- Skolkovo Institute of Science and Technology, Moscow, Russia

Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-*r* tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.

## 1 Introduction

Tensor network methods provide the contemporary state of the art in the classical simulation of quantum systems. A range of numerical and analytical tools have now emerged, including tensor network algorithms, to simulate quantum systems classically; these algorithms are based in part on powerful insights related to the area law [1–9]. The area law places bounds on quantum entanglement that a many-body system can generate, which translates directly to the amount of memory required to store a given quantum state; see, e.g., [8].

The leading classical methods to simulate random circuits for quantum computational supremacy demonstration are also based on tensor network contractions. Additionally, classical machine learning has been merged with matrix product states and other tensor network methods [10–14]. How might quantum computing accelerate tensor network algorithms?

Although tensor network tools have traditionally been developed to simulate quantum systems classically, we propose a quantum algorithm to approximate an eigenvector of a unitary matrix with bounded rank tensor network states. The algorithm works given only black-box access to a unitary matrix. In general, tensor network contraction can simulate any quantum computation.

We focus on 1D chains of tensors (matrix product states) due to some associated analytical simplifications; indeed, matrix product states can be approximated classically which offers an attractive gold standard to compare the quantum algorithm against. The general framework we develop applies equally well to 2D and, e.g., sparse networks (projected entangled pair states, etc.). However, an early merger between these topics is better situated to focus on 1D.

Even in 1D, tensor networks offer certain insights into quantum algorithms. For example, the maximal degree of entanglement can often be bounded in the description of the tensor network state itself. In other words, the bond dimension (the dimension of the wires) in the tensor network acts to bound the maximal entanglement. Merging quantum computation with ideas from tensor networks provides new tools to quantify the entanglement that a given quantum circuit can generate [15, 16].

For the sake of simplicity, we work in the black-box setting and assume access to a provided unitary *Q*. The black-box setting does not consider the implementation of *Q*. Prima facie, this appears to be a limitation; in practice, however, the restriction can easily be lifted. For example, in QAOA, the problem Hamiltonian can be applied for varying times, offering a natural extension of the oracle idea by giving *Q* a simple time dependence [17].

In Discussion, we drop the black-box access restriction and cast the steps needed to perform a meaningful near-term demonstration of our algorithm on a quantum computer, providing a low-rank approximation to eigenvectors of the quantum computers free- (or effective) Hamiltonian. The presented algorithm falls into the class of variational quantum algorithms [18–25]. It returns a classical description, in the form of a tensor network, of an eigenvector of an operator found through an iterative classical-to-quantum optimization process.

We present a general framework to determine tensor networks using quantum processors. We focus on 1D, which enables several results related to the maximum amounts of entanglement required to demonstrate these methods. This analysis is followed by a discussion focused on applications of these techniques and what might be required for a meaningful near-term experimental demonstration.

## 2 Methods

The algorithm we propose solves the following problem: *given black-box access to a unitary Q, find any eigenvector of Q.*

We work in the standard mathematical setting of quantum computing. We define *n* qubits arranged on a line and fix the standard canonical (computational) basis. We consider the commutative Hermitian subalgebra generated by the *n*-projectors:

where the subscript *i* denotes the corresponding *i*th qubit acted on by

Rank is the maximum Schmidt number (the nonzero singular values) across any of the *r* state has at most *k* ebit approximations, where the *k*’th approximation can be used to seed the

An ebit is the amount of entanglement contained in a maximally entangled two-qubit (Bell) state. A quantum state with *q* ebits of entanglement (quantified by any entanglement measure) has the same amount of entanglement (in that measure) as *q* Bell states. If a task requires *l* ebits, it can be done with *l* or more Bell states, but not with fewer. Maximally entangled states in

have

We parameterize a circuit family generating matrix product states with *θ*, a real vector with entries in

and

both of yet to be specied rank.

