Quantum system for performing a classical error-correcting code using physical qubits

Abstract

There is provided a system (1 ) for performing a classical error-correcting code using physical qubits. The system comprises a command circuit (5) for selectively applying control signals; a qubit-hosting circuit (103) comprising: at least three data resonators, each data resonator being coupled to the command circuit for stabilizing a respective data cat qubit (502,503) having a rate κ2 at which pairs of bosons are exchanged with the environment, and at least two ancilla qubit-hosting structures, each ancilla qubit-hosting structure being coupled to the command circuit for preparing a respective ancilla qubit (501 ); a first stabilizer (102), wherein the first stabilizer comprises a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer (102), wherein the second stabilizer comprises a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit (503) which is one of the at least two data cat qubits of the first stabilizer. The command circuit is configured to apply control signals to the qubit-hosting circuit so as to: perform a first stabilizer measurement on the first stabilizer comprising the operations of: (i) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and (ii) performing a respective CNOT gate between the first ancilla qubit as control and each of the other at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time; perform a second stabilizer measurement on the second stabilizer comprising the operations of: (iii) performing a CNOT gate between the second ancilla qubit as control and the shared data cat qubit as target, and (iv) performing a respective CNOT gate between the second ancilla qubit as control and each of the other at least two data cat qubits of the second stabilizer as target, wherein the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time; and stabilize the shared data cat qubit for a re-stabilization time period (I) between the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement, wherein the re-stabilization time period is greater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit (II) at which pairs of bosons are exchanged with the environment, (III). The is also provided a method for performing a classical error-correcting code using qubits on a classical error-correcting code arrangement and corresponding computer program or computer-readable medium.

Description

[0001] QUANTUM SYSTEM FOR PERFORMING A CLASSICAL ERROR-CORRECTING CODE USING PHYSICAL QUBITS

[0002] FIELD OF THE INVENTION

[0003] The invention concerns a quantum system for performing a classical errorcorrecting code using physical qubits, for instance a repetition error-correcting code comprised of physical cat qubits as the physical data qubits.

[0004] BACKGROUND

[0005] Realizing a large-scale quantum computer is challenging due to noise from uncontrolled interactions between its components and the environment, which destroys the fragile quantum characteristics required to usefully encode and manipulate information in a quantum state.

[0006] Fault-tolerant quantum computing addresses this issue with quantum errorcorrecting codes (QECC). These codes protect quantum information by encoding it in non-local entangled states, making it less likely for local errors to corrupt it. The surface code is the most popular QECC.

[0007] The threshold theorem, central to quantum fault tolerance, states that reliable, long quantum calculations can be performed if noise (i.e. physical error rates on the qubits and gates used in the QECC) is below a constant value called the fault-tolerant error threshold. Below this threshold, QECCs theoretically provide arbitrarily good protection against noise with increasing the distance of the error correction (related to the number of qubits required to implement the error correction), solving the decoherence problem. However, implementing QECCs requires significant physical resources, creating a trade-off known as the "resource overhead problem."

[0008] Realistic quantum computing approaches must address this problem. Continuous variable systems, such as harmonic oscillators with infinite-dimensional Hilbert spaces, offer an advantage over discrete variable systems with finite-dimensional spaces. Continuous variable encodings, involving superpositions of specific harmonic oscillator states (e.g., GKP qubits, Fock states, cat qubits), show promise for protecting and processing quantum information efficiently.

[0009] In particular, pumped or stabilized cat qubits (which can be hosted in superconducting resonators) benefit from a noise bias, where bit-flip errors are exponentially suppressed with the average number of bosons (normally photons) in the so-called Schrodinger cat states hosted in a resonator. This suppression is effective for a wide range of physical noise processes, including photon loss, thermal excitations, photon dephasing, and nonlinearities from i.e. Josephson junctions. Recent experiments in quantum superconducting circuits have demonstrated this exponential suppression of bit-flip errors.

[0010] Given the exponential suppression of bit-flips, quantum error correction for such cat qubits can take the form of a repetition code (or other 1 D classical error correction codes) to only correct phase-flip errors, and thus becomes similar in complexity to classical error correction. This involves encoding logical information in cat qubits and detecting errors through repeated measurements of stabilizers using ancilla qubits, which preferably can also be cat qubits. Key operations include preparing the cat states, measuring photon parity, and applying CNOT gates.

[0011] For example, to correct phase-flips using a phase-flip repetition code formed from a linear 1 D array of data cat qubits, the repetition code measures the parity of neighbouring data cat qubits using a so-called stabilizer measurements (also known as parity checks). The code space can be defined as the common +1 eigenspace of the N — 1 stabilizers Sj = Xj ® X;+1, j e [[1,1V — 1]], where N is the number of data cat qubits. The logical operators for the repetition cat qubit are XL= XltZL= jZj , YL= i XLZL. The logical states |+)tand |-)tare given by |±)L:= |±)®w, where |±)cdenotes the physical states |+)cand |— )cof the data cat qubits.

[0012] Due to the principles and restraints of quantum mechanics, each of the aforementioned parity checks are typically performed using a so-called ancilla qubit (i.e. used in the code, but not for hosting information). The ancilla is prepared in a known state (typically for an exemplary phase-flip correcting code this could be a | — ) or | +)), and subsequently entangled with the data qubits whose parity is to be checked (the data cat qubits of the so-called “stabilizer”). The entangling may involve a quantum CNOT gate between a first data cat qubit as the control and the ancilla qubit as the target, and another quantum CNOT gate between a second data cat qubit (neighbouring the first) as the control and the ancilla qubit as the target, and similarly further CNOT gates if the stabilizer is comprised of more than two data cat qubits. The ancilla is then measured in the X-basis, and the result of the measurement is attributed as the ±1 of the parity check.

[0013] Guillaud, Jérémie, and Mazyar Mirrahimi. "Repetition cat qubits for fault-tolerant quantum computation." Physical Review X 9.4 (2019): 041053, discloses generally a repetition code using cat qubits, wherein a linear array of n data cat qubits are interspersed by n - 1 ancilla cat qubits. The parity checks of all the stabilizers are performed simultaneously (each being comprised of a pair of data cat qubits and the corresponding interspersed ancilla qubit for using in the CNOT gates) and repeated an optimal number r times, such that an optimal decoding may be implemented based on the outcome of the (n — l)r parity checks (ancilla qubit measurements).

[0014] Guillaud, Jérémie, and Mazyar Mirrahimi. "Error rates and resource overheads of repetition cat qubits." Physical Review A 103.4 (2021 ): 042413, more explicitly models error rates for a repetition code using cat qubits. However, again the parity checks of all the stabilizers are performed simultaneously. The circuit is divided into time-steps wherein every qubit (both data and ancilla) in the circuit is acted upon at every time step. Said otherwise, for the data cat qubit shared between two stabilizers, as soon as the CNOT gate is completed between the ancilla of one stabilizer and the shared data cat qubit, the next CNOT gate is immediately implemented between the ancilla of the other stabilizer and the shared data cat qubit.

[0015] Both of these prior disclosures thus consider performing immediately successive CNOT gates on the data cat qubits shared between two stabilizers. Said otherwise, the majority of the data cat qubits act as target during two consecutive CNOTs before being re-confined with 2-photon dissipation (the exceptions being the end-most data cat qubits delimiting the linear array). This means that the majority of the data cat qubits are operated on over a duration 2TCX(where TCXis the time taken to perform a CNOT gate), during which leakage errors may accumulate in a non-trivial way arising from the dynamics involved in implemented the CNOT gate.

[0016] Le Régent, Francois-Marie, et al. "High-performance repetition cat code using fast noisy operations." Quantum ? (2023): 1198, discloses a repetition code using cat qubits optimized using fast parity measurements via accelerated but noisy CNOT gates and fast ancilla parity-check qubits. A qubit refreshing time is added between two CNOT gates in one cycle to avoid leakage induced errors, where the qubit refresh time is the same as the CNOT gate time TCX. However, this is at the expense of a wait time on the ancilla qubit in between the CNOT gates with the data cat qubits of the stabilizers on either side, which increases the QECC cycle time. This may lead to increased phase- and bit-flip errors on both the ancilla and data cat qubits, thus degrading the logical qubit performance.

[0017] Moreover, all of the above prior art disclosures explicitly consider that the parity checks of all the stabilizers are performed simultaneously, which may result in issues such as frequency-crowding problems resulting from simultaneously applying the necessary electromagnetic pulses inducing quantum operations.

[0018] Furthermore, all of the above prior art disclosures consider only CNOT gates between two cat qubits requiring engineering the target qubit stabilization such that its phase is conditioned on the state if the control qubit to thereby maintain a rotating data cat qubit stabilization for the duration of the CNOT gate. This requires exquisite calibration of increasing complexity with increasing numbers of data cat qubits in the repetition code.

[0019] This invention aims to address one or more of the aforementioned problems to enable the implementation of a cat qubit repetition code compatible with the faulttolerance error threshold.

[0020] SUMMARY

[0021] In a first aspect, there is provided a system for performing a classical errorcorrecting code using physical qubits. The system comprises a command circuit for selectively applying control signals; a qubit-hosting circuit comprising: at least three data resonators, each data resonator being coupled to the command circuit for stabilizing a respective data cat qubit having a rate K2at which pairs of bosons are exchanged with the environment, and at least two ancilla qubit-hosting structures, each ancilla qubit-hosting structure being coupled to the command circuit for preparing a respective ancilla qubit; a first stabilizer, wherein the first stabilizer comprises a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer, wherein the second stabilizer comprises a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer.

[0022] The command circuit is configured to apply control signals to the qubit-hosting circuit so as to: perform a first stabilizer measurement on the first stabilizer comprising the operations of: (i) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and (ii) performing a respective CNOT gate between the first ancilla qubit as control and each of the other at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time; perform a second stabilizer measurement on the second stabilizer comprising the operations of: (iii) performing a CNOT gate between the second ancilla qubit as control and the shared data cat qubit as target, and (iv) performing a respective CNOT gate between the second ancilla qubit as control and each of the other at least two data cat qubits of the second stabilizer as target, wherein the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time; and stabilize the shared data cat qubit for a re-stabilization time period trcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement, wherein the re-stabilization time period is greater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit at which pairs of bosons are exchanged with the environment, Trc> I / IOK^ .

[0023] Thus, the shared data cat qubit is stabilized for the period of timercbetween the end of the CNOT gate of the first syndrome / stabilizer measurement (between the first ancilla cat qubit as control and the shared data cat qubit as target) and the beginning of the CNOT gate of the second syndrome / stabilizer measurement (between the second ancilla cat qubit as control and the shared data cat qubit as target).

[0024] By stabilizing the shared data cat qubit between these CNOT gates for the restabilization time period trc, the present inventors have recognized that the rate of bitflips of the shared data cat qubit is substantially reduced compared to being subject to these CNOT gates in immediate succession.

[0025] In particular, the present inventors have discovered that the rate of bit-flips of the shared data cat qubit during immediately consecutive CNOT gates (i.e. without being subject to re-stabilization for a period of time) is surprisingly much greater than simply twice the rate of bit-flips of the shared data cat qubit during just a single CNOT.

[0026] That is, by implementing the re-stabilization for a time periodrcwhich is greater than or equal to a third of the reciprocal of the rate of the shared data cat qubit at which pairs of bosons are exchanged with the environment, the present Inventors have recognized that the leakage errors which accumulate during the CNOT gate of operation (i) of the first stabilizer measurement (operation (i) is the CNOT gate of the first stabilizer which involves the shared data cat qubit as target) may be substantially reduced or brought back to zero, thereby minimizing the subsequent rate of bit-flips of the shared data cat qubit during the CNOT gate of operation (iii) of the second stabilizer measurement (operation (iii) is the CNOT gate of the second stabilizer which involves the shared data cat qubit as target). Moreover, the start of the second stabilizer measurement does not occur at the same time as the start of the first stabilizer measurement, but rather is staggered so as to start afterwards. This ensures that the second ancilla cat qubit experiences no excess idling time (which itself would increase the rate of both phase and bit-flips on the second ancilla cat qubit) whilst enabling the aforementioned re-stabilization on the shared data cat qubit for the time period Trc> l / lOic *-

[0027] Thus, the present invention minimizes the QEC cycle duration and the propagation of bit-flips through the repetition code via the synergetic combination of both performing the re-stabilization on the shared data cat qubit and staggering the starts of stabilizer measurements.

[0028] The system may be configured to perform the error-correcting code by repeating the first stabilizer measurement and subsequent second stabilizer measurement a plurality of times.

[0029] The command circuit is configured to selectively applying control signals to the qubit-hosting circuit so as to dissipatively stabilize a data cat qubit in each of the data resonators and prepare an ancilla qubit in each of the ancilla qubit-hosting structures.

[0030] In a preferred embodiment, each of the ancilla qubit-hosting structures is an ancilla resonator being coupled to the command circuit for preparing a respective ancilla cat qubit.

[0031] Specifically, for each data cat qubit, pairs of bosons are exchanged between the respective data resonator and the environment.

[0032] For the avoidance of doubt, by performing a respective CNOT gate between the first (or respectively second) ancilla qubit as control and each of the other at least two data cat qubits of the first (or respectively second) stabilizer as target, the first (or respectively second) stabilizer measurement comprises one CNOT gate per data cat qubit in the first (or respectively second) stabilizer.

[0033] The first stabilizer measurement may comprise: prior to operations (i) and (ii), preparing the first ancilla qubit in a first state, and after operations (i) and (ii), measuring the state of the first ancilla in the basis of an observable whose set of eigenstates contains the first state. The second stabilizer measurement may comprise: prior to operations (iii) and (iv), preparing the second ancilla qubit in a second state, and after operations (iii) and (iv), measuring the state of the second ancilla qubit in the basis of an observable whose set of eigenstates contains the second state. As will be appreciated, in embodiments comprising further stabilizer measurements, the further stabilizer measurements may also comprise: prior to the CNOT operations thereof, preparing the corresponding ancilla qubit in an initial state, and after the CNOT operations thereof, measuring the state of the ancilla qubit in the basis of an observable whose set of eigenstates contains the initial state.

[0034] Each of the ancilla qubit-hosting structures may be an ancilla resonator being coupled to the command circuit for preparing a respective ancilla cat qubit; wherein the first stabilizer measurement may comprise: prior to operations (i) and (ii), preparing the first ancilla cat qubit in a | +) state or a | — ) state, and after operations (i) and (ii), measuring the state of the first ancilla cat qubit in the X-basis. The second stabilizer measurement may comprise: prior to operations (iii) and (iv), preparing the second ancilla cat qubit in a | +) state or a | — ) state, and after operations (iii) and (iv), measuring the state of the second ancilla cat qubit in the X-basis.

[0035] As will be appreciated, any further stabilizer measurement may also comprise preparing of the corresponding ancilla cat qubit in a | +) state or a | — ) state, and subsequently, after the operations of the CNOT gates of the further stabilizer measurement are performed, measuring the state of the corresponding ancilla cat qubit in the X-basis.

[0036] In embodiments, the CNOT gate of operation (i) is the CNOT gate of the first stabilizer measurement which is performed first in time, and the CNOT gate of operation (iii) is the CNOT gate of the second stabilizer measurement which is performed first in time; or the CNOT gate of operation (i) is the CNOT gate of the first stabilizer measurement which is performed last in time, and the CNOT gate of operation (iii) is the CNOT gate of the second stabilizer measurement which is performed last in time.

[0037] Said otherwise, the order of the operations of CNOT gates in one of the first or second stabilizer measurements is reversed with respect to the other of the first or second stabilizer measurements. Thus, if the CNOT gate of the first stabilizer measurement which is performed first in time is at first time t±, and if the CNOT gate of the second stabilizer measurement which is performed first in time is at second time t2, then second time t2is given as t2= ti + Trc+Tc > whereCXis the time taken to perform the CNOT gate of the first stabilizer measurement which is performed first in time.

[0038] The command circuit may be configured to apply control signals to the qubithosting circuit so as to: perform a third stabilizer measurement on the first stabilizer comprising the operations of: (v) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and (vi) performing a respective CNOT gate between the first ancilla qubit as control and each of the other at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the third stabilizer measurement which is performed first in time is performed at a third time; and stabilize the shared data cat qubit for a further re-stabilization time period between the end of the CNOT gate of operation (iii) of the second stabilizer measurement and the beginning of the CNOT gate of operation (v) of the third stabilizer measurement which is equal to the re-stabilization time periodrcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement.

