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Repetition Cat Qubits for Fault-Tolerant Quantum Computation

Jérémie Guillaud, Mazyar Mirrahimi

arXiv:1904.09474v4quant-phcond-mat.supr-con

TL;DR

Fault-tolerant quantum computation needs protection against environmental decoherence without prohibitive hardware overhead. This paper combines noise-biased cat qubits, bias-preserving operations, and a 1D repetition code to construct universal protected logical gates, including Toffoli, without magic-state processing. Numerical analysis indicates threshold-compatible effective error probabilities in experimentally relevant regimes, with small logical error rates potentially reaching 10^-9 using a few tens of cat qubits and mean photon number about 10.

  • Problem

    Fault-tolerant quantum computation requires protection against decoherence, while surface-code implementations carry substantial experimental overhead and non-Clifford gates require magic-state processing.

  • Method

    The paper combines cat qubits with preserved noise bias, bias-preserving operations, and a 1D repetition code to construct protected logical operations.

  • Results

    The construction provides a universal set of fully protected logical gates, including NOT, CNOT, and Toffoli, without magic-state preparation, distillation, or injection.

  • Takeaways & Limitations

    Numerical analysis indicates threshold-compatible effective error probabilities in experimentally relevant superconducting-circuit regimes and supports experimental realization with minor modifications.

Abstract

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We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardware-efficient universal and fault-tolerant quantum computation. The cat qubits that are stabilized by a two-photon driven-dissipative process, exhibit a tunable noise bias where the effective bit-flip errors are exponentially suppressed with the average number of photons. We propose a realization of a set of gates on the cat qubits that preserve such a noise bias. Combining these base qubit operations, we build, at the level of the repetition cat qubit, a universal set of fully protected logical gates. This set includes single-qubit preparations and measurements, NOT, controlled-NOT, and controlled-controlled-NOT (Toffoli) gates. Remarkably, this construction avoids the costly magic state preparation, distillation, and injection. Finally, all required operations on the cat qubits could be performed with slight modifications of existing experimental setups.

I. INTRODUCTION

Cat codes offer a promising route to hardware-efficient fault-tolerant quantum computation, but prior protection was limited and universal protected gates remained unresolved. The paper proposes cat qubits as repetition-code base qubits whose preserved noise bias enables a universal protected gate set.

  • Cat codes use coherent-state encodings in a harmonic oscillator and have reached break-even quantum-memory error correction.
  • Prior cat-code proposals suggested protected logical gates, but their protection remained limited to first-order photon-loss errors.
  • The paper asks whether cat encoding can support fully fault-tolerant universal computation with hardware-efficient scaling.
  • Using cat qubits as base qubits modifies the noise structure so quantum error correction can use a simple repetition code with parity measurements between neighboring qubits.
  • Bias-preserving operations retain the cat qubits’ noise structure and yield a universal set of protected logical gates at the repetition-code level.
  • The proposed roadmap covers bias-preserving operations, realistic-noise performance analysis, and experimental realization using existing components or minor modifications.

II. PUMPED/STABILIZED CATS AS QUBITS WITH BIASED NOISE

Two-photon stabilization creates a cat-qubit manifold with exponentially suppressed bit flips and a tunable noise bias. The paper then uses repetition encoding to suppress phase flips without reintroducing bit flips.

  • Cat-qubit construction: Two-photon driven dissipation stabilizes a two-dimensional manifold spanned by coherent states |α⟩ and |−α⟩, with α set by drive and dissipation parameters.
  • Cat-qubit construction: The coherent-state manifold is equivalently described by even and odd Schrödinger cat states formed from superpositions of |α⟩ and |−α⟩.
  • Biased noise: Local phase-space perturbations remain within the attraction domain of the corresponding coherent-state equilibrium, explaining suppression of bit flips.
  • Biased noise: Effective bit-flip errors are exponentially suppressed with 2|α|2, while the cat size is tunable through the two-photon drive strength.
  • Biased noise: Increasing the mean photon number raises phase-flip rates linearly with |α|2, producing a tunable noise bias exp(−2|α|2)/|α|2.
  • Alternative stabilization: Kerr-based protection can require friction to prevent leakage from excursions within each well, whereas combining Kerr dynamics with two-photon dissipation can avoid higher phase-flip rates.
  • Repetition encoding: The proposed repetition encoding extends this partial protection to both phase and bit flips without reintroducing bit flips.