We will construct an objective function (Eq. 6) to minimize and hence to recover our approximate eigenvector. The choice of this function provides a desirable degree of freedom to further tailor the algorithm to the particular quantum processor at hand. We choose

and call

the log-likelihood function of the *n*-point correlator

The minimization of Eq. 6 corresponds to maximizing the probability of measuring each qubit in

Algorithm 1: Find successive tensor network approximations of an eigenvector of *Q*.

Choose the maximum number of ebits

Choose the maximum number of optimization iterations

**for****do**

Construct the ansatz *k* ebit MPS

Set

**for****do**

Evaluate

Evaluate

Update

**end for**

Store

**end for**

**return**

The algorithm begins with rank-1 qubit states as

Minimization of the objective function Eq. 6 returns

**FIGURE 1**. Example of a tensor network as a quantum circuit: **(left)** quantum circuit realization of a matrix product state with open boundary conditions; **(right)** using standard graphical rewrite rules—or by manipulating equations—one readily recovers the familiar matrix product state depiction as a “train of tensors.”

## 3 Results

The algorithm works given only oracle access to a unitary *Q*. The spectrum of *Q* is necessarily contained on the complex unit circle and so we note immediately that

with equality of the left-hand side if and only if *Q*. One advantage of the presented method is that it terminates when the measurement reaches a given value. This implies that the system is in an eigenstate. Such a certificate is not directly established using other variational quantum algorithms.

Importantly, the maximization over *θ* on the right-hand side of Eq. 9 corresponds to the minimization of the log-likelihood Eq. 6. We will then parameterize *k* denotes a *k* ebit matrix product state of interest. Learning this matrix product state recovers an approximation to an eigenvector of *Q*. With a further promise on *Q* that all eigenvectors have a rank-*p* matrix product state representation, we conclude that *r*’th singular value of the state takes the value ε. It then follows that the one-norm error scales with

which is valid for

Indeed, increasing the rank of the matrix product state approximation can improve the eigenvector approximation. Yet, it should be noted that ground state eigenvectors of physical systems are in many cases known to be well approximated with low-rank matrix product states [1–9]. This depends on the further properties of *Q* and is a subject of intensive study in numerical methods, further motivating the quantum algorithm we present here. We will develop our algorithm agnostic to *Q*, leaving a more specific near-term demonstration (in which *Q* is implemented); e.g., we will express any

In Eq. 11, the rank-*r* of the representation is embedded into the realization of the *A*’s. Quantum mechanics allows the deterministic generation of a class of isometries, where an isometry *U* that is also an endomorphism on a particular space is called unitary.

Matrix product states (Eq. 11) are not isometries, though correlation functions are readily calculated from them. Furthermore, matrix product states can be deterministically generated by the uniform quantum circuit given in Figure 1. Other isometric structures of interest include trees and the so-called Multiscale Entanglement Renormalization Ansatz (MERA) networks [3, 26–28].

Consider then a rank-*r* approximation to an eigenvector of *Q*. The blocks in Figure 1 represent unitary maps. These circuits act on at most

CNOT gates, where the method in [29] requires *n*. Hence, the implementation complexity scales as *l* steps (perhaps in a local minimum).

Instead of preparing

where

where *T* orders the sequence by time and superscript *k* indexes the *k*th operator

We then consider vertical partitions of a quantum circuit with the *n* qubits positioned horizontally on a line. For an *m*-depth quantum circuit (where *m* is presumably bounded above by a low-order polynomial in *n*), the maximum number of two-qubit gates crossed in a vertical partition is never more than *m*. The maximum number of ebits generated by a fully entangling two-qubit CNOT gate is never more than a single ebit. We then consider the *m*-depth quantum circuit on *n* qubits never applies more than

ebits of entanglement. This immediately puts a lower bound of *Q* to potentially drive a system into a state supporting the maximum possible ebits of entanglement.

In Figure 2, we demonstrate our algorithm for finding an eigenstate of randomly generated 5-qubit unitary matrices. For minimizing the function Eq. 6, we used the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization method [30]. For each *k* ebit MPS, we place the *k*-layered hardware-efficient ansatz as the operators in blocks [21].

