Quantum Algorithms for Testing Hamiltonian Symmetry

Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.

Figure 1

Quantum circuit to test for the covariance of a unitary Hamiltonian evolution. The unitary $V^{\mathrm{\Phi }}$ generates the state $|\mathrm{\Phi }{⟩}_{RA}$, the maximally entangled state on $RA$. The evolution of the system is given by $e^{-iHt}=W_1{W}_2^{†}$, and the $U(g)$ gates are controlled on a superposition over all of the elements $g\in G$, as in (4).

Figure 2

Quantum circuit to test for the covariance of a unitary Hamiltonian evolution. Here, $\pi$ denotes the maximally mixed state $\mathbb{I}/d$.

Figure 3

Results of symmetry tests for the transverse Ising model for $N=3$ and $N=4$, using IBM Quantum’s noisy simulator. The symmetries in question are given by acting simultaneously on all systems by either the cyclic group of order $N$ or a conjugation by $({\sigma }^X{)}^{\otimes N}$.