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Toward bootstrapping tensor-network contractions

Seishiro Ono, Yanbai Zhang, Hoi Chun Po

Abstract

Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. Existing approaches typically rely on approximate contraction schemes and do not provide certified error bars. We introduce a numerical bootstrap framework which casts the problem of tensor-network contractions into a convex optimization problem, thereby yielding certified lower and upper bounds on expectation values of physical observables. As a proof-of-principle, we construct such constraints explicitly for translationally invariant matrix product states and demonstrate that, assuming a canonical form, second-order-cone relaxation can provide tight bounds on the contraction result. We further demonstrate that when the requirement on canonical form is lifted, a more general semidefinite-programming approach could yield similar tight bounds at higher but still polynomial computational cost. Our work suggests numerical bootstrap could be a possible way forward for the rigorous contractions of tensor networks.

Toward bootstrapping tensor-network contractions

Abstract

Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. Existing approaches typically rely on approximate contraction schemes and do not provide certified error bars. We introduce a numerical bootstrap framework which casts the problem of tensor-network contractions into a convex optimization problem, thereby yielding certified lower and upper bounds on expectation values of physical observables. As a proof-of-principle, we construct such constraints explicitly for translationally invariant matrix product states and demonstrate that, assuming a canonical form, second-order-cone relaxation can provide tight bounds on the contraction result. We further demonstrate that when the requirement on canonical form is lifted, a more general semidefinite-programming approach could yield similar tight bounds at higher but still polynomial computational cost. Our work suggests numerical bootstrap could be a possible way forward for the rigorous contractions of tensor networks.
Paper Structure (9 sections, 22 equations, 4 figures)

This paper contains 9 sections, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results
  3. General idea
  4. Warm up: Single qubit example
  5. Linear matrix inequality and second-order cone
  6. Bootstrap approach for contracting MPS
  7. MPS in canonical form
  8. MPS without canonical form
  9. Discussion

Figures (4)

  • Figure 1: Illustration of an example of two-dimensional tensor networks. (a) A transfer matrix defined by a local tensor of PEPS. (b) An operator acting on the local bond space. (c) A diagrammatic expression of the expectation value of a local operator $O$ with respect to a given environment $E$. (d) An expectation value of a local operator $O$ for a given boundary environment $\rho$. Tensors surrounding the target site form a linear map $\mathcal{E}:\mathbb{M}_{\chi^{4N}}(\mathbb{C})\rightarrow \mathbb{M}_{\chi^{4}}(\mathbb{C})$.
  • Figure 2: Parameter space $\mathcal{B}_{n}$ of density matrices for a single qubit for $n=0,1,2,3,4$. For comparison, $\{\mathcal{B}_{m}\}_{m=0}^{n-1}$ are also shown semi-transparently. The parameter space becomes increasingly constrained as $n$ increases.
  • Figure 3: Projected parameter space of environments at the target site for a random MPS with $\chi = 20$. The full parameter space is defined by $\mathcal{C}_{\Lambda_{R}}\ (1 \leq R \leq 20)$ and the normalization slice $x_0 = 1/\chi$, whose intersection forms a hyper-ellipsoid. For visualization purposes, the parameter space is projected onto the $x_1$-$x_2$ plane. The right panel shows an enlarged view of the region $-1.5 \times 10^{-4} \leq x_1 \leq 5 \times 10^{-5}$ and $-2 \times 10^{-4} \leq x_2 \leq 1 \times 10^{-5}$. Similar tightening can be observed for other projections of the parameter space.
  • Figure 4: Numerical bounds on the expectation value of a randomly chosen observable for a randomly generated MPS with bond dimension $\chi=8$. Here, we take $L=R$ and compute the bounds for $0\leq R \leq 30$. The result for $R \to \infty$ is computed by left- and right-fixed point environments.