• Open Access

Yijia Xu (许逸葭)1,2, Yixu Wang (王亦许)3,4,1, Christophe Vuillot5, and Victor V. Albert1

• 1Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
• 2Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
• 3Institute for Advanced Study, Tsinghua University, Beijing 100084, China
• 4Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China
• 5Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Abstract

Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode’s occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a Calderbank-Shor-Steane-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional code spaces and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of χ-squared encodings, while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogs of repetition codes, the Shor code, and a surface-code-like construction that is not a concatenation of a known cat code with the qubit surface code. Code words are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.

• Quantum error correction
• Quantum information processing
• Quantum information processing with continuous variables

Popular Summary

Quantum error correction is essential for protecting fragile quantum information from environmental noise. While codes for qubits—the standard two-level quantum systems—are well developed, constructing codes for harmonic oscillators, where each oscillator has infinitely many energy levels, has been piecemeal and limited. Previous approaches often relied on combining qubit and oscillator codes, leaving much of the oscillator’s natural structure unused. In this work, we address this gap by introducing tiger codes, a unified framework for designing quantum codes directly in oscillator systems.

Tiger codes are defined using two integer matrices defining in turn a homology group—an algebraic object capturing the global structure of the system—which determines the logical operations of the code. By extending homology-based methods from qubit codes, which rely on binary arithmetic, to an integer setting, we construct multimode bosonic codes without resorting to concatenation. This approach allows us to exploit the oscillator’s full potential while maintaining favorable error-correcting properties. Importantly, tiger codes provide efficient, linear-algebraic tools for analyzing their robustness, and each code is associated with a unique generating function that captures its performance characteristics.

This framework not only unifies previously scattered examples of oscillator codes but also generates entirely new ones with favorable error-correcting properties and resource overhead. By connecting quantum error correction with deeper ideas in algebra and geometry, tiger codes chart a systematic path for harnessing harmonic oscillators in scalable quantum information processing. Looking forward, this approach may also illuminate the role of oscillator codes in many-body quantum physics, further expanding their potential applications.

Article Text

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