach intrinsically does not rely on active quantum error correction for operation. A quantum computer based on our approach would therefore require drastically reduced, or potentially no, active error correction, significantly reducing or removing the expensive measurement and auxiliary-qubit apparatus required by standard approaches.

[0043] In a second aspect of the disclosure herein is a system comprising:

[0044] - a superconducting resonator comprising a first Josephson junction coupled to an electromagnetic resonator;

[0045] - a voltage source configured to apply a voltage tone across the first Josephson junction to drive the electromagnetic resonator; - wherein the superconducting resonator is coupled to a thermodynamic environment; and P7918PC00

[0046] 5

[0047] wherein the system is configured to generate wavefunction representations of Gottesman-Kitaev-Preskill (GKP) codes in quadratures of the superconducting resonator.

[0048] In a third aspect of the disclosure herein is a method of executing a read-out protocol for reading a logical Gottesman-Kitaev-Preskill (GKP) state stored in a superconducting resonator, the method comprising:

[0049] a. coupling a Josephson junction to the superconducting resonator at a circuit node located at an intermediate position along an inductance of the superconducting resonator;

[0050] b. applying an auxiliary voltage tone across the Josephson junction; c. measuring a time-averaged supercurrent across the secondary Josephson junction; and

[0051] d. distinguishing between the even and odd well GKP states based on the sign of the measured time-averaged supercurrent.

[0052] This is a new way of reading out the state of a GKP qubit. It avoids the need for homodyne detection or mapping out the Wigner function, which can be costly. The time-averaged supercurrent can, for example, be detected via the frequency shift of a transmon-qubit magnetometer, in which a superconducting quantum interference device (SQUID) and a capacitor are coupled in parallel. If a background flux (e.g., from a flux line) is already threading the SQUID loop, the time-averaged flux generated by the supercurrent produces a shift in the magnetometer’s resonance frequency that depends on the sign of the supercurrent. This frequency shift can be efficiently measured using standard techniques such as dispersive readout. A challenge is ensuring that the Josephson junction connected to the auxiliary node remains effectively decoupled during normal qubit operation to avoid unintended dephasing. This can be achieved by applying a DC bias larger than the Josephson energy of the auxiliary junction when the readout is inactive.

[0053] The auxiliary voltage tone in step (b) may be used to ensure that the auxiliary junction becomes active only when the GKP state in the qubit resonator (which oscillates approximately at a rational multiple of the drive frequency close to w / 2) is aligned such that the encoded Z logical observable corresponds to sign(cos(φ / 2)). In embodiments stabilizing the square or rectangular-lattice GKP encoding, the nodes of the auxiliary P7918PC00

[0054] 6

[0055] voltage tone across the auxiliary junction may be spaced with a periodicity approximately equal to T0 / 2 or T0, where T0is the resonator oscillation period, and may be synchronised with the voltage tone driving the main resonator so that the nodes of the auxiliary tone coincide with one or more nodes of the main-junction tone. The principle of this readout protocol is described in further detail in the referenced arXiv preprint (https: / / arxiv.org / pdf / 2412.03650).

[0056] Description of the drawings

[0057] The following detailed description references the accompanying drawings which form a part this application, and which show, by way of illustration, specific example implementations. Other implementations may be made without departing from the scope of the disclosure.

[0058] Fig. 1 is a circuit diagram of a system according to the present disclosure, the circuit diagram of the superconducting qubit design.

[0059] Fig. 2 is a circuit diagram of another system according to the present disclosure, the system executing a read-out protocol for reading a logical Gottesman-Kitaev-Preskill (GKP) state stored in a superconducting resonator.

[0060] Fig. 3 is a circuit diagram of another system according to the present disclosure, the system using a driven auxiliary resonator and a loop designed to reduce flux noise.

[0061] Fig. 4 is a circuit diagram of another system according to the present disclosure, the system comprising an array of auxiliary resonators.

[0062] Detailed description of the disclosure

[0063] As used herein, the term “phase-space-local excitations”, when referring to a GKP-encoded state, may be understood, within the context of the overall disclosure, as encompassing any of the following non-exclusive categories:

[0064] a. perturbations of the state of the resonator (e.g. representable via a Kraus map or a dissipative term in the master equation) that do not displace the phase-space support P7918PC00

[0065] 7

[0066] of the Wigner function (expressed in the dimensionless quadratures φ / 2π and 2n, with φbeing the phase difference across a Josephson junction and nits canonical conjugate) by more than a characteristic threshold D, which may be smaller than 1 / 2, preferably smaller than 1 / 8, more preferably smaller than 1 / 20;

[0067] b. modifications to the evolution of the resonator induced by terms in the Liouvillian of the resonator mode given by low-order polynomials in φand n, preferably of order 10 or smaller, more preferably of order 5 or smaller, and most preferably of order 2 or smaller;

[0068] c. modifications to the evolution of the resonator induced by terms in the Liouvillian of the resonator mode given by linear combinations of phase-space displacement operators generating displacements (with respect to the Euclidean norm in the phasespace coordinates specified in (a)) smaller than 1 / 4, preferably smaller than 1 / 16, more preferably smaller than 1 / 40;

[0069] d. modifications to the evolution of the resonator arising from terms in its Liouvillian that generate continuous flow of its Wigner function in phase space without large displacements of its support; or

[0070] e. perturbations arising from terms in the Liouvillian of the resonator mode generating phase-space displacements of its Wigner function smaller than 1 / 4, preferably smaller than 1 / 16, more preferably smaller than 1 / 40.

[0071] Certain features and their benefits are:

[0072] 1. Realization of the GKP error correcting code.

[0073] The code protects against both bit-flip and phase errors due to phase space-local noise, including capacitive and inductive fluctuations and photon loss, which are dominant error sources in superconducting qubits.

[0074] 2. Simple realization of dissipative error correction.

[0075] The autonomous error-correction mechanism based on dissipation significantly reduces technological overhead, which is otherwise required for error correction; in particular the need for fast readout and feedback control.

[0076] 3. Natural implementation of protected single-qubit Clifford gates. P7918PC00

[0077] 8

[0078] The device admits simple-to-perform single-qubit Clifford gates that are protected against control noise. Small errors from imperfect control are corrected by the dissipative error-correction mechanism.

[0079] 4. Efficient read-out scheme.

[0080] We proposed an efficient read-out scheme that is near-native to the device, involving only a simple ancilla element.

[0081] 5. Realizability with current materials and technologies.

[0082] We identified NbN as a promising candidate material for realization of the device. The remaining device elements are either recently demonstrated or require minor engineering improvements over current achieved elements that we believe are achievable.

[0083] After having identified the mechanism for dissipative generation of GKP states via an oscillating voltage bias in a Josephson junction connected to an LC resonator early in the process, we encountered and overcame the following challenges:

[0084] 1. Identification of an efficient dissipative stabilization mechanism.

[0085] We tried out several ways of connecting the resonator to baths before concluding that phase-space local coupling, such as, e.g., capacitive coupling leads to dissipative error correction.

[0086] 2. Development of a new numerical code to simulate this system.

[0087] We developed a numerical simulation package to verify the coherence properties of our design. The numerical simulation is an extension of the recently developed Lindblad formalism for driven-dissipative open quantum systems.

[0088] 3. Identification of a reasonable parameters for a realistic realization of our proposal. With initial parameters, we noticed that the bias voltage amplitude across the Josephson junction needs to be larger than typical pair-breaking voltages. This challenge was solved by two developments: (i) defining an optimized waveform that reduced the maximal voltage amplitude by a factor -0.34, and (ii) identifying the sideband cooling mechanism that allowed to increase the environmental temperature of the P7918PC00

[0089] 9

[0090] cooling bath by one order of magnitude. Overcoming these challenges is what allowed our design to be achievable with current materials such as NbN.

[0091] It should be understood that the subject matter defined in the appended claims is not necessarily limited to the specific implementations described above. The specific implementations described below are disclosed as examples only.

[0092] Specific features described with reference to one specific implementation may provide particular technical advantages and may be employed independently or in combination with one or more other features described with reference to another implementation described herein, unless the overall disclosure explicitly indicates otherwise. Features described in connection with any one aspect of the disclosure may, where technically compatible, be applied equally to the other aspects, and combinations of features across the various aspects are considered to fall within the scope of the present disclosure.

[0093] As used herein, the term “coupling,” when referring to the coupling between any two subsystems of the circuit, such as the superconducting resonator, the Josephson junction(s), the thermodynamic environment, or any other circuit element, may be understood as encompassing any interaction representable by a term in the Hamiltonian of the combined system that acts non-trivially on both subsystems. By way of non-limiting examples, such coupling may include capacitative, inductive, or galvanic coupling, or coupling mediated via one or more Josephson junctions to which a voltage tone may be applied. Capacitative, inductive, or galvanic coupling may be realised using any suitable components, including but not limited to capacitors, inductive couplers, conducting or superconducting elements, metallic structures, or transmission lines, in any operative combination.

[0094] Preferably, the method further comprises:

[0095] a. inserting a band-pass filter between a quantum system (qubit), such as the superconducting resonator, and the thermodynamic environment, the band-pass filter being configured to admit dissipative processes involving relaxation by emission of a single resonator photon and to suppress dissipative processes involving zero-photon or multi-photon P7918PC00

[0096] 10

[0097] emission or absorption processes, which may involve resonator photons; and

[0098] b. providing photon-assisted side-band cooling by step (a).

[0099] The side-band cooling mechanism typically emerges when the temperature of the environment is smaller than ℏω / kB, and may preferably be smaller than ℏω / 3kB, more preferably smaller than ℏω / 5kB, and most preferably smaller than ℏω / 10kB, for example if the following holds:

[0100] We consider that the combination of a suitable drive and a filtered environment as described above can provide a relatively simple way to implement a powerful sideband cooling mechanism. By way of illustration, in an embodiment where a square- or rectangular-lattice GKP code is stabilized via a periodic driving signal with angular frequency Ωclose to 2ω, the thermal environment may preferentially absorb energies around Ω / 2, and may be more likely to absorb energies above Ω / 2 than below Ω / 2. More precisely, the spectral density S(ωB) of the environmental observable(s) coupled to the resonator can, for example, have effective support only within a frequency interval around Ω / 2, and may take larger values above Ω / 2 than below Ω / 2.

