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Quantum Annealing and Tensor Networks: a Powerful Combination to Solve Optimization Problems

Miquel Albertí Binimelis

TL;DR

The work investigates a synergistic framework that combines tensor networks with quantum annealing to tackle optimization problems, exemplified by the quadratic knapsack problem. It develops methods to represent quantum systems with matrix product operators via finite automata, employs DMRG to study annealing dynamics and estimate minimum gaps, and applies a QUBO/QKP formulation with unbalanced penalization to produce MPOs suitable for simulated annealing and DP benchmarking. Key technical contributions include an automata-based MPO construction for the annealing Ising Hamiltonian, corrections and optimizations to the MPO tables, and a scheduling strategy informed by gap estimates for potential quantum annealing improvements. The results demonstrate the potential of tensor networks to model and analyze quantum optimization processes and provide a practical, self-contained tutorial with Python implementations, offering a foundation for future QA-TN hybrids in large-scale optimization.

Abstract

Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While both technologies can address similar problems, the primary aim of this thesis is not to compare them. Such comparison would be unfair, as quantum devices are still in an early stage, whereas tensor network algorithms represent the state-of-the-art in quantum simulation. Instead, we explore a potential synergy between these technologies, focusing on how two flagship algorithms from each paradigm, the Density Matrix Renormalization Group (DMRG) and quantum annealing, might collaborate in the future. Furthermore, a significant challenge in the DMRG algorithm is identifying an appropriate tensor network representation for the quantum system under study. The representation commonly used is called Matrix Product Operator (MPO), and it is notoriously difficult to obtain for certain systems. This thesis outlines an approach to this problem using finite automata, which we apply to construct the MPO for our case study. Finally, we present a practical application of this framework through the quadratic knapsack problem (QKP). Despite its apparent simplicity, the QKP is a fundamental problem in computer science with numerous practical applications. In addition to quantum annealing and the DMRG algorithm, we implement a dynamic programming approach to evaluate the quality of our results. Our results highlight the power of tensor networks and the potential of quantum annealing for solving optimization problems. Moreover, this thesis is designed to be self-explanatory, ensuring that readers with a solid mathematical background can fully understand the content without prior knowledge of quantum mechanics.

Quantum Annealing and Tensor Networks: a Powerful Combination to Solve Optimization Problems

TL;DR

The work investigates a synergistic framework that combines tensor networks with quantum annealing to tackle optimization problems, exemplified by the quadratic knapsack problem. It develops methods to represent quantum systems with matrix product operators via finite automata, employs DMRG to study annealing dynamics and estimate minimum gaps, and applies a QUBO/QKP formulation with unbalanced penalization to produce MPOs suitable for simulated annealing and DP benchmarking. Key technical contributions include an automata-based MPO construction for the annealing Ising Hamiltonian, corrections and optimizations to the MPO tables, and a scheduling strategy informed by gap estimates for potential quantum annealing improvements. The results demonstrate the potential of tensor networks to model and analyze quantum optimization processes and provide a practical, self-contained tutorial with Python implementations, offering a foundation for future QA-TN hybrids in large-scale optimization.

Abstract

Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While both technologies can address similar problems, the primary aim of this thesis is not to compare them. Such comparison would be unfair, as quantum devices are still in an early stage, whereas tensor network algorithms represent the state-of-the-art in quantum simulation. Instead, we explore a potential synergy between these technologies, focusing on how two flagship algorithms from each paradigm, the Density Matrix Renormalization Group (DMRG) and quantum annealing, might collaborate in the future. Furthermore, a significant challenge in the DMRG algorithm is identifying an appropriate tensor network representation for the quantum system under study. The representation commonly used is called Matrix Product Operator (MPO), and it is notoriously difficult to obtain for certain systems. This thesis outlines an approach to this problem using finite automata, which we apply to construct the MPO for our case study. Finally, we present a practical application of this framework through the quadratic knapsack problem (QKP). Despite its apparent simplicity, the QKP is a fundamental problem in computer science with numerous practical applications. In addition to quantum annealing and the DMRG algorithm, we implement a dynamic programming approach to evaluate the quality of our results. Our results highlight the power of tensor networks and the potential of quantum annealing for solving optimization problems. Moreover, this thesis is designed to be self-explanatory, ensuring that readers with a solid mathematical background can fully understand the content without prior knowledge of quantum mechanics.

Paper Structure

This paper contains 45 sections, 1 theorem, 91 equations, 26 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical framework of quantum mechanics and quantum computing
  3. Hilbert space
  4. Entanglement
  5. Operators
  6. Unitary and hermitian operators
  7. Projection operators
  8. Hamiltonians
  9. QUBO formulation
  10. Adiabatic quantum computing
  11. Quantum annealing
  12. Introduction to tensor networks
  13. Basic concepts: notation, operations, and gauge freedom
  14. Special types of tensors
  15. Tensor operations
  16. ...and 30 more sections

Key Result

Theorem 3.3

(Schmidt decomposition) Given a bipartition of a Hilbert space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$, any state $|\psi\rangle \in \mathcal{H}$ can be written as a Schmidt decomposition into orthonormal vectors $|\alpha_1 \rangle, \ldots , |\alpha_n \rangle \in \mathcal{H}_A$ and $|\bet where $s_i$ are called Schmidt coefficients. These coefficients can be chosen as real and non-negat

Figures (26)

  • Figure 1: (a) shows the diagrammatic representation of a vector (1-rank tensor), a matrix (2-rank tensor), and a 3-rank tensor, while (b) illustrates the notation for the identity matrix in tensor notation. Images are from Ref. TNorg.
  • Figure 2: Example of the contraction of a 2-rank tensor with a 3-rank tensor over a common index. Image from Ref. TNorg.
  • Figure 3: Image from Or_s_2014 showing the importance of the order in which indices are contracted in the contraction of multiple tensors, assuming each index has a dimension $D$. See how case (a) is more efficient because it starts the contraction with less interconnected tensors, but this heuristic is not always right (in fact, determining the optimal ordering is an NP-hard problem GlenEvenbly2022practical).
  • Figure 4: Decomposition of the initial tensor $C_{i_1 i_2 \ldots i_N}$ (left), which is an $N$-rank tensor, into its MPS form (right), which contains $N-2$ rank-3 tensors and two boundary rank-2 tensors.
  • Figure 5: Right-orthogonal tensor in an MPS.
  • ...and 21 more figures

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • ...and 3 more