Scaling of symmetry-restricted quantum circuits

TL;DR

This paper explores the mathematical structure and properties of symmetry-restricted quantum circuits, specifically focusing on subspaces of the special unitary group invariant under certain symmetries, and provides methods to compute their dimensions.

Contribution

It introduces a combinatorial approach to determine the dimensions of symmetry-invariant subspaces of $SU(2^N)$ for permutation and Hamiltonian symmetries, extending understanding of symmetric quantum circuit structures.

Findings

Symmetry-invariant subspaces inherit topological and group properties from $SU(2^N)$.

A combinatorial method for computing subspace dimensions is developed.

Numerical results support the theoretical framework.

Abstract

The intrinsic symmetries of physical systems have been employed to reduce the number of degrees of freedom of systems, thereby simplifying computations. In this work, we investigate the properties of $\mathcal{M}SU(2^N)$, $\mathcal{M}$-invariant subspaces of the special unitary Lie group $SU(2^N)$ acting on $N$ qubits, for some $\mathcal{M}\subseteq M_{2^N}(\mathbb{C})$. We demonstrate that for certain choices of $\mathcal{M}$, the subset $\mathcal{M}SU(2^N)$ inherits many topological and group properties from $SU(2^N)$. We then present a combinatorial method for computing the dimension of such subspaces when $\mathcal{M}$ is a representation of a permutation group acting on qubits $(GSU(2^N))$, or a Hamiltonian $(H^{(N)}SU(2^N))$. The Kronecker product of $\mathfrak{su}(2)$ matrices is employed to construct the Lie algebras associated with different permutation-invariant groups $GSU(2^N)$. Numerical results on the number of dimensions support the the developed theory.