Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers: A quantum many-body perspective

Abstract

We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.

Revides version, submitted to Eur. Phys. J. A in the topical issue on "Quantum computing in low-energy nuclear theory"

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• Quantum Magic and Multi-Partite Entanglement in the Structure of Nuclei
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• Stabilizer-Accelerated Quantum Many-Body Ground-State Estimation
• Semicoherent Symmetric Quantum Processes: Theory and Applications
• Toward scalable quantum computations of atomic nuclei
• Simplified projection on total spin zero for state preparation on quantum computers
• Controlled Gate Networks: Theory and Application to Eigenvalue Estimation
• Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions
• TabularQGAN: A quantum generative model for tabular data synthesis
• Accelerated spin-adapted ground state preparation with non-variational quantum algorithms