Kerr-cat Qubit Operations Below the Fault-tolerant Threshold
Bingcheng Qing, Ahmed Hajr, Ke Wang, Gerwin Koolstra, Long B. Nguyen, Jordan Hines, Irwin Huang, Bibek Bhandari, Zahra Pedramrazi, Larry Chen, Ziqi Kang, Christian Jünger, Noah Goss, Nikitha Jain, Hyunseong Linus Kim, Kan-Heng Lee, Akel Hashim, Nicholas Frattini, Justin Dressel, Andrew N. Jordan, David Santiago, Irfan A. Siddiqi
Abstract The ubiquitous noise in quantum system hinders the advancement of quantum information processing and has driven the emergence of different quantum error correction protocols. Among them, quantum error correction codes tailored for noise-biased qubits exhibit comparatively high error thresholds, making them a promising platform to achieve fault-tolerance. Nevertheless, their quantum operations are challenging and the demonstration of their performance beyond the fault-tolerant threshold remains incomplete. Here, we leverage Schrödinger cat states in scalable planar superconducting circuits to thoroughly characterize the high-fidelity single-qubit quantum operations on noise-biased qubits with systematic quantum tomography and benchmarking tools, demonstrating their state-of-the-art performance beyond the fault-tolerant threshold of the surface code. These results thus embody a transformative milestone in the exploration of quantum systems with structured noises. Notably, our framework is extensible to other structured-noise systems, paving the way for systematic characterization and validation of novel quantum platforms with structured noise.
Abstract The ubiquitous noise in quantum system hinders the advancement of quantum information processing and has driven the emergence of different quantum error correction protocols. Among them, quantum error correction codes tailored for noise-biased qubits exhibit comparatively high error thresholds, making them a promising platform to achieve fault-tolerance. Nevertheless, their quantum operations are challenging and the demonstration of their performance beyond the fault-tolerant threshold remains incomplete. Here, we leverage Schrödinger cat states in scalable planar superconducting circuits to thoroughly characterize the high-fidelity single-qubit quantum operations on noise-biased qubits with systematic quantum tomography and benchmarking tools, demonstrating their state-of-the-art performance beyond the fault-tolerant threshold of the surface code. These results thus embody a transformative milestone in the exploration of quantum systems with structured noises. Notably, our framework is extensible to other structured-noise systems, paving the way for systematic characterization and validation of novel quantum platforms with structured noise.
Ubiquitous noise sources in quantum systems remain a key obstacle to building quantum computers, necessitating the use of quantum error correction codes. Recently, error-correcting codes tailored for noise-biased systems have been shown to offer high fault-tolerance thresholds and reduced hardware overhead, positioning noise-biased qubits as promising candidates for building universal quantum computers. However, quantum operations on these platforms remain challenging, and their noise structures have not yet been rigorously benchmarked to the same extent as those of conventional quantum hardware. In this work, we develop a comprehensive quantum control toolbox for a scalable noise-biased qubit, detuned Kerr-cat qubit, including initialization, universal single-qubit gates, and quantum nondemolition readout. We systematically characterize the noise structure of these operations using gate set tomography and dihedral randomized benchmarking, achieving high local gate fidelities, with [Formula: see text] and [Formula: see text]. Notably, the noise bias of the detuned Kerr-cat qubit approaches 250 with a phase-flip time of [Formula: see text], which outperforms its resonant-Kerr-cat qubit counterparts as reported previously, representing a state-of-the-art performance benchmark for Kerr-cat qubits. Moreover, our results reveal a critical overestimation of operational noise bias inferred from bit-flip and phase-flip times alone, highlighting the necessity of a precise and direct benchmarking for noise-biased qubit operations. Our work thus establishes a framework for systematically characterizing and validating the performance of quantum operations in structured-noise architectures, which lays the groundwork for implementing efficient quantum error correction in next-generation architectures.
