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arXiv:2309.00774 · PRX Quantum

Learning Conservation Laws in Unknown Quantum Dynamics

Yongtao Zhan, Andreas Elben, Hsin-Yuan Huang, Yu Tong

Published Mar 22, 2024• 7 citations
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Abstract

We present a learning algorithm for discovering conservation laws given as sums of geometrically local observables in quantum dynamics. This includes conserved quantities that arise from local and global symmetries in closed and open quantum many-body systems. The algorithm combines the classical shadow formalism for estimating expectation values of observable and data analysis techniques based on singular value decompositions and robust polynomial interpolation to discover all such conservation laws in unknown quantum dynamics with rigorous performance guarantees. Our method can be directly realized in quantum experiments, which we illustrate with numerical simulations, using closed and open quantum system dynamics in a Z2 gauge theory and in many-body localized spin chains. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.010350

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We present a learning algorithm for discovering conservation laws given as sums of geometrically local observables in quantum dynamics. This includes conserved quantities that arise from local and global symmetries in closed and open quantum many-body systems. The algorithm combines the classical shadow formalism for estimating expectation values of observable and data analysis techniques based on singular value decompositions and robust polynomial interpolation to discover all such conservation laws in unknown quantum dynamics with rigorous performance guarantees. Our method can be directly realized in quantum experiments, which we illustrate with numerical simulations, using closed and open quantum system dynamics in a Z2 gauge theory and in many-body localized spin chains. Published by the American Physical Society 2024
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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.
Measuring properties of quantum systems is a fundamental problem in quantum mechanics. We provide a simple method for estimating the expectation value of observables with an unknown quantum state. The idea is to use a data structure to sample the terms of observables based on the Pauli decomposition proportionally to their importance. We call this technique operator shadow as a shorthand for the procedure of preparing a sketch of an operator to estimate properties. Only when the numbers of observables are small for multiple local observables, the sample complexity of this method is better than the classical shadow technique. However, if we want to estimate the expectation value of a linear combination of local observables, for example the energy of a local Hamiltonian, the sample complexity is better on all parameters. The time complexity to construct the data structure is 2O(k)2^{O(k)}2O(k) for kkk-local observables, similar to the post-processing time of classical shadows.
Interacting many-body quantum systems and their dynamics, while fundamental to modern science and technology, are formidable to simulate and understand. However, by discovering their symmetries, conservation laws, and integrability one can unravel their intricacies. Here, using up to 124 qubits of a fully programmable quantum computer, we uncover local conservation laws and integrability in one- and two-dimensional periodically-driven spin lattices in a regime previously inaccessible to such detailed analysis. We focus on the paradigmatic example of disorder-induced ergodicity breaking, where we first benchmark the system crossover into a localized regime through anomalies in the one-particle-density-matrix spectrum and other hallmark signatures. We then demonstrate that this regime stems from hidden local integrals of motion by faithfully reconstructing their quantum operators, thus providing a detailed portrait of the system's integrable dynamics. Our results demonstrate a versatile strategy for extracting the hidden dynamical structure from noisy experiments on large-scale quantum computers.