[0101] Via the side-band cooling mechanism described in Appendix I, dissipation from such an environment may preferentially drive processes in which the system relaxes its quasienergy by fix, with I x I smaller than the width of the support of S(coB). The relative probabilities of quasienergy-lowering (x > 0) versus quasienergy-raising (x < 0) processes with I x I within this support are approximately given by the ratio

[0102] s(x + n / 2)

[0103] S(-x + fl / 2)’

[0104] As a result, dissipation may generate relaxative dynamics equivalent to those of a time-independent system governed by the effective (Floquet) Hamiltonian of the driven resonator (with the quasienergy zone obtained from a high-frequency expansion in the comoving frame, as set out in the cited manuscript), and coupled to a bath with an effective spectral density S'(ωB) = S(ωB- Ω / 2). These dynamics can correspond to those of a time-independent system coupled to a thermodynamic environment at an effective temperature P7918PC00

[0105] 11

[0106] AE

[0107] T

[0108]

[0109] 'ff =^ 3=i'

[0110] with AE « 41 / 2 a characteristic excitation energy of the effective Hamiltonian. Notably, this effective temperature may be controlled primarily by the sharpness of the environmental spectral density rather than its physical temperature. Because this spectral sharpness can often be made large with comparatively simple circuit engineering, the effective temperature of the resonator can be made much lower than the ambient temperature.

[0111] This feature can be highly advantageous, as it may significantly reduce or avoid the need for deep cooling of the ambient environment, which is a major technological challenge in most superconducting-qubit approaches. For instance, the barrier height Ej of the stabilizer Hamiltonian (approximately equal to the effective Hamiltonian of the driven resonator, as defined below Eq. 4 of Appendix I; page 3, col. 2) is typically much smaller than Ej. For the parameter regimes with Al- and NbN-based junctions that were estimated feasible, embodiments not relying on the side-band cooling mechanism could require temperatures on the order of 10-30 mK, which can present substantial engineering difficulties. The side-band cooling mechanism may therefore constitute a key innovation, significantly extending the utility and impact of the approach.

[0112] Identification of this mechanism involved analysing driven-dissipative quantum dynamics of a nonlinear resonator, which is generally a technically demanding task. However, the Universal Lindblad Equation (ULE) [Nathan and Rudner, PRB 2020; https: / / journals.aps. Org / prb / abstract / 10.1103 / PhysRevB.102.115109] provided a simplified and systematic way to perform this analysis. Even so, notable mathematical and numerical effort was required to apply the ULE framework to this system and uncover the side-band cooling mechanism. This application of the ULE appears not to have been carried out previously and was crucial in enabling the discovery of the effect.

[0113] Preferably, the band-pass filter absorbs only energies in a window approximately between ftc and 3 / 2 ftco, wherein co is an angular resonance frequency of the electromagnetic resonator. P7918PC00

[0114] 12

[0115] Alternatively, the method comprises

[0116] c. inserting a low-pass filter between the superconducting resonator and the thermodynamic environment, the low-pass filter being configured to suppress photon assisted sideband transitions and to admit direct dissipation processes that do not involve exchange of resonator photons; and

[0117] d. providing direct cooling of the superconducting resonator.

[0118] Compared to the realization with the band-pass filter, the application of a low-pass filter may be technologically easier to implement because it does not rely on constructing additional on-chip resonators required for the band-pass filter. The environment should preferably be maintained at lower temperatures. For the square- or rectangular-lattice GKP code described above, the temperature is preferably smaller than:

[0119] II

[0120]

[0121] kB

[0122] with Ej defined in Appendix I (Ej = Ej cos ( (0) -ir / 4) / (T (ce |V'(0)| / h))

[0123] ). We estimate this to be on the order of 10-30 mK for the most feasible parameters identified using NbN junctions, preferably 100 mK or less, more preferably 50 mK or less.

[0124] Preferably, the steps (c) and (d) of the method involving the low-pass filter above comprises

[0125] a. maintaining a temperature of the thermodynamic environment below Ej / kB, preferably one or more orders of magnitude below Ej / kB, such as less than 0.1 Ej / kB, such as less than 0.05Ej / kB, such as less than 0.01Ej / kB, such as less than 0.005 Ej / kB, such as less than 0.001 Ej / kB,

[0126] wherein Ej is the Josephson energy of a Josephson function within the superconducting resonator, and kBis the Boltzmann constant.

[0127] In the implementation, which, in the preferred approach, is carried out without sideband cooling, the temperature of the environment is preferably smaller, such as one or more orders of magnitude smaller, than l / kBtimes the largest of the renormalized barrier heights, defined as the largest of the prefactors P7918PC00

[0128] 13

[0129] Ej f2e|V'(ti)|p

[0130]

[0131] 14qT \ h ) '

[0132] as set out in Eq. A7 of Appendix I, with the quantities above defined therein. For the embodiment employing the square or rectangular-lattice, the relevant barrier height is of order

[0133] _! L

[0134] VA’

[0135] where A is the amplitude of the phase oscillation induced by the voltage drive. This temperature requirement is not apply applicable when the side-band cooling configuration is used as the mechanism for dissipative error correction.

[0136] Additionally, or alternatively, the thermodynamic environment may be maintained at a temperature below ftco / kB, where co is the angular resonance frequency of the superconducting resonator and kBis the Boltzmann constant. Preferably, the temperature is less than 0.25 ftco / kB, more preferably less than 0.1 ftco / kB, and most preferably less than 0.02 ftco / kB.

[0137] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator satisfies any one of the following:

[0138] a. the drive frequency is a single harmonic tone with an angular frequency Q approximately equal to twice the angular resonance frequency co of the superconducting resonator;

[0139] b. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to 2co, such that nodes of the voltage tone occur at uniform spacing TT / (2CO);

[0140] c. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal 2co, such that the nodes of the voltage tone occur at substantially uniform spacing TT / (2CO).

[0141] The key idea underlying this implementation is that a voltage tone can effectively activate or deactivate the Josephson junction, giving an efficient way P7918PC00

[0142] 14

[0143] of controling it. When the instantaneous voltage amplitude is large compared to the Josephson energy, the junction is active only in the vicinity of the nodes of the oscillation. This enables generation of an effective GKP stabilizer Hamiltonian in an appropriate rotating frame of the resonator phase space, i.e., a frame rotating synchronously with a rational multiple of the drive frequency and approximately synchronously with the natural oscillation of the resonator. Prior theoretical proposals showed that rapidly swtiching on and off a Josephson junction in a high-impedance resonator could lead to generatin of the GKP stabilizer Hamiltonian, but there was no known way of achieving sufficiently high control resoluiton (rise / fall time and on-off-ratio) of the Josephson coupling with known technology. This changed with our paper, where found and demonstrated that such control can be achieved relatively easily via a simple voltage bias tone of sufficiently high amplitude. Below we give all the different incarnations of the tone where we think the scheme may work for the version with impedance near h / 2e2. Later we generalize to arbitrary impedances above this threshold.

[0144] An approach with a cos(2cp) junction and a flux junction was suggested this year by Ifran Siddiqi’s group, https: / / arxiv.org / pdf / 2509.14656, but this approach is not useful for qubits, since the inevitably imperfect balancing of the interferometric cos(2cp) will cause phase flip errors. Through the driving scheme here, and the architecture above, we propose a way of doing it which is relatively simple, through Floquet engineering.

[0145] Furthermore, earlier proposals to generate GKP states relied on fast, repeated switching of the Josephson coupling, a strategy that requires unconventional hardware, is difficult to stabilise, and produces unwanted heating. The present approach avoids these difficulties by employing a harmonic voltage tone that uses established hardware, affords high-precision control, and avoids heating because the drive is smooth and its peak amplitude remains well below the superconducting gap A / e.

[0146] Additionally, or alternatively, wherein the voltage tone used to drive the superconducting resonator satisfies any one of the following:

[0147] a. the drive frequency has a fundamental angular frequency Q approximately equal to w / (N + 1 / 2), where N is any positive integer, P7918PC00

[0148] 15

[0149] such that nodes of the voltage tone occur at substantially uniform spacing TT / Q;

[0150] b. the drive frequency has a fundamental angular frequency Q approximately equal to w / (N + 1 / 2), where N is any positive integer; c. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to co / (N + 1 / 2), where N is any positive integer; or

[0151] d. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to co / (N + 1 / 2), where N is any positive integer;

[0152] such that the nodes of the voltage tone occur at substantially uniform spacing TT / Q.

[0153] Additionally, or alternatively, wherein the voltage tone used to drive the superconducting resonator satisfies any one of the following:

[0154] a. the drive frequency has a fundamental angular frequency Q approximately equal to w / (N + p / q), where N is any integer, and p / q is an irreducible rational fraction with p and q integers preferably smaller than 100 such that nodes of the voltage tone occur at substantially uniform spacing TT / Q;

[0155] b. the drive frequency has a fundamental angular frequency Q approximately equal to w / (N + p / q), where N is any integer and p / q is an irreducible rational fraction with p and q integers preferably smaller than 100;

[0156] c. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to w / (N + p / q), where N is any positive integer and p / q is an irreducible rational fraction with p and q integers each preferably smaller than 10Or; or

[0157] d. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to w / (N + p / q), where N is any positive integer and p / q is an irreducible rational fraction with p and q integers each preferably smaller than 100; such that the nodes of the voltage tone occur at substantially uniform spacing TT / Q. P7918PC00

[0158] 16

[0159] By providing any one of the above voltage tones, the drive signal can be implemented using a single harmonic or a small number of dominant harmonics, which allows for practical generation using standard tone-generation circuitry. This stands in contrast to all other presently theorised approaches to dissipative stabilisation of GKP states, which rely on technology that is not yet available. In certain embodiments, the drive signal may even be generated on-chip by resonantly driving an auxiliary circuit, as described in one of the methods of the present disclosure.

[0160] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is configured such that the time-interval between nodes of the voltage tone is approximately equal to half-integer multiples of T0 / 2, where Tois the oscillation period of the superconducting resonator.

[0161] This embodiment allows for generation of the square- or rectangular-lattice GKP code. Moreover, by allowing more flexibility in the driving tone, this gives more freedom in how to realize the qubit, in particular the driving tone does not necessarily have to be periodic with this type of voltage tone. As above, by allowing the spacings between voltage nodes to vary, this more flexibility in the driving tone, and thus more freedom in how to realize the qubit. In particular the driving tone does not necessarily have to be periodic with this type of voltage tone. An advantage of the avoidance of periodicity could be the avoidance of resonant excitations of other modes in the circuit than the qubit resonator more, which could give rise to unwanted noise and limit the performance. This implementation may be achieved with arbitrary-waveform generators (AWGs).

[0162] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is configured such that the time-interval between adjacent nodes of the voltage tone is approximately equal to an integer multiple of To / 6, where Tois the oscillation period of the superconducting resonator, the integer multiple optionally varying between different adjacent node spacings.