[0039] Thus, the order of the operations of CNOT gates of the third stabilizer measurement is in the same order as those of the first stabilizer measurement.

[0040] The qubit-hosting circuit may comprise: at least three four-wave mixing nonlinear elements respectively coupled to each of the at least three data resonators; and control lines coupled to the command circuit so as to selectively apply control signals to each of the four-wave mixing non-linear elements. The command circuit may be configured to perform any one of the CNOT gates between a given ancilla qubit and a given data cat qubit by: during a CNOT gate time window, delivering in the control lines only a radiation at a frequency equal to a resonant frequency of the given ancilla qubit; and outside the CNOT gate time window, delivering control signals to the qubit-hosting circuit to dissipatively stabilize the given data cat qubit.

[0041] For the avoidance of doubt, during the CNOT gate time window, the control signals which dissipatively stabilize the given data cat qubit are not applied, the only signals applied are those corresponding to the radiation at a frequency equal to the resonant frequency of the given ancilla qubit.

[0042] The four-wave mixing non-linear elements may be used to dissipatively stabilize the given cat qubit. That is, the command circuit may be configured to dissipatively stabilize the given cat qubit by delivering control signals to the corresponding four-wave mixing non-linear element.

[0043] Alternatively, the qubit-hosting circuit may further comprise at least three three- wave mixing non-linear elements also respectively coupled to each of the at least three data resonators, wherein the four-wave mixing non-linear element is used to perform any one of the CNOT gates, whereas the three-wave mixing non-linear element is used to dissipatively stabilize the given cat qubit. The given qubit-hosting structure may be configured for preparing the given ancilla qubit having a confinement rate, and may comprise a resonator portion having a mode (q) with the resonant frequency for hosting the given ancilla qubit and is arranged such that a Rabi oscillation of the mode (q) is induced when it is subject to a radiation having said resonant frequency of the given ancilla qubit with a strength which is less than the confinement rate of the given ancilla qubit.

[0044] The mode (q) of the given ancilla qubit may be linearly coupled with the four- wave mixing element or may be non-linearly coupled with the four-wave mixing element.

[0045] Each of the four-wave mixing non-linear element may be an asymmetrically threaded superconducting quantum interference device (ATSs), and the control lines may comprise flux lines through which radiation can be sent to modulate a common flux and / or a differential flux through each of the ATSs. Each of the data resonators may be comprised of at least one resonant portion having a first mode (a) with a first resonant frequency and a second mode (b) with a second resonant frequency, and each ATS may be arranged such that when the command circuit delivers a radiation having the second resonant frequency to the a least one resonant portion to drive the second mode (b) and delivers radiation in the flux lines to modulate said common flux at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency, the qubit-hosting circuit stabilizes a data cat qubit hosted in the first mode (a) having the first resonant frequency.

[0046] The command circuit may thus be configured to perform any one of the CNOT gates between a given ancilla qubit and a given data cat qubit by, during the CNOT gate time window, delivering in said flux lines only a radiation at a frequency equal to the resonant frequency of the mode (q) of said given ancilla qubit operate the CNOT gate between said given ancilla qubit and said given data cat qubit, and arranged outside of the CNOT gate time window to deliver a radiation having the second resonant frequency to the a least one resonant portion corresponding to the given data cat qubit to drive said second mode (b) and to deliver radiation in said flux lines to modulate said common flux through the corresponding ATS at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency.

[0047] Said given data cat qubit and said given ancilla qubit may be linearly coupled in such a way that the phase difference across said ATS corresponding to said given data cat qubit writes <p = + (pq(q + q^) with a being the photon annihilation operator of said first mode (a) of said given data cat qubit, <pabeing the zero-point fluctuation of the phase of said first mode (a) across said ATS of said given data cat qubit, b being the photon annihilation operator of said second mode (b) of said given data cat qubit, (pbbeing the zero-point fluctuation of the phase of said second mode (b) across said ATS of said given data cat qubit, q being the photon annihilation operator of said mode (q) of said given ancilla qubit, (pqbeing the zero-point fluctuation of the phase of said mode (q) of said given ancilla qubit across said ATS of said given data cat qubit, and <pq< pa / 2.

[0048] Said command circuit may be arranged to pump said flux lines with only a radiation at a frequency equal to the resonant frequency of the mode (q) of said given ancilla qubit during the CNOT gate time window with an amplitude ecx, which induces a Hamiltonian having the formula HATS= Hcx— Ejecx(pq + where Hcxis a CNOT Hamiltonian with the formula Hcx= hgcx(q + q^Ça^a — a2) and gcx= Ejecx(pq(pa / h, with a being the photon annihilation operator of said first mode (a) of said given data cat qubit, <pabeing the zero-point fluctuation of the phase of said first mode (a) across said ATS of said given data cat qubit, b being the photon annihilation operator of said second mode (b) of said given data cat qubit, <pbbeing the zero-point fluctuation of the phase of said second mode (b) across said ATS of said given data cat qubit, q being the photon annihilation operator of said mode (q) of said given ancilla qubit, (pqbeing the zero-point fluctuation of the phase of said mode (q) of said given ancilla qubit across said ATS of said given data cat qubit, Ej is the Josephson energy of the side junctions of the ATS, and a2is the photon number of said first mode (a) of said given data cat qubit.

[0049] Said command circuit may be further arranged to induce a compensation Hamiltonian having the formula HCmp= Ejecx(pqq + q+)(l — (p a2during said CNOT gate time window.

[0050] Said command circuit may be further arranged to induce the compensation Hamiltonian by delivering in said flux lines only a radiation at a frequency equal to the resonant frequency of the mode (q) of said given ancilla qubit to modulate said differential flux (p* substantially having the formula

[0051] <psis the common flux and ELis the inductive energy of the central inductance of the. Said command circuit may be further arranged to induce the compensation Hamiltonian by delivering radiation at a frequency equal to the resonant frequency of the mode (q) of said given ancilla qubit to the corresponding given ancilla qubit hosting structure.

[0052] Each of the ancilla qubit-hosting structures may be an ancilla resonator being coupled to the command circuit for preparing a respective ancilla cat qubit; and wherein the control circuit is configured to, for at least one of the stabilizer measurements, perform two or more of the CNOT gates of said at least one of the stabilizer measurements simultaneously.

[0053] In embodiments: (i) the rate of the shared data cat qubit falls within the range 10-100 MHz; and / or (ii) the re-stabilization time-period Trcfalls within the range 20 ns - 2 microseconds; and / or (iii) each CNOT gate has a CNOT gate time TCXwhich falls within the range 10 ns - 1 microsecond.

[0054] Preferably, falls within the range 40-60 MHz.

[0055] Preferably, the CNOT gate timeCXfalls within the range 20-80 ns, and most preferably 40-60 ns.

[0056] Preferably, the re-stabilization time period trcfalls within the range 100-700 ns, and most preferably 300-500 ns.

[0057] The qubit-hosting circuit may comprise: at least three four-wave mixing nonlinear elements respectively coupled to each of the at least three data resonators; and control lines coupled to the command circuit so as to selectively apply control signals to each of the four-wave mixing non-linear elements. Each of the ancilla qubit-hosting structures may be an ancilla resonator being coupled to the command circuit for stabilizing a respective ancilla cat qubit having an ancilla resonant frequency; wherein each of the ancilla resonators is respectively linearly coupled to at least two of the four- wave mixing non-linear elements of the data resonators such that each of the ancilla cat qubits is connected to at least two data cat qubits. The command circuit may be configured to apply control signals to the qubit-hosting circuit to perform any given one of the CNOT gates between a given ancilla cat qubit as control and a given data cat qubit as target by: (I) during a CNOT gate time window of the given CNOT gate having CNOT gate time TCX, delivering in the control lines only a radiation at a frequency equal to the ancilla resonant frequency; and (II) outside the CNOT gate time window, delivering control signals to the qubit-hosting circuit to dissipatively stabilize the given data cat qubit; and wherein the rate of the shared data cat qubit Kdfalls within the range 10-100 MHz, the re-stabilization time-period Trcfalls within the range 20 ns - 2 microseconds, and the given CNOT gate has CNOT gate time TCXwhich falls within the range 10 ns - 1 microsecond.

[0058] The present inventors have discovered that, in embodiments wherein the ancillla qubits are also cat qubits (and most preferably dissipatively stabilized cat qubits wherein the command circuit is configured to stabilize a respective ancilla cat qubit via dissipative stabilization wherein pairs of bosons are exchanged between the ancilla resonator and the environment), the most optimal combination of parameters comprises: Kfdfalling within the range 10-100 MHz (and most preferably 40-60 MHz); the CNOT gate time TCXfalls within the range 20-80 ns (and most preferably 40-60 ns); and the re-stabilization time period rrcfalls within the range 100-700 ns (and most preferably 300-500 ns). This combination may yield a particularly effective classical error-correcting code formed from dissipatively stabilized data cat qubits and dissipatively stabilized ancilla cat qubits, especially wherein the CNOT gates performed therebetween (between a given ancilla cat qubit as control and a given data cat qubit as target) are done by performing a so-called “Hamiltonian” CNOT gate implementation as described herein, which does not require rotation of the stabilization on the data cat qubit during the CNOT gate.

[0059] For the avoidance of doubt, by “given ancilla cat qubit” it is meant any one of the ancilla cat qubits stabilized in the respective one of the ancilla resonators, and by “given data cat qubit” it is meant any one of the data cat qubits stabilized in the respective one of the data resonators and which is one of the data cat qubits connected to the given ancilla cat qubit.

[0060] Each of said four-wave mixing non-linear element may be an ATS, and the control lines may be flux lines.

[0061] Said given data cat qubit and said given ancilla qubit may be linearly coupled in such a way that the phase difference across said ATS corresponding to said given data cat qubit writes tp = <pa(a + a+) + <pb(b + ô+) + (pq(q + q+) with the terms being as described above.

[0062] Said command circuit may be arranged to pump said flux lines with only a radiation at a frequency equal to the ancilla resonant frequency during the CNOT gate time window with an amplitude ecx, which induces a Hamiltonian having the formula the terms being as described above. Said command circuit may be further arranged to induce a compensation Hamiltonian having the formula HCmp= Ejecx(pqq + qt)(l — ^a2) during said CNOT gate time window, with the terms being as described above..

[0063] Said command circuit may be further arranged to induce the compensation Hamiltonian by delivering in said flux lines only a radiation at a frequency equal to the ancilla resonant frequency to modulate said differential flux <pàsubstantially having the formula ®A (C) = —

[0064] EL with the terms being as described above.

[0065] Said command circuit may be further arranged to induce the compensation Hamiltonian by delivering radiation at a frequency equal to the ancilla resonant frequency to the corresponding given ancilla resonator.

[0066] Each of the ancilla qubit-hosting structures is an ancilla resonator being coupled to the command circuit for stabilizing a respective ancilla cat qubit having an ancilla resonant frequency.

[0067] Each of the data cat qubits may be selected from the group consisting of: a parametrically pumped dissipatively stabilized cat qubit, a resonant dissipatively stabilized cat qubit, a DC dissipatively stabilized cat qubit, a dissipatively stabilized squeezed cat qubit.

[0068] Each of the ancilla qubits may be selected from the group consisting of: a parametrically pumped dissipatively stabilized cat qubit, a resonant dissipatively stabilized cat qubit, a DC dissipatively stabilized cat qubit, a dissipatively stabilized squeezed cat qubit, a cat qubit stabilized by a Kerr Hamiltonian, a cat qubit stabilized by a detuned Kerr Hamiltonian, and a cat qubit stabilized by two-photon exchange Hamiltonian.

[0069] In embodiments, each of the data cat qubits is a parametrically pumped dissipatively stabilized cat qubit, and each of the ancilla qubits is a parametrically pumped dissipatively stabilized cat qubit or a resonant dissipatively stabilized cat qubit.

[0070] The present Inventors have recognized that this combination of cat qubits benefits from high functionality whilst needing the least amount of additional control signals. In particular, for ancilla qubits which are resonant dissipatively stabilized cat qubits, the present Inventors have recognized that this specific combination is particular synergetic in combining the aforementioned functionality and reduced number of control signals with particularly high quality and long lifetimes of resonant dissipative stabilized cat qubits to be used as the ancilla qubits.

[0071] The system may comprise a plurality of first stabilizers and a plurality of second stabilizers wherein the first and second stabilizers are arranged in an alternating manner to form a repetition error-correcting code arrangement; wherein the stabilizer measurement on each of the first stabilizers start at substantially the same time; and wherein the stabilizer measurement on each of the second stabilizers start at substantially the same time.

[0072] Herein, by an ancilla qubit being “connected” to a data cat qubit, it is meant that the ancilla qubit-hosting structure is coupled to the data resonator via a coupling. The coupling may be a direct galvanic connection, or may be non-galvanically coupled (such as capacitively or inductively). The command circuit is configured to operate a two-qubit gate between the ancilla qubit prepared in the ancilla qubit-hosting structure and the data cat qubit stabilized in the corresponding data resonator, wherein the two- qubit gate is mediated via the coupling, to thereby entangle the two qubits.

[0073] The classical error-correcting code may be a repetition error-correcting code, wherein each of the stabilizers comprises an ancilla qubit connected to two data cat qubits.

[0074] Alternatively, the classical error-correcting code may be an LDPC errorcorrecting code, wherein at least one of the stabilizers, and preferably each of the stabilizers, comprises an ancilla qubit connected to more than two data cat qubits.

[0075] In a second aspect, there is provided a method for performing a classical errorcorrecting code using qubits on a classical error-correcting code arrangement, wherein the classical error-correcting code arrangement comprises: a first stabilizer comprising a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer comprising a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer; wherein each of the data cat qubits of the first and second stabilizers has a rate K2at which pairs of bosons are exchanged with the environment.

[0076] The method of second aspect comprises: performing a first stabilizer measurement on the first stabilizer comprising the operations of: (i) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and (ii) performing a CNOT gate between the first ancilla qubit as control and another of the at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time; performing a second stabilizer measurement on the second stabilizer comprising the operations of: (iii) performing a CNOT gate between the second ancilla qubit as control and the shared data cat qubit as target, and (iv) performing a CNOT gate between the second ancilla qubit as control and another of the at least two data cat qubits of the second stabilizer as target, wherein the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time; and stabilizing the shared data cat qubit for a re-stabilization time period Trcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement, wherein the re-stabilization time period is greater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit at which pairs of bosons are exchanged with the environment, Trc> I / IOK^ .

[0077] The second aspect may comprise one or more of the optional features described above in relation to the first aspect.

[0078] In a third aspect, there is provided a computer program or computer-readable medium comprising instructions which, when the program is executed by a classical computer coupled to a quantum processor comprising a classical error-correcting code arrangement, cause the classical computer coupled to the quantum processor to carry out the method of the second aspect, wherein the classical error-correcting code arrangement comprises: a first stabilizer comprising a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer comprising a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer; wherein each of the data cat qubits of the first and second stabilizers has a rate K2at which pairs of bosons are exchanged with the environment.