III. FROM CAT QUBITS TO PROTECTED LOGICAL QUBITS

Cat qubits provide a tunable noise bias that repetition coding extends to phase-flip protection, while their expanded bias-preserving gate set enables universal protected computation at the repetition-code level without magic-state operations.

  • Noise bias: Cat qubits suppress effective bit-flip errors exponentially with cat size, enabling a path toward full protection with minimal hardware overhead.The relevant size is the mean photon number of the cat states.
  • Experimental layout: The proposed layout connects each data cat qubit to two ancilla cat qubits for joint parity measurement and uses low-Q stripline resonators for readout.Each cat qubit is continuously stabilized by a two-photon driven-dissipative process; Josephson-mediated couplings and extra microwave drives support bias-preserving CNOT operations.
  • Protected logical gates: Unlike regular two-level systems, cat qubits support a universal bias-preserving gate set at the repetition-code level without magic-state preparation, distillation, or injection.The cat-qubit construction uses continuous distortion of the encoded manifold while preserving exponential bit-flip suppression.
  • Repetition-code protection: The repetition code C1 protects against phase-flip errors but does not detect or correct physical bit-flip errors.Its phase-flip correcting capacity is (n−1)/2.
  • Repetition-code protection: For mean photon numbers n̄ = 10 and n̄ = 15, the achievable accuracy can be as low as 10−9 and 10−13, respectively.The lower bound is set by the logical bit-flip probability, which decreases exponentially with cat size.
  • Protected logical gates: The encoded set includes preparation, X-basis measurement, NOT, CNOT, and Toffoli, with logical CNOT implemented transversally.Universality follows because the set contains Toffoli and supports construction of a logical Hadamard.

IV. BIAS-PRESERVING OPERATIONS

The paper constructs bias-preserving cat-qubit operations and combines them into fault-tolerant operations for repetition cat qubits. The operations include preparation, measurement, single-qubit gates, entangling gates, and a scalable controlled gate.

  • Bias-preserving operations: The cat-qubit operation set S includes preparation of |±⟩c, X-basis measurement, X, Z, CNOT, and Toffoli gates.These operations are described as sufficient to build the universal logical-gate set for the repetition code.
  • State preparation: Even-cat preparation uses two-photon driven dissipation from the vacuum, while odd-cat preparation can be obtained by applying a Z operation.Alternative optimal-control strategies can improve preparation fidelity relative to passive two-photon-dissipation preparation.
  • Measurement: X-basis measurement is implemented as a cavity photon-number parity measurement using either destructive or QND protocols on disposable ancilla qubits.The QND protocol can improve fidelity by repeating measurements, and the ancilla can be discarded after each measurement.
  • Single-qubit gates: An X operation is realized by slowly rotating the complex cat amplitude so that |α⟩ and |−α⟩ exchange positions while the two-photon dissipation remains active.For sufficiently slow evolution, the cat states follow the moving stabilized subspace; finite-time errors are phase flips while bit flips remain exponentially suppressed.
  • Single-qubit gates: In the lossless, infinitely slow limit, the X-operation fidelity is 1, and finite-time errors preserve the noise bias because bit flips remain exponentially suppressed.A Hamiltonian rotation can reduce phase-flip errors caused by nonadiabaticity, while deterministic geometric phases can be compensated by a local Z(θ) operation or suitable path choice.
  • Entangling gates: The same bias-preserving construction extends from CNOT and Toffoli to an n-qubit entangling gate C^(n−1)X when the required multimode couplings are available.At the cat-qubit level, Toffoli implementation is described as similar in complexity to CNOT and avoids the usual difficulty of realizing a non-Clifford gate.