[0163] This embodiment allows for generation of the hexagonal GKP code, as described in Appendix I. By permitting the spacings between voltage nodes to vary, the driving tone gains additional flexibility and, accordingly, offers greater freedom in how the qubit may P7918PC00

[0164] 17

[0165] be realised. In particular, the driving tone need not be strictly periodic under this implementation. Avoiding strict periodicity may help prevent resonant excitation of circuit modes other than the qubit resonator, thereby reducing unwanted noise and potential performance limitations. Such an implementation may, for example, be achieved using arbitrary-waveform generators (AWGs).

[0166] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is configured such that the spacing between adjacent nodes of the voltage tone is approximately given by an integer multiple of pT0 / q, where p / q is an irreducible rational fraction, p and q are optionally integers smaller than 100, and Tois the oscillation period of the superconducting resonator. The integer multiple may optionally vary between different adjacent node spacings.

[0167] This implementation allows for generation of the quasicrystal GKP code, that we describe below. Moreover, by allowing more flexibility in the driving tone, this gives more freedom in how to realize the qubit, in particular the driving tone does not necessarily have to be periodic with this type of voltage tone. As above, by allowing the spacings between voltage nodes to vary, this more flexibility in the driving tone, and thus more freedom in how to realize the qubit. In particular the driving tone does not necessarily have to be periodic with this type of voltage tone. An advantage of the avoidance of periodicity could be the avoidance of resonant excitations of other modes in the circuit than the qubit resonator more, which could give rise to unwanted noise and limit the performance. Such a voltage tone may, for example, be generated using arbitrary-waveform generators (AWGs).

[0168] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is periodic.

[0169] Periodic voltage tones can be generated using simple hardware, including standard tone generators and linear resonators. Periodic tones may also be produced on-chip by (near-)resonantly driving an auxiliary resonator circuit whose resonance frequency is close to the desired driving frequency. Such implementations are relatively straightforward and compatible with standard hardware. In particular, generating the tone by driving an on-chip resonator can be advantageous for minimising flux noise, as described below. P7918PC00

[0170] 18

[0171] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is non-periodic.

[0172] An advantage of non-periodic tones could be the avoidance of resonant excitations of other modes in the circuit than the qubit resonator more, which could give rise to unwanted noise and limit the performance. A non-periodic voltage tone may be achieved with arbitrary-waveform generators (AWGs).

[0173] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator has a maximum amplitude less than A / 2, where A is a superconducting energy gap of a Josephson junction within the superconducting resonator.

[0174] Limiting the maximum drive amplitude to below A / 2 may help suppress quasiparticle generation, as excessive drive strength can induce quasiparticle poisoning, which is expected to limit the achievable qubit lifetime.

[0175] Additionally, or alternatively, the autonomous error correction of step (d) is achieved by dissipation processes which relax phase-space-local excitations of a GKP-encoded state within a single GKP potential well.

[0176] Here, relaxation within a GKP potential well may be understood as relaxation of phase-space-local excitations during which, for any unitary phase-pace displacement operator defining a stabilizer for the GKP encoding of the qubit, U, the state of the resonator does not acquire significant support in the subspace spanned by eigenstates of U with eigenvalues whose real part is smaller than some threshold value y > -1. As a result, under relaxation within a GKP potential well, the logical qubit state encoded in the resonator via the GKP encoding does not change. For instance, for the implementation with the square- or rectangular-lattice GKP encoding with stabilizers

[0177]

[0178] = cos (4>), S2= cos (4ml),

[0179] in the comoving frame of the resonator rotating in phase space with angular velocity fl / 2, as specified in the second paragraph of Sec. III. C of Appendix I, relaxation of phase-space-local excitations within a GKP potential well may refer to relaxation processes where the state of the resonator in the comoving frame does not gain P7918PC00

[0180] 19

[0181] support on eigenstates of S, and S2with eigenvalues smaller than some value y larger than -1, i.e., the support remains within the “potential wells” of the stabilizers -Stand -S2that appear in the effective Hamiltonian.

[0182] Dissipation from a thermodynamic environment will generically cause phase-space-local excitations of the GKP-encoded state in our proposed approach to take place within a single GKP potential well. This occurs once the system is stabilized in a GKP-encoded state. In this way, the dissipation stabilizes the encoded quantum information without corrupting it, thereby realising genuine dissipative error correction. In most other qubit approaches, dissipation will in general affect the logical state encoded in the qubit, making genuine dissipative error correction impossible. Using the GKP encoding circumvents this obstacle. The fact that dissipation takes place within a single well is a defining feature and advantage of our approach.

[0183] Additionally, or alternatively, the coupling is configured to admit dissipative processes involving relaxation by emission of a single resonator photon and to suppress dissipative processes involving zero-photon and multi-photon transitions. It may be the bath spectral density that determines which dissipative processes are permitted and which are suppressed. This allows the resonator to be cooled to sub-ambient temperatures with respect to the effective (Floquet) Hamiltonian of the mode, which is identical to the GKP stabilizer Hamiltonian in the scheme described. In contrast, conventional superconducting qubits rely on cooling the entire ambient environment and therefore require costly dilution refrigerators to reach operating temperatures on the order of ~50 mK, where transmons typically function. Cooling capacity also limits how many control lines can be connected to a device. In the present approach, the qubit may in principle operate at any ambient temperature at which the resonator remains superconducting, thereby potentially circumventing these limitations.

[0184] Additionally, or alternatively, further comprising executing an initialisation routine for magic-state preparation, the initialisation routine comprising:

[0185] a. preparing the superconducting resonator in a ground state while a Josephson junction of the superconducting resonator is decoupled, the decoupling being implemented using a tunable Josephson-energy element, a tunable coupler, or by applying a voltage bias across the junction that continuously remains larger, such as at least one order of P7918PC00

[0186] 20

[0187] magnitude larger than the Josephson energy of the junction divided by 2e; and

[0188] b. switching on the Josephson coupling and applying a resonant voltagedrive tone configured to realise the dissipative error correction of any method according to the first aspect of the disclosure, preferably with the amplitude of the voltage tone selected such that an effective GKP potential comprises a potential well located at an origin of the resonator phase space (4> = 0, n = 0), the “GKP potential” referring to the terms in the effective Hamiltonian of the system generated by the stabilisation protocol that approximate stabilisers of the particular class of GKP encoding used to encode the qubit.

[0189] Additionally, or alternatively, the band-pass filter is configured to be more likely to admit photons with angular frequency above X than photons with angular frequency below X, where X = (πN + π / 2) / τ, with N being the positive integer for which X is closest to the angular resonance frequency co of the superconducting resonator, and where T is either the spacing between nodes of the voltage tone used to drive the superconducting resonator (e.g., when those spacings are substantially identical), or, additionally or alternatively, in the case of non-periodic or non-uniform voltage tones, a characteristic time Tofor which the time instances of a large set of consecutive nodes of the voltage tone (e.g., 10, 50, 100, or 1000 nodes, or any multiples thereof) can be partitioned into a smaller number of subsets (e.g., 2, 3, 4, 6, 10, or 20 subsets, or any multiples thereof), such that within each subset the mutual spacings are approximately equal to integer multiples of the same duration To.

[0190] Additionally, or alternatively, the band-pass filter is configured to more readily admit photons with angular frequency above X than photons with angular frequency below X, where X = (πN + π / 2) / τ, with N being an integer and p / q an irreducible fraction of integers p,q < 100, and where T is either the spacing between nodes of the voltage tone used to drive the superconducting resonator (when those spacings are substantially identical), or, additionally or alternatively, in the case of non-periodic or non-uniform voltage tones, a characteristic time Tofor which the time instances of a large set of consecutive nodes of the voltage tone (for example 10, 50, 100 or 1000 nodes, or any multiples thereof) can be partitioned into a significantly smaller number of subsets (for example 2, 3, 4, 6, 10 or 20 subsets, or any multiples thereof) such that, P7918PC00

[0191] 21

[0192] within each subset, the mutual spacings are approximately integer multiples of the same duration To.

[0193] This configuration allows the dissipative side-band-cooling mechanism to operate with the more general drive waveforms and encodings described above, including hexagonal and quasicrystalline encodings, and is compatible with both periodic and non-periodic voltage tones.

[0194] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within a frequency interval within ±25% of X, where X = (πN + π / 2) / τ, preferably within ±7% of X, more preferably within ±2% of X, while suppressing photons outside the frequency interval, with photons above the frequency interval being admitted to a greater extent than photons below the frequency interval, where X = (πN + π / 2) / τ and T is the uniform spacing between nodes of the voltage tone. Alternatively, T is the spacing between nodes of the voltage tone used to drive the superconducting resonator when those spacings are substantially uniform; or alternatively, in the case of non-periodic or non-uniform voltage tones, Tmay be taken as a characteristic time Tofor which the time instances of a large set of consecutive nodes of the voltage signal (for example 10, 50, 100 or 1000 nodes, or any multiples thereof) can be partitioned into a significantly smaller number of subsets (for example 2, 3, 4, 6, 10 or 20 subsets, or any multiples thereof) such that, within each subset, the mutual spacings are approximately integer multiples of To.

[0195] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within a frequency interval lying within ±25%, more preferably within 15%, more preferably within 10%, of X = (πN + π / 2) / τ, where N is an integer and p / q is an irreducible fraction of integers p, q < 100, preferably within ±7% of X, and more preferably within ±2% of X, while suppressing photons outside the interval, and with photons above the interval being admitted to a greater extent than photons below the interval. Here, T is the spacing between nodes of the voltage tone used to drive the superconducting resonator when those spacings are substantially uniform; additionally, or alternatively, in the case of non-periodic signals, Tmay be taken as a characteristic time Tofor which the time instances of a large set of consecutive nodes of the voltage signal (for example 10, 50, P7918PC00

[0196] 22

[0197] 100 or 1000 nodes, or any multiples thereof) can be partitioned into a significantly smaller number of subsets (for example 2, 3, 4, 6, 10 or 20 subsets, or any multiples thereof) such that, within each subset, the mutual spacings are approximately integer multiples of To.

[0198] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit dissipative processes in which the superconducting resonator absorbs photons from the driving field induced by the voltage tone and emits energy into the thermodynamic environment greater than the energy absorbed from the driving field.

[0199] This describes a more general environmental configuration under which side-band cooling may be implemented. For example, such behaviour can arise when the environment exhibits spectral support outside the principal interval discussed above, provided this additional support does not significantly modify the dynamics, for instance because it is far detuned from the characteristic frequencies of the system, or because the associated dissipative processes do not adversely affect qubit operation. In particular, this may occur when the spectral density contains multiple peaks near integer multiples of the resonator frequency, or other characteristic frequencies noted herein, with each peak taking a substantially larger value above the relevant multiple than below it. Such spectra may arise naturally in certain environments or may be engineered, for example using auxiliary resonators with multiple modes coupled to dissipative elements, or arrays of coupled auxiliary resonators in which one or more resonators is coupled to a thermodynamic environment.

