Efficient classical algorithms for simulating symmetric quantum systems

Eric R. Anschuetz1, Andreas Bauer2, Bobak T. Kiani3, and Seth Lloyd4,5

1MIT Center for Theoretical Physics, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
2Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
3MIT Department of Electrical Engineering and Computer Science, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
4MIT Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
5Turing Inc., Cambridge, MA 02139, USA

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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.

We investigate whether the presence of symmetries in quantum systems can make them more amenable to analysis by classical algorithms. We show that classical algorithms can efficiently compute a variety of static and dynamical properties of quantum models with large symmetry groups; we focus on the permutation group as a specific example of such a symmetry group. Our algorithms, which run in time polynomial in the system size and are adaptable to various quantum state inputs, challenge the perceived necessity of using quantum computation to study these models and open new avenues for using classical computation to study quantum systems.

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