[0079] BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Various embodiments of the invention will now be described, by way of example only, and with reference to the accompanying drawings in which:

[0081] Figure 1 shows a diagram of a generic quantum system comprising a non-linear superconducting quantum circuit and a command circuit which may provide the data cat qubit and / or ancilla qubit of a classical error-correcting code, according to an embodiment of the invention;

[0082] Figure 2 shows components of a quantum system for stabilizing a cat qubit by implementing a parametric dissipative stabilization with an ATS, according to an embodiment of the invention; Figure 3 shows components of a quantum system for stabilizing a cat qubit arranged to perform a DC dissipative stabilization of a cat qubit, according to an embodiment of the invention;

[0083] Figure 4A shows components of a quantum system for stabilizing a cat qubit arranged to perform a resonant dissipative stabilization of a cat qubit, according to an embodiment of the invention;

[0084] Figure 4B shows an exemplary three-wave mixing nonlinear element which may be used in the embodiment of Figure 4A;

[0085] Figure 5 shows an exemplary quantum circuit diagram of the operations required to perform a parity check, according to an embodiment of the invention;

[0086] Figure 6 shows the influence of a CNOT gate in a symmetrized picture on different coherent state components of a control “ancilla” cat qubit and a target “data” cat qubit, according to an embodiment of the invention;

[0087] Figure 7 shows components of a quantum system on which a CNOT gate may be implemented, according to an embodiment of the invention;

[0088] Figure 8 shows two longitudinal coupling establishment schemes for a CNOT gate, according to an embodiment of the invention;

[0089] Figure 9A shows a schematic of the pulse sequence for performing a CNOT gate between two dissipative stabilized cat qubits, according to an embodiment of the invention;

[0090] Figure 9B shows a schematic of the pulse sequence for performing a CNOT gate between a resonant dissipative stabilized cat qubit as ancilla qubit and a parametric dissipative stabilized cat qubit as data cat qubit, according to an embodiment of the invention;

[0091] Figure 10 shows a physical circuit diagram of a repetition code, according to an embodiment of the invention;

[0092] Figure 11 shows a quantum system for performing a repetition error-correcting code with data cat qubits and the corresponding temporal circuit diagram of the operations required to perform the parity checks on the physical qubits, according to an embodiment of the invention;

[0093] Figure 12 shows a temporal circuit diagram of the operations required to perform the parity checks on the physical qubits, according to an embodiment of the invention;

[0094] Figure 13 shows a quantum system for performing a repetition error-correcting code with data cat qubits and the corresponding temporal circuit diagram of the operations required to perform the parity checks on the physical qubits, according to an embodiment of the invention;

[0095] Figure 14 shows a generic diagram of a CXX gate between two data cat qubit and an ancilla cat qubit, according to an embodiment of the invention;

[0096] Figure 15 shows a diagram explaining how the quantum gate of Figure 14 is executed, according to an embodiment of the invention;

[0097] Figure 16 shows a temporal circuit diagram of the operations required to perform the parity checks on the physical qubits with simultaneous CXX gates, according to an embodiment of the invention; and

[0098] Figure 17 shows a quantum system for performing an LDPC error-correcting code with data cat qubits which has stabilizers of weight greater than 2 and the corresponding temporal circuit diagram of the operations required to perform the parity checks on the physical qubits, according to an embodiment of the invention.

[0099] DETAILED DESCRIPTION

[0100] Quantum systems providing physical qubits

[0101] Figure 1 shows an exemplary quantum system 1 which is arranged to stabilize a so-called cat qubit. A two-legged cat qubit is defined as a two-dimensional manifold spanned by the so-called cat states which are superpositions of two coherent states

[0102] Stabilized cat qubits are known to benefit from a high noise bias, which means that the bit-flip probability is exponentially smaller than the phase-flip probability. More precisely, an effective error channel (e.g., bit errors or “bit-flips”) is suppressed in an exponential way with the “size” - i.e. the average number of photons n = | a |2- of the Schrodinger cat states of the cat qubits. As previously mentioned, this exponential suppression of bit-flips is only at the cost of linear increase of phase-flips.

[0103] The cat qubits can be stabilized or confined by the following exemplary schemes:

[0104] (A) A parametric dissipative stabilization, with jump operator L2= / ïc^ a2— a2), where K2is the two-photon dissipation rate, a is the photon annihilation operator of the memory mode a and a is a complex number defining the cat qubit. This jump operator can be realized by coupling a lossy buffer mode b with dissipation rate Kb, and a four- wave mixing non-linear element - typically a Josephson junction or an asymmetrically threaded SQUID (ATS) - to the memory mode a and by engineering the Hamiltonian + h.c., where b is the photon annihilation operator of the buffer mode b and g2is the two-photon coupling rate, by applying to the four-wave mixing nonlinear element a pump at frequency |2 / a— fb\ and a drive of the buffer mode b at frequency fbprovided g2< Kb.

[0105] (B) A Kerr Hamiltonian — a2) (a2— a2), where K is the amplitude of the Kerr Hamiltonian, a is the photon annihilation operator, and |a|2is the mean photon number.

[0106] (C) A detuned Kerr Hamiltonian where K is the amplitude of the Kerr Hamiltonian, a is the photon annihilation operator, a is a complex number defining the cat qubit, and A is the detuning factor.

[0107] (D) A two-photon exchange (TPE) Hamiltonian ^ / fl= g2^a2~ a2)cr++ h.c., where g2is the complex two-photon coupling rate, a is the photon annihilation operator, a is a complex number defining the cat qubit, and cr±are the lowering and raising operators of the two-level system. This Hamiltonian can be engineered in the same way as the parametric dissipative stabilization (A).

[0108] (E) A dissipative squeezing stabilization, with jump operator Lsc— ((cosh(r)a + sinh(r)eiea+)2- a2^, where KSCis the squeezed two-photon dissipation rate, a is the photon annihilation operator of the memory mode a, a is a complex number defining the cat qubit, r and 0 are the modulus and argument of the complex squeezing parameter f = re10. This jump operator can be realized by coupling a lossy buffer mode b with dissipation rate Kb, and a four-wave mixing non-linear element - typically a Josephson junction or an ATS -, to the memory mode a and by engineering the Hamiltonian = gsc((cosh(r)a + sinh(r)eiea+)2— a2) + h.c., where b is the photon annihilation operator of the buffer mode b and gscis the squeezed two-photon coupling rate, with several pumps at frequencies \2fa— fb\, fband 2fa+ fb, and a drive of the buffer mode b at frequency fbprovided gsc< Kb.

[0109] (F) A variant of the previous stabilization scheme e), for which the Applicant filed the European patent application EP 23175147.0, in which a bosonic qubit -called “moon cat qubif since the two blobs of the Wigner function have a crescent moon shape - is stabilized by engineering the Hamiltonian = g2(a2+ Aa+a — a2)M + h. c., where g2is g2is the two-photon coupling rate, a is the annihilation operator of the memory mode a, 2 is a complex number which phase and amplitude result from the amplitude of longitudinal coupling produced by a pump at frequency fb, a is a complex number resulting from a drive of the buffer mode b at frequency fband b is the annihilation operator of the buffer mode b; a comparison between the moon cat qubit and the squeezed cat qubit could be established by expressing 2 as a function of the complex squeezing parameter f = re10as follows: 2 = 2tanh (r).

[0110] (G) A DC dissipative stabilization, for which the Applicant filed the European patent application EP 23306839.4, in which a cat qubit is stabilized in the spirit of the previous stabilization scheme a), except that the two-photon pump that engineers the non-linear conversion between two photons of the memory mode a and one photon of the buffer mode b is replaced with a DC voltage source which biases a non-linear element formed exclusively of one or more Josephson junctions, such that the DC-biased non-linear element acts as a voltage-to-frequency converter which, at the appropriate voltage bias, provides the required parametric interaction at the frequency \2fa— fb\ necessary to achieve dissipative stabilization. In particular, the two-photon coupling rate g2is therefore not limited by the amplitude of the two-photon pump: g2= — <p <pb, where Ej is the Josephson energy of the one or more Josephson junctions, <pais the zero-point fluctuation of the phase of the memory mode a, and <pbis the zero-point fluctuation of the phase of the buffer mode b.

[0111] (H) A resonant dissipative stabilization, for which the Applicant filed the European patent application EP 21306965.1 , with jump operator L2= K^O2— a2), where K2is the two-photon dissipation rate, a is the photon annihilation operator of the memory mode a and a is a complex number defining the cat qubit. This jump operator can be realized by coupling a lossy buffer mode b with dissipation rate Kb, and a three-wave mixing nonlinear element to the memory mode a which engineers the Hamiltonian = + h. c., where b is the photon annihilation operator of the buffer mode b provided the mode frequencies verify substantially 2fa= fband g2< Kbto which a drive of the buffer mode at frequency fbis added.

[0112] The present invention is generic and is not limited to a specific stabilization scheme. Indeed, one or more of the above schemes can be used in isolation or in combination to stabilize a two-legged or a multi-legged cat qubit.

[0113] As illustrated in Figure 1 , the quantum system 1 comprises a non-linear superconducting quantum circuit 3 and a command circuit 5. The non-linear superconducting quantum circuit 3 is arranged to make possible non-linear wave mixing (e.g., three- or four-wave mixing) between a first mode a and a second mode b. In the following, the first mode a is used as a memory hosting a cat qubit, while the second mode b is used as a buffer in between the cat qubit and the external environment.

[0114] The first mode a and the second mode b each correspond to natural resonant frequencies of the non-linear superconducting quantum circuit 3. Consequently, the first mode a and the second mode b each have a respective resonant frequency. The first mode a has a resonant frequency fa= and the second mode b has a resonant frequency fb= where o)aand cobare the respective angular frequencies of the first mode a and the second mode b.

[0115] By “having” a first mode and a second mode, it should be understood here that the non-linear superconducting quantum circuit 3 comprises components operating in a superconducting regime which host the modes independently of each other or concurrently. In other words, the first mode a and the second mode b may be hosted in different subsets of components of the superconducting circuit or on the same subset of components.

[0116] The memory mode a has a high-quality factor Qawhile the buffer mode b has a low-quality factor Qb. As will be appreciated, the quality factor (Q-factor) can be determined in various ways, for instance by: (a) spectroscopic linewidth measurement, wherein the quality factor is given by Q = / A / , wherein f is resonant frequency and A / is the spectroscopic linewidth; or (b) via a time-domain measurement wherein a tone is sent in, and a return signal is measured after a pre-determined time, wherein Q = f * T with T the characteristic decay time. Of course, the skilled person would be aware of various other methods for determining the quality factor of a particular mode.

[0117] The non-linear superconducting quantum circuit 3 is intended to be subject to electromagnetic radiation delivered by the command circuit 5 in order to engineer various non-linear interactions between the first mode a and the second mode b. The frequency of each electromagnetic radiation is tuned to select specific terms within the rotating wave approximation. Typically, for superconducting materials such as aluminium, the electromagnetic radiation used herein may be in the microwave regime.

[0118] The non-linear superconducting quantum circuit 3 comprises a non-linear element 7 and at least one resonant portion 9. The non-linear element 7 may be a four- wave mixing non-linear element, such as an ATS or Josephson junction suitable for parametric dissipative stabilization schemes such as scheme (A) above, or one or more Josephson junctions suitable for a DC dissipative stabilization scheme corresponding to scheme (G) above. The non-linear element 7 can also be a three-wave mixing non-linear element, for example a superconducting non-linear asymmetric inductive element (SNAIL) or other superconducting non-linear element in the case of a resonant dissipative stabilization which corresponds to stabilization scheme (H) above.

[0119] The resonant portion 9 is arranged to be connected to the non-linear element 7 to provide the non-linear superconducting quantum circuit 3 with the first mode a and the second mode b having respective resonant frequencies faand fb. More particularly, the first mode a and the second mode b “participate” in the non-linear element 7, which means that a portion or the entirety of the mode magnetic energy is stored in the nonlinear element 7. Such a participation can be quantified by the zero-point fluctuation of the superconducting phase across the non-linear element 7, noted <pafor the first mode a and (pbfor the second mode b.

[0120] In the schematic diagram of the quantum system 1 illustrated in Figure 1 , the non-linear superconducting quantum system 3 comprises only one resonant portion, i.e. the resonant portion 9. This may be achieved via design of the resonator (e.g. various capacitances and inductances as defined for instance by the resonator geometry) such that it hosts modes of different frequencies. The design may also result in the modes having different engineered quality factors.

[0121] However, in general it should be understood here that the non-linear superconducting quantum system 3 comprises at least one resonant portion, and typically two or more resonant portions to form the two electromagnetic modes, which act as a cat qubit “memory” mode and a buffer mode.

[0122] The first resonator potion may be provided by a 3D resonator, such as a cavity machined out of high purity aluminium (e.g. above 99.99% purity). Similarly, the second resonator portion may provided as an antenna-like cavity. Alternatively, the first and second resonator potions may be provided by a 2D resonator, such as a co-planar waveguide design. Some resonator portions may be 3D, some may be 2D.

[0123] Although electromagnetic resonators are convenient for design and control purposes, the resonator portions need not necessarily be electromagnetic resonators. In particular, for the first resonator portion which hosts the cat qubit, it could be a nanomechanical resonator, such as phononic-crystal-defect resonators (PCDRs), similar to those described in P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Péchai, W. Jiang, T. P. McKenna, J. D. Witmer, R. Van Laer, and A. H. Safavi-Naeini's work, "Resolving the energy levels of a nanomechanical oscillator," published in Nature 571 , 537 (2019). These PCDRs are periodically patterned suspended nanostructures that support localized acoustic resonances in the gigahertz range. They are made from a piezoelectric material like LiNbO3, enabling the coupling of these resonances to superconducting circuits with nearly the same efficiency as standard electromagnetic cavities. Specifically, quasi-one-dimensional PCDRs made from lithium niobate, a piezoelectric crystalline material, could be used, with modes localized within a volume less than 1 pm3 of a suspended nanostructure.

[0124] In such acoustic resonators, the cat qubit is encoded in the phonons of the resonator.

[0125] Other types of acoustic resonators, such as acoustic membranes, also exist. The primary requirement is an acoustic-to-electromagnetic coupler, to couple the acoustic first resonator portion to the non-linear element 7.

[0126] Furthermore, other types of resonators may be used, such as a magnetic resonator wherein the quanta of oscillations are magnons (collective spin excitations). An example could be a designed yttrium-ion-garnet (YIG) particle, as described in Kounalakis, Marios, Gerrit EW Bauer, and Yaroslav M. Blanter. "Analog quantum control of magnonic cat states on a chip by a superconducting qubit." Physical review letters 129.3 (2022): 037205. Again, there is a requirement to couple magnetic first resonator portion to the non-linear element 7. Yet another type of resonator suitable for hosting cat qubits could be semiconductor double quantum dots embedded in a cavity such as a split-ring resonator cavity as disclosed in Kozin, Valerii K., et al. "Quantum phase transitions and cat states in cavity-coupled quantum dots." Physical review research 6.3 (2024): 033188.

[0127] Thus, herein a cat qubit may be encoded in the bosons of the resonator. Herein, for the sake of simplification, the 2-to-1 boson exchange is a 2-to-1 photon exchange, however it will be appreciated that all reference to photons and photon exchange regarding the first mode which hosts the cat qubit can be replaced with bosons or boson exchange (e.g. acoustic phonons, or magnons, or indeed plasmons).

[0128] The command circuit 5 is arranged to deliver control signals, such as electromagnetic radiation, current biases, or DC voltage biases. To this end, as illustrated in figure 1 , the command circuit 5 comprises at least a first mode drive 11 and a second mode drive 13. For instance, the command circuit 5 may be arranged at least to drive the second mode b by delivering radiation at a frequency substantially equal to the second resonant frequency fbto the resonant portion 9.

[0129] By “a frequency substantially equal to”, it should be understood that, ideally, the frequency is exactly equal to the desired value. However, in practice, the frequency value deviates from the desired value, typically by 1 or even 5%, due to the inherent precision of the hardware used.

[0130] Moreover, as will be understood, the frequency of a mode may be shifted in use. For instance, under certain dynamical interactions such as application of an electromagnetic pump to the resonant portion 9, the frequencies of the first mode a and the second mode b may be effectively shifted (e.g. Stark shifted) due to the presence of a static potential or dynamical interaction. However, the skilled person would be well aware of characterising such effective frequencies, and for simplicity herein, by the first mode a and the second mode b “having” respective resonant frequencies fa and fb, it will be understood as static meaning “un-dressed” frequencies or shifted “dressed” frequencies when in the presence of a dynamical interaction, depending on the situation.

[0131] Figure 2 illustrates an embodiment in which the quantum system 1 is arranged to perform the parametric dissipative stabilization to stabilize a cat qubit, wherein the nonlinear element 7 is an ATS. An ATS is an inductive dipole element formed by a pair of Josephson junctions (indicated by the crosses and Ej label in Figure 2) being shunted by an inductance (indicated by the ELlabel in Figure 2). The inductance may be formed for instance from a linear array of Josephon junctions. The Josephson junctions are each arranged on a different superconducting path such that they each form a distinct loop with the inductance, such that the ATS can be viewed as two superconducting loops connected by the shared edge which comprises the inductance. The ATS is thus sensitive to the magnetic flux threaded through each of the two loops.

