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Block-Encoding Tensor Networks and QUBO Embeddings

Sebastian Issel

TL;DR

The paper develops a constructive method to convert any tensor network into a sequence of local unitaries whose product block-encodes the TN contraction, enabling direct use in QET/QSVT workflows. It introduces per-site Unitary-SVD dilations, a linearization into tensor-train form, and a local-SVD concentration canonicalization to manage nonunitary weight, along with a deterministic QUBO embedding sweep whose coupling scales with sweep pathwidth. The framework supports arbitrary TN geometries with nonuniform bond dimensions, provides explicit resource and error bounds, and includes pseudocode, complexity analyses, and post-selection bookkeeping. This work bridges structured classical TN representations with quantum block-encoding primitives, offering practical pathways for state preparation, operator learning, and hardware-aware embeddings in quantum algorithms.

Abstract

We give an algorithm that converts any tensor network (TN) into a sequence of local unitaries whose composition block-encodes the network contraction, suitable for Quantum Eigenvalue / Singularvalue Transformation (QET/QSVT). The construction embeds each TN as a local isometry and dilates it to a unitary. Performing this step for every site of the tensor, allows the full network to be block-encoded. The theory is agnostic to virtual-bond sizes; for qubit resource counts and examples we assume global power-of-two padding. Further, we present a deterministic sweep that maps Quadratic Unconstrained Binary Optimization (QUBO) / Ising Hamiltonians into Matrix Product Operators (MPOs) and general TN. We provide formal statements, pseudo-code, resource formulae, and a discussion of the use for state preparation and learning of general quantum operators.

Block-Encoding Tensor Networks and QUBO Embeddings

TL;DR

The paper develops a constructive method to convert any tensor network into a sequence of local unitaries whose product block-encodes the TN contraction, enabling direct use in QET/QSVT workflows. It introduces per-site Unitary-SVD dilations, a linearization into tensor-train form, and a local-SVD concentration canonicalization to manage nonunitary weight, along with a deterministic QUBO embedding sweep whose coupling scales with sweep pathwidth. The framework supports arbitrary TN geometries with nonuniform bond dimensions, provides explicit resource and error bounds, and includes pseudocode, complexity analyses, and post-selection bookkeeping. This work bridges structured classical TN representations with quantum block-encoding primitives, offering practical pathways for state preparation, operator learning, and hardware-aware embeddings in quantum algorithms.

Abstract

We give an algorithm that converts any tensor network (TN) into a sequence of local unitaries whose composition block-encodes the network contraction, suitable for Quantum Eigenvalue / Singularvalue Transformation (QET/QSVT). The construction embeds each TN as a local isometry and dilates it to a unitary. Performing this step for every site of the tensor, allows the full network to be block-encoded. The theory is agnostic to virtual-bond sizes; for qubit resource counts and examples we assume global power-of-two padding. Further, we present a deterministic sweep that maps Quadratic Unconstrained Binary Optimization (QUBO) / Ising Hamiltonians into Matrix Product Operators (MPOs) and general TN. We provide formal statements, pseudo-code, resource formulae, and a discussion of the use for state preparation and learning of general quantum operators.

Paper Structure

This paper contains 30 sections, 3 theorems, 20 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries and Notation
  4. Brief Introduction to Tensor Networks
  5. Used Notation and Conventions
  6. Unitary SVD Decomposition
  7. Overview
  8. Preparing a Tensor Site
  9. Singular Value Decomposition
  10. Padding and Dropping Dimensions
  11. Single‑site dilation
  12. Tensor Block-Encoding
  13. Sequential Composition
  14. Remark.
  15. Sequential Composition: Canonicalization Sweep
  16. ...and 15 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a graph with site tensors $A^{(v)}$ (physical space $P_v$ and bond registers $X_e$ for incident edges). For each $v\in V$ let $\beta_v:=\|A^{(v)}\|_2$ (spectral norm of the unfolding used in the per‑site routine) and suppose there exists a unitary $B^{(v)}$ acting on $P_v\otimes\big as an operator on $P_v\otimes\bigl(\bigotimes_{e\ni v} X_e\bigr)$. Fix boundary bond states $\ket{r

Figures (7)

  • Figure 1: Two undirected site tensors ("spiders") with many open legs and an undirected bond dimension between them.
  • Figure 2: MPO / tensor-trains: virtual bonds undirected, physical in/out directed.
  • Figure 3: Per‑site conversion pipeline. (a) undirected tensor $T$ with many virtual legs. (b) choose a flow and collect virtual legs into top/bottom directed bundles. (c) unfold the site to a matrix $A$.
  • Figure 4: (d)--(f) SVD and core operations: (d) SVD stack; (e) pad/drop to square core; (f) ancilla dilation to unitary $C$.
  • Figure 5: Two‑step sweep schematic: (a) before processing the current site; (b) after processing, the current site emits a directed flow.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: Sequential block‑encoding on graphs
  • proof : Sketch
  • Lemma 1: Local error to global operator bound
  • Lemma 2: Sequential dilation preserves block product
  • proof : Proof sketch