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arXiv:2502.01464

Predicting symmetries of quantum dynamics with optimal samples

Masahito Hayashi, Yu-Ao Chen, Chenghong Zhu, Xin Wang

Published Feb 3, 2025• 0 citations
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Abstract

Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by $δ$, we establish the explicit optimal sample complexity which scales as $\mathcal{O}(δ^{-1/3})$ for identity testing and $\mathcal{O}(δ^{-1/2})$ for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.

DOI
10.48550/arxiv.2502.01464

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Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by δδδ, we establish the explicit optimal sample complexity which scales as O(δ−1/3)\mathcal{O}(δ^{-1/3})O(δ−1/3) for identity testing and O(δ−1/2)\mathcal{O}(δ^{-1/2})O(δ−1/2) for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.
Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (TTT symmetry) and diagonal symmetry (ZZZ symmetry) with limited queries. Contrasting with the advantages of indefinite causal-order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal-order strategies have equal power for our tasks. We establish optimal protocols for TTT symmetry testing and ZZZ symmetry testing for six and five queries, respectively, from which we infer that the type-II error exhibits a decay rate of $O({m}^{\ensuremath{-}2})$ with respect to the number of queries mmm. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $O({m}^{\ensuremath{-}1})$.
Can a symmetry be tested optimally without using the quantum resource naturally associated with it? For parallel subgroup-versus-Haar testing, we derive a quantitative representation-theoretic converse for every protocol with zero type-I error: the excess type-II error is bounded below by the sum of two nonnegative penalties, one for weight on suboptimal subgroup types and one for deviations of within-type Schur profiles from their typewise extremal profiles. At optimality, both penalties vanish, yielding support and weight locking for every optimal system marginal. Disjointness of the resulting locked set and the free marginal set rules out resource-free optimality, whereas overlap alone does not establish attainability. For the diagonal torus in dimension d≥2d\geq 2d≥2, optimal testing without coherence is possible only with a single query. We determine both the exact optimal type-II error over all protocols with incoherent system marginals and the minimum system-marginal coherence required to attain the unrestricted optimum. For the orthogonal subgroup in dimension d≥2d\geq 2d≥2, a real protocol attains the optimum at every query number. For the qutrit Clifford group, Wigner-positive optimal protocols exist at three and four queries but not from five through nine; whether they exist beyond nine queries remains open. Together, these results relate the subgroup- and query-dependent resource requirements of optimal symmetry testing to structural constraints on optimal system marginals.
We give collective tomography protocols for quantum states and channels with known symmetries, using random purification and dilation to reduce learning to pure-state estimation. For states commuting with a compact-group representation with multiplicities mλm_λmλ​, the optimal copy complexity is Θ((∑λmλ2+log⁡η−1)/ε2)Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)Θ((∑λ​mλ2​+logη−1)/ε2) for sufficiently small trace-distance error ε\varepsilonε and failure probability ηηη for the nontrivial case ∑λmλ2>1\sum_λm_λ^2>1∑λ​mλ2​>1. For GGG-covariant channels with finite-dimensional unitary representations of a compact group GGG, parallel O~(DG/ε2)\widetilde{O}(D_G/\varepsilon^2)O(DG​/ε2) queries achieve a diamond-distance error $\varepsilon $ at fixed success probability 2/32/32/3, where DGD_GDG​ counts the real parameters of covariant Choi operators before imposing trace preservation. For permutation-covariant channels on kkk qudits of fixed local dimension ddd, at sufficiently small fixed error and fixed success probability 2/32/32/3, we obtain the optimal query scalings Θd(kd4−1)Θ_d(k^{d^4-1})Θd​(kd4−1) for diamond distance and Θd(kd4−d2)Θ_d(k^{d^4-d^2})Θd​(kd4−d2) for Choi trace distance. Furthermore, we resolve an open problem in quantum tomography of constructing efficient quantum circuits that approximate the Hayashi measurement for optimal pure-state estimation. We encode states in the symmetric subspace into bosonic occupation modes, realize the measurement by heterodyne detection and normalization, and approximate this procedure on qubits using the quantum Hermite transform. Combined with our symmetry-compatible purification and dilation circuits, this yields query-optimal and gate-efficient learners of permutation-covariant states and channels with gate complexity Od(poly(k,ε−1,log⁡η−1))O_d(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))Od​(poly(k,ε−1,logη−1)).