[0132] The at least one resonant portion 9 is coupled via a linear coupler 15 to the ATS 7 which acts as an inductive element, such that the first mode a and the second mode b participate in the ATS 7. Example of embodiments for the linear networks which form the at least one resonant portion are known. For instance, among these embodiments, the non-linear superconducting quantum circuit 3 can be a two-mode hybridized system (also known as "galvanic cat", for which the Applicant has filed patent application EP22306815.6 and EP22306816.4) for which two lumped modes couple strongly via the ATS.

[0133] It is known to the skilled person that an ATS can be used to engineer in general a 2-to-1 boson conversion, i.e. the first term a2of the jump operator 'norder to perform a parametric dissipative stabilization More particularly, a 2-to-1 photon conversion between the first mode a and the second mode b is obtained by parametrically pumping the ATS at the frequency fp= \2fa— fb\, as successfully shown by Lescanne, Raphaël, et al. "Exponential suppression of bit-flips in a qubit encoded in an oscillator." Nature Physics 16.5 (2020): 509-513. Advantageously, the pump frequency satisfies fp» g2to make this parametric pumping work as well as possible.

[0134] It is known to the skilled person that, in the case of the non-linear element 7 being an ATS 7, when biased at its flux working point (0 — it, namely 0 flux threaded through one of the loops and it flux in unit of the flux quantum threaded through the other of the lopps), the Hamiltonian of the ATS 7 has the following “sin-sin” form: where <p = <pa(a + a+) the total superconducting phase difference across the ATS 7, <pais the zero-point fluctuation of the phase of the first mode a across the ATS 7 and <pbis the zero-point fluctuation of the phase of the second mode b across the ATS 7, £) is the Josephson energy of the side junctions and ELis the inductive energy of a central inductance, <ps(t') corresponds to a common flux modulation of the two loops of the ATS 7 and corresponds to a differential flux modulation of the two loops of the ATS 7. The former can be implemented by delivering electromagnetic radiations through flux lines of the ATS 7 out of phase, the latter can be implemented by delivering electromagnetic radiations through the two flux lines of the ATS 7 in phase.

[0135] Parametric pumping of the ATS 7 is typically done by pumping the common flux as pumping the differential flux merely displaces the modes coupled to the ATS 7. As an example, by pumping the common flux at the frequency fp= \2fa— fb\, (ps(t) = e2p7lcos (2nfpt), the non-linear resonant part of the Hamiltonian writes in the rotating frame: HATS= + h. c.), which is typically the two-to-one photon exchange Hamiltonian needed to engineer the two-photon stabilization.

[0136] When an external DC magnetic field is set such that a 0 mod 2it magnetic flux threads one of the loop and a it mod 2ÎT flux threads the other loop, it is ensured that the ATS Hamiltonian has its “sin-sin” form. For clarity, the set-up applying the external DC magnetic field is not drawn on Figure 2 but can be applied via the two bottom mutual inductances of the ATS 7. A typical implementation consists in interleaving a bias-tee connected to a DC current source to input DC current into the system while letting the electromagnetic radiations go through. As will be described below, the components which generate the required magnetic fields are current-carrying flux bias lines close to the ATS 7.

[0137] An electromagnetic source 17 is set-up to modulate the common flux in the ATS 7. For this purpose, an electromagnetic network 19 is used to split the radiation emitted by the electromagnetic source 17 and apply it to each node of the ATS 7 with the correct phase. Alternatively, two different electromagnetic sources could be used, each being simply coupled to a single node of the ATS 7 and their relative phase and amplitude being set so as to achieve the desired flux modulation.

[0138] When the electromagnetic source 17 is set at the frequency \2fa— fb\, the nonlinear superconducting quantum circuit 3 performs the 2-to-1 photon conversion between the first mode a and the second mode b. To convert this 2-to-1 photon conversion into two-photon dissipation, the second mode b is selectively coupled to a load 21 via a linear coupler 23 and an electromagnetic filter 25 configured as a band pass filter with a frequency fb.

[0139] Alternatively, the electromagnetic filter 25 may be configured as a band stop filter at a frequency faand may be placed in between, on the one hand, the external environment and, on the other hand, the first mode a and the second mode b to isolate the first mode a and thus prevent the first mode a from suffering additional losses coming from unwanted coupling to the load 21.

[0140] Alternatively, it may be configured as a low-pass (respectively high-pass) filter if fa > fb (resP / b > fa)- 'nother embodiments, the electromagnetic filter 25 can be omitted when coupling between the load 21 and substantially only the second mode b can be established. The skilled person thus understands that the first mode a has a high-quality factor while the second mode b has a low-quality factor.

[0141] As previously explained, the second mode b is driven at its resonant frequency fb. This two-photon drive is performed by an electromagnetic source 13 set at frequency f -

[0142] In the above, the load 21 can be seen as part of the command circuit 5 of figure 2, while the linear coupler 23 and the electromagnetic filter 25 can be seen as part of the non-linear superconducting quantum circuit 3.

[0143] The command circuit 5 may further comprise an electromagnetic source 11 and a electromagnetic source 27. In addition to stabilizing the cat qubit, the command circuit 5 also enables measurement an observable of the cat qubit or application of a quantum gate to the cat qubit.

[0144] The electromagnetic source 11 may be arranged to drive the first mode a by delivering electromagnetic radiation at the frequency fato the linear electromagnetic network b / a. Such a drive of the first mode a causes the non-linear superconducting quantum circuit 3 to engineer a Hamiltonian Hz expressed = eza + h. c., where the complex rate ezresults from the amplitude and phase of the drive of the first mode a. An additional electromagnetic source may be set-up, in addition to the electromagnetic source 17, to contribute to the modulation of the common flux in the ATS 7. The additional electromagnetic source may be arranged to deliver electromagnetic radiation for causing the non-linear superconducting quantum circuit 3 to engineer a Hamiltonian. For instance, such a Hamiltonian may yield a so-called longitudinal coupling term between the first mode a and the second mode b, which may be used to apply a Z gate.

[0145] Finally, the electromagnetic source 27 is set-up to modulate the differential flux in the ATS 7 through the electromagnetic network 19. The electromagnetic source 27 can be used to reduce spurious Hamiltonian terms induced by the additional electromagnetic source. It is to be noted that the electromagnetic network 19 is used for convenience but can be omitted and the electromagnetic source 27 and any additional electromagnetic source could be applied directly to both nodes of the ATS 7, their relative phase and amplitude being set so as to achieve the desired flux modulation.

[0146] For the sake of completeness, it may also be noted that the electromagnetic source 27 can be used, instead of the electromagnetic source 13, to deliver electromagnetic radiation at a frequency fbto the linear electromagnetic network b / a to drive the second mode b.

[0147] Typically, the circuit has flux lines through which radiation can be delivered to provide (and optionally modulate) the common flux (herein also referred to as <psor “sigma flux”) and provide (and optionally modulate) the differential flux (herein also referred to as <pAor “delta flux”). These are generally referred to as the two bias modes. As such, the flux lines may deliver both a DC-bias (setting the working point described above by applying DC current to thread the ATS loops with the relevant magnetic fluxes) and an RF-bias (superposed modulation to permit the time-dependent magnetic flux terms to “pump” the ATS as discussed above).

[0148] Figure 3 illustrates an embodiment in which the quantum system 1 is arranged to perform the DC dissipative stabilization to stabilize a cat qubit.

[0149] The non-linear element 7 is a set of one or more Josephson junctions. In the example of Figure 3, the non-linear element 7 includes only one Josephson junction 49 for the sake of simplification.

[0150] Moreover, the at least resonant portion 9 includes a first resonant portion 29 and a second resonant portion 31 which confer the first mode a and the second mode b respectively to the non-linear superconducting quantum circuit 3. Compared to the quantum system 1 of Figure 2, the same reference signs have been kept for ease of understanding.

[0151] The first and second resonant portions 29, 31 take the form of an electromagnetic network which can be compared to that of the quantum system 1 illustrated in Figure 2 and is such that, when coupled via the linear coupler 15 to the non-linear element 7, the non-linear superconducting quantum circuit 3 has the first mode a and the second mode b at respective resonant frequencies faand fbwhich participate in the non-linear element 7.

[0152] Similarly to the quantum system 1 of Figure 2, the electromagnetic radiation source 11 - which can be a microwave radiation source - is arranged to deliver electromagnetic radiation at the resonant frequency fato the first resonant portion 29 to drive the first mode a; and the electromagnetic radiation source 13 - which can be a microwave radiation source - is arranged to deliver electromagnetic radiation at the resonant frequency fbto the second resonant portion 31 to drive the second mode b.

[0153] The second mode b is selectively coupled to the load 21 via the linear coupler 23 and the electromagnetic filter 25 - which can be an electromagnetic filter - configured as a band pass filter with a frequency fb. Again, the electromagnetic filter 25 may be configured in a similar manner as described above for Figure 2.

[0154] Compared to the quantum system 1 of figure 2, the command circuit 5 does not comprise anything functionally comparable to the two-photon pump 17, the electromagnetic network 19 nor electromagnetic radiation source 27. Instead, the command circuit 5 comprises the DC voltage source 51 and the electromagnetic radiation source 53 - which can be a microwave radiation source - to perform the 2-to- 1 photon exchange between the first mode a and the second mode b.

[0155] The DC voltage source 51 is arranged to bias the non-linear element 7 with the delivered DC voltage having a value inducing an electromagnetic oscillation having a frequency substantially equal to \2fa— fb\ in the non-linear superconducting quantum circuit 3.

[0156] The electromagnetic radiation source 53 - which can be a microwave radiation source - is arranged to deliver electromagnetic radiation at a frequency substantially equal to the frequency of the electromagnetic oscillation induced by the DC voltage source 51 to cause injection locking thereof. Such an injection locking is particularly useful when the DC voltage source 51 is not an ideal DC voltage source, i.e. when the DC voltage source 51 induces voltage noise. The DC dissipative stabilization is only one of the stabilization schemes concerned by the invention; consequently, the principle of the DC dissipative stabilization is explained briefly below.

[0157] The DC voltage source 51 delivers a constant voltage VDCacross the Josephson junction 49. The phase of the Josephson junction 49 is denoted <pj. At this stage, for simplicity, we can consider that the DC voltage source 13 is ideal, which means that the operation of the DC voltage source 51 does not generate voltage noise. An ideal DC voltage source can itself be made with a Josephson junction.

[0158] The Cooper pairs - also known as BCS pairs (Bardeen-Cooper-Schrieffer pairs) -, which are pairs of electrons bound together at low temperatures, travel the circuit with an energy equal to 2eVDC, where e is the elementary charge. In the environment set up by the Josephson junction 49, such an energy 2eVDCis converted into photons with an angular frequency wDCequal to ^eV°c / fL- It can thus be considered that the Josephson junction 49 acts as a “voltage-to-frequency converter”. Consequently, the equation of 2e motion can be described with the following differential equation: <pj = —VDC = <^DC-

[0159] Since the DC voltage VDCis constant over the time, the time evolution of the phase (pj can be expressed as follows: cpj(t) = d)DCt + <Pj0, where: <p}ois the initial value of the phase <pj, <pJo- (pj(O). The phase (pj thus varies linearly with time.

[0160] Nonetheless, the non-linear superconducting quantum circuit 3 of the quantum system 1 further comprises, in addition to the Josephson junction 49, the first resonant portion 27 corresponding to the memory mode a having a zero-point phase fluctuation <paas well as the second resonant portion 29 corresponding to the buffer mode b having a zero-point phase fluctuation <pb.

[0161] The value VDCof the DC voltage induces an electromagnetic oscillation having 2e an angular frequency roDCsuch that a)DC= -^VDC- Yet, since the DC voltage source 13 is intended to perform the 2-to-1 photon exchange which is for instance obtained with parametric pumping at a frequency \2fa— fb\ in the embodiment of Figure 2, the value VDCmust be set such that the frequency fDCis equal to \2fa— fb\. Therefore: fDC= a)DC / 2n = (l / 2it)(2e / K)VDC. Then, since the reduced Planck constant h is defined as h = h / 2n, we have fDC= (2e / h)7DC= \2fa- fb\. Thus, VDC= \2fa- fb\(h / 2e).

[0162] The DC dissipative stabilization allows to obtain a two-photon coupling rate which is not determined - and thus limited - by the amplitude of any parametric pump: g2= (£} / 4)^a^h, where Ej is the Josephson energy of the Josephson junction 49. In the case where the non-linear element 7 includes N Josephson junctions, where N is a strictly positive natural number, the value VDCof the DC voltage in order to induce an electromagnetic oscillation having a frequency substantially equal to \2fa— fb\ in the non-linear superconducting quantum circuit 3 is the following: VDC= N\2fa— fb\(h / 2e).

[0163] Figure 4A illustrates an embodiment in which the quantum system 1 is arranged to perform resonant dissipative stabilization to stabilize a cat qubit, i.e. wherein the nonlinear element 7 is a three-wave mixing non-linear element 7. Thus, in comparison with Figure 1 , the non-linear element 7 and the at least one resonant portion 9 are brought together to form part of the non-linear superconducting quantum circuit 3, which explains why the reference signs “7” and “9” are absent from Figure 4A. Here, the non-linear superconducting quantum circuit 3 is arranged to perform intrinsically the 2-to-1 photon exchange - symbolized here by a back-and-forth single arrow 57 and double arrows 59 - between the first mode a and the second mode b.

[0164] Like reference numerals indicate features which are the same as described above in Figures 2 or 3. For instance, the electromagnetic radiation sources 11 and 13 are configured in the same manner as in Figures 2 or 3 to respectively drive the first mode a and the second mode b.

[0165] The command circuit 5 further comprises a current source 61 or applying a current to the non-linear element 7 and the at least one resonant portion 9. The current source 61 may be connected by wires (i.e. galvanically) to the non-linear element 7 and the at least one resonant portion 9, or may be a flux line for flux biasing a loop (e.g. of the non-linear element 7 as in Figure 4B) to thereby induce a current flowing around the loop due to the magnetic flux therethrough. The current source 61 is arranged to deliver current which flows through one or more components of the non-linear superconducting quantum circuit 3. The current source 61 is configured to both allow three-wave mixing interaction and to tune the frequency matching condition 2fa= fb, in particular by inducing a particular current flow across the three-wave mixing non-linear element 7 which enables tuning of the resonant frequencies of first mode a and the second mode b as they participate in the non-linear element 7.

[0166] For instance, Figure 4B shows a possible circuit to realize the three-wave mixing non-linear element 7, which in this embodiment is formed by at least one loop 63 including a first Josephson junction 65, a central inductive element 67 and a second Josephson junction 69. Alternative embodiments of the three-wave mixing non-linear element 7 are described in EP21306965.1 filed by the Applicant. Returning to Figures 4A and 4B, when a predetermined current of a constant intensity lcis applied by the current source 61 , the resonant frequency fbis substantially equal to twice the resonant frequency fa. Said otherwise, what is important is that the superconducting circuit 3 has these structural features so linked, and that (regardless of the specific shape of the components) the resonance matching condition is satisfied when the constant current is applied. This current can be determined by the skilled person any number of known ways, for instance as described in Marquet, Antoine, et al. "Autoparametric resonance extending the bit-flip time of a cat qubit up to 0.3 s." Physical Review X 14.2 (2024): 021019, or in Marquet, Antoine, et al. "Harnessing two-photon dissipation for enhanced quantum measurement and control." arXiv preprint arXiv:2403.07744 (2024), or in European patent application EP 21306965.1. In particular, what is essential is that the non-linear element comprises at least one loop 63 including at least one Josephson junction therein, and that the resonant modes a and b at least partially participate in the loop.

[0167] The circuit shown in Figure 4B is particularly configured to discriminate symmetrically the memory mode a and the buffer mode b. The high symmetry of such a circuit achieves an improved quality of the 2-to-1 photon exchange, as well as to make the function of filter 25 integral to the flux line used to flux bias the loop 63 (which is also the second mode electromagnetic source 13). The relative position with respect to the loop 63, and the geometry, of the flux line which acts as the second mode electromagnetic source 13 enables coupling to the second mode whilst inherently filtering the first mode to prevent or substantially minimize coupling of the first mode a to the load (environment) 21 .

