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Nearly tight bounds for testing tree tensor network states

Benjamin Lovitz, Angus Lowe

TL;DR

The paper addresses the problem of testing whether an unknown pure state on $n$ qudits has a tree tensor network state (TTNS) description with bond dimension $r$ or is $\varepsilon$-far in trace distance. It develops symmetry-based testing strategies grounded in Schur-Weyl duality and Weak Schur Sampling to obtain dimension-free copy complexities, proving an upper bound of $O\big(\frac{nr^2}{\varepsilon^2}\big)$ copies for one-sided TTNS tests and a matching $\Omega\big(\frac{nr^2}{\varepsilon^2\log n}\big)$ lower bound for $r \ge 2+\log n$. The work also analyzes a special Prod$_2$ class with constant Schmidt rank $r=2$ on $3$-level local systems, establishing a tight $\Theta(\sqrt{n})$ copy complexity, and investigates practical few-copy measurement regimes with explicit, closed-form error expressions. Together, these results substantially close the previous quadratic gap in copy complexities for TTNS/MPS testing, illuminate the role of symmetry in pure-state property testing, and offer guidance for verifying tensor-network descriptions in quantum many-body systems. Open questions include constant-bond TTNS testing, extensions to general graphs, and potential improvements for two-sided error regimes.

Abstract

Tree tensor network states (TTNS) generalize the notion of having low Schmidt-rank to multipartite quantum states, through a parameter known as the bond dimension. This leads to succinct representations of quantum many-body systems with a tree-like entanglement structure. In this work, we study the task of testing whether an unknown pure state is a TTNS on $n$ qudits with bond dimension at most $r$, or is far in trace distance from any such state. We first establish that, independent of the dimension of the state, $O(nr^2)$ copies suffice to accomplish this task with one-sided error. We then prove that $Ω(n r^2/\log n)$ copies are necessary for any test with one-sided error whenever $r\geq 2 + \log n$. In particular, this closes a roughly quadratic gap in the previous bounds for testing matrix product states in this setting. On the other hand, when $r=2$ we show that $Θ(\sqrt{n})$ copies are both necessary and sufficient for the related task of testing whether a state is a product of $n$ bipartite states having Schmidt-rank at most $r$, for some choice of the qudit dimensions. We also study the performance of tests using measurements performed on a small number of copies at a time. Here, we obtain new bounds for testing rank, Schmidt-rank, and TTNS when the tester is restricted to making measurements on $r+1$ copies of the state.

Nearly tight bounds for testing tree tensor network states

TL;DR

The paper addresses the problem of testing whether an unknown pure state on qudits has a tree tensor network state (TTNS) description with bond dimension or is -far in trace distance. It develops symmetry-based testing strategies grounded in Schur-Weyl duality and Weak Schur Sampling to obtain dimension-free copy complexities, proving an upper bound of copies for one-sided TTNS tests and a matching lower bound for . The work also analyzes a special Prod class with constant Schmidt rank on -level local systems, establishing a tight copy complexity, and investigates practical few-copy measurement regimes with explicit, closed-form error expressions. Together, these results substantially close the previous quadratic gap in copy complexities for TTNS/MPS testing, illuminate the role of symmetry in pure-state property testing, and offer guidance for verifying tensor-network descriptions in quantum many-body systems. Open questions include constant-bond TTNS testing, extensions to general graphs, and potential improvements for two-sided error regimes.

Abstract

Tree tensor network states (TTNS) generalize the notion of having low Schmidt-rank to multipartite quantum states, through a parameter known as the bond dimension. This leads to succinct representations of quantum many-body systems with a tree-like entanglement structure. In this work, we study the task of testing whether an unknown pure state is a TTNS on qudits with bond dimension at most , or is far in trace distance from any such state. We first establish that, independent of the dimension of the state, copies suffice to accomplish this task with one-sided error. We then prove that copies are necessary for any test with one-sided error whenever . In particular, this closes a roughly quadratic gap in the previous bounds for testing matrix product states in this setting. On the other hand, when we show that copies are both necessary and sufficient for the related task of testing whether a state is a product of bipartite states having Schmidt-rank at most , for some choice of the qudit dimensions. We also study the performance of tests using measurements performed on a small number of copies at a time. Here, we obtain new bounds for testing rank, Schmidt-rank, and TTNS when the tester is restricted to making measurements on copies of the state.

Paper Structure

This paper contains 37 sections, 29 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Copy complexity of testing trees
  4. Tests using few-copy measurements
  5. Overview of techniques
  6. Open questions
  7. Preliminaries
  8. Notation
  9. Sets, ordering
  10. Bra-ket notation, adjoints
  11. Random variables, distributions
  12. Property testing of quantum states
  13. Optimal measurements of pure states from symmetries
  14. Pure state tests with perfect completeness
  15. Schur-Weyl duality applied to bipartite systems
  16. ...and 22 more sections

Key Result

Theorem 1.1

Fix a sequence $(G_n)_n$ of tree graphs on $n\geq 2$ vertices. There exists an algorithm for testing whether $\psi \in \mathsf{TTNS}(G_n,r)$ with one-sided error using $O(nr^2/\varepsilon^2)$ copies of $\psi$. Moreover, in the regime where $\varepsilon\in (0,1/\sqrt{6}]$ and $r\geq 2+\log n$, any al

Figures (3)

  • Figure 1: (a) Depiction of a TTNS on a graph $G$ with $n=6$ vertices and bond dimension $r$. Physical indices are represented by thin lines and bond indices by thick lines. Here, each empty vertex has physical dimension $d$ and each filled vertex has physical dimension $d^3$. (b) A product of $5$ bipartite states where all the physical dimensions are equal to $d$. The dotted lines are a visual guide to identify Hilbert spaces. If each of the Schmidt-ranks is at most $r$ then this state is an element of $\mathsf{Prod}_2(10,r)\subseteq\mathsf{TTNS}(G,r)$ and the TTNS test accepts. On the other hand, our hard instance for the lower bound takes each of the $5$ bipartite states to be slightly far from Schmidt-rank $r$ (as in \ref{['eq:hard_phi']}), and a test with one-sided error struggles to reject.
  • Figure 3: The error probability $\beta(\varepsilon)$ for $d$ large and $r=1$ (red), $r=2$ (orange), $r=3$ (yellow), $r=4$ (green), and $r=5$ (blue), along with the previous best known upper bound for $r=1$ (grey) soleimanifar2022testingmps, and the previous best known lower bound for $r=1$ (black) harrow2013testing. The red and grey plots are equal for $\varepsilon\leq 1/2$.
  • Figure 4: Illustration of the orthogonal projection operator $\Gamma_2 = P^{(v)}_{u_1}P^{(v)}_{u_2}$ corresponding to the set of vertices at height $2$, which consists of only the root node $v$. The two projections in the product commute since they act non-trivially on disjoint subsystems.

Theorems & Definitions (57)

  • Theorem 1.1: Testing trees with one-sided error
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4: Testing trees with few-copy measurements
  • Lemma 2.1
  • proof
  • Theorem 2.2: Implied by Theorem 3.2 in chen2024local
  • Proposition 2.3
  • Theorem 2.4: PC-optimal Schmidt-rank test
  • Definition 2.5: Schur polynomials
  • ...and 47 more