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

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.