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arXiv:2302.08586 · Physical Review Research

Automated detection of symmetry-protected subspaces in quantum simulations

Caleb Rotello, Eric B. Jones, Peter A Graf, Eliot Kapit

Published Aug 4, 2023• 2 citations
arXivHTMLPDF

Abstract

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 ${T}_{6}$ and ${F}_{4}$ 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.

DOI
10.1103/physrevresearch.5.033082

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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 kkk-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{T}_{6}T6​ and F4{F}_{4}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.
Symmetry-protected subspaces in quantum systems are subspaces demarcated by a conserved quantity of that quantum system. 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. This software is paired with the paper of the same name at https://arxiv.org/pdf/2302.08586.pdf.
Simulating the dynamics of quantum systems is an important application of quantum computers and has seen a variety of implementations on current hardware.We show that by introducing quantum gates implementing unitary transformations generated by the symmetries of the system, one can induce destructive interference between the errors from different steps of the simulation, effectively giving faster quantum simulation by symmetry protection.We derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection.In particular, when the symmetry transformations are chosen as powers of a unitary, the error of the algorithm is approximately projected to the so-called quantum Zeno subspaces.We prove a bound on this approximation error, exponentially improving a recent result of Burgarth, Facchi, Gramegna, and Pascazio.We apply the symmetry-protection technique to the simulations of the XXZ Heisenberg interactions with local disorder and the Schwinger model in quantum field theory.For both systems, the technique can reduce the simulation error by several orders of magnitude over the unprotected simulation.Finally, we provide numerical evidence suggesting that the technique can also protect simulation against other types of coherent, temporally correlated errors, such as the 1/f noise commonly found in solid-state experiments.
Quantum many‑body systems often display emergent stability that cannot be explained by any obvious symmetry of the microscopic Hamiltonian. In this work we propose a systematic framework, the Hidden Conservation Law Search in Quantum Matter (HCLQM), for uncovering unknown conserved quantities QhQ_hQh​ that may underlie such anomalous behavior. We develop an algebraic criterion that any operator QhQ_hQh​ must satisfy in order to commute with the full Hamiltonian HHH while not belonging to the algebra generated by known symmetries. Using this criterion we design a numerical search algorithm based on tensor‑network representations of the Hilbert space, enabling us to explore large Hilbert spaces efficiently. We apply HCLQM to three representative systems: a one‑dimensional spin‑1/2 XXZ chain, a two‑dimensional Chern insulator with disorder, and a Hubbard‑type strongly correlated lattice model. In each case we discover non‑trivial conserved operators that explain the presence of long‑lived excited states, protected edge modes, and anomalous transport properties. These findings demonstrate that hidden conserved quantities can be systematically identified and may provide a route to discovering new symmetries in quantum materials, potentially leading to Nobel‑level breakthroughs.
In closed systems, dynamical symmetries lead to conservation laws. However, conservation laws are not applicable to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. The usual approach to the problem of finding such rules relies heavily on group theory and involves a detailed study of the structure of the respective symmetry group. We approach the problem in a completely new way by using entanglement to investigate the asymmetry properties of quantum states. To this end, we embed the space state of the system in a tensor product Hilbert space, whereby symmetric transformations between two states are replaced with local operations on their bipartite images.The embedding enables us to use the well-studied theory of entanglement to investigate the consequences of dynamic symmetries. Moreover, under reversible transformations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators. Our method is not group-specific and applies to general symmetries associated with any semi-simple Lie group.
