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Gauging tensor networks with belief propagation

by Joseph Tindall, Matt Fishman

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Submission summary

Authors (as registered SciPost users): Joseph Tindall
Submission information
Preprint Link:  (pdf)
Code repository:
Date accepted: 2023-11-13
Date submitted: 2023-11-09 15:34
Submitted by: Tindall, Joseph
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


Effectively compressing and optimizing tensor networks requires reliable methods for fixing the latent degrees of freedom of the tensors, known as the gauge. Here we introduce a new algorithm for gauging tensor networks using belief propagation, a method that was originally formulated for performing statistical inference on graphical models and has recently found applications in tensor network algorithms. We show that this method is closely related to known tensor network gauging methods. It has the practical advantage, however, that existing belief propagation implementations can be repurposed for tensor network gauging, and that belief propagation is a very simple algorithm based on just tensor contractions so it can be easier to implement, optimize, and generalize. We present numerical evidence and scaling arguments that this algorithm is faster than existing gauging algorithms, demonstrating its usage on structured, unstructured, and infinite tensor networks. Additionally, we apply this method to improve the accuracy of the widely used simple update gate evolution algorithm.

Author comments upon resubmission

Dear Editor and Referees,

We thank the referees for their reviews and thorough comments on our article 'Gauging tensor networks with belief propagation’. The referees found our work 'timely’ and 'worthy of publication’. Referee 2 directly recommended publication with no changes whilst Referee 1 has several requested changes which we have addressed and responded to.

We hope that the manuscript is now fit for publication in SciPost Physics.

The authors,

Joseph Tindall and Matt Fishman

List of changes

1) We have updated Figure 4 to fix the update schedule when comparing algorithms. We have also changed Fig. 4d to a benchmark the timings of belief propagation gauging using various update schedules.
2) We have discussed, in section 3.2, the importance of update schedules and its effect on the runtime of the algorithms. We have provided extensive references on the subject of scheduling in belief propagation.
3) We have changed the value of O(C) targeted in Fig. 6b to $10^{ -6}$ from $10^{-3}$. We have also changed the update schedule for BP to a sequential one based on a custom sequence. This has significantly improved timings whilst maintaining the same fidelity.
4) We have avoided the term 'generalized BP gauging’ and instead referred to it as 'BP gauging on a partitioned network'. We have also pointed out explicitly that this method does not consider overlapping partitions, which is possible in 'generalized BP' as defined in the literature.
5) We have added a section title 'Using square root belief propagation to gauge a tensor network state' to the discussion on square root belief propagation.
6) Minor grammatical changes throughout.

Published as SciPost Phys. 15, 222 (2023)

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