Letting the tiger out of its cage: bosonic coding without concatenation

TL;DR

This paper introduces a versatile framework for bosonic quantum codes using linear constraints and homological conditions, enabling fault-tolerant quantum computation with various code families and without concatenation.

Contribution

It develops a general construction of multimode bosonic codes based on integer matrices satisfying homological conditions, expanding the types of codes and stabilizing mechanisms available.

Findings

Constructed multimode codes with linear occupation-number constraints.

Demonstrated syndrome extraction compatible with superconducting circuits.

Included examples like analogues of repetition, Shor, and surface-like codes.

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 CSS-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 codespaces, 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 chi-squared encodings while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogues of repetition codes, the Shor code, and a surface-like code that is not a concatenation of a known cat code with the qubit surface code. Codewords 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.