R. Rousseau, Diego Ruiz, E. Albertinale, P. d’Avezac, D. Banys, U. Blandin, N. Bourdaud, G. Campanaro, G. Cardoso, N. Cottet, C. Cullip, S. Deléglise, L. Devanz, A. Devulder, Antoine Essig, P. Février, A. Gicquel, Élie Gouzien, A. Gras, J. Guillaud, E. Gümüş, M. Hallén, A. Jacob, P. Magnard, A. Marquet, S. Miklass, T. Peronnin, S. Polis, F. Rautschke, U. Réglade, J. Roul, Jeremy Stevens, Jeanne Solard, A. Thomas, J.-L. Ville, P. Wan-Fat, Raphaël Lescanne, Z. Leghtas, J. Cohen, S. Jezouin, A. Murani
Bosonic qubits implement error correction by harnessing the high-dimensional Hilbert space of a harmonic oscillator to redundantly encode logical quantum information. Squeezing of quantum states — a cornerstone of precision metrology — redistributes uncertainty between conjugate quadratures to enhance sensitivity in one variable at the expense of the other. In bosonic codes, decoherence can be seen as the environment performing an unwanted measurement on a noise-sensitive quadrature, collapsing superpositions and inducing errors. By redistributing uncertainty to that quadrature, squeezing can be leveraged to reduce the environment’s measurement strength, thereby prolonging coherence. Here, we exploit this insight to implement a squeezing deformation of the simplest bosonic code — the cat qubit — extending the bit-flip time with minimal impact on the phase-flip rate. We observe a dramatic reduction in bit-flip errors, achieving a 160-fold improvement over standard cat qubits and bringing multi-cat architectures closer to fault-tolerant quantum computation. In bosonic codes, decoherence can be seen as the environment performing an unwanted measurement on a noise-sensitive quadrature, collapsing superpositions and inducing errors. Here, the authors show how to use squeezing to further protect bosonic codes from noise, extending the bit-flip times of the cat qubit by two orders of magnitude without reducing phase-flip times.
Bosonic qubits implement error correction by harnessing the high-dimensional Hilbert space of a harmonic oscillator to redundantly encode logical quantum information. Squeezing of quantum states — a cornerstone of precision metrology — redistributes uncertainty between conjugate quadratures to enhance sensitivity in one variable at the expense of the other. In bosonic codes, decoherence can be seen as the environment performing an unwanted measurement on a noise-sensitive quadrature, collapsing superpositions and inducing errors. By redistributing uncertainty to that quadrature, squeezing can be leveraged to reduce the environment’s measurement strength, thereby prolonging coherence. Here, we exploit this insight to implement a squeezing deformation of the simplest bosonic code — the cat qubit — extending the bit-flip time with minimal impact on the phase-flip rate. We observe a dramatic reduction in bit-flip errors, achieving a 160-fold improvement over standard cat qubits and bringing multi-cat architectures closer to fault-tolerant quantum computation. In bosonic codes, decoherence can be seen as the environment performing an unwanted measurement on a noise-sensitive quadrature, collapsing superpositions and inducing errors. Here, the authors show how to use squeezing to further protect bosonic codes from noise, extending the bit-flip times of the cat qubit by two orders of magnitude without reducing phase-flip times.
Dissipative cat-qubits are a promising architecture for quantum processors due to their built-in quantum error correction. By leveraging two-photon stabilization, they achieve an exponentially suppressed bit-flip error rate as the distance in phase-space between their basis states increases, incurring only a linear increase in phase-flip rate. This property substantially reduces the number of qubits required for fault-tolerant quantum computation. Here, we implement a squeezing deformation of the cat qubit basis states, further extending the bit-flip time while minimally affecting the phase-flip rate. We demonstrate a steep reduction in the bit-flip error rate with increasing mean photon number, characterized by a scaling exponent γ=4.3, rising by a factor of 74 per added photon. Specifically, we measure bit-flip times of 22 seconds for a phase-flip time of 1.3 μs in a squeezed cat qubit with an average photon number nˉ=4.1, a 160-fold improvement in bit-flip time compared to a standard cat. Moreover, we demonstrate a two-fold reduction in Z-gate infidelity, with an estimated phase-flip probability of εX=0.085 and a bit-flip probability of εZ=2.65⋅10−9 which confirms the gate bias-preserving property. This simple yet effective technique enhances cat qubit performances without requiring design modification, moving multi-cat architectures closer to fault-tolerant quantum computation.