V. UNIVERSAL SET OF LOGICAL GATES

The paper constructs a universal set of logical gates for repetition cat qubits from bias-preserving cat-qubit operations, with error correction integrated into the gate procedures. Preparations, measurements, NOT, CNOT, Toffoli, and a derived Hadamard are addressed, while pieceable correction makes the Toffoli construction fault tolerant.

  • Universal logical gate set: The logical gate set is built from bias-preserving operations on individual cat qubits and is universal through a constructed Hadamard and the Toffoli gate.The proposed logical set includes preparations, X-basis measurements, NOT, CNOT, and Toffoli operations.
  • Quantum error correction: Repetition-code error correction measures neighboring-cat-qubit parity checks repeatedly and decodes the ancilla outcomes to correct effective phase-flip errors.The code has n−1 joint-parity stabilizers, measurements are repeated r times, and optimal decoding uses (n−1)r ancilla measurements.
  • Preparations and measurements: Logical preparations are transversal, followed by error correction, while fault-tolerant X-basis measurement measures all cat qubits and applies a majority vote.Error correction reduces preparation infidelity according to the repetition-code distance.
  • Logical CNOT: The logical CNOT is implemented transversally with n physical CNOT gates, preventing forward error propagation that could create an uncorrectable logical error.For illustration, the circuit uses n = 3 cat qubits per repetition cat qubit.
  • Logical Toffoli: The logical Toffoli uses n^2 physical Toffoli gates and realizes a controlled-CNOT through parity of conditional logical CNOT operations.Each cat-level Toffoli is bias preserving, so bit-flip errors remain exponentially suppressed at the repetition-code level.
  • Fault tolerance and limitations: Because the Toffoli circuit is nontransversal, phase-flip errors from the third logical qubit can spread into an uncorrectable error, requiring intermediate QEC stages and additional blocks as code distance grows.The construction is pieceable fault tolerant: correction is inserted after one-third and two-thirds of the circuit, while threshold benchmarking remains future work.

VI. ERROR ANALYSIS

The error analysis finds that photon loss and nonadiabaticity primarily produce phase-flip errors, while bit-flip errors remain exponentially suppressed with cat size. Gate fidelity is optimized at a finite duration and remains high under additional thermal noise and dephasing, although Toffoli simulations are omitted because of their computational cost.

  • Photon-loss errors: Photon loss during the implemented CNOT primarily induces phase flips rather than bit flips on the cat qubits.A loss on the control causes a phase flip on that qubit without affecting the target, while target loss produces phase-flip errors.
  • Photon-loss errors: The CNOT error matrix contains only Z1, Z2, or Z1Z2 errors from photon loss and nonadiabaticity in the analyzed model.The remaining errors containing X or Y operators are exponentially suppressed by cat size, demonstrating bias preservation.
  • Gate fidelity and optimal time: A finite optimal gate time minimizes infidelity because photon-loss errors increase with duration while nonadiabatic phase-flip errors decrease.The highest achievable fidelity is set by the ratio κ1ph/κ2ph.
  • Gate fidelity and optimal time: 98.2% CNOT fidelity is predicted and numerically obtained for κ1ph/κ2ph = 10^-3.Adding thermal excitations with nth = 10% and photon dephasing lowers fidelity from 98.2% to 97.8%, through increased phase-flip errors while bit flips remain suppressed.
  • Toffoli gate: Toffoli numerical simulations are omitted because the corresponding simulation would be about 2000 times longer than the approximately 13-hour CNOT simulation.The paper instead gives an analytical discussion of the expected Toffoli error mechanisms and rates.
  • Toffoli gate: 97.3% fidelity is predicted for the Toffoli gate at κ1ph/κ2ph = 10^-3.The optimal gate time for both CNOT and Toffoli decreases as the mean photon number increases.