[0200] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons with angular frequency above the angular resonance frequency co of the superconducting resonator to a greater extent than photons with angular frequency below co.

[0201] This describes a more general environmental configuration under which side-band cooling may be implemented. For example, such a situation can arise when the environment exhibits spectral support outside the principal interval discussed above, provided this additional support does not significantly modify the system dynamics, for instance because it is far detuned from the characteristic frequencies of the resonator, P7918PC00

[0202] 23

[0203] or because the associated dissipative processes are not detrimental to qubit operation. Such spectral structures may occur naturally or may be engineered, for example using ancillary resonators with multiple modes coupled to dissipative elements, or arrays of coupled ancillary resonators in which one or more resonators are coupled to a thermodynamic environment

[0204] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within an interval around the angular resonance frequency co of the superconducting resonator while suppressing photons outside that interval, preferably with photons above co being admitted to a greater extent than photons below co.

[0205] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within intervals approximately equal to one or more integer multiples of the angular resonance frequency of the superconducting resonator, while preferably suppressing photons outside those intervals, and preferably admitting photons within a given interval above the associated integer multiple of the resonator frequency to a greater extent than photons below that multiple.

[0206] Additionally, or alternatively, the step of coupling the superconducting resonator to the thermodynamic environment in step (b) of claim 1 comprises coupling the superconducting resonator to the thermodynamic environment via an auxiliary electromagnetic resonator having a resonance frequency selected from:

[0207] a. a resonance frequency approximately equal to X, where X = (πN + π / 2) / τ, where T is the spacing between nodes of the voltage tone used to drive the superconducting resonator; or

[0208] b. a resonance frequency approximately equal to co, where co is an angular resonance frequency of the superconducting resonator; and the auxiliary electromagnetic resonator being coupled to a thermal reservoir or transmission line so as to generate a substantially Lorentzian spectral density in the coupling.

[0209] In this implementation, the centre frequency of the Lorentzian, i.e., the resonance frequency of the auxiliary resonator, is preferably chosen to be at co + 6 (or X + 6), P7918PC00

[0210] 24

[0211] where 6 > 0 and 6 is smaller than, for example, 5, 20, or 200 times the Lorentzian peak width r) (the decay rate of the auxiliary resonator). Moreover, the 6 is preferably adjusted such that 6 + 5r), 6 + 20r), or 6 + 50r) exceeds the excitation energy (quasienergy gap) of the effective Hamiltonian of the driven circuit or GKP stabilizer Hamiltonian.

[0212] Advantageously, this implementation may provide a simpler realisation of the filtered environment. A single lossy resonator gives rise to a Lorentzian power spectral density and can readily be integrated on-chip. When the centre and width of this power spectral density are suitably adjusted, it exhibits precisely the features required for the dissipative error correction, as described in the previous claims.

[0213] Additionally, or alternatively, wherein step of the method coupling the superconducting resonator to the thermodynamic environment in step (b) comprises coupling the superconducting resonator to the thermodynamic environment via an array of auxiliary electromagnetic resonators, at least one or more auxiliary resonator of the array being coupled to a thermal reservoir or transmission line, the array exhibiting a transmission spectrum corresponding to the coupling properties defined in any method according to the first aspect of the disclosure.

[0214] This is another way of generating the environment, which provides greater flexibility in adjusting the power spectral density of the filter than a single Lorentzian, to further optimise performance. For example, this network of resonators can be tuned to ensure more rapid decay of the asymptotes of the power spectral density, or a profile which is locally flat. This implementation may be implemented on a chip.

[0215] Additionally, or alternatively, the method comprises maintaining a temperature of the thermodynamic environment within a working temperature range, the working temperature range being 20 mK or less, such as 40 mK or less, such as 100 mK or less, such as 200 mK or less, such as 500 mK or less.

[0216] The ability to operate at comparatively high temperatures provides an advantage over standard circuit approaches such as transmon qubits. The side-band-cooling mechanism may allow the qubit to function as long as the rate of thermal excitation of quasiparticles in the superconductor remains low compared to the inverse lifetime of P7918PC00

[0217] 25

[0218] the qubit and the superconducting state of the circuit is preserved, rather than melting into the normal state.

[0219] Additionally, or alternatively, the method comprises

[0220] a. modulating of voltage tone driving a superconducting resonator; and thereby

[0221] i. implementing high fidelity single-qubit Clifford gates; and providing

[0222] ii. correcting small errors from imperfect control by a dissipative error-correction mechanism; and

[0223] iii. providing an efficient read-out and initialization scheme, and optionally, producing error-corrected Clifford gates.

[0224] The methods to perform Clifford gates, where small errors are autonomously corrected by the dissipative quantum-error-correction mechanism, are as follows.

[0225] (i) For the square or rectangular-lattice GKP code, a Hadamard (H) gate operation is generated by a TT / 2 phase-space rotation. Such a rotation can be implemented in the circuit by adiabatically shifting the phase of the voltage tone by IT (see Fig. 4(a) of Appendix I). Provided that the phase shift is performed sufficiently slowly, the system remains confined to the low-energy Hef^, resulting in strong suppression of control errors.

[0226] (ii) A protected S gate can be implemented for the square- or rectangular lattice implementation as follows. First, an additional driving tone at frequency flis adiabatically turned on so that < X>(t) -> ft) + TTCOS fit. This shifts the minima of Heffby A4> = TT, such that the logical I 0)and I 1)GKP states acquire support centered around 4> G 4TTZ and 4> G 4TT(Z + 1 / 2), respectively. Second, the system is abruptly shifted to a driving frequency f' = f0+ 6f' and voltage amplitude such that 0= 2TTI< - 3TT / 4, while the minima of the potential remain located at 4> G 2TTZ. This step effectively turns off the term proportional to cos(2n^n) in Heffthat stabilizes inter-well phase coherence, thereby allowing a relative phase to accumulate between the two logical sectors. I, = Z / (h / 2e2). By waiting a time Ats= l / (27i6f'), states with support near 4> = 2TTI< acquire a phase factor

[0227] e-2ik2TT26f'Ats / 2 _ (— l)k

[0228]

[0229] P7918PC00

[0230] 26

[0231] which is equivalent to the action of an S-gate. Finally, quenching back to the resonantly modulated driving tone and adiabatically turning off the modulation at frequency flreactivates the inter-well-coherence-stabilizing term proportional to cos (2TT< ), which corrects small relative-phase errors arising from an imperfect choice of waiting time Ats.

[0232] Realizing high accuracy quantum gates is among the key challenges of quantum computation. For example, it motivates exotic approaches such as Majorana based quantum computation. Here, we propose a way to achieve Clifford gates where errors are corrected by the dissipative quantum error correction mechanism. Our approach enables error-corrected Clifford gates in a single physical device and without feedback control, which enormously reduces hardware overhead while at the same time being achievable with demonstrated superconducting circuit elements. Our unique approach to perform Clifford gates is enabled by our approach to realize the GKP qubit as set out in the present disclosure.

[0233] Additionally, or alternatively, the step of driving the superconducting resonator with a voltage tone comprises oscillating the voltage tone at a frequency that is a rational multiple of a resonance frequency of the superconducting resonator, such as twice the resonance frequency. In some implementations, the rational multiple is two, such that the superconducting resonator is driven at twice its resonance frequency to generate a square or rectangular-lattice Gottesman-Kitaev-Preskill code. Alternatively, in other implementations, 3 / N, where N is any integer not divisible by three, such that the superconducting resonator is driven at a frequency equal to (3 / N) co to generate a hexagonal-lattice Gottesman-Kitaev-Preskill code. For instance, the rational multiple may be three-halves, such that the superconducting resonator is driven at one-and-a-half times its resonance frequency to generate a hexagonal-lattice Gottesman-Kitaev-Preskill code.

[0234] The hexagonal lattice GKP code is anticipated to offer slightly improved performance compared with the square-lattice version, as a larger phase-space displacement is required to induce a logical error. This encoding can be obtained straightforwardly by adjusting the voltage tone and resonator impedance, illustrating the versatility of the present approach. Advantageously, the square- or rectangular-lattice implementation P7918PC00

[0235] 27

[0236] can be achieved with a lower resonator impedance than the hexagonal implementation; as high impedance is expected to be the most demanding fabrication element of the qubit, the square- or rectangular-lattice setup may in some cases offer a less costly implementation.

[0237] Other rational multiples corresponding to irreducible fractions distinct from those required for the square, rectangular, and hexagonal encodings can result in a yet undescribed and un-characterized class of bosonic encodings beyond the paradigm of GKP encodings, which we refer to as quasicrystal encodings. Such encodings were observed in our simulations and are yet undescribed in the prior art. For these quasicrystal codes, the effective Hamiltonian consists of multiple independent phasespace displacement terms related by phase-space rotations by angles 2n / q for integer qdistinct from 1,2, 3, 4, 6 in the phase space of appropriately normalised conjugate quadrature variables of the resonator. With appropriate choice of the resonator impedance, the displacement operators are nearly commuting. The Hamiltonian may moreover comprise a confining potential of the form acf>2+ bn2, with a,b > 0, that increases with distance from the phase-space origin. This confining potential may ensure that residual non-commutativity acts only as a small perturbation to the first several low-energy states, such that each displacement operator in the Hamiltonian has those low-energy states as superpositions of approximate eigenstates with eigenvalues close to 1 (e.g., within 1 or within 1 / 2). In our simulations, the system supports a manifold of near-degenerate low-energy states, with spacings within the manifold several orders of magnitude smaller than spacings to higher-energy states. The degeneracy appears to persist for the first several excitations, implying that these degenerate manifolds can in principle encode a qubit.

[0238] Interestingly, for the quasicrystal encodings, phase space rotations by an angle 2n / q map the degenerate manifold to itself. If the qubit transforms nontrivially under this operation, as is the case for conventional GKP codes where q = 1,2, 3, 4, 6, the operation corresponds to a Zqsubgroup in the logical space of the qubit, and thus must be a non-Clifford operation. Quasicrystal codes may therefore support ways of achieving robust non-Clifford gates via simple Gaussian operations, such as phasespace rotation. This could be highly advantageous, since most standard quantum computing architectures do not support efficient non-Clifford gates and instead rely on costly magic-state distillation protocols that are widely regarded as a major bottleneck. P7918PC00

[0239] 28

[0240] In this context, a non-Clifford gate may in particular be implemented by a simple phase shift of the voltage tone or a simple shift of the time frame, operations that are extremely cheap in hardware. By providing a technologically simple way to implement a robust magic gate, quasicrystal encodings in our qubit may form a very valuable resource for magic-state generation or universal gate operation. We are actively conducting research in this direction.

