The advancement of quantum simulators motivates the development of a theoretical framework to assist with efficient state preparation in quantum many-body systems. Generally, preparing a target entangled state via unitary evolution with time-dependent couplings is a challenging task and very little is known about the existence of solutions and their properties. In this work we develop a constructive approach for preparing matrix product states (MPS) via continuous unitary evolution. We provide an explicit construction of the operator that exactly implements the evolution of a given MPS along a specified direction in its tangent space. This operator can be written as a sum of local terms of finite range, yet it is in general non-Hermitian. Relying on the explicit construction of the non-Hermitian generator of the dynamics, we demonstrate the existence of a Hermitian sequence of operators that implements the desired MPS evolution with an error that decreases exponentially with the operator range. The construction is benchmarked on an explicit periodic trajectory in a translationally invariant MPS manifold. We demonstrate that the Floquet unitary generating the dynamics over one period of the trajectory features an approximate MPS-like eigenstate embedded among a sea of thermalizing eigenstates. These results show that our construction is not only useful for state preparation and control of many-body systems, but also provides a generic route towards Floquet scars—periodically driven models with quasilocal generators of dynamics that have exact MPS eigenstates in their spectrum. Published by the American Physical Society 2024
The advancement of quantum simulators motivates the development of a theoretical framework to assist with efficient state preparation in quantum many-body systems. Generally, preparing a target entangled state via unitary evolution with time-dependent couplings is a challenging task and very little is known about the existence of solutions and their properties. In this work we develop a constructive approach for preparing matrix product states (MPS) via continuous unitary evolution. We provide an explicit construction of the operator that exactly implements the evolution of a given MPS along a specified direction in its tangent space. This operator can be written as a sum of local terms of finite range, yet it is in general non-Hermitian. Relying on the explicit construction of the non-Hermitian generator of the dynamics, we demonstrate the existence of a Hermitian sequence of operators that implements the desired MPS evolution with an error that decreases exponentially with the operator range. The construction is benchmarked on an explicit periodic trajectory in a translationally invariant MPS manifold. We demonstrate that the Floquet unitary generating the dynamics over one period of the trajectory features an approximate MPS-like eigenstate embedded among a sea of thermalizing eigenstates. These results show that our construction is not only useful for state preparation and control of many-body systems, but also provides a generic route towards Floquet scars—periodically driven models with quasilocal generators of dynamics that have exact MPS eigenstates in their spectrum. Published by the American Physical Society 2024
Exactly solvable models in quantum many-body dynamics provide valuable insights into many interesting physical phenomena and serve as platforms for rigorously investigating fundamental theoretical questions. Nevertheless, they are extremely rare and existing solvable models and solution techniques have serious limitations. In this paper, we present a family of exactly solvable unitary circuits that model quantum many-body dynamics in discrete space and time. Unlike many previous solvable models, one can exactly compute the full quantum dynamics initialized from any matrix product state in this family of models. The time evolution of local observables and correlations, the linear growth of Rényi entanglement entropy, spatiotemporal correlations, and out-of-time-order correlations are all exactly computable. A key property of these models enabling the exact solution is that any time-evolved local operator is an exact matrix product operator with a finite bond dimension, even at arbitrarily long times, which we prove using the underlying C * -(weak) Hopf algebra structure along with tensor network techniques. We lay down the general framework for construction and solution of this family of models, and give several explicit examples. In particular, we study in detail a model constructed out of a C * -weak Hopf algebra that is very close to a Floquet version of the PXP model, and the exact results we obtain may shed light on the phenomenon of quantum many-body scars, and more generally, Floquet quantum dynamics in constrained systems.
Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states. Published by the American Physical Society 2024
We study the conditions under which Matrix Product States (MPS) or Matrix Product Operators are exact eigenvectors of an extensive local operator, such as a Hamiltonian. By suitably choosing the local operator, this covers a wide range of settings: Exact eigenstates of Hamiltonians, including scar states, exact MPS trajectories for driven quantum systems, steady states of local Lindbladians, generalized symmetries of either Hamiltonians or density matrices, and many more. Our key result is that a local, fixed-size equation---namely, how a single term in the operator acts on a block of tensors---provides a necessary and sufficient condition for exact solutions. This local characterization allows us to obtain the full space of MPS solutions to all of the aforementioned problems; in particular, we exemplify the power of our results by recovering the quantum group symmetries of the XXZ model. We also discuss applications to numerical algorithms with MPS and the generalization of our results to 2D, i.e., projected entangled pair states (PEPS).
Quantum many-body scars (QMBSs) are a novel paradigm for the violation of the eigenstate thermalization hypothesis—Hamiltonians of these systems exhibit mid-spectrum eigenstates that are equidistant in energy and which possess low entanglement and evade thermalization for long times. We present a novel approach to understanding the anomalous dynamical behavior in these systems. Specifically, we postulate that QMBS Hamiltonians H can generically be partitioned into a set of terms O a which do not commute over the entire Hilbert space, but commute to all orders within the subspace of scar states. All states in the scar subspace thus evolve according to a “broken unitary” U s ( t ) = ∏ a e − i O a t , where H = ∑ a O a , which provides a simple interpretation of the anomalous dynamical features exhibited by quantum scars; they evolve in time according to a simpler unitary operator allowing for revivals. It is found that the observed dynamical decoupling is often a direct consequence of simple local conditions, which when satisfied lead to recurring aspects of QMBSs, including equidistant eigenvalues, many-body revivals, and sub-volume-law entanglement entropy. Two classes of scar models emerge in this picture—those with a finite set of O a , as pertaining of, for instance, scars in the AKLT model, and those with an extensive set of such operators, such as eta pairing scar states in the Hubbard model. Besides discussing how many well-known scar models fit into the above picture, we show how the broken unitary formalism captures many known scar construction methods, known in the literature and generalizes others such as quasiparticles matrix product state (MPS) based methods, and the Shiraishi-Mori approach. Published by the American Physical Society 2024
Quantum many-body scar states are highly excited eigenstates of many-body systems that exhibit atypical entanglement and correlation properties relative to typical eigenstates at the same energy density. Scar states also give rise to infinitely long-lived coherent dynamics when the system is prepared in a special initial state having finite overlap with them. Many models with exact scar states have been constructed, but the fate of scarred eigenstates and dynamics when these models are perturbed is difficult to study with classical computational techniques. In this work, we propose state preparation protocols that enable the use of quantum computers to study this question. We present protocols both for individual scar states in a particular model, as well as superpositions of them that give rise to coherent dynamics. For superpositions of scar states, we present both a system-size-linear depth unitary and a finite-depth nonunitary state preparation protocol, the latter of which uses measurement and postselection to reduce the circuit depth. For individual scarred eigenstates, we formulate an exact state preparation approach based on matrix product states that yields quasipolynomial-depth circuits, as well as a variational approach with a polynomial-depth ansatz circuit. We also provide proof of principle state-preparation demonstrations on superconducting quantum hardware.
One-dimensional fracton systems can exhibit perfect localization, failing to reach thermal equilibrium under arbitrary local unitary time evolution. We investigate how this nonergodic behavior manifests in the dynamics of a driven fracton system, specifically a one-dimensional Floquet quantum circuit model featuring conservation of a U(1) charge and its dipole moment. For a typical basis of initial conditions, a majority of states heat up to a thermal state at near-infinite temperature. In contrast, a small number of states flow to a localized steady state under the Floquet time evolution. We refer to these athermal steady states as ``dynamical scars, in analogy with the scar states observed in the spectra of certain many-body Hamiltonians. Despite their small number, these dynamical scars are experimentally relevant due to their high overlap with easily-prepared product states. Each scar state displays a single agglomerated fracton peak, in agreement with the steady-state configurations of fractonic random circuits. The details of these scars are insensitive to the precise form of the Floquet operator, which is constructed from random unitary matrices. Rather, dynamical scar states arise directly from fracton conservation laws, providing a concrete mechanism for the appearance of scars in systems with constrained quantum dynamics.