In strongly correlated quantum systems, the presence of conserved quantities beyond the obvious symmetries often leads to unexpected robustness of certain phases and excitations. We propose a systematic framework to search for hidden conserved operators, denoted Q_h, that commute with the full Hamiltonian H of a complex material. The core idea is that such hidden conservation laws may underpin anomalously stable states that defy conventional classification, potentially opening new avenues for topological quantum computation and material design. We formulate the problem as a constrained eigenvalue optimization, derive analytic expressions for candidate Q_h in representative lattice models, and introduce an iterative operator construction algorithm that exploits the algebraic structure of the spin, fermionic, and bosonic operators involved. By applying the method to a set of benchmark systems—spin chains with next‑nearest‑neighbour interactions, Hubbard‑type models with spin–orbit coupling, and coupled spin‑boson lattices—we demonstrate the emergence of non‑trivial conserved operators that are not linear combinations of known symmetries. The discovery of these hidden laws not only clarifies the stability of exotic phases such as quantum spin liquids and fractional topological insulators but also suggests that a deeper symmetry principle may underlie many unexplained experimental observations. Finally, we discuss the implications of our findings for future theoretical developments and the pursuit of a Nobel‑level breakthrough in the field of quantum materials.
Our works [1–7] focus on algorithm development for current and near-term quantum computing platforms. Symmetries, which are mathematically described by groups, divide the full space of possibilities, the Hilbert space, into smaller sectors. If a symmetry is conserved within an application, then one of these smaller symmetry sectors will be sufficient to model it. Therefore, our works incorporate symmetries like translational invariance [1, 2, 5, 7], time-periodicity [3, 4] or electron number conservation [6] to improve algorithms for relevant system classes compared to symmetry-unaware standard methods. In the first direction, we utilise symmetries for Hamiltonian simulation, where one aims to approximate the time evolution of a reproduced quantum system of interest on a quantum computer and predict the time-dependent properties of the system, such as correlations. Concretely, we improve Hamiltonian simulation algorithms for time-periodic systems by taking advantage of high-frequency expansions in works [3, 4]. In Ref. [5], we reduce the approximation error of product formulas by classical variational optimisation, and translational symmetry to scale up the system size n. The second direction is state preparation, where one targets the recreation of a quantum state, typically a ground state, and the prediction of its properties that are too difficult to compute classically. In Ref. [6], we take advantage of the electron number conservation of fermionic systems for a symmetry-aware circuit ansatz and self-consistent configuration recovery. Paired with warm-starting and classical subsystem diagonalisation, it allows us to estimate the Fe(III)-NTA ground state energy within chemical accuracy via samples obtained from an ion-trap quantum computer. The third direction is to use symmetry for the reduction of statistical uncertainty of quantum estimators, which are computed from measured bitstring values at the end of a quantum computation. Concretely, in Refs. [1, 2], we improve the sampling complexity for energy estimation of quantum lattice systems, which can be mapped efficiently to current quantum hardware. Our approach leverages translational invariance to construct more sample-efficient estimators, where their minimal improvement compared to local Pauli bases is guaranteed by an analytical lower bound. Further in Refs. [3, 4], we take advantage of the translational invariance of both the quantum state and observable to boost the number of observed bitstring values per ion trap hardware sample by the number of qubits. [1] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, in Quantum Computing Theory in Practice 2025 (Freie Universität Berlin, 2025). [2] T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, arXiv preprint arXiv:2410.21251 10.48550/arXiv.2410.21251 (2024). [3] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, in Quantum Computing Theory in Practice 2024 (University of Edinburgh, 2024). [4] T. Eckstein, R. Mansuroglu, P. Czarnik, J.-X. Zhu, M. J. Hartmann, L. Cincio, A. T. Sornborger, and Z. Holmes, npj Quantum Information 10, 84 (2024). [5] R. Mansuroglu, T. Eckstein, L. N¨utzel, S. A. Wilkinson, and M. J. Hartmann, Quantum Science and Technology 8, 025006 (2023). [6] L. Nützel, A. Gresch, L. Hehn, L. Marti, R. Freund, A. Steiner, C. D. Marciniak, T. Eckstein, N. Stockinger, S. Wolf, et al., Quantum Science and Technology 10, 015066 (2025). [7] S. Tasler, J. Old, L. Heunisch, V. Feulner, T. Eckstein, M. Müller, and M. J. Hartmann, arXiv preprint arXiv:2506.09028 10.48550/arXiv.2506.09028 (2025).