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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-

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 [

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 [

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 [

For the sake of simplicity, we work in the black-box setting and assume access to a provided unitary

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 [

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.

The algorithm we propose solves the following problem:

We work in the standard mathematical setting of quantum computing. We define

Rank is the maximum Schmidt number (the nonzero singular values) across any of the

An ebit is the amount of entanglement contained in a maximally entangled two-qubit (Bell) state. A quantum state with

We parameterize a circuit family generating matrix product states with

We will construct an objective function

Algorithm 1: Find successive tensor network approximations of an eigenvector of

Choose the maximum number of ebits

Choose the maximum number of optimization iterations

Construct the ansatz

Set

Evaluate

Evaluate

Update

Store

The algorithm begins with rank-1 qubit states as

Example of a tensor network as a quantum circuit:

The algorithm works given only oracle access to a unitary

Importantly, the maximization over

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 [

Matrix product states (

Consider then a rank-

Instead of preparing

We then consider vertical partitions of a quantum circuit with the

In

Algorithm demonstration on randomly generated 6-qubit unitaries

We now turn to the realization of

Let

When

For a near-term demonstration, we envision

The interaction graph of the Hamiltonian generating

The first interesting demonstrations of the quantum algorithm we have presented should realize rank-

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,

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.

AK and JB acknowledge support from agreement No. 014/20,

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.