[0168] The central inductive element 67 can be an inductance, a single Josephson junction or an array of Josephson junctions. The central inductive element 67 may thus be arranged between the first Josephson junction 65 and the second Josephson junction 69 as a loop in series. The arrangement in series may comprise a first inner node connecting a pole of the first Josephson junction 65 with a pole of the central inductive element 67. The arrangement in series may also comprise a second inner node connecting a pole of the second Josephson junction 69 with another pole of the central inductive element 67. The arrangement in series may also comprise a closed-loop node connecting another pole of the first Josephson junction 65 with another pole of the second Josephson junction 69.

[0169] The at least one loop 63 may be connected to a common ground via the closed- loop node. The circuit may also comprise a first capacitive element 71 and a second capacitive element 73. The first capacitive element 71 may be connected in parallel with the first Josephson junction 65 between the common ground and the first inner node of the loop. The second capacitive element 73 may be connected in parallel with the second Josephson junction 69 between the common ground and the second inner node of the loop.

[0170] The first Josephson junction 65 and the second Josephson junction 69 are substantially identical and the capacitive elements 71 and 73 are also substantially identical. Hence, the symmetry of the circuit implies that the memory mode a is the symmetric superposition of the two resonators (as shown by the full arrows) and the buffer mode b is the anti-symmetric superposition of the two resonators (as shown by the dashed arrows). It can be noticed that only the buffer mode b has a contribution across the central inductive element 67 which is advantageously used to preferentially couple the external environment to this buffer mode b while isolating the memory mode a from the external environment via the design of the electromagnetic source 13 as described above.

[0171] As will be appreciated, for any dissipative stabilization schemes suitable for realizing cat qubits, such as those described above in reference to Figures 1-4, a key metric is the so-called two-photon dissipation rate K2which is a measure of the rate at which pairs of photons are exchanged with the environment. This rate governs the relevant timescales, or instance dictates the time periods required to stabilize a cat state of a given fidelity, or how fast single- or multi-qubit gates should be applied.

[0172] There are various ways to measure K2of a dissipatively stabilized cat qubit. For instance, the two-photon interaction strength g2and the second mode b decay rate Kb(also known as linewidth) can be experimentally measured, and K2is then given by K2= 4^2 b ■

[0173] The second mode b decay rate Kbcan be directly measured by performing a standard spectroscopy of the second mode b using any known technique (e.g., done by sending in a signal having the second resonant frequency fband measuring the return signal).

[0174] The two-photon interaction strength g2may be measured by the following. (I) First, prepare a coherent state |a) in the first mode a (or a low-number Fock state, such as Fock 1 or Fock 2). (II) Turn on the two-photon interaction having strength g2(e.g. turning on the parametric pump in the context of Figure 2 or the DC voltage bias in the context on Figure 3), for a given time period. (Ill) At the end of the time period, measure the populations in Fock states 0, 1 , and 2 (or alternatively, the size of the coherent state, namely photon number (IV) Iterate operations (l)-(lll) over different time periods to create a dataset of measured populations vs time periods . (V) Fit the data set with a model of the system given by a Lindblad master equation so as to extract g2. The skilled person would be well aware of such exemplary ways determine g2, for instance as explicitly described in Section 4, C, 2. “Two-photon exchange rate calibration" in Réglade, Ulysse, et al. "Quantum control of a cat qubit with bit-flip times exceeding ten seconds." Nature (2024): 1-6.

[0175] Alternatively, the two-photon dissipation rate K2can be measured directly. To this end, one starts from vacuum, starts the cat stabilization to prepare an even Cat state, then apply a Zeno drive (drive resonant with the first mode, orthogonal to the cat axis) of varying amplitude ezand time period duration, and measures the parity of the resulting state. For each Zeno drive amplitude ezthere is a measured oscillation frequency between positive and negative parity values (as a function of time period duration of the Zeno drive). A decay rate rzof the oscillations can be extracted for each Zeno drive amplitude ez. The measured decay rate of the oscillations rzis then fitted with the following model rz(ez) = 2|a|2rc1+ 2ez / la2fK2. The single-photon loss from the first mode K S independent from the Zeno drive and thus easily identifiable as the offset of the curve of the fit. Accordingly, K2is extracted via fitting the model.

[0176] Stabilizer measurements

[0177] Due to the noise bias of stabilized cat qubits, quantum error correction becomes similar in complexity to classical error correction. For instance, a certain level of quantum error correction can be achieved by using a phase-flip correcting repetition code.

[0178] Specifically, to encode a logical qubit state \i )L= c+| +)L+ c_| — )L(with | +)L= (|0)L+ |1)L) / 2 and | -)L= (|0)L- |1)L) / V2 with c+i_ the corresponding complex probability amplitudes, where |0)Land |1)Lare the canonical computational states of the logical qubits) a phase-flip repetition code can duplicate the state across several physical qubits, for instance in 3-qubit repetition code we may wish to encode the logical state are the code words respectively for logical “+” and

[0179] During computation of an algorithm or information storage, a phase-flip error may affect some of the physical qubits, e.g. to change one of the qubits encoded in the state | + + +) to | -I - F). To detect such an error, the repetition code measures the parity of neighbouring data cat qubits using a so-called stabilizer measurement (also known as parity checks). A parity measurement may be performed on a data cat qubit by entangling it with an ancilla qubit (via an appropriate two-qubit gates such as a CNOT gate) and measuring the parity of the ancilla to deduce the parity of the data cat qubit. In this example of the 3-qubit repetition code, we measure in the X-basis the parity of the first and second qubits, and the second and third qubits: Sr= Xr0X2and S2= X2® X3, where X is the Pauli-X operator. The parity check returns a +1 if the two qubits involved have the same parity (e.g. both | — ) or both | +)) or a -1 if they have differing parity (e.g. one is | +) and the other is | — )).

[0180] Based on the result of the parity check, we can identify which qubit has succumbed to a phase-flip error. For instance, if both S and S2are +1 then no error is detected, if S±is -1 and S2is +1 the error is on the first qubit, if Sris +1 and S2is -1 the error is on the third qubit, whereas if both and S2return -1 then the error is on the second qubit. Once an erroneous qubit is identified, a corrective operation can be applied to restore the correct logical state (for instance, in our example simply applying the appropriate Z-gate on the offending qubit to flip the phase back). Decoding may be applied after error correction, in the above example simply by performing a majority vote on the physical qubits in the X-basis.

[0181] It will also be appreciated that other classical error correction codes could be used, such as so-called LDPC classical error correcting codes which use stabilizers of weight greater than 2 (which means the parity check involves more than 2 data cat qubits) - as detailed in Ruiz, Diego, et al. "LDPC-cat codes for low-overhead quantum computing in 2D." arXiv preprint arXiv:2401.09541 (2024).

[0182] By “classical” it will be understood to mean that the error correcting codes work to correct only one error (e.g. phase-flip errors as discussed above), as opposed to correcting both types of errors as done in a surface-code or other variant of two- dimensional error correcting codes. Thus, the terminology “classical error-correcting code” refers to a code which is one-dimensional, in that it corrects only a single error (in this context correcting phase-flip errors occurring on data cat qubits). Of course, such a classical error-correcting code may itself be concatenated within a further error correcting code, however at the level of the classical error-correcting code itself, only a single error can be corrected.

[0183] The parity checks are repeated multiple times over the course of a computation or information storage, which is crucial to maintain the integrity of the encoded quantum information over time (by detecting and correcting accumulated errors, and reducing the probability of error propagation). As mentioned, due to the principles and restraints of quantum mechanics, the aforementioned parity checks are typically performed using a so-called ancilla qubit (i.e. used in the code, but not for hosting information).

[0184] Figure 5 shows an exemplary quantum circuit diagram of the operations required to perform a parity check. The middle line denotes operations involving the ancilla qubit 501 , whereas the top and bottom lines denote operations involving data cat qubits 503, wherein the qubits are connected such that the ancilla qubit is able to be entangled with each of the two data cat qubits via respective CNOT gates, as discussed in more detail below.

[0185] The ancilla 501 is prepared in a known state (typically for an exemplary phaseflip correcting code this could be a | — ) or | +)), and subsequently entangled with the data cat qubits 503 whose parity you want to check (the data cat qubits of the so-called “stabilizer”). The entangling may involve a quantum CNOT gate between the ancilla qubit 501 as the control and the first data cat qubit 503 as the target, and another quantum CNOT gate between the ancilla qubit 501 as the control and the second data cat qubit 503 as the target, and similarly further CNOT gates if the stabilizer is comprised of more than two data cat qubits 503.

[0186] The ancilla qubit 501 is then measured in the X-basis, and the result of the measurement is attributed as the ±1 of the parity check. The measurement outcome of the ancilla qubit 501 thus provides information about the parity of the data cat qubits without directly measuring them, thereby providing an indirect method for detecting errors while preserving the coherence of the data cat qubits.

[0187] More generally, the ancilla qubit 501 could be prepared in a state defined on the equator of the Bloch sphere (e.g. 1 is a phase comprised in the range [0;2TT]) SO long as the ancilla qubit 501 is measured with an observable whose set of eigenstates contains the state the ancilla was prepared in.

[0188] CNOT gate implementations

[0189] A CNOT gate requires coupling the control qubit (here, the ancilla 501 ) to the target cat qubit (here, the data cat qubit 503), such that, under a particular dynamical interaction, the state of the target cat qubit is conditionally altered depending on the state of the control qubit.

[0190] The action of a CNOT gate (also known as a CX or controlled-X gate) can be understood from the following truth table, where the first bit is the state of the control qubit and the second bit is the state of the target qubit. Should the first bit be 0, the gate keeps both bits unchanged. Should the second bit be 1 , the gate keeps the first bit unchanged and performs an X gate on the second bit, i.e. changes the second bit to its opposite:

[0191] A natural representation of the CNOT is to write that the target stays idle when the control qubit is in the coherent state |cr)cand rotates by it (in the complex plane) when the control qubit is in the coherent state | — a)c.

[0192] Otherwise, it is convenient to adopt a symmetrized picture by performing a frame rotation of the target qubit so that it always rotates, but by +TT / 2 or — TT / 2 in the complex plane around the origin depending on the state of the control qubit, as shown in Figure 6. In this picture, the global phase of the target cat basis has been rotated by n / 2 after the CNOT but can be taken into account in software. Accordingly, the corresponding truth table is the following:

[0193] For the avoidance of doubt, the actual a term in |a)cand |a)rmay be different.

[0194] Figure 7 shows a physical system on which the CNOT may be implemented. Here, the quantum system 30 comprises a target cat qubit device 300 (which is similar to that of Figure 2) and a control qubit device 302 connected by a linear electromagnetic coupler 304. In the example described herein, the target cat qubit device 300 comprises a non-linear superconducting circuit 306 to which are connected several microwave sources 310, 311 , 316 and a load 314.

[0195] In a similar manner as described above in relation to Figure 2, the non-linear superconducting circuit 306 comprises at least one resonant portion (denoted by reference numerals 320 and 322 to indicate hosting of the two modes) such that when coupled via a linear coupler 309 to an ATS 308 which acts as an inductive element the non-linear superconducting circuit comprises at least 2 normal modes (or eigenmodes) a and b at frequency faand fbwhich participate in the ATS. As above, this participation means that a portion or the entirety of the mode magnetic energy is stored in the ATS. This participation can be quantified by the zero-point fluctuation of the superconducting phase across the ATS, noted cpafor first mode a and (pbfor second mode b.

[0196] In the context of the present embodiment, the physical realization of a CNOT gates between a control qubit with annihilation operator q and a stabilized cat qubit with annihilation operator a usually relies on the use of the following two ingredients.

[0197] (1 ) The confinement of the control qubit such that a microwave drive at its resonant frequency results in a Rabi oscillation. This is typically native in two-level system qubits (e.g. such as transmons) and engineered via parametric interactions for cat qubits.

[0198] (2) The addition in the circuit of a ‘CNOT’ Hamiltonian or ‘longitudinal’ Hamiltonian with the formula Hcx / h = gcxq + q^a^a — a2) where gcxis the amplitude of the Hamiltonian (it is chosen real without loss of generality). This longitudinal coupling can be seen as a drive on the control qubit (first factor) which amplitude depends on the photon number of the target cat qubit (second factor). For this Hamiltonian to be effective on the target cat qubit, one also needs to turn-off the confinement on the target cat qubit, hence the target cat qubit confinement strength should be controllable.

[0199] To engineer the CNOT Hamiltonian between the control qubit and the target cat qubit one needs to: (i) couple the control qubit to the ATS such that the phase difference across the ATS writes (p + (pb(b + &+) + (pq[q + q+) ; and (ii) pump the common flux at the control qubit frequency fq, <px(t = eCxC0Sq ) withecx the amplitude of the pump drive.

[0200] In the rotating frame, the parametric part of the Hamiltonian writes HATS= ~Ejecx<Pq(ci + This Hamiltonian can be written to highlight the desired dynamics Since the second mode b (the "buffer”) is a lossy mode coupled to a cold environment, one can further assume that b^b = 0, such that the engineered Hamiltonian is

[0201] This Hamiltonian is close to the CNOT Hamiltonian except for 2 additional terms. The first corresponds to a linear drive with strength Ej6cx(pq(1 - ^a2) / ^. This linear drive can readily be compensated by sending a counter drive directly on the control qubit with the correct relative phase and amplitude that can both be tuned experimentally. The Applicant also found that the accuracy and scope of the compensation can be greatly increased by pumping the differential flux of the ATS with the correct phase and amplitude which writes, at first order in ecx, <pA(t) = — (pz(t)(l — (Paa2). Although the EL amplitude and phase of the drive can be computed analytically, it is fine-tuned experimentally by ensuring the control qubit remains undergo no drive in the right circumstances. Experimentally, this compensation is much more appropriate than the counter drive on the control qubit because, it directly compensates the spurious linear drive where it originates from (i.e. , at the ATS) and does not only try to compensate its main consequences (i.e., the displacement of the control qubit).

[0202] The second term cannot typically be fully compensated in a simple manner in view of the design choices made for the present embodiment. Instead, the system may be defined such that the amplitude of this term is much smaller than the amplitude of the CNOT Hamiltonian. In other words, Ejecx^- < E}ecx(pq(pa, which simplifies into (pq / 2 « <Pa- Typically, the Applicant has found that <pq< <pa / is sufficient minimize the detrimental impacts of this second term.

[0203] In summary, provided these two conditions (compensation and smaller amplitude for the second term) are met, the ATS Hamiltonian HATScan be engineered such that it is close enough to the perfect CNOT Hamiltonian.

[0204] For clarity, the set-up applying the external DC magnetic field is not drawn on Figure 7, but can be applied via the 2 bottom mutual inductances of the ATS. A typical implementation consists in interleaving a bias-tee connected to a DC current source to input DC current into the system while letting the electromagnetic radiations go through. The electromagnetic sources 310 and 311 are set-up to modulate respectively the common and differential flux in the ATS which is required to activate parametric interactions. To clearly distinguish the roles of the two sources in the Hamiltonian, an electromagnetic network 324 which applies the correct phase offset is provided in the schematic. Alternatively, each electromagnetic source can be simply coupled to a single node of the ATS and their relative phase and amplitude can be set so as to get the desired flux modulation. In that case, to modulate the common flux the two sources need to address the circuit out of phase and to modulate the differential flux, the two sources need to address the circuit in phase. The 2-to-1 photon conversion necessary to dissipatively stabilize a cat qubit is performed by setting the frequency of the electromagnetic source 310 to fp= \2fa— fb\, and stabilization is further realised by driving the electromagnetic source 316 at the second mode b frequency fb.

[0205] When the CNOT gate is not performed, i.e., in a so-called "idle mode", the command circuit controlling the entire system is configured to perform the data (target) cat qubit stabilization exclusively.

[0206] The control qubit and the configuration required to perform the CNOT gate according to an embodiment will now be described.

