Error Correction of Quantum Algorithms: Arbitrarily Accurate Recovery Of Noisy Quantum Signal Processing

TL;DR

This paper introduces a novel quantum algorithm error correction method using quantum signal processing, capable of arbitrarily suppressing certain errors at the algorithmic level with theoretical bounds on recovery sequence length.

Contribution

It presents the first approach to error correction at the quantum algorithm level via quantum signal processing, including error models, suppression techniques, and bounds on recovery sequence length.

Findings

Pauli Z-errors are not recoverable without extra resources.

Pauli X and Y errors can be arbitrarily suppressed with coherent recovery QSP.

The method is demonstrated on Grover's fixed-point search algorithm.

Abstract

The intrinsic probabilistic nature of quantum systems makes error correction or mitigation indispensable for quantum computation. While current error-correcting strategies focus on correcting errors in quantum states or quantum gates, these fine-grained error-correction methods can incur significant overhead for quantum algorithms of increasing complexity. We present a first step in achieving error correction at the level of quantum algorithms by combining a unified perspective on modern quantum algorithms via quantum signal processing (QSP). An error model of under- or over-rotation of the signal processing operator parameterized by $\epsilon < 1$ is introduced. It is shown that while Pauli $Z$-errors are not recoverable without additional resources, Pauli $X$ and $Y$ errors can be arbitrarily suppressed by coherently appending a noisy `recovery QSP.' Furthermore, it is found that a recovery QSP of length $O(2^k c^{k^2} d)$ is sufficient to correct any length-$d$ QSP with $c$ unique phases to $k^{th}$-order in error $\epsilon$. Allowing an additional assumption, a lower bound of $\Omega(cd)$ is shown, which is tight for $k = 1$, on the length of the recovery sequence. Our algorithmic-level error correction method is applied to Grover's fixed-point search algorithm as a demonstration.