[0241] The non-Clifford gate may in particular be implemented by a simple phase shift of the voltage tone, or a simple shift of time frame, which are operations that are very cheap or free with most hardware. By providing a technologically simple way to implement a robust magic gate, this could make quasicrystal encodings in our qubit an extremely valuable resource for magic state generation or universal gate operation.

[0242] Additionally, or alternatively, the superconducting resonator is driven by a microwave voltage bias tone. Microwave tones are advantageous because the associated control and read-out electronics are well developed in the microwave domain. Although radiofrequency or infrared tones could be used, radio-frequency operation may slower for practical error correction, and infrared operation may require materials and components not readily available. A voltage bias tone here may simply refer to an applied electromagnetic wave.

[0243] The present disclosure also provides a method of increasing qubit coherence time using any method disclosed herein.

[0244] A method of generating GKP states using any method disclosed herein, wherein the GKP codes protect against both bit-flip and phase errors due to phase space-local noise.

[0245] Additionally, or alternatively, the method comprises protecting against both bit-flip and phase errors comprises protecting against noise that generates a continuous flow of the Wigner function of the resonator mode.

[0246] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is modified so as to generate a quasicrystalline, an oblique or hexagonal Gottesman-Kitaev-Preskill (GKP) code. P7918PC00

[0247] 29

[0248] The advantages of the quasicrystalline and hexagonal encodings are described above. Oblique GKP codes may likewise be generated by suitably modifying the voltage tone. An advantage of this encoding is that it may be implemented with an impedance different from the specific values (e.g., h / 2e2) required to generate square, rectangular, or hexagonal GKP codes. In particular, for any impedance greater than h / 2e2, it is possible to identify a family of voltage tones such that the method stabilizes a dissipatively corrected qubit encoded via an oblique GKP encoding. Accordingly, it is not necessary to target a specific impedance with high precision when fabricating the device, simplifying fabrication and prototyping. Instead, the voltage drive signal can be adjusted for a given device to identify a tone that stabilizes an oblique-GKP-encoded qubit. Since adjusting a tone or waveform is considerably simpler, and essentially cost-free, compared to fabricating a resonator with impedance falling within a narrow target window, this implementation may substantially reduce the complexity and cost of building and testing the qubit.

[0249] Additionally, or alternatively, the modification comprising adjusting temporal positions of nodes of the voltage tone within each period T, where T is approximately equal to To / 2 with Tobeing a natural oscillation period of the superconducting resonator. In some implementations, the temporal positions of nodes occur approximately at times t = 0,t = ^arcsin (i), where z = Z / (h / 2e2) is a normalised impedance of the resonator or the resonator mode having impedance Z, any branch of the arcsine function being admissible, such that a resulting encoded state corresponds to an oblique GKP lattice with primitive-vector angle arcsin (1 / z).

[0250] Here any branch of the arcsine function may be used, provided the same convention is applied for all node spacings. This modification avoids the need for fine-tuning the resonator impedance by instead adjusting the drive tone and resulting encoding. This can be a significant improvement, as it allows the device to be fabricated with a given, non-fine-tuned impedance, after which the driving tone may be tuned to stabilise an oblique-GKP-encoded qubit. Since tuning the waveform of a voltage drive is straightforward with standard microwave sources, whereas precise impedance targeting during fabrication is costly and technically demanding, this approach may greatly simplify both prototyping and device realisation by shifting the fine-tuning burden from fabrication to waveform engineering. The waveform required for the P7918PC00

[0251] 30

[0252] oblique implementation may, for example, be generated by adding a DC voltage bias to the tone or by introducing additional harmonic components with relative phase shifts.

[0253] Additionally, or alternatively, the voltage tone used to drive the superconducting resonator is modified such that the spacing between adjacent nodes of the voltage tone is approximately equal to: At = k y +

[0254]

[0255] s arcsin (i), where Tois the natural oscillation period of the superconducting resonator, z = Z / (h / 2e2) is a normalised impedance of the resonator having impedance Z, k is an integer, and s is a binary parameter taking values 0 or 1. the parameters k and s may differ between different pairs of adjacent nodes; in particular allowing for both periodic and non-periodic modulation of the voltage signal.

[0256] The advantage of this implementation is the greater flexibility in the choice of driving signal, in combination with the support of oblique GKP encodings (whose advantages were established above). In particular, the driving signal can be non-periodic, allowing for potential avoidance of adversarial resonant excitations of non-qubit modes in the device. The fact that the method works with non-periodic signal means that the mode encoding the qubit can in principle can be addressed individually through driving with a non-periodic signal of the right character. Other modes in the device may in general not have a resonant response to this kind of driving.

[0257] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is configured such that the thermodynamic environment admits photons having angular frequencies more than or equal to 2n / T0, where Tois the natural oscillation period of the superconducting resonator. Preferably, the admitted frequency interval extends up to 20% above 2n / T0, more preferably up to 15% above 2n / T0, and most preferably up to 10% above 2n / T0, where Tois as defined the preceding disclosure or the natural oscillation period of the superconducting resonator.

[0258] Additionally, or alternatively, the superconducting resonator has an impedance approximately equal to (4 / V3) • (h / 4e2); and wherein the voltage tone used to drive the superconducting resonator has a fundamental driving frequency close to a rational multiple p / q of the resonance frequency of the superconducting resonator. In some implementations, p = 3 and q is an even integer, so as to generate a hexagonal P7918PC00

[0259] 31

[0260] Gottesman-Kitaev-Preskill code. In other implementations, q = 4 and p is an odd integer, so as to generate an oblique variant of a quasicrystalline Gottesman-Kitaev-Preskill code. In further alternative implementations, q = 5 and p is an even integer, so as to generate a quasicrystalline Gottesman-Kitaev-Preskill code having a Penrosetiling structure.

[0261] The quasicrystalline codes described above are new results that are not yet disclosed elsewhere. This encoding represents a new paradigm of bosonic codes not previously described in the literature. Advantageously, these encodings may enable efficient implementation of non-Clifford gates.

[0262] The non-Clifford gates may be realized because of the higher (e.g. C5, C8, or C10) fold rotation symmetry of the quasicrystalline code. Previously, for the C4-symmetric square or rectangular GKP code, we showed that phase-space rotations followed by a quench can realize a unitary rotation U = exp (−iπn̂ / 2). Specifically, for the C8oblique quasicrystalline code, the system exhibits a two-fold degenerate ground state at impedance Z « 4.1 (h / 4e2).

[0263] There, a phase-space rotation by TT / 4 rotates between the two qubit states, and we anticipate its effect as

[0264] U = exp (−iπn̂ / 4),

[0265] thereby realizing the square-root of a Hadamard gate, which lies outside the Clifford group. The exponential protection is achieved because small deviations from the ideal TT / 4 rotation are corrected by the autonomous error-correcting mechanism. This provides a reliable way to realize accurate non-Clifford gates, which enables universal quantum computation in combination with a two-qubit gate.

[0266] Additionally, or alternatively, the method comprises selecting or adjusting an impedance of the superconducting resonator to be approximately equal to (D / 2) • (h / 2e2), for any integer D > 2, so as to generate a D-dimensional square or rectangular-lattice Gottesman-Kitaev-Preskill (GKP) qudit.

[0267] Replacing qubits with qudits may provide increased versatility for quantum information processing. Using qudits rather than qubits has the advantage that more quantum information can in principle be stored or processed in a single physical hardware P7918PC00

[0268] 32

[0269] element (e.g., a resonator mode). For example, a qudit with D = 4 can encode the same amount of information as two qubits. This may be advantageous for reducing hardware footprint or for implementing alternative quantum error-correction schemes that benefit from the underlying physical system storing quantum information as a qudit rather than a qubit. In particular, certain classes of active quantum error detection or correction may, in principle, be performed using a single logical qudit without combining it with other qudits, thereby providing an additional layer of protection for the encoded information. Depending on the relative hardware costs in an eventual implementation, such qudit-based embodiments could yield significant performance improvements under realistic resource constraints

[0270] Preferable and / or optional features of the second aspect of the present disclosure are described herein, providing further advantageous implementations of the system.

[0271] Preferably, the superconducting resonator is capacitatively coupled to the thermodynamic environment through a band-pass filter having a bandwidth approximately confined between one-half and three-quarters of a frequency of the voltage tone.

[0272] With this implementation, the system is driven at approximately twice the resonator frequency, implying that the resonator has impedance h / 2e2to operate. The relatively large bandwidth of the filter allows rapid dissipation of photons into the surrounding environment, minimizing complications and adverse effects arising from non-Markovian dynamics and / or backscattering of photons into the qubit resonator. The filter may be realised as a single highly lossy resonator with an eigenfrequency at the centre of the interval specified above, or as an array of resonators (e.g., serially connected) whose transmission spectrum is engineered to exhibit the desired features. The capacitive coupling is simple to design and fabricate to the required strength and specifications.

[0273] Additionally, or alternatively, the coupling between the superconducting resonator and the thermodynamic environment is provided via a filtered reservoir configured to provide frequency-selective dissipation suitable for autonomously error correcting phase- and bit-flip errors of the encoded qubit,

[0274] the filtered reservoir comprising: P7918PC00

[0275] 33

[0276] a. a single auxiliary electromagnetic resonator having a resonance frequency approximately equal to X, where X = (πN + π / 2) / τ, or approximately equal to co, where co is an angular resonance frequency of the superconducting resonator, the auxiliary resonator being coupled to a thermal reservoir or transmission line so as to generate a Lorentzian spectral density in the coupling and where T is either the spacing between nodes of the voltage tone used to drive the superconducting resonator (e.g., when those spacings are substantially identical), or, additionally or alternatively, in the case of non-periodic or non-uniform voltage tones, a characteristic time TO for which the time instances of a large set of consecutive nodes of the voltage tone (e.g., 10, 50, 100, or 1000 nodes, or any multiples thereof) can be partitioned into a smaller number of subsets (e.g., 2, 3, 4, 6, 10, or 20 subsets, or any multiples thereof), such that within each subset the mutual spacings are approximately equal to integer multiples of the same duration TO; or b. an array of auxiliary resonators, at least one auxiliary resonator of the array being coupled to a thermal reservoir or transmission line; or c. a network of auxiliary resonators arranged as an arbitrary graph of coupled resonators, the resonators of the network being coupled capacitatively, inductively, galvanically, or via Josephson junctions, with one or more auxiliary resonators being coupled to a thermal reservoir or transmission line.

[0277] Engineering single resonators or arrays of resonators is relatively standard, and thus we anticipate this to be a relatively low-cost implementation that achieves the required spectral features for optimal performance. In addition to fabricating the resonator or resonator array with a desired spectral function, the resonance frequencies can be tuned by substituting their inductances with SQUIDs that are flux-controlled via a flux line, allowing on-demand adjustment of the spectral function of the environment.