[0207] In the example of Figure 7, the control qubit device 302 comprises a mode q 305 with resonant frequency fqwhich hosts the control qubit. In various embodiments, the control qubit device 302 can be any superconducting qubit hosted in a qubit-hosting structure such as a transmon qubit, a flux-qubit or a fluxonium qubit or any bosonic qubit encoded in a resonator such as a Kerr cat qubit (detuned or not), another dissipative stabilized cat qubit device (squeezed or not), or a cat qubit confined via a two-photon exchange Hamiltonian.

[0208] As described above, the control qubit 305 is coupled to the target cat qubit device 300 via a small linear coupler 304. The linear coupler 304 is arranged such that the control qubit 305 slightly hybridizes with the cat qubit device 300 which leads to a small participation of the control qubit in the cat qubit device ATS 308. This participation is denoted (pq. As described above with respect to the second spurious member of the engineered Hamiltonian which is not compensated, the fact that this participation remains small compared to the participation <paof the target cat qubit mode is critical to accurately implement the CNOT Hamiltonian. In various embodiments, coupler 304 can be capacitive, inductive, galvanic or mediated via a resonating bus coupler or an additional linear electromagnetic network. In the embodiment shown in Figure 7, the control qubit device 302 is coupled to the first mode a 320. In other embodiments, control qubit device 302 can be coupled to the second mode b 322. This coupling location is not critical as mode a and b are typically delocalized in the electromagnetic network comprising the at least one resonant portions a / b 320-322 and coupling to a specific location does not necessarily mean coupling to a specific mode unless the linear microwave network and the ATS are specifically designed to do so.

[0209] The control qubit device 302 also comprises a control signal source 303 which is coupled to the control qubit 305 in order to have the ability to drive it. For instance, source 303 may be an electromagnetic radiation source, which may be used to compensate the linear drive arising from the CNOT Hamiltonian engineering. As explained above, a further limitation on the control qubit device 302 is that the non-linear superconducting circuit 306, the first mode (hereinafter cat qubit mode) a participates strongly in the ATS 308 as compared to the control qubit mode 305. When the control qubit mode q participates into the ATS via a small linear coupling with the cat qubit mode a in order to perform the CNOT gate - typically, capacitive coupling with capacitance value small compared to the either mode capacitance, inductive coupling with inductance value small compared to the either mode inductance or coupling mediated by a detuned bus resonator -, this coupling is moderate in general. More precisely, the two coupled modes (here a and q, alternatively b and q) detuning (i.e. , the frequency difference A = \fa— fq\) also plays a role. Indeed, if the two modes have the same resonant frequency, any slight linear coupling would lead to full hybridization and to <pa« <pq. However, with typical detunings 21 / 2 it greater than a few tens of MHz, the participation asymmetry is achieved with standard linear couplings.

[0210] As explained earlier, the weaker coupling of the control qubit q to the ATS 308 (compared to the target cat qubit mode a coupling to the ATS 308) is key. If the linear coupling is characterized by a strength g as known in the field, then one also needs to ensure the detuning A between the control qubit mode q and the mode it is coupled to (a, 320 or b,322) is such that g < 21. Although this provides design rules, a full electromagnetic (e.g. microwave) simulation or circuit diagonalization of the circuit layout is required to precisely compute the values of (Pa tPb’ Vq- This feature helps to ensure that the CNOT gate can be performed accurately and that the circuit uses a single ATS in its elements for the CNOT gate, and that parametric pumping of the ATS of the target cat qubit has a single purpose at all times.

[0211] Figure 8 shows exemplary signal timings for operations 800, 810 and 820. During idle mode, the data cat qubit is stabilized with two-photon dissipation K2. When the gate starts, this dissipation is turned off such that the CNOT Hamiltonian can be effective. Finally, after a time TCX= n / (4\ac\gcx) (such that the target rotates by the requisite amount, e.g. as described in the above truth-tables) the CNOT Hamiltonian is turned off and the two-photon dissipation is turned back on again. At no point in time does the ATS serve two purposes at once, which guarantees experimental robustness. The limitations for the CNOT pulse strength and shape (dotted or dashed lines) will now be explained.

[0212] The CNOT Hamiltonian effectively acts as a linear drive on the control qubit q which strength depends on the number of photons in the target cat qubit a. The nature of the control qubit q sets upper bounds on the maximal effective drive strength gcxa.

[0213] There exist two main cases. In the first case, the control qubit q is defined by a real or Hamiltonian gap - for a two-level system this gap is the anharmonicity |ro12— | where a)qis the qubit frequency or the frequency of ground to first excited state transition and c12is the frequency of first excited to second excited state transition (this may for instance be a transmon). In that case, the adiabatic theorem applies and states that the qubit stays exponentially confined despite the action of the effective drive provided gcxa < For instance, if the ancilla qubit is a cat qubit with amplitude acconfined by a Kerr Hamiltonian (scheme (B) above) the condition writes gcxa < Ka . If it is confined by a detuned Kerr Hamiltonian (scheme (C) above) the condition writes it is confined by a TPE Hamiltonian (scheme (D) above), the condition writes gcxa < acg2.

[0214] In the second case, the control qubit q is defined by an imaginary gap or equivalently is stabilized by dissipation. In that case, the adiabatic theorem does not apply and the gate strength has to be much smaller than the imaginary gap in order to avoid undesired decoherence of the ancilla qubit during the gate. In practice, for an ancilla cat qubit stabilized by two-photon dissipation with rate K2(a) the condition writes gcxa « K2«C , which applies also to schemes (G) and (H) above. For a squeezed ancilla cat qubit (scheme (E) above), the condition writes gcxa « K2a.ce2rwhere r is the squeezing parameter, and a similar condition may be derived for the “moon cat” stabilization (scheme (F) above).

[0215] On top of the gate strength requirement, the nature of the control qubit confinement sets constraints on how the gate should be applied. In the case of a dissipative ancilla qubit, the CNOT Hamiltonian can be turned on instantaneously as shown by the dashed lines of the above figure as soon as the data cat qubit confinement is turned off (solid line K2(t)). In the case of a Hamiltonian ancilla qubit, the CNOT Hamiltonian may be turned on smoothly such that the spectral content of the pulse gCx(t) (dotted lines) does not contain frequency components above the gap. Typically, a gaussian pulse can be used. On the above figure, a cosine shape is used for its finite temporal envelope. In the case of a time dependent gcx, the pulse amplitude should be such that J 4agcx(t)dt = n.

[0216] It is noted that the CNOT may also be implemented using another four-wave mixing non-linear element in place of the ATS 308. What is important is that: (i) the control qubit mode q at least partially participates in the non-linear element 7; and (ii) the non-linear element 7 is able to provide an oscillation at the chip-level having frequency of the control qubit fq. Alternatively, an additional four-wave mixing non-linear element may be used, in particular for coupling a resonantly stabilized data cat qubit (e.g., as described above in relation to Figures 4A-4B) to control qubit. In such a scenario, the target cat qubit device 300 in Figure 7 may be replaced with a device for resonantly stabilizing a cat qubit as described in relation to Figure 4A, and the linear coupler 304 shown in Figure 7 may be replaced with a four-wave mixing nonlinear element (such as an ATS, or a Josephson junction).

[0217] Here, the CNOT gate can similarly be implemented by: (i) turning off the stabilization of the target data cat qubit (e.g. by turning off the buffer drive and shifting in flux away from the flux working-point where the modes are at the resonant condition); (ii) driving the four-wave mixing nonlinear element at a frequency equal to the control qubit frequency (e.g. for similar gate time TCX= n / (4\ac\gcx) as described above; and (iii) turning back on the stabilization of the target data cat qubit, but with a buffer drive phase shifted by n such that the stabilization sub-space is wherein again the stabilization has to be kept during a time larger than where x2,t is the two- photon dissipation rate of the target cat such that its memory state is projected onto the correct cat-qubit manifold.

[0218] Figure 9A shows the pulses for a CNOT gate, wherein both the control and the target as parametric dissipatively stabilized cat qubits. Here the instance the four-wave mixing non-linear element (4WM) may be an ATS as in Figures 2 and 7, or a DC-biased Josephson junction as in Figure 3). For instance, the 4WM may be an ATS for both the control and the target, an ATS for the control and a DC-biased Josephson junction for the target, a DC-biased Josephson junction for the target for the control and an ATS for the target, or a DC-biased Josephson junction for the target for both the control and the target.

[0219] Figure 9B shows the pulses for a CNOT gate, wherein the target is parametric dissipatively stabilized cat qubits as discussed regarding Figure 9A (i.e. the 4WM may be an ATS as in Figures 2 and 7, or a DC-biased Josephson junction as in Figure 3), whereas the control is a resonantly stabilized cat qubit. For instance, the control cat qubit may be realized as discussed above in relation to Figures 4A and 4B.

[0220] Figure 10 shows an example circuit coupling 4 dissipatively stabilized cat qubits (for simplicity here, each is shown parametrically stabilized by an ATS, although other 4WM nonlinear elements may be used instead of the ATS, such as a parametrically pumped JJ or a DC-biased JJ etc.).

[0221] In this non-limiting exemplary implementation 4 dissipative cat qubits 1101 , 1102, 1103 and 1104 with similar parameters (except for their frequencies which are slightly detuned from one another in order to selectively address each mode with the radiation and not having the modes delocalized over the entire circuit) are laid on a superconducting chip in circular pattern with nearest neighbor connection.

[0222] This is a partial schematic version of the practical implementation. In this schematic, the choice has been made to make 1101 and 1103 the two data cat qubits and 1102 or 1104 the ancilla cat qubit for readability, hence the labels “a” and “q”. Both the data and the ancilla cat qubits are dissipative cat qubits. The circuit representation is similar to figure 5 (except for the control qubit) with a capacitive bus coupling 304 between the data 1101 and the ancilla 1102 qubit modes. The microwave radiation sources are not represented but the CPW (co-planar waveguide) transmission lines that connect the circuit to the outside rest of the quantum system are represented 1120, 1122, 1124. The lines 1120, 1122 are responsible for the DC current bias and the parametric flux modulation of the ATS 308, the first one addressing mostly the right loop and the second one addressing mostly le left loop. By pumping any linear combination of both, the control circuit can pump fa or fa. The line 1124 is connecting the buffer to a 50Q environment to enable losses and drive. The ancilla qubit 1102 have the same input transmission lines.

[0223] Alternative CNOT gates may be implemented using a dissipatively stabilized cat qubit as the target, for instance as disclosed in Gautier, Ronan, Alain Sarlette, and Mazyar Mirrahimi. "Combined dissipative and hamiltonian confinement of cat qubits." PRX Quantum 3.2 (2022): 020339; or in Guillaud, Jérémie, and Mazyar Mirrahimi. "Repetition cat qubits for fault-tolerant quantum computation." Physical Review X 9.4 (2019): 041053.

[0224] Classical error-correcting codes using physical qubits

[0225] Figure 11 shows a quantum system 100 for performing a repetition errorcorrecting code with data cat qubits. The left-hand side of Figure 11 shows the physical quantum system 100, whereas the right-hand side of Figure 11 shows an exemplary quantum circuit diagram of the operations required to perform the parity checks on the physical qubits corresponding to the left-hand side, which results in encoding a logical qubit in the data cat qubits of the repetition error-correcting code.

[0226] Quantum system 100 is shown as being comprised of 5 physical qubits arranged in a repetition code arrangement, which is formed from coupling ancilla qubits 501 in an alternating manner with data cat qubits 502,503 in a linear array. Data cat qubits 503 at either end of the linear array are each coupled to a different ancilla qubit 501 . The data cat qubit 502 in the middle of the linear array is coupled to both ancilla qubits 501 positioned adjacent and either side of the data cat qubit 502. The repetition code arrangement thus comprises a first stabilizer 102 formed from one of the ancilla qubits 501 and the immediately adjacent data cat qubits 502,503 either side, and a second stabilizer 102 formed from the other of the ancilla qubits 501 and the immediately adjacent data cat qubits 502,503 either side (wherein the stabilizers 102 are indicated by the dashed ovals). As shown, the middle data cat qubit 502 is shared between the two stabilizers 102, and thus can be termed a shared data cat qubit 502.

[0227] More precisely, quantum system 100 comprises a qubit-hosting circuit 103 and a command circuit 5 configured to selectively apply control signals to the qubit-hosting circuit 103.

[0228] The qubit-hosting circuit 103 comprises data resonators each being coupled to the control circuit 5. Upon receiving the appropriate control signals from the control circuit 5, a respective data cat qubit 502,503 is stabilized in each of the data resonators. Preferably, each of the data cat qubits 502,503 is a dissipatively stabilized cat qubit (e.g., as described above in relation to Figures 1-4), and in particular stabilized via parametric dissipative stabilization with a four-wave mixing non-linear element (e.g., as described above in relation to Figures 2-3). Most preferably, the data cat qubits 502,503 are stabilized using an ATS (e.g., as described above in relation to Figure 2). That is, the data resonator is thus the provided by the at least one resonant portion having the first mode a in which the data cat qubit is hosted, as described above. As such, the control signals may be electromagnetic pulses / drives, and / or DC voltage biases, and / or constant current biases, as appropriate. Each of the data cat qubits will thus be stabilized by the command circuit 5 so as to have a particular two-photon dissipation rate K2as described above.

[0229] The qubit-hosting circuit 103 also comprises ancilla qubit-hosting structures each being coupled to the control circuit 5. Upon receiving the appropriate control signals from the control circuit 5, a respective ancilla qubit 501 is prepared in each of the ancilla qubithosting structures. The qubit-hosting structure may for instance be a Josephson junction connected between two capacitive pads, such that the prepared ancilla qubit 501 is a so-called transmon. Alternatively, the qubit-hosting structure may be the physical components required such that the prepared ancilla qubit 501 is a flux-qubit or a fluxonium qubit or any bosonic qubit encoded or stabilized in a resonator such as a Kerr cat qubit or a GKP qubit. Preferably, the ancilla qubit-hosting structure is an ancilla resonator for hosting a cat qubit. Preferably, each of the data cat qubits 502,503 is a dissipatively stabilized cat qubit (e.g., as described above in relation to Figures 1-4). Most preferably, the data cat qubits 502,503 are stabilized using an ATS (e.g., as described above in relation to Figure 2) or via resonant stabilization with a three-wave mixing non-linear element (e.g., as described above in relation to Figure 4).

[0230] Indeed, the Applicant has discovered that using resonantly stabilized cat qubits as the ancilla qubits 501 is particularly effective in the context of the present invention when combined with parametrically stabilized cat qubits as the data cat qubits 502,503. This advantageously enables an optimal classical error-correcting code described below, because a resonantly stabilized cat qubit requires no time-dependent pump in order to stabilize a ancilla cat qubit, making frequency crowding issues simpler to solve. Additionally, a resonant-cat can reach higher values of the two-photon loss rate K2in general, which may allow for faster and higher fidelity implementation of the CNOT gates.

[0231] Thus, the command circuit 5 is configured to selectively apply control signals to the qubit-hosting circuit 103 for stabilizing or preparing qubits in the data resonators and ancilla qubit-hosting structures therein (i.e. the appropriate control signals to each of the data resonators and ancilla qubit-hosting structures to achieve stabilization or preparation of the corresponding qubit).

[0232] The command circuit 5 is also configured to apply the appropriate control signals to perform stabilizer measurements (also referred to as parity checks) of stabilizers 102 at particular times. The stabilizer measurements may comprise the operations as described above in relation to Figure 5. These control signals may thus be the appropriate electromagnetic pulses / drives, and / or DC voltage biases, and / or constant current biases delivered to the qubit-hosting circuit 103 known in the art or described herein which enable: (i) the ancilla qubit 501 of a given stabilizer 102 to be prepared in a known state; (ii) CNOT gates to be performed between the data cat qubits 502,503 adjacent to the ancilla qubit 501 of that stabilizer 102; and (iii) the ancilla qubit 501 to be measured so as to extract the syndrome. As an example, the control signal for performing CNOT gates in certain embodiments may be those as described above in relation to Figures 6-10. Such control signals result in the timing of operations as shown in the quantum circuit diagram of the operations in the right-hand side of Figure 11.

