Quantum computing in spin-adapted representations for efficient simulations of spin systems

TL;DR

This paper introduces a new quantum algorithm formalism that efficiently exploits non-Abelian symmetries, like total-spin, to simulate spin systems with reduced computational complexity and improved accuracy.

Contribution

It develops a novel formalism for quantum algorithms in total-spin eigenbasis using symmetric group methods and truncation, enabling efficient simulation of spin systems.

Findings

Hierarchy of spin-adapted Hamiltonians converges quickly to exact solutions.

Sparse, local qubit Hamiltonians suitable for quantum simulation are constructed.

Shallow quantum circuits in total-spin basis effectively approximate ground states.

Abstract

Exploiting inherent symmetries is a common and effective approach to speed up the simulation of quantum systems. However, efficiently accounting for non-Abelian symmetries, such as the $SU(2)$ total-spin symmetry, remains a major challenge. In fact, expressing total-spin eigenstates in terms of the computational basis can require an exponentially large number of coefficients. In this work, we introduce a novel formalism for designing quantum algorithms directly in an eigenbasis of the total-spin operator. Our strategy relies on the symmetric group approach in conjunction with a truncation scheme for the internal degrees of freedom of total-spin eigenstates. For the case of the antiferromagnetic Heisenberg model, we show that this formalism yields a hierarchy of spin-adapted Hamiltonians, for each truncation threshold, whose ground-state energy and wave function quickly converge to their exact counterparts, calculated on the full model. These truncated Hamiltonians can be encoded with sparse and local qubit Hamiltonians that are suitable for quantum simulations. We demonstrate this by developing a state-preparation schedule to construct shallow quantum-circuit approximations, expressed in a total-spin eigenbasis, for the ground states of the Heisenberg Hamiltonian in different symmetry sectors.