Reducing measurements in quantum erasure correction by quantum local recovery

TL;DR

This paper formalizes the minimal set of stabilizer measurements needed for quantum erasure correction, reducing measurement costs by leveraging quantum local recovery techniques, with practical implications for surface codes.

Contribution

It introduces a formal framework for identifying relevant stabilizers in quantum erasure correction, minimizing measurements using quantum local recovery, and applies this to generalized surface codes.

Findings

Minimal stabilizer measurements are sufficient for erasure correction.

Correction of δ erasures requires at most δ vertex and δ face measurements.

Measurement reduction is independent of code parameters.

Abstract

As measurements are costly and prone to errors on certain quantum computing devices, we should reduce the number of measurements and the number of measured qudits as small as possible in quantum erasure correction. It is intuitively obvious that a decoder can omit measurements of stabilizers that are irrelevant to erased qudits, but this intuition has not been rigorously formalized as far as the author is aware. In this paper, we formalize relevant stabilizers sufficient to correct erased qudits with a quantum stabilizer code, by using a recent idea from quantum local recovery. The minimum required number of measured stabilizer observables is also clarified. As an application, we also show that correction of $\delta$ erasures on a generalized surface code proposed by Delfosse, Iyer and Poulin requires at most $\delta$ measurements of vertexes and at most $\delta$ measurements of faces, independently of its code parameters.