qSWIFT: High-order randomized compiler for Hamiltonian simulation

TL;DR

qSWIFT is a high-order randomized quantum algorithm for Hamiltonian simulation that significantly reduces the number of gates needed for high-precision results, outperforming previous methods like qDRIFT especially at very high accuracy levels.

Contribution

The paper introduces qSWIFT, a novel high-order randomized algorithm for Hamiltonian simulation with gate complexity independent of Hamiltonian size and exponentially reduced systematic error.

Findings

qSWIFT reduces gate count compared to qDRIFT for high-precision simulations.

The systematic error in qSWIFT decreases exponentially with the order parameter.

For a relative error of 10^{-6}, third-order qSWIFT requires 1000 times fewer gates than qDRIFT.

Abstract

Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in Hamiltonian, while the systematic error is exponentially reduced with regards to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), in which the number of gates scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. Particularly, the advantage is significant for problems where high precision is required; for example, to achieve a systematic relative propagation error of $10^{-6}$, the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT.