Quantum Error Correction Exploiting Quantum Spatial Distribution and Gauge Symmetry

TL;DR

This paper investigates how quantum spatial distribution and gauge symmetry can be combined within stabilizer formalism to enhance quantum error correction, demonstrating resilience against various noise types and flexible system architecture.

Contribution

It introduces a novel error correction scheme leveraging quantum spatial distribution and gauge symmetry, improving noise resilience and architectural flexibility.

Findings

Gauge symmetry offers resilience against multiple noise types.

The scheme corrects a unified noise model involving decoherence and dephasing.

Implementations include error detection, logical gates, and quantum addition with local interactions.

Abstract

We explore what the integrated use of quantum spatial distribution (QSD), or more specifically, superposition of both spin and position states of particles, and gauge symmetry (GS) within stabilizer formalism provides for quantum error correction. The exploration employs $3+2$ particles on nested squares proposed in the companion letter (arXiv:2504.07941), where three of them encode Shor's nine-qubit code and the remaining two detect errors in this code through their spin state measurements (unlike the letter's quantum walk model, each particle evolves by gate operations acting exclusively on either its spin or position state). The first result is that the GS offers resilience against three types of noise acting on a particle: arbitrary decoherence of its spin or position state, and dephasing of both states, which partly or completely destroys its QSD. To show that, we formulate a noise model unifying the above noise and prove the correctability of this unified model under our error-correcting scheme. The second result is that QSD provides architectural flexibility allowing us to stack the error-correcting systems both vertically and horizontally. Indeed, we show implementations of the error detection (stabilizer measurement), logical Hadamard and Toffoli gates, and a quantum adder with the required interactions only between nearest-neighbor and next-nearest-neighbor particles.