Constant-Overhead Addressable Gates via Single-Shot Code Switching
Louis Golowich, Kathleen Chang, Guanyu Zhu
TL;DR
This work tackles the challenge of performing fault-tolerant, addressable and parallel logical operations on constant-rate quantum LDPC codes without incurring growing space-time overhead. The authors introduce a single-shot code-switching framework that transitions between r-dimensional and (r−1)-dimensional product codes built from lossless expanders, enabling versatile logical permutations and targeted gates while preserving encoded information. A new small-set flip decoder for high-dimensional product codes under locally stochastic noise yields a constant-threshold fault-tolerance regime and underpins state preparation, error correction, and switching gadgets. The construction supports constant-overhead implementations of Clifford gates (including Hadamard and CNOT) and scalable state preparation, with universality prospects via magic states or transversal non-Clifford gates discussed in extensions and appendices. Overall, the approach promises substantial overhead reductions for fault-tolerant quantum computation on qLDPC codes and advances toward practical, scalable universal quantum fault tolerance.
Abstract
It is a major challenge to perform addressable and parallel logical operations on constant-rate quantum LDPC (qLDPC) codes. Indeed, the overhead of targeting specific logical qubits represents a crucial bottleneck in many quantum fault-tolerance schemes. We introduce fault-tolerant protocols for performing various addressable as well as parallel logical operations with constant space-time overhead, on a family of constant-rate and polynomial-distance qLDPC codes. Specifically, we construct gadgets for a large class of permutations of logical qubits. We apply these logical permutations to construct gadgets for applying a targeted Hadamard (or $CNOT$) gate on any chosen logical qubit (pair). We also construct gadgets for preparing logical code states, and for applying Hadamard gates on all logical qubits in a codeblock. All of our gadgets use constant quantum space-time overhead along with polynomially bounded classical computation. Prior protocols for such operations required larger overhead, or else relied on codes with certain symmetries that lack known asymptotic constructions. Our codes are given by tensor products of classical codes constructed from lossless expander graphs. Our core technical contribution is a constant-overhead code-switching procedure between 2- and 3-dimensional product codes, which generalizes Bombin's dimensional jump (arXiv:1412.5079). We prove that all of our gadgets exhibit a constant threshold under locally stochastic noise. Along the way, we develop a small-set flip decoder for high-dimensional product codes from lossless expanders. Our techniques yield additional interesting consequences, such as single-shot state preparation of 2-dimensional product codes with constant space-time overhead. We also propose a method for performing parallel non-Clifford gates by extending our techniques to codes supporting transversal application of such gates.