Bosonic quantum codes redundantly encode quantum information in the states of a quantum harmonic oscillator, making it possible to detect and correct errors. Schr\"odinger cat codes---based on the superposition of two coherent states with opposite displacements---can correct phase-flip errors induced by dephasing, but they are vulnerable to bit-flip errors induced by particle loss. Here, we develop a bosonic quantum code relying on squeezed cat states, i.e., cat states made of a linear superposition of displaced-squeezed states. Squeezed cat states allow to partially correct errors caused by particle loss, while at the same time improving the protection against dephasing. We present a comprehensive analysis of the squeezed cat code, including protocols for code generation and elementary quantum gates. We characterize the effect of both particle loss and dephasing and develop an optimal recovery protocol that is suitable to be implemented on currently available quantum hardware. We show that with moderate squeezing, and using typical parameters of state-of-the-art quantum hardware platforms, the squeezed cat code has a resilience to particle loss errors that significantly outperforms that of the conventional cat code.
Noise-biased qubits are a promising route toward significantly reducing the hardware overhead associated with quantum error correction. The squeezed-cat code, a nonlocal encoding in phase space based on squeezed coherent states, is an example of a noise-biased (bosonic) qubit with exponential error bias. Here we propose and analyze the error correction performance of a dissipatively stabilized squeezed-cat qubit. We find that for moderate squeezing the bit-flip error rate gets significantly reduced in comparison with the ordinary cat qubit while leaving the phase-flip rate unchanged. Additionally, we find that the squeezing enables faster and higher-fidelity gates.
To solve problems of practical importance1,2, quantum computers probably need to incorporate quantum error correction, in which a logical qubit is redundantly encoded in many noisy physical qubits3–5. The large physical-qubit overhead associated with error correction motivates the search for more hardware-efficient approaches6–18. Here, using a superconducting quantum circuit19, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance d = 5 (ref. 10). A stabilizing circuit passively protects cat qubits against bit flips20–24. The repetition code, using ancilla transmons for syndrome measurement, corrects cat qubit phase flips. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below the threshold. The logical bit-flip error is suppressed with increasing cat qubit mean photon number, enabled by our realization of a cat-transmon noise-biased CX gate. The minimum measured logical error per cycle is on average 1.75(2)% for the distance-3 code sections, and 1.65(3)% for the distance-5 code. Despite the increased number of fault locations of the distance-5 code, the high degree of noise bias preserved during error correction enables comparable performance. These results, where the intrinsic error suppression of the bosonic encodings enables us to use a hardware-efficient outer error-correcting code, indicate that concatenated bosonic codes can be a compelling model for reaching fault-tolerant quantum computation. Bosonic qubits can be engineered to feature intrinsic protection against certain kinds of errors, which makes quantum error correction across many bosonic qubits possible with less overhead.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Quantum computing crucially relies on maintaining quantum coherence for the duration of a calculation. Bosonic quantum error correction protects this coherence by encoding qubits into superpositions of noise-resilient oscillator states. In the case of the Kerr-cat qubit (KCQ), these states derive their stability from being the quasi-degenerate ground states of an engineered Hamiltonian in a driven nonlinear oscillator. KCQs are experimentally compatible with on-chip architectures and high-fidelity operations, making them promising candidates for a scalable bosonic quantum processor. However, their bit-flip time must increase further to fully leverage these advantages. Here, we present direct evidence that the bit-flip time in a KCQ is limited by leakage out of the qubit manifold and experimentally mitigate this process. We coherently control the leakage population and measure it to be > 9%, twelve times higher than in the undriven system. We then cool this population back into the KCQ manifold with engineered dissipation, identify conditions under which this suppresses bit-flips, and demonstrate increased bit-flip times up to 3.6 milliseconds. By employing both Hamiltonian confinement and engineered dissipation, our experiment combines two paradigms for Schrödinger-cat qubit stabilization. Our results elucidate the interplay between these stabilization processes and indicate a path towards fully realizing the potential of these qubits for quantum error correction.