Efficient classical algorithms for simulating symmetric quantum systems

TL;DR

This paper presents classical algorithms that efficiently simulate certain symmetric quantum systems by leveraging tensor-network methods and symmetry properties, achieving polynomial runtime for ground states and dynamics.

Contribution

It introduces classical algorithms that emulate quantum systems with permutation symmetry using tensor-network transformations and block-diagonalization techniques.

Findings

Classical algorithms achieve polynomial runtime for symmetric quantum systems.

Tensor-network methods enable efficient basis transformations.

Applicable to various input and output state representations.

Abstract

In light of recently proposed quantum algorithms that incorporate symmetries in the hope of quantum advantage, we show that with symmetries that are restrictive enough, classical algorithms can efficiently emulate their quantum counterparts given certain classical descriptions of the input. Specifically, we give classical algorithms that calculate ground states and time-evolved expectation values for permutation-invariant Hamiltonians specified in the symmetrized Pauli basis with runtimes polynomial in the system size. We use tensor-network methods to transform symmetry-equivariant operators to the block-diagonal Schur basis that is of polynomial size, and then perform exact matrix multiplication or diagonalization in this basis. These methods are adaptable to a wide range of input and output states including those prescribed in the Schur basis, as matrix product states, or as arbitrary quantum states when given the power to apply low depth circuits and single qubit measurements.