Abstract
Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.
Highlights
Special purpose quantum simulators permit the efficient implementation of unitary dynamics governed by physically meaningful families of Hamiltonians, while the general task is BQP-hard – since we can implement any quantum computation by a sequence of Hamiltonian evolutions
Recall that in a strong quantum simulation one computes the amplitude of a particular outcome, whereas in a weak simulation one only samples from the output distribution of a quantum circuit
Our approach is tackling a problem that is harder than the one solved by quantum computers
Summary
Special purpose quantum simulators permit the efficient implementation of unitary dynamics governed by physically meaningful families of Hamiltonians, while the general task is BQP-hard – since we can implement any quantum computation by a sequence of Hamiltonian evolutions. An important class of methods exploits the entanglement structure of the states to obtain efficient representations, known as tensor networks [VMC08; Orú14], that can be evolved either via Trotterisation [Vid04] or with a time-dependent variational principle [VGC04; Hae+11]. We conclude this introduction by noting that, from a computational perspective, the problem of simulating Hamiltonian dynamics is related, but different, to the problem of simulating quantum circuits. Recent techniques developed for this problem involve the notion of stabilizer rank [Bra+18] or neural network quantum states [JBC18]
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