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 \(Q_h\) that may underlie such anomalous behavior. We develop an algebraic criterion that any operator \(Q_h\) must satisfy in order to commute with the full Hamiltonian \(H\) 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.
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 (Q_h) that may underlie such anomalous behavior. We develop an algebraic criterion that any operator (Q_h) must satisfy in order to commute with the full Hamiltonian (H) 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 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.
We review recent progress in understanding the notion of locality in integrable quantum lattice systems. The central concept are the so-called quasilocal conserved quantities, which go beyond the standard perception of locality. Two systematic procedures to rigorously construct families of quasilocal conserved operators based on quantum transfer matrices are outlined, specializing on anisotropic Heisenberg XXZ spin-1/2 chain. Quasilocal conserved operators stem from two distinct classes of representations of the auxiliary space algebra, comprised of unitary (compact) representations, which can be naturally linked to the fusion algebra and quasiparticle content of the model, and non-unitary (non-compact) representations giving rise to charges, manifestly orthogonal to the unitary ones. Various condensed matter applications in which quasilocal conservation laws play an essential role are presented, with special emphasis on their implications for anomalous transport properties (finite Drude weight) and relaxation to non-thermal steady states in the quantum quench scenario.
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.
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.
Strongly correlated systems exhibit phenomena -- such as high-T_c superconductivity or the fractional quantum Hall effect -- that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime. Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems. It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems. Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases -- symmetry-protected topological (SPT) order, -- and tensor networks provide the right language to classify these phases. In one dimension and for certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.
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 more 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. Local integrals of motion are useful for understanding emergent integrability in many-body localized systems. Here the authors use a large-scale superconducting quantum processor with up to 124 qubits to simulate many-body dynamics in 1D and 2D systems and demonstrate extraction of local integrals of motion.