pith. sign in

arxiv: 1906.10156 · v1 · pith:G45J6626new · submitted 2019-06-24 · ⚛️ physics.pop-ph · cond-mat.str-el· quant-ph

Pushing Tensor Networks to the Limit

Anastasiia A. Pervishko , Jacob Biamonte This is my paper

Pith reviewed 2026-05-25 16:46 UTC · model grok-4.3

classification ⚛️ physics.pop-ph cond-mat.str-elquant-ph
keywords tensor networkscontinuous matrix product statescMPScontinuum limithigher-dimensional quantum systemsquantum many-body physics
0
0 comments X

The pith

Tensor networks can now be generalized to the continuum in two and higher dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This viewpoint discusses work that removes barriers which had blocked tensor networks from applying to continuous quantum systems beyond one dimension. Tilloy and Cirac introduce continuous matrix product states, called cMPS, that operate directly in the continuum for spatial dimensions of two and higher. The piece presents this as a way to carry the computational advantages of tensor networks into new regimes of quantum many-body physics. A reader would care if the extension holds because it could let the same efficient contraction techniques describe field theories and higher-dimensional models that were previously out of reach.

Core claim

Those authors overcame several past limitations in the generalization of tensor networks to the continuum and proposed a new class of continuous tensor network states (cMPS) which apply to spatial dimensions of two and higher.

What carries the argument

continuous matrix product states (cMPS), the proposed generalization of tensor networks that works in continuous space for dimensions two and above.

Load-bearing premise

The viewpoint accepts that the referenced work has resolved earlier limitations without re-deriving or independently verifying those results here.

What would settle it

Direct computation showing that the proposed cMPS fail to represent or simulate a known solvable continuous 2D system with the expected accuracy or scaling would falsify the claim that the limitations have been overcome.

Figures

Figures reproduced from arXiv: 1906.10156 by Anastasiia A. Pervishko, Jacob Biamonte.

Figure 1
Figure 1. Figure 1: FIG. 1. Tilloy and Cirac have extended the application of tensor networks from a 2D lattice case (left) to a continuous case [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

This $Physics$ viewpoint considers recent work by Tilloy and Cirac [Phys. Rev. X 9, 021040 (2019), arXiv:1808.00976]; those authors overcame several past limitations in the generalization of tensor networks to the continuum and proposed a new class of continuous tensor network states (cMPS) which apply to spatial dimensions of two and higher.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. This Physics viewpoint article summarizes recent work by Tilloy and Cirac (Phys. Rev. X 9, 021040, 2019) claiming that those authors overcame prior limitations in extending tensor networks to the continuum and introduced continuous matrix product states (cMPS) applicable in spatial dimensions d ≥ 2.

Significance. If the summary is accurate, the viewpoint usefully draws attention to a technical advance in continuous tensor-network constructions for higher-dimensional quantum systems. Viewpoints of this type can aid dissemination, but the manuscript itself contains no new derivations, numerical results, or independent checks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; viewpoint summarizes external work

full rationale

This Physics viewpoint article summarizes the Tilloy-Cirac cMPS construction from the cited Phys. Rev. X 9, 021040 (2019) without originating derivations, equations, predictions, or parameter fits. Its claims reduce to reporting of independent external results; no self-definitional steps, fitted inputs called predictions, or load-bearing self-citations exist within the document itself. The paper is self-contained as accurate reporting against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The document introduces no new mathematical content, parameters, axioms, or entities; it is a commentary on external research.

pith-pipeline@v0.9.0 · 5583 in / 1077 out tokens · 33836 ms · 2026-05-25T16:46:01.669891+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 2 internal anchors

  1. [1]

    Ignacio Cirac

    Antoine Tilloy and J. Ignacio Cirac. Continuous ten- sor network states for quantum fields. Phys. Rev. X , 9:021040, May 2019

    work page 2019

  2. [2]

    P. A. M. Dirac. The Principles of Quantum Mechanics

    work page

  3. [3]

    A practical introduction to tensor net- works: Matrix product states and projected entangled pair states

    Romn Ors. A practical introduction to tensor net- works: Matrix product states and projected entangled pair states. Annals of Physics , 349:117 – 158, 2014

    work page 2014

  4. [4]

    Evenbly and G

    G. Evenbly and G. Vidal. Tensor Network States and Ge- ometry. Journal of Statistical Physics , 145(4):891–918, Nov 2011

    work page 2011

  5. [5]

    J. Eisert. Entanglement and tensor network states. Mod- eling and Simulation , page 520, Aug 2013

    work page 2013

  6. [6]

    Tensor Networks in a Nutshell

    Jacob Biamonte and Ville Bergholm. Tensor Networks in a Nutshell. arXiv e-prints, page arXiv:1708.00006, Jul 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions

    Frank Verstraete and J Ignacio Cirac. Renormalization algorithms for quantum-many body systems in two and higher dimensions. arXiv preprint cond-mat/0407066 , 2004

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  8. [8]

    G. Vidal. