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ADoSPSiQS (Automated Detection of Symmetry-Protected Subspaces in Quantum Simulations ) [SWR-23-55]

Caleb Rotello, Peter A Graf

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.11578/dc.20240119.30

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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.
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.
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.
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).
Besides potentially delivering a huge leap in computational power, quantum computers also offer an essential platform for simulating properties of quantum systems. Consequently, various algorithms have been developed for approximating the dynamics of a target system on quantum computers. But generic quantum simulation algorithms—developed to simulate all Hamiltonians—are unlikely to result in optimal simulations of most physically relevant systems; optimal quantum algorithms need to exploit unique properties of target systems. The aim of this dissertation is to study two prominent properties of physical systems, namely locality and symmetry, and subsequently leverage these properties to design efficient quantum simulation algorithms. In the first part of the dissertation, we explore the locality of quantum systems and the fundamental limits on the propagation of information in power-law interacting systems. In particular, we prove upper limits on the speed at which information can propagate in power-law systems. We also demonstrate how such speed limits can be achieved by protocols for transferring quantum information and generating quantum entanglement. We then use these speed limits to constrain the propagation of error and improve the performance of digital quantum simulation. Additionally, we consider the implications of the speed limits on entanglement generation, the dynamics of correlation, the heating time, and the scrambling time in power-law interacting systems. In the second part of the dissertation, we propose a scheme to leverage the symmetry of target systems to suppress error in digital quantum simulation. We first study a phenomenon called destructive error interference, where the errors from different steps of a simulation cancel out each other. We then show that one can induce the destructive error interference by interweaving the simulation with unitary transformations generated by the symmetry of the target system, effectively providing a faster quantum simulation by symmetry protection. We also derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection.
Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.
We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing U(1)U(1)U(1) spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by 8 ∥H∥(1−αDT)1/2\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}8​∥H∥(1−αDT​​​)1/2, where αDTα_{D_T}αDT​​ is the probability weight captured by the DTD_TDT​ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-1/21/21/2 Heisenberg XXZ chain (up to N=62N=62N=62 qubits on the IBM Heron processor), the algorithm achieves a 3.5%3.5\%3.5% energy error in the gapped Neel phase (Δ=2.0Δ=2.0Δ=2.0) and below 0.5%0.5\%0.5% at the ferromagnetic boundary (Δ=−1.0Δ=-1.0Δ=−1.0). The accuracy degrades to 28.7%28.7\%28.7% in the strongly quasi-long-range-ordered regime (Δ=0.5Δ=0.5Δ=0.5). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.