**FIGURE 2**. Algorithm demonstration on randomly generated 6-qubit unitaries *Q*: the value of Eq. 9**(upper)**, overlap between the variational state and the closest eigenstate of *Q***(middle)**, and the von Neumann entropy of the subsystem of the first three qubits **(lower)**. The vertical solid lines indicate the iteration numbers after which *k*, the number of ebits that the MPS ansatz can support, increases by 1. The plot is obtained by averaging over 10 randomly generated unitaries *Q*.

## 4 Discussion

We now turn to the realization of *Q* and sketch a possible demonstration for a near-term device. Polynomial-time simulation of Hamiltonian evolution is well known to be **BQP**-hard. This provides an avenue for *Q* to represent a problem of significant computational interest, as simulating quantum evolution and quantum factoring are in **BQP**. We aim to bootstrap properties of the quantum processor as much as possible to reduce resources for a realization; see, for example, [21].

Let

we can minimize the overall eigenvectors, which is **NP**-hard. Hence, finding even rank-1 states can be **NP**-hard. This provides a connection between our method and QAOA [31]. Similarly, we can also use this external minimization to connect our method to VQE [20]. However, our method provides a certificate that, on proper termination, the system is indeed in such a desired eigenstate.

When

is in turn **QMA**-hard. For example, pairing our procedure with an additional procedure (quantum phase estimation) to minimize *Q*, the overall eigenvectors would hence provide rank-*k* variational states and hence our methods provide a research direction which incorporates tensor network methods in works such as, e.g., [19–21]. It should however be noted that phase estimation adds significant experimental difficultly compared with the algorithm presented here and the algorithm is closer to VQE (with evident differences as listed above and in the main text).

For a near-term demonstration, we envision *Q* to be realized by bootstrapping the underlying physics of the system realizing *Q*, e.g., using the hardware-efficient ansatz [21]. For instance, one can realize *Q* as a modification of the systems free Hamiltonian using effective Hamiltonian methods (modulating local gates). This greatly reduces practical requirements on *Q*.

The interaction graph of the Hamiltonian generating *Q* can be used to define a PEPS tensor network (as it will have the same structure as the layout of the chip, it will no longer have the contractable properties of matrix product states, and yet is still of interest) [4]. The algorithm works otherwise unchanged, but the circuit acts on this interaction graph (instead of a line) to create a corresponding tensor network state (a quantum circuit in the form of, e.g., the variational ansatz). Tailored free evolution of the system Hamiltonian generates *Q*. Our algorithm returns a tensor network approximation of an eigenstate of *Q*.

The first interesting demonstrations of the quantum algorithm we have presented should realize rank-*k* tensor networks (matrix product state), and the corresponding tensor network can be realized with a few hundred gates for a system with a few hundred qubits.

## Data Availability Statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found: GitHub, https://git.io/JkjvV.

## Author Contributions

All authors conceived and developed the theory and design of this study and verified the methods. AK developed and deployed the code to collect numerical data. All authors contributed to interpreting the results and writing the manuscript.

## Acknowledgements

AK and JB acknowledge support from agreement No. 014/20, *Leading Research Center on Quantum Computing*. AU acknowledges RFBR project No. 19-31-90159 for financial support. This manuscript has been released as a preprint as arXiv:1804.02398 (32).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: quantum computing, quantum algorithms and circuits, tensor network algorithms, ground state, properties, machine learning, quantum information

Citation: Kardashin A, Uvarov A and Biamonte J (2021) Quantum Machine Learning Tensor Network States. *Front. Phys.* 8:586374. doi: 10.3389/fphy.2020.586374

Received: 23 July 2020; Accepted: 08 December 2020;

Published: 01 March 2021.

Edited by:

Sabre Kais, Purdue University, United StatesReviewed by:

Peter McMahon, Cornell University, United StatesKathleen Hamilton, Oak Ridge National Laboratory (DOE), United States

Copyright © 2021 Kardashin, Uvarov and Biamonte. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Andrey Kardashin, andrey.kardashin@skoltech.ru