[0278] Additionally, or alternatively, the superconducting resonator is capacitatively coupled to the thermodynamic environment through a low-pass filter coupling the superconducting resonator to the thermodynamic environment, the low-pass filter having a spectral density that is substantially confined to energies | E |≲ ½ħω and that decreases rapidly P7918PC00

[0279] 34

[0280] for energies | E |> ½ħω, where ω is an angular resonance frequency of the electromagnetic resonator.

[0281] The low-pass filter may be implemented in the same way as the thermodynamic reservoir described above. This implementation may be simpler than the filtered-reservoir configuration, since the spectral density of the low-pass filter does not require a particular effective slope near co, allowing for greater tolerance to manufacturing inconsistencies.

[0282] Additionally, or alternatively, the system is configured to operate within a working operating temperature range extending up to at least 10 mK, such as up to at least 30 mK, such as up to at least 50 mK, such as up to at least 100 mK, such as up to at least 200 mK, such as up to at least 500 mK.

[0283] Temperature ranges down to 10 mK can be achieved with state-of-the-art dilution refrigerators. Lower temperatures enable the low-pass-filter solution, and in general will inevitably lead to less noise, and hence slower decoherence of the qubit (even if the DEC it supports may make thermal noise a relatively insignificant limiting factor for the qubit). The higher temperatures can be compatible with the qubit using the filtered-reservoir configuration. These temperatures are achievable with less costly refrigeration infrastructure than dilution refrigerators, and thus provide a significant advantage in smaller resource cost of the approach

[0284] Additionally, or alternatively, the electromagnetic resonator has an impedance approximately equal to or greater than h / 2e2.

[0285] An impedance of approximately h / 2e2(circa 12.9 kQ) is advantageous, since achieving a high resonator impedance is anticipated to be one of the more costly elements of the approach. The value h / 2e2is special because an electromagnetic resonator with this impedance undergoing free oscillation exhibits a phase ( ) oscillation amplitude, in units of 2?r, that is identical to its dimensionless charge (n) oscillation amplitude. Equivalently, its flux oscillation amplitude, in units of the superconducting flux quantum h / 2e, is identical to its charge oscillation amplitude in units of the Cooper-pair charge 2e. This property is crucial to the emergence of the square- or rectangular-lattice GKP stabilizer Hamiltonian in the present protocol. P7918PC00

[0286] 35

[0287] Achieving such a large impedance can be realised using arrays of Josephson junctions (superinductors), possibly stacked in the third dimension as in

[0288]

[0289] https: / / arxiv.org / abs / 2505.02764 to minimise the area to ground geometric inductors; inductors based on superconductors with low Cooper-pair density (and therefore high kinetic inductance); granular-aluminium inductors; or suitable combinations of these elements.

[0290] Additionally, or alternatively, the superconducting resonator has an impedance approximately equal to (D / 2) • (h / 2e2), for any integer D > 2, so as to realise a D-dimensional square or rectangular-lattice Gottesman-Kitaev-Preskill (GKP) qudit.

[0291] This implementation advantageously provides access to a D-level qudit using the minimal impedance required for that dimensionality, as described herein in the present disclosure.

[0292] Additionally, or alternatively, the system comprises a readout circuit including:

[0293] a. a secondary Josephson junction coupled to the electromagnetic resonator at a position part way along an inductance of the resonator; and

[0294] b. a voltage source configured to apply an auxiliary voltage tone across the secondary Josephson junction.

[0295] The advantages and implementation details of the readout protocol is disclosed herein.

[0296] Additionally, or alternatively, the system comprises an auxiliary resonator having a ground node electrically connected to a ground node of the superconducting resonator; and wherein the electrical connection between the ground nodes forms a closed circuit loop having an effective enclosed area less than 1000 pm2, preferably less than 400 pm2, more preferably less than 50 pm2, and most preferably less than 20 pm2, thereby reducing susceptibility of the system to flux noise.

[0297] The method according to the first aspect of the disclosure may further comprise the step of providing a system according to the second aspect of the disclosure. P7918PC00

[0298] 36

[0299] Preferable and / or optional features of the third aspect of the present disclosure are described herein, providing further advantageous implementations of the system.

[0300] Preferably, the circuit node is located between two inductive segments of the superconducting resonator, the two inductive segments having approximately equal inductance. This configuration may enable readout of the square- or rectangular-lattice / 2TT, as described in Appendix I.

[0301] Additionally, or alternatively, the superconducting resonator is a superconducting resonator of a system according to the second aspect of the disclosure, or wherein the logical GKP state stored in the superconducting resonator is prepared according to any suitable method of the first aspect of the disclosure.

[0302] Our system and method can be used to generate GKP states even in implementations where the quality of operation is insufficient to achieve substantial suppression of logical error rates via dissipative error correction. GKP states produced in this manner may nevertheless represent a valuable resource for quantum computing or sensing in other contexts, even when not autonomously error-corrected via the present scheme. Our simulations indicate that generating GKP states with the approaches disclosed herein is significantly simpler than achieving strong logical-error suppression through dissipative error correction, providing a practical stepping stone and a high-value application for lower-cost implementations of the invention.

[0303] In one example, the disclosed method further comprises:

[0304] a. inserting a bad-pass filter between a quantum system (qubit) and its environment to absorb only energies in a window approximately between 1 and 3 / 2 times the resonator frequency;

[0305] b. providing photon-assisted side-band cooling by (a); and

[0306] c. increasing a maximal operating temperature of the cooling bath by tenfold.

[0307] In one example, the disclosed method further comprises:

[0308] a. modulating voltage tone driving a superconducting resonator; and thereby

[0309] b. implementing high fidelity single-qubit Clifford gates; and providing P7918PC00

[0310] 37

[0311] i. correcting small errors from imperfect control by a dissipative error-correction mechanism; and

[0312] c. providing an efficient read-out and initialization scheme.

[0313] In one example, the method further comprises increasing qubit coherence time.

[0314] In one example, the method further comprises generating GKP states, wherein the GKP codes protect against both bit-flip and phase errors due to phase space-local noise.

[0315] In one example, the method further comprises producing error-corrected Clifford gates.

[0316] Examples

[0317] An example implementation is provided in Appendix I of U. S. Provisional Application No. 63 / 724,408, which is hereby incorporated by reference in its entirety.

[0318] The invention is a superconducting qubit design based on resonant driving of an LC resonator, with an example of a circuit diagram shown in Fig. 1. In this example, the LC circuit in the centre represents the qubit resonator, and the variables and n at the node denote the quantum-mechanical degrees of freedom used to encode the GKP qubit. An oscillating voltage bias with base frequency close to a rational multiple of the resonance frequency is applied across a Josephson junction to the resonator. The system is connected to a thermodynamic reservoir, such as a resistor, via a band-pass or Purcell filter. The filter may for instance be comprised of a single resonator, or a network or array of coupled resonators, with one or more of these coupled to a thermodynamic reservoir.

[0319] When the bias oscillation amplitude is much larger than the resonance frequency of the LC circuit and the Josephson energy of the junction, the stationary states of the resonator are “Gottesman-Kitaev-PreskiH” (GKP)-states [Gottesman2001] for certain values of the resonator impedance; GKP states are wave function representations of the GKP error correcting codes that can encode a qubit or qudit in the wavefunction of the resonator quadratures.

[0320] The stabilisation of a GKP qubit emerges because the oscillating voltage bias essentially deactivates the Josephson junction at all times except near the nodes of the bias oscillation, allowing efficient control of the Josephson coupling. With the driving P7918PC00

[0321] 38

[0322] specified above, the Josephson junction is essentially only active at n evenly spaced instances per oscillation period of the LC resonator. In the comoving frame of the resonator, this activation pattern appears as sequential brief activation of the GKP stabilizers when the resonator impedance takes a suitable value. For instance, this occurs if the impedance is given by h / 2e2and the drive frequency is close to twice the resonator frequency. As a consequence of the brief activation, the effective (approximately time-averaged) Hamiltonian of the LC resonator in the comoving frame approaches the GKP stabilizer Hamiltonian, allowing for a realisation of the GKP stabilizer Hamiltonian via Floquet engineering. The GKP qubit is encoded in the low-energy states of this effective Hamiltonian in the comoving frame, which coincide with GKP states. When suitably engineered, the resistor and band-pass filter cause the resonator to thermally relax with respect to this Hamiltonian, leading to initialization, stabilization, and dissipative error correction of the encoded qubit. Moreover, a side-band-cooling mechanism described herein allows this relaxation to occur in principle independently of the ambient temperature, provided it is cold relative to the frequency of the LC resonator.

[0323] Fig. 2 illustrates a schematic example of how the qubit circuit may be modified to support the read-out protocol. The circuit is identical to that of Fig. 1 except that an additional circuit node is inserted halfway along the inductor of the qubit resonator, dividing it into two serially connected inductors of inductance L / 2 each (with L the inductance of the inductor in Fig. 1). A Josephson junction is attached to this intermediate node and connected to ground through an oscillating voltage bias. In one implementation, where the voltage signal across the main junction oscillates at approximately twice the resonance frequency of the resonator and the resonator impedance is near h / 2e2, the voltage bias on the read-out branch may oscillate at frequency X2 / 4, synchronized with the voltage signal of the main branch such that each node of the voltage signal on the read-out branch occurs effectively simultaneously with a node of the signal across the main junction. An amperemeter is inserted in this branch to schematically illustrate that the mean current through the read-out branch is measured to read out the state of the qubit, for example via a flux-biased SQUID-based magnetometer positioned near the read-out branch.

[0324] Capacitive coupling the resonator to a cold bath autonomously corrects phase and bit flip errors of the encoded qubit by dissipation. P7918PC00

[0325] 39

[0326] Using a band-pass filter to absorb only energies in a window approximately between 1 and 3 / 2 times the resonator frequency leads to a side-band cooling mechanism that allows the system to operate with environmental temperatures of the order of 100 mK.

[0327] We provide an optimized waveform of the oscillating voltage bias that reduces the maximal voltage amplitude across the Josephson junction, thereby suppressing quasiparticle excitations and heating.

[0328] Modulating the oscillating voltage bias tone allows one to perform single-qubit Clifford gates with high fidelity due to the passive error-correcting properties of the circuit.

[0329] The quantum information of the circuit can be read by connecting a weak Josephson junction at a node half-way along the inductor. This is achieved by driving the read-out junction with an oscillating voltage bias at a fraction of the frequency of the main driving tone. The read-out signal is the mean current across the Josephson junction, which is opposite for the two logical states of the qubit.

