TY - JOUR
T1 - Automated Detection of Symmetry-Protected Subspaces in Quantum Simulations
T2 - Article No. 033082
AU - Rotello, Caleb
AU - Jones, Eric
AU - Graf, Peter
AU - Kapit, Eliot
PY - 2023
Y1 - 2023
N2 - The analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and difficult to accomplish in general. Symmetries imply conservation laws, which partition Hilbert space into invariant subspaces of the time-evolution operator, each of which is demarcated according to its conserved quantity. We show that, starting from a chosen basis, any invariant, symmetry-protected subspaces which are diagonal in that basis are discoverable using transitive closure on graphs representing state-to-state transitions under k-local unitary operations. Importantly, the discovery of these subspaces relies neither upon the explicit identification of a symmetry operator or its eigenvalues nor upon the construction of matrices of the full Hilbert space dimension. We introduce two classical algorithms, which efficiently compute and elucidate features of these subspaces. The first algorithm explores the entire symmetry-protected subspace of an initial state in time complexity linear to the size of the subspace by closing local basis state-to-basis state transitions. The second algorithm determines, with bounded error, if a given measurement outcome of a dynamically generated state is within the symmetry-protected subspace of the state in which the dynamical system is initialized. We demonstrate the applicability of these algorithms by performing postselection on data generated from emulated noisy quantum simulations of three different dynamical systems: the Heisenberg-XXX model and the T6 and F4 quantum cellular automata. Due to their efficient computability and indifference to identifying the underlying symmetry, these algorithms lend themselves to the postselection of quantum computer data, optimized classical simulation of quantum systems, and discovery of previously hidden symmetries in quantum mechanical systems.
AB - The analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and difficult to accomplish in general. Symmetries imply conservation laws, which partition Hilbert space into invariant subspaces of the time-evolution operator, each of which is demarcated according to its conserved quantity. We show that, starting from a chosen basis, any invariant, symmetry-protected subspaces which are diagonal in that basis are discoverable using transitive closure on graphs representing state-to-state transitions under k-local unitary operations. Importantly, the discovery of these subspaces relies neither upon the explicit identification of a symmetry operator or its eigenvalues nor upon the construction of matrices of the full Hilbert space dimension. We introduce two classical algorithms, which efficiently compute and elucidate features of these subspaces. The first algorithm explores the entire symmetry-protected subspace of an initial state in time complexity linear to the size of the subspace by closing local basis state-to-basis state transitions. The second algorithm determines, with bounded error, if a given measurement outcome of a dynamically generated state is within the symmetry-protected subspace of the state in which the dynamical system is initialized. We demonstrate the applicability of these algorithms by performing postselection on data generated from emulated noisy quantum simulations of three different dynamical systems: the Heisenberg-XXX model and the T6 and F4 quantum cellular automata. Due to their efficient computability and indifference to identifying the underlying symmetry, these algorithms lend themselves to the postselection of quantum computer data, optimized classical simulation of quantum systems, and discovery of previously hidden symmetries in quantum mechanical systems.
KW - quantum computing
KW - quantum information
KW - quantum physics
UR - http://www.scopus.com/inward/record.url?scp=85167868636&partnerID=8YFLogxK
U2 - 10.1103/PhysRevResearch.5.033082
DO - 10.1103/PhysRevResearch.5.033082
M3 - Article
SN - 2643-1564
VL - 5
JO - Physical Review Research
JF - Physical Review Research
IS - 3
ER -