Constant-Overhead Injection into Quantum Codes

2026-09-30

AI Summary

This study addresses the high-overhead bottleneck of state injection and extraction in quantum error-correcting codes by proposing the first constant-overhead fault-tolerant scheme. Methodologically, we construct quantum codes via high-dimensional hypergraph products of classical LDPC codes, integrated with inner code concatenation. This approach achieves, for the first time, efficient bidirectional conversion between physical qubits and logical quantum states under constant space and time overheads. The proposed scheme supports single-shot execution and exhibits linear scalability, performing fault-tolerant operations at constant circuit depth under local stochastic noise models while maintaining extremely low logical error rates. Ultimately, this work establishes an efficient paradigm for state initialization and readout, advancing the practical realization of large-scale fault-tolerant quantum computation.

Abstract

We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side. We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.

Problem

• quantum error-correcting codes
• fault-tolerant state injection
• constant overhead
• locally stochastic noise
• LDPC codes

Innovation

• fault-tolerant state injection
• constant overhead
• quantum LDPC codes
• hypergraph product
• single-shot error correction