[0330] Fig. 3 illustrates an example system for generating the oscillating voltage bias used to drive the superconducting resonator according to the present disclosure. In this example, an auxiliary electromagnetic resonator is driven so as to produce a voltage tone on-chip across the Josephson junction of the qubit resonator. The closed loop formed by the two inductive arms, the Josephson junction, and the shared ground return path is kept small to mitigate flux noise. Preferably, the effective loop area is below 1000 pm2, more preferably below 400 pm2, more preferably below 50 pm2, and most preferably below 20 pm2, to minimise dephasing from flux noise.

[0331] Fig. 4 illustrates example implementations of the thermodynamic environment according to the present disclosure. Panel A shows a simplified representation of an environment modelled as an effective resistor at temperature TE. Panel B shows an embodiment in which the environment is realised as a single lossy resonator, which may provide a Lorentzian spectral density suitable for frequency-selective dissipation. Panel C shows a more general embodiment in which the environment comprises an arbitrary network of auxiliary resonators, which may be coupled in any combination of P7918PC00

[0332] 40

[0333] serial and, or, parallel configurations. In this arrangement, the individual resonators may be dissipative or non-dissipative (i.e., each Rf imay be zero or non-zero), allowing the network to be engineered to exhibit customised spectral properties. Likewise each and may be zero, infinite, or nonzero. Such networks may advantageously allow more precise tailoring of the effective power spectral density compared to a single-resonator implementation.

[0334] Appendix I contains a fully detailed description of the operating principle, including numerical simulations demonstrating the autonomous error-correction mechanism leading to a lifetime-enhancement by a factor of ~1000.

[0335] The invention provides a novel design for an operational superconducting qubit where an error correction mechanism is achieved with minimal technical overhead. This principle increases qubit coherence time by several orders of magnitude at potentially minimal cost.

[0336] As used herein, the quantity denotes the angular resonance frequency of the superconducting resonator, and Todenotes its natural oscillation period. The quantity denotes the angular frequency of a driving tone applied to the superconducting resonator. A “node” of a voltage tone across a Josephson junction refers to an instant or narrow time interval (compared to the resonator oscillation period) at which the voltage passes through zero or is effectively smaller than the Josephson energy of the junction, and “node spacing” refers to the temporal separation between consecutive nodes. The symbol Z denotes the impedance of the superconducting resonator, and z = Z / (h / 2e2) is a normalised impedance used to characterise conditions for generating square, rectangular, hexagonal, oblique, or quasicrystalline GKP lattices. The symbol denotes the Josephson energy of a Josephson junction in the resonator, 21 denotes the superconducting energy gap of that junction, and kBdenotes the Boltzmann constant.

[0337] As used herein, references to a parameter being “approximately equal to”, “close to”, “near”, “within”, or “substantially equal to” a stated value are intended to encompass deviations that still preserve the technical advantages of the relevant embodiment. For frequencies or angular frequencies, such wording may correspond to tolerances within ±20%, preferably ±10%, more preferably ±5%, and most preferably ±0.5% of the P7918PC00

[0338] 41

[0339] nominal value. For impedances, it may correspond to variations within ±10%, preferably ±5%, more preferably ±1%, and most preferably ±0.5%. For temporal node positions or node spacings of a voltage tone, it may correspond to variations within ±10%, preferably ±5%, more preferably ±2%, and most preferably ±1%.

[0340] The systems and methods described herein may comprise or be implemented by a computer program or a plurality of computer programs, which may exist in a variety of forms both active and inactive in a single computer system or across multiple computer systems. For example, they may exist as software program(s) comprised of program instructions in source code, object code, executable code or other formats for performing some of the steps. Any of the above may be embodied on a computer readable medium, which include storage devices and signals, in compressed or uncompressed form.

[0341] The term "computer" refers to any electronic device comprising a processor, such as a general-purpose central processing unit (CPU), a specific purpose processor or a microcontroller. A computer is capable of receiving data (an input), of performing a sequence of predetermined operations thereupon, and of producing thereby a result in the form of information or signals (an output). Depending on context, the term "computer" will mean either a processor in particular or can refer more generally to a processor in association with an assemblage of interrelated elements contained within a single case or housing.

[0342] As used herein, a “computer-readable medium” or “storage medium” can be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. The computer readable medium can be, for example but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (a non-exhaustive list) of the computer-readable medium can include the following: an electrical connection having one or more wires, a portable computer diskette, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fibre, and a portable compact disc read-only memory (CDROM). P7918PC00

[0343] 42

[0344] Further details of drawings

[0345] 100, 200, 300, 400a, 400b, 400c- Systems according to the present disclosure 101, 201, 301 - Drive source (Q source)

[0346] 102, 202, 302 - Josephson junction J with Josephson energy Ej

[0347] 202a - Auxiliary Josephson junction J'

[0348] 103 - Node with (4>, n)

[0349] 203a - Central node

[0350] 104, 204 - Capacitor C

[0351] 105, 205a, 205b - Inductor L or L / 2

[0352] 106, 206 - Filter capacitor CF

[0353] 107, 207, 407 - Environment resistor R (optional)

[0354] 108, 208, 308, 408 - Thermal environment

[0355] 209 - Auxiliary current meter A

[0356] 210 - Auxiliary tone source (Q / 4)

[0357] 311 - Input coupler (T1)

[0358] 312a, 312b - First and second resonator capacitor

[0359] 313a, 313b - First and second coupling capacitor

[0360] 314a, 314b - First and second resonator inductor

Claims

P7918PC0043Claims1. A method comprisinga. driving a superconducting resonator with a voltage tone;b. coupling the superconducting resonator to a thermodynamic environment;c. generating wave function representations of Gottesman-Kitaev-Preskill error correcting codes (GKP states) that that can encode a qubit or qudit in the wavefunction of the resonator quadratures; andd. autonomously error correcting phase and bit flip errors of the encoded qubit by dissipation.

2. The method of claim 1, wherein step (b) comprises capacitatively coupling the superconducting resonator to the thermodynamic environment.

3. The method of claim 1 or 2, further comprisinga. inserting a band-pass filter between a quantum system (qubit), such as the superconducting resonator, and the thermodynamic environment, the band-pass filter being configured to admit dissipative processes involving relaxation by emission of a single resonator photon and to suppress dissipative processes involving zero-photon or multi-photon emission or absorption processes; andb. providing photon-assisted side-band cooling by step (a).

4. The method of claim 3, wherein the band-pass filter absorbs only energies in a window approximately between hc and 3 / 2 ftto, wherein to is an angular resonance frequency of the electromagnetic resonator.

5. The method of claim 1 or 2, further comprisinga. inserting a low-pass filter between the superconducting resonator and the thermodynamic environment, the low-pass filter being configured to suppress photon assisted sideband transitions and to admit direct dissipation processes that do not involve exchange of resonator photons; andb. providing direct cooling of the superconducting resonator.P7918PC00446. The method of claim 5, further comprisinga. maintaining a temperature of the thermodynamic environment below Ej / kB, preferably one or more orders of magnitude below Ej / kB, such as less than 0.1 Ej / kB, such as less than 0.05Ej / kB, such as less than 0.01Ej / kB, such as less than 0.005 Ej / kB, such as less than 0.001 Ej / kB,wherein Ej is the Josephson energy of a Josephson function within the superconducting resonator, and kBis the Boltzmann constant.

7. The method of any one of claims 1-6, wherein the voltage tone used to drive the superconducting resonator satisfies any one of the following:a. the drive frequency is a single harmonic tone with an angular frequency Q approximately equal to twice the angular resonance frequency co of the superconducting resonator;b. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to 2co, such that nodes of the voltage tone occur at uniform spacing IT / Q;c. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal 2co, such that the nodes of the voltage tone occur at substantially uniform spacing TT / Q.

8. The method of any one of claims 1-6, wherein the voltage tone used to drive the superconducting resonator satisfies any one of the following:a. the drive frequency has a fundamental angular frequency Q approximately equal to co / (N + 1 / 2), where N is any integer, such that nodes of the voltage tone occur at substantially uniform spacing TT(N + 1 / 2) / co;b. the drive frequency has a fundamental angular frequency Q approximately equal to co / (N + 1 / 2), where N is any integer; c. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to co / (N + 1 / 2), where N is any integer; orP7918PC0045d. the voltage tone comprises multiple frequency components given by integer multiples of a fundamental frequency Q approximately equal to co / (N + 1 / 2), where N is any integer;such that the nodes of the voltage tone occur at substantially uniform spacing IT / Q.

9. The method of any one of claims 1-6, wherein the voltage tone used to drive the superconducting resonator is configured such that the time-interval between adjacent nodes of the voltage tone is approximately equal to an integer multiple of To / 6, where Tois the oscillation period of the superconducting resonator, the integer multiple optionally varying between different adjacent node spacings.

10. The method according to any one of the preceding claims, wherein the voltage tone used to drive the superconducting resonator is configured such that the time-interval between nodes of the voltage tone is approximately equal to halfinteger multiples of T0 / 2, where Tois the oscillation period of the superconducting resonator, the integer multiple optionally varying between different adjacent node spacings.

11. The method of any one of the preceding claims, wherein the voltage tone used to drive the superconducting resonator is periodic.

12. The method of any one of the preceding claims, wherein the voltage tone used to drive the superconducting resonator is non-periodic.

13. The method according to any one of the preceding claims, wherein the voltage tone used to drive the superconducting resonator has a maximum amplitude less than A / 2, where A is a superconducting energy gap of a Josephson junction within the superconducting resonator.

14. The method according to any one of the preceding claims, wherein the autonomous error correction of step (d) is achieved by dissipation processes which relax phase-space-local excitations of a GKP-encoded state within a single GKP potential well.P7918PC004615. The method according to any one of the preceding claims, wherein the coupling is configured to admit dissipative processes involving relaxation by emission of a single resonator photon and to suppress dissipative processes involving zerophoton and multi-photon transitions.

16. The method according to any one of the preceding claims,further comprising executing an initialisation routine for magic-state preparation, the initialisation routine comprising:a. preparing the superconducting resonator in a ground state while a Josephson junction of the superconducting resonator is decoupled, the decoupling being implemented using a tunable Josephson-energy element, a tunable coupler, or by applying a voltage bias across the junction that continuously remains larger, such as at least one order of magnitude larger, than the Josephson energy of the junction divided by 2e; andb. switching on the Josephson coupling and applying a resonant voltage-drive tone configured to realise the dissipative error correction of any one of claims 1-16, with the amplitude of the voltage tone selected such that an effective GKP potential comprises a potential well located at an origin of the resonator phase space ( = 0,n = 0).

17. The method of any one of claims 1-4 or 7-16, wherein the band-pass filter is configured to be more likely to admit photons with angular frequency above X than photons with angular frequency below X, whereX= (jtN +with N being the positive integer for which X is closest to the angular resonance frequency of the superconducting resonator, and T being the spacing between nodes of the voltage tone used to drive the superconducting resonator.

