Abstract
Product formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation for product formulas of any given order, and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that the randomized approach has better empirical performance as well.
Highlights
Simulating quantum dynamics is one of the major potential applications of quantum computers
We have shown that randomization can be used to establish better performance for quantum simulation algorithms based on product formulas
By randomizing how the summands in the Hamiltonian are ordered, we introduce terms in the average evolution that could not appear in any deterministic product formula approximation of the same order, and thereby give a more efficient algorithm
Summary
Simulating quantum dynamics is one of the major potential applications of quantum computers. This bound follows by using the mixing lemma to combine an error bound for the average evolution operator with standard product formula error bounds for the error of the individual terms. In light of the large gap between proven and empirical performance of product formulas, it is natural to ask whether randomized product formulas still offer an improvement under the best possible error bounds
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