The minimal canonical form of a tensor network
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Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for equality of PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.
Originalsprog | Engelsk |
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Titel | Proceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023 |
Forlag | IEEE |
Publikationsdato | 2023 |
Sider | 328-362 |
ISBN (Elektronisk) | 9798350318944 |
DOI | |
Status | Udgivet - 2023 |
Begivenhed | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, USA Varighed: 6 nov. 2023 → 9 nov. 2023 |
Konference
Konference | 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 |
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Land | USA |
By | Santa Cruz |
Periode | 06/11/2023 → 09/11/2023 |
Sponsor | IEEE, IEEE Computer Society, IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, NSF |
Bibliografisk note
Funding Information:
DPG acknowledges support by the Spanish Ministry of Science and Innovation (“Severo Ochoa Programme for Centres of Excellence in R&D” CEX2019-000904-S and grant PID2020-113523GB-I00), the Spanish Ministry of Economic Affairs and Digital Transformation (project QUANTUM ENIA, as part of the Recovery, Transformation and Resilience Plan, funded by EU program NextGenerationEU), Comunidad de Madrid (QUITEMAD-CM P2018/TCS-4342), and the CSIC Quantum Technologies Platform PTI-001. HN and MW acknowledge NWO grant OCENW.KLEIN.267. MW also acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA -390781972, by the BMBF through project QuBRA, and by the European Research Council (ERC) through ERC Starting Grant 101040907-SYMOPTIC.
Publisher Copyright:
© 2023 IEEE.
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