[0233] In particular, at a first time tl ta first stabilizer measurement on one of the stabilizers 102 is started. This includes performing a CNOT gate between the ancilla qubit 102 as control and the shared data cat qubit 502 as target, and performing a CNOT gate between the ancilla qubit 102 as control and the other data cat qubit 503 of the stabilizer 102 as target. At a second time t2, a second stabilizer measurement on the other of the stabilizers 102 is started. This includes performing a CNOT gate between the ancilla qubit 102 as control and the shared data cat qubit 502 as target, and performing a CNOT gate between the ancilla qubit 102 as control and the other data cat qubit 503 of the stabilizer 102 as target.

[0234] The first time is shown as when the first of the CNOT gates of the first stabilizer measurement is performed. The second time t2is shown as when the first of the CNOT gates of the second stabilizer measurement is performed. Other operations of the stabilizer measurements may occur prior to these times, such as preparing the ancilla qubits as discussed below, however what is important is that the first CNOT gate of the second stabilizer measurement occurs some time after the first CNOT gate of the first stabilizer measurement.

[0235] In the embodiment of Figure 11 , the first CNOT gate of the first stabilizer measurement is the CNOT gate performed between the ancilla qubit 102 as control and the shared data cat qubit 502 as target, whereas the first CNOT gate of the second stabilizer measurement is the CNOT gate performed between the ancilla qubit 102 as control and the other data cat qubit 503 of the second stabilizer 102 as target (i.e. not the shared data cat qubit 502).

[0236] As will be appreciated, it is the data cat qubits which are the targets in each CNOT, such that for all intents and purposes a “target cat” can be interpreted as a data cat qubit in the context of a classical error-correcting code.

[0237] For each of the stabilizer measurements, the ancilla 501 is prepared in a known state, which typically for an exemplary phase-flip correcting code is a | — ) or | +) state, and subsequently entangled with the data cat qubits 503 whose parity is to be checked (the data cat qubits of the respective stabilizer measurement) via the quantum CNOT gates described above. As known in the art, the ancilla qubit may be prepared in any state, entangled via a two-qubit gate with the data cat qubit, and subsequently measured in the correct basis, to be equivalent to the stabilizer measurements operations described herein.

[0238] If the ancilla qubit 501 was initially prepared in a | — ) or | +) state, then the syndrome-extraction step comprises measuring in the X-basis, and the result of the measurement is attributed as the ±1 of the parity check. More generally, for each of the stabilizer measurements, the ancilla qubit 501 could be prepared in a state defined on the equator of the Bloch sphere (e.g. (|0) + el<^1|l)) / 2 where 1 is a phase comprised in the range [0;2TT]) SO long as the ancilla qubit 501 is measured with an observable whose set of eigenstates contains the state the ancilla was prepared in. Furthermore, each of the stabilizer measurements can be prepared in a different basis (of initial states and measurement basis) compared to others of the stabilizer measurements. What is important is that the ancilla qubit is measured with an observable whose set of eigenstates contains the state the ancilla was prepared in..

[0239] Crucially, the shared data cat qubit is subject to a re-stabilization for a time period Trcbetween the CNOT between itself (as target) and the ancilla qubit (as control) of the first stabilizer and the CNOT between itself (as target) and the ancilla qubit (as control) of the second stabilizer. The present Inventors have discovered that this vastly reduces the amount of leakage errors accumulating during the course of the CNOT gates involving the shared data cat qubit 502.

[0240] In particular, in embodiments the data cat qubits are dissipatively stabilized cat qubits and the CNOT gates are implemented as described herein in relation to Figures 8-9 and 14-15 (namely, that the data cat qubit is not stabilized during the CNOT duration by two-photon dissipation). As such, in a conventional repetition code, the shared data cat qubit would be a CNOT gate target during two consecutive CNOTs before being reconfined with 2-photon dissipation. This means that they would be in “free-flight” for a duration of 2TCX, during which leakage errors accumulate (as distorting effects like Kerr and memory dephasing are not actively corrected by the two-photon dissipation), leading to bit-flip errors reducing the efficacy of the classical error-correcting code approach. However, surprisingly, the present Inventors have recognised that the bit-flip erorrs accumulated during free-flight increase with a high power law, and as a result the bit flip probability during a pair of CNOT gates each of duration TCXplayed consecutively is bigger (and can be orders-of-magnitude bigger) than twice the bit-flip probability during a single CNOT of duration TCX. This is because the second CNOT gate starts with the shared data cat qubit 502 already in a (partially) leaked state, from which much more bitflip can happen.

[0241] Thus, the present invention remedies this issue (to reach acceptable bit-flip errors during a CNOT involving the shared data cat qubit 502) by subjecting the shared data cat qubit to a re-stabilization time period Trcbetween the end of the CNOT gate of involving the shared data cat qubit 502 of the first stabilizer measurement and the beginning of the CNOT gate involving the shared data cat qubit 502 of the second stabilizer measurement, wherein the re-stabilization time period is greater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit 502 at which pairs of bosons are exchanged with the environment, namely trc> I / IOK^ ■ The present Inventors have discovered that this brings the leakage outside the shared data cat qubit subspace, accumulated due to the first CNOT involving the shared data cat qubit 502, to a low enough level before starting its next CNOT gate such that the classical errorcorrecting code is improved.

[0242] In particular, in some embodiments the re-stabilization time period is greater than or equal to the reciprocal of rate namely Trc> I / K^, as this re-stabilization time brings the leakage to extremely low levels, such as substantially back to zero. As clear from the above, in the context of dissipatively stabilized cat qubits subject to CNOT gates described herein in relation to Figures 8-9 and 14-15, the re-stabilization comprises at least turning back on the two-to-one photon exchange.

[0243] Furthermore, given the unavoidable imperfections in implementing any physical CNOT gate between a data cat qubit and an ancilla qubit, the present Inventors have further recognised that the advantages of the present invention may also apply systems configured to implement other implementations of CNOT gates different to those described herein in relation to Figures 8-9 and 14-15, and may further apply to systems for performing a classical error-correcting code comprised of data cat qubits other than the dissipatively stabilized varieties (e.g., the so-called Kerr-cat qubits of (B) and (C) or the TPE stabilized cat of (D) described above). Here, the re-stabilization comprises at performing the operations which only stabilize a cat qubit, but not applying other operations in addition, e.g. applying the operations required to stabilize a cat qubit according to schemes (B), (C), or (D) described above

[0244] However, the present Inventors have recognised that subjecting the shared data cat qubit 502 to a re-stabilization time period Trcbetween the successive CNOT gates is not enough in isolation to provide an improved classical error-correcting code, as this would require including a wait time TWon each of the ancilla qubits 501 between the CNOTs involving each ancilla qubit 501 in order to accommodate the re-stabilization time period Trcon the shared data cat qubit 502. This increases the QEC cycle time and the ancilla and data phase and bit- flip errors by r / zTw, where Yx / Zis the characteristic rate of exponential decay of the X or Z observable of the said cat qubit, thus degrading the performance of the error-correcting code.

[0245] Therefore, so as to reduce the idling time of the ancilla qubit 501 between its two CNOTs of a given stabilizer 102 and still subject the shared data cat qubit 501 to the restabilization for time period Trc, the present Inventors have further recognised that the time t2must be after the time This enables the shared data cat qubit 501 to be re- stabilized whilst reducing or minimising the idling time of the ancilla qubit 501 between its two CNOTs of a given stabilizer 102.

[0246] Figure 12 shows a quantum circuit diagram of the operations required to perform a repetition error-correcting code, according to an embodiment. The system 100 for performing the repetition error-correcting code may be the same as that of the left-hand side of Figure 11 , but with the command circuit 5 configured to apply the appropriate control signals to perform the operations of the first and second stabilizer measurements in the order shown in Figure 12.

[0247] In particular, in Figure 12, the first CNOT gate of the first stabilizer measurement is the CNOT gate performed between the ancilla qubit 102 as control and the shared data cat qubit 502 as target, whereas the first CNOT gate of the second stabilizer measurement is now the CNOT gate performed between the ancilla qubit 102 as control and the shared data cat qubit 502 as target.

[0248] Indeed, flipping the order of the CNOT gates performed in one stabilizer measurement compared to the adjacent stabilizer measure has been found to be the most optimal configuration because this permits the same re-stabilization period trcto be repeated on the same shared data cat qubit during repeated error correction rounds. It has been discovered that this may provide the most optimal compromise between reducing the QEC cycle time TQEC, reducing the amount of leakage between successive CNOTs on the shared data cat qubit, and not having to increase the idle times between successive CNOTs on the ancilla qubit 501 of a given stabilizer 102 during a given stabilizer measurement.

[0249] For instance, as shown in Figure 13, which shows a quantum system 100 on the left-hand side and a corresponding quantum circuit diagram of the operations required to perform a distance 7 repetition error-correcting code on the right-hand side, each shared data cat qubit 502 is subject to consecutive re-stabilization periods having the same re-stabilization time periodrc. In embodiments having such flipped order of stabilizer CNOT gate operations, the second time t2is then given as t2= +CX+RC.

[0250] In particular, quantum system 100 of Figure 13 comprises a qubit-hosting circuit 103 and a command circuit 5 configured to selectively apply control signals to the qubithosting circuit 103. Qubit-hosting circuit 103 comprises 13 physical qubits arranged in a repetition code arrangement, which is formed from coupling ancilla qubits 501 in an alternating manner with data cat qubits 502,503 in a linear array. Similar to Figures 11- 12, data cat qubits 503 at either end of the linear array are each coupled to a different ancilla qubit 501 . In contrast however, there is now a plurality of shared data cat qubits 502 each coupled to both ancilla qubits 501 positioned adjacent and either side of the shared data cat qubit 502.

[0251] The repetition code arrangement thus comprises 6 stabilizers 102 formed from pairs of data cat qubits 502,503 each coupled to an ancilla qubit 501 therebetween. Moreover, in general for a distance d repetition code, the repetition code arrangement comprises d - 1 stabilizers.

[0252] As shown on the left-hand side of Figure 13, the qubit-hosting circuit 103 comprises a plurality of first stabilizers 102 and a plurality of second stabilizers 102 which are arranged in an alternating manner to form the repetition code arrangement. That is, starting from an end stabilizer 102 at one end of the repetition code arrangement (i.e. comprising one of the data cat qubits 503 at an end of the linear array), the plurality of first stabilizers 102 may be defined as the group comprising that end stabilizer 102 and every other stabilizer 102 selected in the direction going towards the other end of the linear array (i.e. the stabilizer adjacent to the end stabilizer and sharing a data cat qubit 502 is not included in the group, the next one along is). The plurality of second stabilizers 102 may be defined as the group comprising all the remaining stabilizers 102.

[0253] As shown on the right-hand side of Figure 13, the stabilizer measurements are staggered. Specifically, the first stabilizer measurement as described above is performed on each of the plurality of first stabilizers 102 simultaneously, and the second stabilizer measurement as described above is performed on each of the plurality of second stabilizers 102 simultaneously and wherein the plurality of second stabilizers 102 starts after the plurality of first stabilizer measurements.

[0254] Moreover, the order of the operations of CNOT gates between adjacent ones of the first and second stabilizer measurements is reversed as described above in relation to Figure 12. That is, if the CNOT gate involving a given shared data cat qubit 502 in one of the first stabilizer measurements of a given first stabilizer 102 is the temporally first CNOT gate of that first stabilizer measurement, then the CNOT gate involving said given shared data cat qubit 502 in the second stabilizer measurement of the second stabilizer 102 adjacent to said given first stabilizer 102 is the temporally first CNOT gate of that second stabilizer measurement. Similarly, if the CNOT gate involving a given shared data cat qubit 502 in one of the first stabilizer measurements of a given first stabilizer 102 is the temporally last CNOT gate of that first stabilizer measurement, then the CNOT gate involving said given shared data cat qubit 502 in the second stabilizer measurement of the second stabilizer 102 adjacent to said given first stabilizer 102 is the temporally last CNOT gate of that second stabilizer measurement. This results in the staggered and optimal order of operations of CNOT gates between adjacent ones of the first and second stabilizer measurements as shown on the right-hand side of Figure 13. As will be appreciated, typically each quantum error correcting round is repeated a multiple number of times so as to operate the repetition error-correcting code.

[0255] As discussed above, the ancilla qubits may be, and preferably are, ancilla cat qubits.

[0256] The present Inventors have further recognized that combining the system configured to perform a classical error-correcting code as described herein a fully catqubit based architecture (both data and ancillas being cat-qubits) presents further advantageous synergies. In particular, by stabilizing an ancilla cat qubit in the ancilla qubit-hosting structure (i.e. ancilla resonntor) of the qubit-hosting circuit 103, the command circuit 5 may be configured to perform two or more CNOT gates respectively between the ancilla cat qubit of a stabilizer and two or more of the data cat qubits in said stabilizer substantially simultaneously. This is particularly advantageous as outlined further below, as it reduces the number of timesteps required to perform the stabilizer measurement. Said otherwise, given that the parity check is usually a sequence of finite time CNOT gates being performed between the ancilla qubit and each of the data cat qubits of the stabilizer, by reducing the sequence (either partially or fully to a single CXSgate), the shorter the parity check takes in time, and thus the shorter the chance an error occurs.

[0257] Figure 14 depicts a general diagram of a quantum system 1402 for performing a pair of simultaneous CNOT gates (a CNOTNOT quantum gate) according to an embodiment of the invention. In the example described here, a quantum system 1402 for performing a CXX gate links two data cat qubits 1404 and one ancilla cat qubit 1406, and thus may be used as a stabilizer 102 as described herein. The two data cat qubits 1404 are connected to the ancilla cat qubit 1406 via a CXX gate 1411. However, contrary to the prior art which uses two sequential CNOT gates, there is a single CXX gate 1411 , and, as shown, both data cat qubits 1404 are connected simultaneously to the CXX gate 1411 , such that the CXX gate is performed simultaneously on both data cat qubits 1404.

[0258] The theoretical realization of CXX gates between three stabilized cat qubits in modes ai and a2 and as relies on the use of the following three ingredients:

[0259] (1 ) The performance of a dissipative stabilization on the control qubit ai, with the jump operator LC1= *2(ai—“2)> where a! is the photon annihilation operator of mode ai and a is a complex number defining the ancilla cat qubit. (2) The addition in the circuit of a ‘feedforward’ Hamiltonian, also called the ‘CXX’ Hamiltonian with the formula , HCXx / ft = ^(ai +ai— 2ai)(a2a2 - a3a3- «2 + «3) where T is the CXX gate duration, ai is the photon annihilation operator of the ancilla cat qubit 6 and a2and a3are the photon annihilation operator of the data cat qubits, and a is a complex number defining the cat qubit. For simplicity, the photon populations of the ancilla and data cat qubits are taken equal but they can be different: at2for ai, a22for a2, a32for a3. The feedforward Hamiltonian can be engineered with a pump at frequency œaiand acts as a pull force and on the data cat qubits 4. During the application of this feedforward Hamiltonian, the stabilization on data cat qubit 4 is turned off.

[0260] (3) One or more dissipative stabilization which acts as a drag force on one or both the data cat qubits 4 which depends on the state of the ancilla cat qubit atcan be used, 12iTIt \ (aj + a) + -aet(a1— a) I and L = 2 / 3

[0261] ( 1 1 \ a3— 2 a(ai + a) + - 2 ae Tt(a1— a) / . The jump operator Lt2 and Lt3 can be realized by providing respective buffer modes b2and b3for the data cat qubits 4 and engineering the Hamiltonian H / ft = g2t>a2b2++ gibaib2++ h. c. and pumps at frequency and a drive at frequency o)bz. The same can be done for jump operator Lt3by replacing a2and b2by a3and b3in the above equations.

[0262] The best implementation theoretically uses the 3 ingredients. However, the experiments of the Applicant have revealed that the two first ingredients offer a good compromise between the ease of realization and quality of the results. As in the case of a single CNOT gate described above, the stabilization schemes (1 ) and (3) can be implemented with any of the aforementioned cat stabilization schemes, and in particular the dissipative stabilization schemes which use an ATS.