18. The method according to any one of the preceding claims, wherein the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within a frequency interval within ±25% of X, whereX = (jiN +P7918PC0047preferably within ±7% of X, more preferably within ±2% of X, while suppressing photons outside the frequency interval, with photons above the frequency interval being admitted to a greater extent than photons below the frequency interval, where X = (nN + TT / 2) / T and T is the uniform spacing between nodes of the voltage tone.

19. The method according to any one of the preceding claims, wherein the coupling between the superconducting resonator and the thermodynamic environment is configured to admit dissipative processes in which the superconducting resonator absorbs photons from the driving field induced by the voltage tone and emits energy into the thermodynamic environment greater than the energy absorbed from the driving field.

20. The method according to any one of the preceding claims, wherein the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons with angular frequency above the angular resonance frequency a> of the superconducting resonator to a greater extent than photons with angular frequency below a>.

21. The method according to any one of the preceding claims, wherein the coupling between the superconducting resonator and the thermodynamic environment is configured to admit photons within an interval around the angular resonance frequency a> of the superconducting resonator while suppressing photons outside that interval, preferably with photons above a> being admitted to a greater extent than photons below a>.

22. The method according to any one of the preceding claims, wherein the step of coupling the superconducting resonator to the thermodynamic environment in step (b) of claim 1 comprises coupling the superconducting resonator to the thermodynamic environment via an auxiliary electromagnetic resonator having a resonance frequency selected from:a. a resonance frequency approximately equal to X, where X = nN + TT / 2) / T, is the spacing between nodes of the voltage tone used to drive the superconducting resonator; orP7918PC0048b. a resonance frequency approximately equal to co, where a> is an angular resonance frequency of the superconducting resonator; and the auxiliary electromagnetic resonator being coupled to a thermal reservoir or transmission line so as to generate a Lorentzian spectral density in the coupling.

23. The method according to any one of the preceding claims, wherein the step of coupling the superconducting resonator to the thermodynamic environment in step (b) comprises coupling the superconducting resonator to the thermodynamic environment via an array of auxiliary electromagnetic resonators,at least one or more auxiliary resonator of the array being coupled to a thermal reservoir or transmission line,the array exhibiting a transmission spectrum corresponding to the coupling properties defined in any one of the preceding claims.

24. The method according to any one of the preceding claims, further comprising maintaining a temperature of the thermodynamic environment within a working temperature range, the working temperature range being 20 mK or less, such as 40 mK or less, such as 100 mK or less, such as 200 mK or less, such as 500 mK or less.

25. The method of any one of the preceding claims, further comprisinga. modulating of voltage tone driving a superconducting resonator; and therebyi. implementing high fidelity single-qubit Clifford gates; and providingii. correcting small errors from imperfect control by a dissipative error-correction mechanism; andiii. providing an efficient read-out and initialization scheme.

26. A method of producing error-corrected Clifford gates using the method of claimP7918PC004927. The method of any one of the preceding claims, wherein step (a) of claim 1 comprises driving the superconducting resonator with a voltage tone oscillating at a frequency that is a rational multiple of a resonance frequency of the superconducting resonator, such as twice the resonance frequency.

28. The method of claim 27, wherein the rational multiple is two, such that the superconducting resonator is driven at twice its resonance frequency to generate a square- or rectangular-lattice Gottesman-Kitaev-Preskill code.

29. The method of claim 27, wherein the rational multiple is 3 / N, where N is any integer not divisible by three, such that the superconducting resonator is driven at a frequency equal to (3 / N) co to generate a hexagonal-lattice Gottesman- Kitaev-Preskill code.

30. The method according to any one of the preceding claims, wherein the superconducting resonator is driven by a microwave voltage bias tone.

31. A method of increasing qubit coherence time using the method of any one of the preceding claims.

32. A method of generating GKP states using the method of any one of the preceding claims, wherein the GKP codes protect against both bit-flip and phase errors due to phase space-local noise.

33. The method according to claim 32, wherein protecting against both bit-flip and phase errors comprises protecting against noise that generates a continuous flow of the Wigner function of the resonator mode.

34. The method according to any one of the preceding claims, wherein the voltage tone used to drive the superconducting resonator is modified so as to generate an oblique or hexagonal Gottesman-Kitaev-Preskill (GKP) code.P7918PC005035. The method of claim 33 or 34, wherein the modification comprising adjusting temporal positions of nodes of the voltage tone within each period T, where T is approximately equal to To / 2 with Tobeing a natural oscillation period of the superconducting resonator,36. The method of claim 33 or 34, wherein the temporal positions of nodes occur at timesTo1t = 0, t = — arcsin (—),2TTwhere z = Z / (h / 2e2) is a normalised impedance of the resonator having impedance Z, any branch of the arcsine function being admissible, such that a resulting encoded state corresponds to an oblique GKP lattice with primitivevector angle arcsin (1 / z).

37. The method according to any one of the preceding claims,wherein the voltage tone used to drive the superconducting resonator is modified such that the spacing between adjacent nodes of the voltage tone is approximately equal to:ToTo1At = k — + s — arcsin (—2 2TTwhere Tois the natural oscillation period of the superconducting resonator, z = Z / (h / 2e2) is a normalised impedance of the resonator having impedance Z, k is an integer, and s is a binary parameter taking values 0 or 1.

38. The method according to any one of the preceding claims, wherein the coupling between the superconducting resonator and the thermodynamic environment is configured such that the thermodynamic environment admits photons having angular frequencies more than or equal to 2n / T0, where Tois the natural oscillation period of the superconducting resonator.

39. The method according to any one of the preceding claims,wherein the superconducting resonator has an impedance approximately equal to (4 / V3) • (h / 4e2);and wherein the voltage tone used to drive the superconducting resonator hasP7918PC0051a fundamental driving frequency close to a rational multiple p / q of the resonance frequency of the superconducting resonator.

40. The method of claim 39, wherein p = 3 and q is an even integer, so as to generate a hexagonal Gottesman-Kitaev-Preskill code.

41. The method of claim 39, wherein q = 4 and p is an odd integer, so as to generate an oblique variant of a quasicrystalline Gottesman-Kitaev-Preskill code.

42. The method of claim 39, wherein q = 5 and p is an even integer, so as to generate a quasicrystalline Gottesman-Kitaev-Preskill code having a Penrosetiling structure.

43. A method according to any one of the preceding claims, further comprising selecting or adjusting an impedance of the superconducting resonator to be approximately equal to (D / 2) • (h / 2e2), for any integer D > 2, so as to generate a D-dimensional square- or rectangular-lattice Gottesman-Kitaev-Preskill (GKP) qudit.

44. A system comprising:a superconducting resonator comprising a first Josephson junction coupled to an electromagnetic resonator;a voltage source configured to apply a voltage tone across the first Josephson junction to drive the electromagnetic resonator;wherein the superconducting resonator is capacitatively coupled to a thermodynamic environment; andwherein the system is configured to generate wavefunction representations of Gottesman-Kitaev-Preskill (GKP) codes in quadratures of the superconducting resonator.

45. The system according to claim 44, wherein the superconducting resonator is capacitatively coupled to the thermodynamic environment through a band-pass filter having a bandwidth approximately confined between one-half and three- quarters of a frequency of the voltage tone.P7918PC005246. The system according to claim 44 or 45, wherein the coupling between the superconducting resonator and the thermodynamic environment is provided via a filtered reservoir configured to provide frequency-selective dissipation suitable for autonomously error correcting phase- and bit-flip errors of the encoded qubit, the filtered reservoir comprising:a. a single auxiliary electromagnetic resonator having a resonance frequency approximately equal to X = (πN + π / 2) / τ, the auxiliary resonator being coupled to a thermal reservoir or transmission line so as to generate a Lorentzian spectral density in the coupling; or b. an array of auxiliary resonators, at least one auxiliary resonator of the array being coupled to a thermal reservoir or transmission line; or c. a network of auxiliary resonators arranged as an arbitrary graph of coupled resonators, the resonators of the network being coupled capacitatively, inductively, galvanically, or via Josephson junctions, with one or more auxiliary resonators being coupled to a thermal reservoir or transmission line.

47. The system according to claim 44, wherein the superconducting resonator is capacitatively coupled to the thermodynamic environment through a low-pass filter coupling the superconducting resonator to the thermodynamic environment, the low-pass filter having a spectral density that is substantially confined to energies IE |< |ftco and that decreases rapidly for energies I E |> iftco, where co is an angular resonance frequency of the electromagnetic resonator.

48. The system according to any one of claims 44-47,wherein the system is configured to operate within a working operating temperature range extending up to at least 10 mK, such as up to at least 30 mK, such as up to at least 50 mK, such as up to at least 100 mK, such as up to at least 200 mK, such as up to at least 500 mK.

49. The system according to any one of claims 44-48, wherein the electromagnetic resonator has an impedance approximately equal to or greater than h / 2e2.P7918PC005350. The system according to any one of claims 44-49, wherein the superconducting resonator has an impedance approximately equal to (D / 2) • (h / 2e2), for any integer D > 2, so as to realise a D-dimensional Gottesman- Kitaev-Preskill (GKP) qudit.

51. The system of according to any one of claims 44-50, further comprising a readout circuit including:a. a secondary Josephson junction coupled to the electromagnetic resonator at a position part way along an inductance of the resonator; andb. a voltage source configured to apply an auxiliary voltage tone across the secondary Josephson junction.

52. The system according to any one of claims 44-51, further comprising an auxiliary resonator having a ground node electrically connected to a ground node of the superconducting resonator; and wherein the electrical connection between the ground nodes forms a closed circuit loop has an effective enclosed area less than 1000 pm2, preferably less than 400 pm2, more preferably less than 50 pm2, and most preferably less than 20 pm.

53. The method according to any one of claims 1-43, comprising the step of providing a system according to any one of claims 44-52.

54. A method of executing a read-out protocol for reading a logical Gottesman- Kitaev-Preskill (GKP) state stored in a superconducting resonator, the method comprising:a. coupling a Josephson junction to the superconducting resonator at a circuit node located at an intermediate position along an inductance of the superconducting resonator;b. applying an auxiliary voltage tone across the Josephson junction; c. measuring a time-averaged supercurrent across the secondary Josephson junction; andd. distinguishing between the even and odd well GKP states based on the sign of the measured time-averaged supercurrent.P7918PC005455. The method of claim 54, wherein the circuit node is located between two inductive segments of the superconducting resonator, the two inductive segments having approximately equal inductance.

56. The method of claim 54 or 55, wherein the superconducting resonator is a superconducting resonator of a system according to any one of claims 45-53, or wherein the logical GKP state stored in the superconducting resonator is prepared according to any one of claims 1-43 or 53.

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