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 promises exponential advantages for simulating quantum systems and solving certain computational problems beyond the reach of classical computers, but the fragility of qubits and the resulting errors present a major challenge for scalable implementations. One promising approach is to engineer qubits whose physical noise processes are intrinsically biased, enabling more efficient quantum error correction. This dissertation investigates quantum computing with superconducting Kerr-cat qubits, a bosonic qubit architecture in which qubit states are encoded in coherent states of a nonlinear oscillator stabilized by a two-photon drive. This stabilization creates an effective double-well pseudo-potential that strongly suppresses bit-flip errors while enabling universal quantum control, leading to a noise-biased qubit. A superconducting device architecture based on SNAIL nonlinear oscillators is developed to realize Kerr-cat qubits with tunable parameters and high coherence, integrating readout resonators, dedicated filters, and microwave control lines for measurement and manipulation. Universal single-qubit gates are implemented using engineered single-photon and two-photon drives to deform the pseudo-potential, and a cat-quadrature readout method is employed to achieve high-fidelity quantum nondemolition measurement. The performance of the qubits is systematically characterized through lifetime measurements and benchmarking protocols, including quantum process tomography, gate set tomography, and dihedral randomized benchmarking, revealing a strongly biased noise structure with high performance. These results establish an experimental framework for the design, control, and characterization of Kerr-cat qubits and demonstrate their potential as hardware-efficient building blocks for scalable fault-tolerant quantum computing.
The development of robust architectures capable of large-scale fault-tolerant quantum computation should consider both their quantum error-correcting codes and the underlying physical qubits upon which they are built, in tandem.Following this design principle, we demonstrate remarkable error-correction performance by concatenating the XZZX surface code with Kerr-cat qubits.We contrast several variants of fault-tolerant systems undergoing different circuit-noise models that reflect the physics of Kerr-cat qubits.Our simulations show that our system is scalable below a threshold gate infidelity of p CX ∼ 6.5% within a physically reasonable parameter regime, where p CX is the infidelity of the noisiest gate of our system, the controlled-NOT gate.This threshold can be reached in a superconducting-circuit architecture with a Kerr nonlinearity of 10MHz, an approximately 6.25-photon cat qubit, single-photon lifetime of 64 μs, and a thermal photon population 8%.Such parameters are routinely achieved in superconducting circuits.
Abstract Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum error correction (FTQEC) to compute reliably. Quantum error correction can protect against general noise if—and only if—the error in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking, which reports a different error rate that is not sensitive to all errors, and cannot be compared directly to diamond norm thresholds. Here we use gate set tomography to completely characterize operations on a trapped-Yb + -ion qubit and demonstrate with greater than 95% confidence that they satisfy a rigorous threshold for FTQEC (diamond norm ≤6.7 × 10 −4 ).
We present a systematic study of Schrödinger cat codes constructed from Kerr-type coherent states, including displaced Kerr coherent states and Barut--Girardello Kerr coherent states, each admitting two distinct families determined by the sign of the Kerr nonlinearity. By tuning the Kerr parameter and coherent-state amplitude, these states interpolate between su(2), su(1,1) coherent states, providing a unified and versatile foundation for this type of bosonic quantum error correction. Unlike standard two-component Schrödinger cat codes, where a single photon-loss event induces an uncorrectable bit-flip, the nonlinear phase-space structure of Kerr cat states enables simultaneous detection and correction of both photon-loss and dephasing errors within a unified recovery framework, with optimal recovery operations determined via convex optimization. We demonstrate that Kerr cat encodings significantly outperform conventional cat codes under combined loss and dephasing noise, and that judicious parameter optimization can suppress both error channels to a level that reduces the overhead of additional error correction layers. We further show that Kerr-deformed coherent-state manifolds under engineered two-photon driving emerge as effective steady states of driven-dissipative dynamics, with single-photon decoherence strongly suppressed and leakage outside the protected manifold appearing only as higher-order corrections in the deformation strength. Our extended formalism identifies generalized Kerr Schrödinger cat codes as promising candidates for fault-tolerant bosonic quantum computation in experimental platforms such as nonlinear photonics.