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arXiv:2105.06866 · Physical Review Research

Preparation and verification of tensor network states

Esther Cruz, F. Baccari, Jordi Tura, Norbert Schuch, J. I. Cirac

Published May 31, 2022• 12 citations
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Abstract

We consider a family of tensor network states defined on regular lattices that come with a natural definition of an adiabatic path to prepare them. This family comprises relevant classes of states, such as injective matrix product and projected entangled-pair states, and some corresponding to classical spin models. We show how uniform lower bounds to the gap of the parent Hamiltonian along the adiabatic trajectory can be efficiently computed using semidefinite programming. This allows one to check whether the adiabatic preparation can be performed efficiently with a scalable effort. We also derive a set of observables whose expectation values can be easily determined and that form a complete set, in the sense that they uniquely characterize the state. We identify a subset of those observables which can be efficiently computed if one has access to the quantum state and local measurements, and analyze how they can be used in verification procedures.

DOI
10.1103/physrevresearch.4.023161

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We consider a family of tensor network states defined on regular lattices that come with a natural definition of an adiabatic path to prepare them. This family comprises relevant classes of states, such as injective matrix product and projected entangled-pair states, and some corresponding to classical spin models. We show how uniform lower bounds to the gap of the parent Hamiltonian along the adiabatic trajectory can be efficiently computed using semidefinite programming. This allows one to check whether the adiabatic preparation can be performed efficiently with a scalable effort. We also derive a set of observables whose expectation values can be easily determined and that form a complete set, in the sense that they uniquely characterize the state. We identify a subset of those observables which can be efficiently computed if one has access to the quantum state and local measurements, and analyze how they can be used in verification procedures.
We propose and study a specific adiabatic path to prepare those tensor network states that are unique ground states of few-body parent Hamiltonians in finite lattices, which include normal tensor network states, as well as other relevant nonnormal states. This path guarantees a gap for finite systems and allows for efficient numerical simulation. In one dimension, we numerically investigate the preparation of a family of states with varying correlation lengths and the one-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) state and show that adiabatic preparation can be much faster than standard methods based on sequential preparation. We also apply the method to the two-dimensional AKLT state on the hexagonal lattice, for which no method based on sequential preparation is known, and show that it can be prepared very efficiently for relatively large lattices.
The preparation of nontrivial states is crucial to the study of quantum many-body physics. Such states can be prepared with adiabatic quantum algorithms, which are restricted by the minimum spectral gap along the path. In this Letter, we propose an efficient method to adiabatically prepare tensor network states (TNSs). We maximize the spectral gap leveraging degrees of freedom in the parent Hamiltonian construction. We demonstrate this efficient adiabatic algorithm for preparing TNSs, through examples of random TNSs in one dimension, AKLT (Affleck-Kennedy-Lieb-Tasaki), and GHZ states. The Hamiltonian optimization applies to both injective and non-injective tensors, in the latter case by exploiting symmetries present in the tensors.
Two-dimensional tensor networks such as projected entangled pairs states (PEPS) are generally hard to contract. This is arguably the main reason why variational tensor network methods in two dimensions are still not as successful as in one dimension. However, this is not necessarily the case if the tensor network represents a gapped ground state of a local Hamiltonian; such states are subject to many constraints and contain much more structure. In this paper, we introduce an approach for approximating the expectation value of a local observable in ground states of local Hamiltonians that are represented by PEPS tensor networks. Instead of contracting the full tensor network, we try to estimate the expectation value using only a local patch of the tensor network around the observable. Surprisingly, we demonstrate that this is often easier to do when the system is frustrated. In such case, the spanning vectors of the local patch are subject to nontrivial constraints that can be utilized via a semidefinite program to calculate rigorous lower and upper bounds on the expectation value. We test our approach in one-dimensional systems, where we show how the expectation value can be calculated up to at least 3 or 4 digits of precision, even when the patch radius is smaller than the correlation length.
We propose a quantum algorithm for many-body state preparation. It is especially suited for injective projected entangled pair states and thermal states of local commuting Hamiltonians on a lattice. We show that for a uniform gap and sufficiently smooth paths, an adiabatic runtime and circuit depth of O(polylogN) can be achieved for O(N) spins. This is an almost exponential improvement over previous bounds. The total number of elementary gates scales as O(NpolylogN). This is also faster than the best known upper bound of O(N^{2}) on the mixing times of Monte Carlo Markov chain algorithms for sampling classical systems in thermal equilibrium.
Tensor networks (TNs) are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of TNs naturally arises: what is then the relation between those sets? The answer to this question in one-dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled pair states in any dimension and lattice geometry (including, for example, the Kagome lattice, hyperbolic lattices, or tree tensor networks), as long as the tensors generating the states are normal, which constitute an important and generic class.