Algorithmic-level Error Correction: Arbitrarily Accurate Recovery Of Noisy Quantum Signal Processing

The fundamentally stochastic behavior of quantum systems means that error correction and noise mitigation strategies are crucial for quantum computation. Most popular existing methods are fine-grained approaches, providing techniques to make more perfect gates from imperfect ones. Remarkably, modern classical computers almost completely eschew such fine-grained error correction, and in their place employ strategies which correct errors at the level of algorithms and protocols. Here we introduce the concept of algorithm-level error correction (ALEC) for quantum information processing, the defining feature of which is the design of subroutines that allow gate-level errors to cancel; therefore requiring a sophisticated understanding of how gate-level errors propagate to the output of algorithms. We demonstrate the first example ALEC for quantum signal processing (QSP) under the simple noise model of a consistent multiplicative under or over rotation in the signal processing operator by a fixed but unknown amount. We construct a recovery sequence, subject to the same noise, capable of suppressing the error to an arbitrary degree. Importantly, our method fits squarely within the noisy QSP model and does not require any additional resources. Finally, we provide an analysis of the query complexity of our recovery procedure.