[0263] Figure 15 shows a block diagram of the operation of the CXX gate of figure 14. In a first operation 1500 which consists the feedforward Hamiltonian (ingredient 2) is turned on on both data cat qubits 1404 and the ancilla cat qubit 1406. Ingredient 1 is always applied as the ancilla cat qubit needs to be stabilized, but the stabilization on data cat qubits 1404 is turned off. Alternatively, operation 1500 could comprise turning on the feedforward Hamiltonian (ingredient 2) and the time dependent dissipation Ltzand Lt3on the data cat qubits (ingredient 3), or only activating the time dependent dissipation Ltzand Lt3on the data cat qubits (ingredient 3). After a chosen duration for operation 1500, which is the gate duration and which will be discussed below, the feedforward Hamiltonian (or its variations of operation 1500) is turned off in an operation 1510 and the stabilization of data cat qubits 4 is restored.

[0264] As will be appreciated, the CXX gate according to an embodiment of the invention can be done either faster with the same error as two CNOT gates performed slower, or it can be done in the same time as two CNOT gates performed but with a better fidelity. Another advantage of implementing CXX in a single step instead of using two consecutive CNOT gates is that the compensation problem mentioned above can be solved by choosing specific phase arrangement for the feedforward Hamiltonian. More precisely, any Hamiltonian of the form Hcx / ft = cx(ai +aî- 2a)((a2a2 - “!) ± (a3a3 - a3)) produces the desired effect. The Applicant has discovered for the specific case of the sign in the last term of the Hex formula, the undesired displacement generated by the respective terms (ai 4-aî)(a2az) and (ai +aJ)(a3a3) of the feedforward Hamiltonian (as explained above) exactly compensate each other if = a3, such that it is not anymore necessary to engineer a compensation. In practice, due to experimental imperfections, a very small compensation may still be needed but, because most of it has been already cancelled, it is much easier to engineer. In alternative embodiments, compensation can be engineered as known conventionally.

[0265] Thus, the measurement of the ancilla cat qubit 1406 allows to measure the joint photon number parity between the data cat qubits 1404.

[0266] Figure 16 thus shows a quantum circuit diagram of the operations required to perform a repetition error-correcting code similar to those described above in relation to Figures 11-13, but wherein for each of the stabilizer measurements, the CNOT operations are performed simultaneously. That is, of a given stabilizer, the operations of (i) performing a CNOT gate between the ancilla qubit as control and the shared data cat qubit as target, and (ii) performing a CNOT gate between the ancilla qubit as control and another of the at least two data cat qubits of the stabilizer as target are performed simultaneously.

[0267] The present inventors have further realized that the present invention may equally be applied to a classical error-correcting code comprised of stabilizers having a weight greater than two.

[0268] Figure 17 shows a so-called classical LDPC error-correcting code having a stabilizer weight greater than 2, for instance as detailed in Ruiz, Diego, et al. "LDPC-cat codes for low-overhead quantum computing in 2D." arXiv preprint arXiv:2401.09541 (2024). The left-hand side of Figure 17 shows a quantum system 100 comprising a qubithosting circuit 103 and a command circuit 5 configured to selectively apply control signals to the qubit-hosting circuit 103. Qubit-hosting circuit 103 comprises 7 physical qubits arranged in an LDPC code arrangement. Specifically a first stabilizer 102 is formed by coupling three data cat qubits 502,503 to an ancilla qubit 501 , and a second stabilizer 102 is formed by coupling three data cat qubits 502,503 to another ancilla qubit 501 , wherein the stabilizers 102 share one of the data cat qubits 502. Thus, the stabilizers are of weight greater than 2 (in this case 3).

[0269] The right-hand side of Figure 17 shows an exemplary quantum circuit diagram of the operations required to perform the parity checks on the physical qubits corresponding to the left-hand side.

[0270] Here, a first stabilizer measurement on one of the stabilizers 102 (herein the first stabilizer 102) comprises the operations of: (i) performing a CNOT gate between the ancilla qubit 501 of the first stabilizer 102 as control and the shared data cat qubit 502 as target, (ii) performing a CNOT gate between the ancilla qubit 501 of the first stabilizer 102 as control and another of the data cat qubits 503 of the first stabilizer as target, and (iii) performing a CNOT gate between the ancilla qubit 501 of the first stabilizer 102 as control and the remaining data cat qubit 503 of the first stabilizer 102 as target.

[0271] A second stabilizer measurement on the other of the stabilizers 102 (herein the second stabilizer 102) comprises the operations of: (i) performing a CNOT gate between the ancilla qubit 501 of the second stabilizer 102 as control and the shared data cat qubit 502 as target, (ii) performing a CNOT gate between the ancilla qubit 501 of the second stabilizer 102 as control and another of the data cat qubits 503 of the second stabilizer as target, and (iii) performing a CNOT gate between the ancilla qubit 501 of the second stabilizer 102 as control and the remaining data cat qubit 503 of the second stabilizer 102 as target.

[0272] Again, the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time, and the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time. Moreover, the shared data cat qubit 502 is again stabilized for a restabilization time periodrcbetween the end of the CNOT gate involving it of the first stabilizer measurement and the start of the CNOT gate involving it of the second stabilizer measurement.

[0273] The order of the various CNOT operations is shown analogous to that described above in relation to Figures 12 and 13 so as to enable the same re-stabilization time period Trcfor successive re-stabilizations of the shared data cat qubit 502 when repeating the stabilizer measurements.

[0274] Although not shown, the CNOT gates of each of the stabilizer measurements may be performed simultaneously in a similar manner as described above in relation to Figure 16. That is, while the embodiment of figures 14 to 15 has been made with only two data cat qubits, the Applicant has discovered that the CXX can be made a CXAN by using N data cat qubits, each connected to the same ancilla cat qubit, which may be applied analogously to the classical LDPC error-correcting codes such as in the embodiment of Figure 17. Thus, in the context of the present disclosure, this is a CXSwhere s is in fact the weight of the stabilizer. By implementing such a multi-qubit gate, the Applicant has recognized that stabilizers having higher weights can advantageously be used. All that is required is, in a first operation, turn on s feedforward Hamiltonian (ingredient 2) on each of the s data cat qubits and the ancilla cat qubit 1406 connected thereto. Ingredient 1 is always applied as the ancilla cat qubit needs to be stabilized. Alternatively, the first operation could comprise turning on the feedforward Hamiltonian (ingredient 2) and the time dependent dissipation Ltito Ltson the data cat qubits (ingredient 3), or only activating the time dependent dissipation Ltiand Ltson the data cat qubits (ingredient 3). After a chosen duration for the first operation, which is the gate duration and which will be discussed below, the feedforward Hamiltonian (or its variations thereof) is turned off.

[0275] The phases of the pumps to turn on the s feedforward Hamiltonians can be chosen so that no compensation is needed. A possible compensation scheme includes choosing pumps of the s feedforward Hamiltonians so that the drives are exactly or closely compensated on the ancilla cat qubit.

Claims

Claims1 . A system for performing a classical error-correcting code using physical qubits, the system comprising: a command circuit for selectively applying control signals; a qubit-hosting circuit comprising: at least three data resonators, each data resonator being coupled to the command circuit for stabilizing a respective data cat qubit having a rate K2at which pairs of bosons are exchanged with the environment, and at least two ancilla qubit-hosting structures, each ancilla qubit-hosting structure being coupled to the command circuit for preparing a respective ancilla qubit; a first stabilizer, wherein the first stabilizer comprises a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer, wherein the second stabilizer comprises a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer; wherein the command circuit is configured to apply control signals to the qubithosting circuit so as to:- perform a first stabilizer measurement on the first stabilizer comprising the operations of:(i) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and(ii) performing a respective CNOT gate between the first ancilla qubit as control and each of the other at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time;- perform a second stabilizer measurement on the second stabilizer comprising the operations of:(iii) performing a CNOT gate between the second ancilla qubit as control and the shared data cat qubit as target, and(iv) performing a respective CNOT gate between the second ancilla qubit as control and each of the other at least two data cat qubits of the second stabilizer as target,wherein the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time; and- stabilize the shared data cat qubit for a re-stabilization time period Trcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement, wherein the re-stabilization time period is greater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit « at which pairs of bosons are exchanged with the environment, Trc> l / 10K2Sd.

2. The system of claim 1 , wherein: the first stabilizer measurement comprises: prior to operations (i) and (ii), preparing the first ancilla qubit in a first state, and after operations (i) and (ii), measuring the state of the first ancilla in the basis of an observable whose set of eigenstates contains the first state; and the second stabilizer measurement comprises: prior to operations (iii) and (iv), preparing the second ancilla qubit in a second state, and after operations (iii) and (iv), measuring the state of the second ancilla qubit in the basis of an observable whose set of eigenstates contains the second state.

3. The system of claim 2, wherein each of the ancilla qubit-hosting structures is an ancilla resonator being coupled to the command circuit for preparing a respective ancilla cat qubit; wherein the first stabilizer measurement comprises: prior to operations (i) and (ii), preparing the first ancilla cat qubit in a | +) state or a | -) state, and after operations (i) and (ii), measuring the state of the first ancilla cat qubit in the X-basis; and wherein the second stabilizer measurement comprises: prior to operations (iii) and (iv), preparing the second ancilla cat qubit in a | +) state or a | -) state, andafter operations (iii) and (iv), measuring the state of the second ancilla cat qubit in the X-basis.

4. The system of any preceding claim, wherein: the CNOT gate of operation (i) is the CNOT gate of the first stabilizer measurement which is performed first in time, and the CNOT gate of operation (iii) is the CNOT gate of the second stabilizer measurement which is performed first in time; or the CNOT gate of operation (i) is the CNOT gate of the first stabilizer measurement which is performed last in time, and the CNOT gate of operation (iii) is the CNOT gate of the second stabilizer measurement which is performed last in time.

5. The system of claim 4, wherein the command circuit is configured to apply control signals to the qubit-hosting circuit so as to:- perform a third stabilizer measurement on the first stabilizer comprising the operations of:(v) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and(vi) performing a respective CNOT gate between the first ancilla qubit as control and each of the other at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the third stabilizer measurement which is performed first in time is performed at a third time; and- stabilize the shared data cat qubit for a further re-stabilization time period between the end of the CNOT gate of operation (iii) of the second stabilizer measurement and the beginning of the CNOT gate of operation (v) of the third stabilizer measurement which is equal to the re-stabilization time period Trcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement.

6. The system of any preceding claim, the qubit-hosting circuit comprises: at least three four-wave mixing non-linear elements respectively coupled to each of the at least three data resonators; and control lines coupled to the command circuit so as to selectively apply control signals to each of the four-wave mixing non-linear elements;wherein the command circuit is configured to perform any one of the CNOT gates between a given ancilla qubit and a given data cat qubit by: during a CNOT gate time window, delivering in the control lines only a radiation at a frequency equal to a resonant frequency of the given ancilla qubit; and outside the CNOT gate time window, delivering control signals to the qubit-hosting circuit to dissipatively stabilize the given data cat qubit.

7. The system of claim 6, wherein each of the ancilla qubit-hosting structures is an ancilla resonator being coupled to the command circuit for preparing a respective ancilla cat qubit; and wherein the control circuit is configured to, for at least one of the stabilizer measurements, perform two or more of the CNOT gates of said at least one of the stabilizer measurements simultaneously.

8. The system of any preceding claim, wherein: (i) the rate of the shared data cat qubit falls within the range 10-100 MHz; and / or (ii) the re-stabilization time-periodrcfalls within the range 20 ns - 2 microseconds; and / or (iii) each CNOT gate has a CNOT gate time TCXwhich falls within the range 10 ns - 1 microsecond.

9. The system of any preceding claim, wherein the qubit-hosting circuit comprises: at least three four-wave mixing non-linear elements respectively coupled to each of the at least three data resonators; and control lines coupled to the command circuit so as to selectively apply control signals to each of the four-wave mixing non-linear elements; wherein each of the ancilla qubit-hosting structures is an ancilla resonator being coupled to the command circuit for stabilizing a respective ancilla cat qubit having an ancilla resonant frequency; wherein each of the ancilla resonators is respectively linearly coupled to at least two of the four-wave mixing non-linear elements of the data resonators such that each of the ancilla cat qubits is connected to at least two data cat qubits; wherein the command circuit is configured to apply control signals to the qubithosting circuit to perform any given one of the CNOT gates between a given ancilla cat qubit as control and a given data cat qubit as target by:(I) during a CNOT gate time window of the given CNOT gate having CNOT gate timeCX, delivering in the control lines only a radiation at a frequency equal to the ancilla resonant frequency; and(II) outside the CNOT gate time window, delivering control signals to the qubit-hosting circuit to dissipatively stabilize the given data cat qubit; and wherein the rate of the shared data cat qubit Kfdfalls within the range 10-100MHz, the re-stabilization time-period trcfalls within the range 20 ns - 2 microseconds, and the given CNOT gate has CNOT gate time TCXwhich falls within the range 10 ns - 1 microsecond.

10. The system of any preceding claim, wherein each of the data cat qubits is selected from the group consisting of: a parametrically pumped dissipatively stabilized cat qubit, a resonant dissipatively stabilized cat qubit, a DC dissipatively stabilized cat qubit, a dissipatively stabilized squeezed cat qubit; and wherein each of the ancilla qubits is selected from the group consisting of: a parametrically pumped dissipatively stabilized cat qubit, a resonant dissipatively stabilized cat qubit, a DC dissipatively stabilized cat qubit, a dissipatively stabilized squeezed cat qubit, a cat qubit stabilized by a Kerr Hamiltonian, a cat qubit stabilized by a detuned Kerr Hamiltonian, and a cat qubit stabilized by two-photon exchange Hamiltonian.11 . The system of claim 10, wherein each of the data cat qubits is a parametrically pumped dissipatively stabilized cat qubit, and each of the ancilla qubits is a parametrically pumped dissipatively stabilized cat qubit or a resonant dissipatively stabilized cat qubit.

12. The system of any preceding claim, comprising a plurality of first stabilizers and a plurality of second stabilizers wherein the first and second stabilizers are arranged in an alternating manner to form a repetition error-correcting code arrangement; wherein the stabilizer measurement on each of the first stabilizers start at substantially the same time; and wherein the stabilizer measurement on each of the second stabilizers start at substantially the same time.

13. A method for performing a classical error-correcting code using qubits on a classical error-correcting code arrangement, wherein the classical error-correcting code arrangement comprises: a first stabilizer comprising a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer comprising a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer; wherein each of the data cat qubits of the first and second stabilizers has a rate K2at which pairs of bosons are exchanged with the environment; the method comprising:- performing a first stabilizer measurement on the first stabilizer comprising the operations of:(i) performing a CNOT gate between the first ancilla qubit as control and the shared data cat qubit as target, and(ii) performing a CNOT gate between the first ancilla qubit as control and another of the at least two data cat qubits of the first stabilizer as target, wherein the CNOT gate of the first stabilizer measurement which is performed first in time is performed at a first time;- performing a second stabilizer measurement on the second stabilizer comprising the operations of:(iii) performing a CNOT gate between the second ancilla qubit as control and the shared data cat qubit as target, and(iv) performing a CNOT gate between the second ancilla qubit as control and another of the at least two data cat qubits of the second stabilizer as target, wherein the CNOT gate of the second stabilizer measurement which is performed first in time is performed at a second time subsequent to the first time; and- stabilizing the shared data cat qubit for a re-stabilization time periodrcbetween the end of the CNOT gate of operation (i) of the first stabilizer measurement and the beginning of the CNOT gate of operation (iii) of the second stabilizer measurement, wherein the re-stabilization time period isgreater than or equal to a tenth of the reciprocal of the rate of the shared data cat qubit Kfdat which pairs of bosons are exchanged with the environment, Trc> l / lO / cf6.

14. A computer program or computer-readable medium comprising instructions which, when the program is executed by a classical computer coupled to a quantum processor comprising a classical error-correcting code arrangement, cause the classical computer coupled to the quantum processor to carry out the method of claim 12, wherein the classical error-correcting code arrangement comprises: a first stabilizer comprising a first ancilla qubit connected to at least two data cat qubits; and a second stabilizer comprising a second ancilla qubit connected to at least two data cat qubits, wherein one of the at least two data cat qubits of the second stabilizer is a shared data cat qubit which is one of the at least two data cat qubits of the first stabilizer; wherein each of the data cat qubits of the first and second stabilizers has a rate K2at which pairs of bosons are exchanged with the environment.

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