# Optimal digital dynamical decoupling for general decoherence via Walsh modulation

## TL;DR

This paper introduces a framework for designing optimal digital dynamical decoupling sequences using Walsh modulation, significantly improving efficiency over previous methods for mitigating qubit decoherence.

## Contribution

It establishes a novel connection between Walsh functions and concatenated projections, identifying a family of optimal sequences that outperform existing digital decoupling methods.

## Key findings

- Optimal Walsh sequences can be exponentially more efficient than known sequences.
- Sequences within the Walsh family can outperform concatenated decoupling under realistic constraints.
- An analytic upper bound on decoupling error is provided, validated by numerical simulations.

## Abstract

We provide a general framework for constructing digital dynamical decoupling sequences based on Walsh modulation --- applicable to arbitrary qubit decoherence scenarios. By establishing equivalence between decoupling design based on Walsh functions and on concatenated projections, we identify a family of {\em optimal Walsh sequences}, which can be exponentially more efficient, in terms of the required total pulse number, for fixed cancellation order, than known digital sequences based on concatenated design. Optimal sequences for a given cancellation order are highly non-unique --- their performance depending sensitively on the control path. We provide an analytic upper bound to the achievable decoupling error, and show how sequences within the optimal Walsh family can substantially outperform concatenated decoupling, while respecting realistic timing constraints. We validate these conclusions by numerically computing the average fidelity in a toy model capturing the essential feature of hyperfine-induced decoherence in a quantum dot.