Automated Detection of Symmetry-Protected Subspaces in Quantum Simulations

Caleb Rotello, Eric Jones, Peter Graf, Eliot Kapit

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Article number033082
Number of pages22
JournalPhysical Review Research
Issue number3
StatePublished - Jul 2023

Bibliographical note

Publisher Copyright:
© 2023 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the ""Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the National Renewable Energy Laboratory (NREL) Library, part of a national laboratory of the U.S. Department of Energy.

NREL Publication Number

  • NREL/JA-2C00-85382


  • quantum computing
  • quantum information
  • quantum physics


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