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Physical Review Research

Efficient adiabatic preparation of tensor network states

Zhi-Yuan Wei, Daniel Malz, J. I. Cirac

Published May 22, 2023• 22 citations
Publisher

Abstract

The Affleck-Kennedy-Lieb-Tasaki (AKLT) states have significant applications in condensed matter physics and quantum computation. An adiabatic path that enables the deterministic preparation of a broad range of many-body states, including AKLT states on arbitrary lattice geometries, is proposed. The numerical results on 1D and 2D AKLT states demonstrate the high efficiency of this approach.

DOI
10.1103/physrevresearch.5.l022037

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The Affleck-Kennedy-Lieb-Tasaki (AKLT) states have significant applications in condensed matter physics and quantum computation. An adiabatic path that enables the deterministic preparation of a broad range of many-body states, including AKLT states on arbitrary lattice geometries, is proposed. The numerical results on 1D and 2D AKLT states demonstrate the high efficiency of this approach.
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.
Affleck-Kennedy-Lieb-Tasaki (AKLT) states are an important class of many-body quantum states that are useful in quantum information processing, including measurement-based quantum computation in particular. Here we propose a general approach for constructing efficient verification protocols for AKLT states on arbitrary graphs with local spin measurements. Our verification protocols build on bond verification protocols and matching covers (including edge coloring) of the underlying graphs, which have a simple geometric and graphic picture. We also provide rigorous performance guarantee that is required for practical applications. With our approach, most AKLT states of wide interest, including those defined on one- and two-dimensional lattices, can be verified with a constant sample cost, which is independent of the system size and is dramatically more efficient than all previous approaches. As an illustration, we construct concrete verification protocols for AKLT states on various lattices and on arbitrary graphs up to five vertices.
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.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.
Abstract Quantum state preparation plays a crucial role in several areas of quantum information science, in applications such as quantum simulation, quantum metrology and quantum computing. However, typically state preparation requires resources that scale exponentially with the problem size, due to their probabilistic nature or otherwise, making studying such models challenging. In this article, we propose a method to prepare the ground state of the Affleck-Lieb-Kennedy-Tasaki (AKLT) model deterministically using an measurement-based imaginary time evolution (MITE) approach. By taking advantage of the special properties of the AKLT state, we show that it can be prepared efficiently using the MITE approach. Estimates based on the convergence of a sequence of local projections, as well as direct evolution of the MITE algorithm suggest a constant scaling with respect to the number of AKLT sites, which is an exponential improvement over the naive estimate for convergence. We show that the procedure is compatible with qubit-based simulators, and show that using a variational quantum algorithm for circuit recompilation, the measurement operator required for MITE can be well approximated by a circuit with a much shallower circuit depth compared with the one obtained using the default Qiskit method.