Dynamical decoupling is a long-established and effective way to suppress unwanted interactions in qubit systems, enabling advances in fields ranging from quantum metrology to quantum computing. For general qudit systems, however, comparable protocols remain rare, mainly because Hamiltonian engineering in higher dimensions lacks the geometric intuition available for qubits. Here we present a general framework for dynamical decoupling in qudit systems, based on Lie group representation theory. By extending the group theory approach to dynamical decoupling, we show how decoupling groups can be systematically identified among the finite subgroups of SU(d) by analyzing their access to the irreducible components of the operator space. As an application, we construct new pulse sequences for interacting qutrit systems based on finite subgroups of SU(3), and show how subgroup factorizations and group orientations can be exploited to obtain shorter and more experimentally practical protocols for spin-1 systems with large zero-field splitting. We further show that the same symmetry-based framework yields quantum error-correcting codes: whenever a finite subgroup of SU(d) acts as a decoupling group for the relevant set of operators, the associated one-dimensional symmetry sectors define codespaces satisfying the Knill-Laflamme conditions, thereby unifying dynamical decoupling and quantum error correction in multi-level quantum systems.
Time-reversal symmetry is a fundamental principle of physics describing the invariance of physical laws under reversal of the direction of time. We formulate a Bell-inequality-like test of this antiunitary symmetry using only forward access and trusted quantum operations: entanglement converts temporal input--output relations into measurable spatial exchange symmetry. For nnn-qubit unitary dynamics, we prove that reliably distinguishing the time-reversal-symmetric circular ensembles from Haar-random dynamics requires Ω(min⁡{2n/2,2n−e})Ω(\min\{2^{n/2},2^{n-e}\})Ω(min{2n/2,2n−e}) queries for any classically adaptive protocol. Here, e=min⁡{es,em}e=\min\{e_{\mathrm s},e_{\mathrm m}\}e=min{es​,em​} with ese_{\mathrm s}es​ and eme_{\mathrm m}em​ representing the probe and measurement logarithmic entanglement negativities, respectively. Maximally entangled probes and SWAP measurements reduce this cost to a constant number of queries. Furthermore, we develop a time-reversal symmetry test for arbitrary fixed, compatible probes and measurements, relate its query complexity to their logarithmic negativities, and match the lower-bound scaling in the high-entanglement regime by optimizing the probe and measurement. Our results establish a quantitative connection between entanglement and time-reversal symmetry, bridging two central concepts in quantum information science and fundamental physics.
Estimating properties of unknown unitary operations is a fundamental task in quantum information science. While full unitary tomography requires a number of samples to the unknown unitary scaling linearly with the dimension (implying exponentially with the number of qubits), estimating specific functions of a unitary can be significantly more efficient. In this paper, we present a unified framework for the sample-efficient estimation of arbitrary square integrable functions f:U(d)→Cf: \mathbf{U}(d) \to \mathbb{C}f:U(d)→C, using only access to the controlled-unitary operation. We first provide a tight characterization of the optimal sample complexity when the accuracy is measured by the averaged bias over the unitary U(d)\mathbf{U}(d)U(d). We then construct a sample-efficient estimation algorithm that becomes optimal under the Probably Approximately Correct (PAC) learning criterion for various classes of functions. Applications include optimal estimation of matrix elements of irreducible representations, the trace, determinant, and general polynomial functions on U(d)\mathbf{U}(d)U(d). Our technique generalize the Hadamard test and leverage tools from representation theory, yielding both lower and upper bound on sample complexity.