Product Formulas for Exponentials of Commutators

Abstract

We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating many-body interaction Hamiltonians.

22 pages, 4 figures. Revised to include discussion of existing approximation building methods and an improved version of the approximation building method of Jean and Koseleff has been added

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• Scaling of errors in digitized counterdiabatic driving
• Double-bracket quantum algorithms for quantum imaginary-time evolution
• Rapid quantum approaches for combinatorial optimisation inspired by optimal state-transfer
• Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators
• Approximating exponentials of commutators by optimized product formulas
• Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas
• Efficient implementation of unitary transformations
• Double-bracket quantum algorithms for high-fidelity ground state preparation