VII. TOWARD EXPERIMENTAL IMPLEMENTATION

The proposed gates can be implemented by extending existing parametrically engineered two-photon dissipation and superconducting-circuit setups. The authors expect experimentally accessible regimes to support below-threshold universal gates with logical error probabilities around 10^-9 using cat qubits in a repetition code.

  • Implementation platform: Cat-qubit proposals exploit an extra phase-space degree of freedom, where locality enables suppression of one error type.The approach targets superconducting circuits but can also apply to other systems with a comparable extra degree of freedom.
  • Gate implementation: The required bias-preserving gates use mandatory dissipation operators and Hamiltonians, while optional terms reduce phase-flip errors from nonadiabaticity.The operator requirements are summarized in Table II.
  • Cat-qubit stabilization: Two-photon driven dissipation is engineered by parametrically coupling a high-Q storage cavity to a low-Q dump mode through a Josephson junction.Adiabatic elimination of the dump mode yields effective storage-mode dynamics κ2phD[â^2 − α^2].
  • Experimental precedent: Experiments have achieved κ2ph about 100 times larger than the natural single-photon loss rate, alongside signatures of exponential bit-flip suppression.These results support the feasibility of the stabilization regime assumed by the proposal.
  • Gate implementation: Photon-number-parity measurements have reached 98.5% fidelity, and the proposed X, CNOT, and Toffoli implementations are described as extensions of existing parametric methods.The X-gate construction varies the resonant-drive phase, while CNOT and Toffoli use time-dependent dissipators and related pump controls.
  • Projected performance: With cat size |α|^2 = 10 and a few tens of repetition modes per logical qubit, the authors expect logical error probabilities of order 10^-9 for a universal gate set.They expect these parameters to place operations below the repetition-code threshold.

VIII. CONCLUSIONS

The paper proposes replacing the surface code with a 1D repetition code built from cat qubits to obtain universal protected logical gates without magic states. Numerical analysis and experimental comparisons indicate promising below-threshold performance, with further error analysis left for future work.

  • Conclusions: A 1D repetition code with cat-qubit physical systems provides a universal set of topologically protected logical gates without magic-state preparation, distillation, or injection.The construction includes the nontrivial operations enabled by exploiting the cat qubits’ two-dimensional phase space.
  • Conclusions: The approach requires only minor modifications of previous experimental realizations and is expected to operate below the repetition-code accuracy threshold.The authors identify it as a candidate for demonstrating universal fully protected logical gates.
  • Conclusions: Logical error probabilities of approximately 10^-9 may be achievable with a few tens of cat qubits and mean photon number about 10.A more thorough effective-error analysis and optimal-decoding study is deferred to future work.

Appendix A: A no-go theorem for bias preserving quantum gates

The appendix defines bias-preserving two-qubit unitaries and proves that CNOT cannot be continuously implemented within the corresponding connected Lie subgroup. This establishes a two-level no-go result that motivates using the oscillator’s extra degree of freedom.

  • Scope and implication: The same analysis applies similarly to the Toffoli gate, while increasing the encoding dimension may approximately preserve the bias.The appendix identifies the finite-dimensional two-level obstruction and notes approximate preservation for higher-dimensional qudit encodings.
  • Bias-preserving gates: A bias-preserving unitary maps phase-flip errors only to combinations of Z1, Z2, and Z1Z2 errors.The definition requires [UZ1,2U †, Z1,2] = 0, equivalently restricting conjugated Z errors to the span of Z1, Z2, and Z1Z2.
  • Lie-group structure: The bias-preserving operations form a topologically closed Lie subgroup B of U(4), with connected identity component C.The proof establishes closure under products and inverses, then invokes topological closedness and Cartan’s theorem.
  • No-go theorem: CNOT is not a member of C, so it cannot be continuously obtained from the identity through bias-preserving processes on two two-level systems.The connected component’s Lie algebra is spanned by I, Z1, Z2, and Z1Z2, which excludes CNOT.
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