Quantum many-body physics simulations with matrix product states can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined beforehand when constructing the tensors for the matrix product state algorithm. In this work, we present a matrix product state algorithm with an adaptive $U(1)$ symmetry. This algorithm can take into account or benefit from $U(1)$ or ${Z}_{2}$ symmetries when they are present or analyze the nonsymmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.
Quantum many-body physics simulations with matrix product states can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined beforehand when constructing the tensors for the matrix product state algorithm. In this work, we present a matrix product state algorithm with an adaptive U(1) symmetry. This algorithm can take into account or benefit from U(1) or Z2 symmetries when they are present or analyze the nonsymmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.
The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g., with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely, (i) [Singh, Pfeifer, and Vidal, Phys. Rev. A 82, 050301 (2010)] and (ii) [Singh, Pfeifer, and Vidal, Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group G that is compact, completely reducible, and multiplicity free, acting as a global internal symmetry. Then, in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper, we describe the implementation of non-Abelian group symmetries in great detail. For concreteness, we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multiscale entanglement renormalization ansatz and apply it to study the low-energy spectrum of a quantum spin chain with a global SU(2) symmetry.
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
This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.
We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size n. Furthermore, we demonstrate that this scaling can be improved to O(log2n) by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.
Abstract Understanding extreme non-locality in many-body quantum systems can help resolve questions in thermostatistics and laser physics. The existence of symmetry selection rules for Hamiltonians with non-decaying terms on infinite-size lattices can lead to finite energies per site, which deserves attention. Here, we present a tensor network approach to construct the ground states of nontrivial symmetric infinite-dimensional spin Hamiltonians based on constrained optimizations of their infinite matrix product states description, which contains no truncation step, offers a very simple mathematical structure, and other minor advantages at the cost of slightly higher polynomial complexity in comparison to an existing method. More precisely speaking, our proposed algorithm is in part equivalent to the more generic and well-established solvers of infinite density-matrix renormalization-group and variational uniform matrix product states, which are, in principle, capable of accurately representing the ground states of such infinite-range-interacting many-body systems. However, we employ some mathematical simplifications that would allow for efficient brute-force optimizations of tensor-network matrices for the specific cases of highly-symmetric infinite-size infinite-range models. As a toy-model example, we showcase the effectiveness and explain some features of our method by finding the ground state of the U(1)-symmetric infinite-dimensional antiferromagnetic XX Heisenberg model.
Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO WI,II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.