Approximating Fractional Time Quantum Evolution

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.