Approximating Fractional Time Quantum Evolution

TL;DR

This paper introduces an algorithm to efficiently approximate fractional powers of an unknown unitary operation, enabling more resource-effective quantum computations involving non-integer powers.

Contribution

It presents a novel algorithm for approximating fractional powers of a black box unitary, with complexity analysis and practical efficiency improvements over direct methods.

Findings

The algorithm approximates $ ext{U}^t$ with controlled error.

For large $t$, it reduces the number of black box calls compared to repeated applications.

An example shows improved efficiency for large integer powers.

Abstract

An algorithm is presented for approximating arbitrary powers of a black box unitary operation, $\mathcal{U}^t$, where $t$ is a real number, and $\mathcal{U}$ is a black box implementing an unknown unitary. The complexity of this algorithm is calculated in terms of the number of calls to the black box, the errors in the approximation, and a certain `gap' parameter. For general $\mathcal{U}$ and large $t$, one should apply $\mathcal{U}$ a total of $\lfloor t \rfloor$ times followed by our procedure for approximating the fractional power $\mathcal{U}^{t-\lfloor t \rfloor}$. An example is also given where for large integers $t$ this method is more efficient than direct application of $t$ copies of $\mathcal{U}$. Further applications and related algorithms are also discussed.