Universality of Quantum Gates in Particle and Symmetry Constrained Subspaces

TL;DR

This paper proves that hardware-efficient quantum gates are universal for state preparation within constrained subspaces, using Lie algebra techniques, with applications to models like Fermi-Hubbard and CFT.

Contribution

It introduces a Lie algebraic framework demonstrating universality of gates in particle and symmetry constrained subspaces, extending to complex phases and providing verification criteria.

Findings

Proves universality of gates in fixed particle number and spin subspaces.

Extends results to complex phases enabling arbitrary state preparation.

Demonstrates applications to Fermi-Hubbard, Bose-Hubbard, and CFT models.

Abstract

Simulating physical systems on near-term quantum computers often requires preparing states within constrained subspaces, like those with fixed particle number or spin. We use Lie algebraic techniques to prove that hardware-efficient gates are universal for state preparation in these subspaces. The key mechanism is Pauli $Z$ dressing: commutators of overlapping gates produce Pauli $Z$ operators on shared qubits, acting as spectator projectors that decompose multi-plane rotations into single-plane generators spanning the full $\mathfrak{so}(w)$ algebra, where $w$ is the dimension of the constrained subspace, thereby guaranteeing universality for real state preparation. Adding independent complex phases extends this to $\mathfrak{su}(w)$, enabling arbitrary complex state preparation. We provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction on the target manifold from almost any parameter configuration. Our findings are applicable to many problem areas, including Fermi-Hubbard models, Bose-Hubbard models, and molecular electronic structure. We apply our framework to two physical settings: we prove the completeness of the binary encoded multi-level particles ansatz on the conserved-particle-number subspace, and we construct symmetry-preserving circuits for the fuzzy sphere regularisation of the 3D Ising conformal field theory (CFT). For the latter, we variationally prepare the ground and excited states to extract CFT scaling dimensions.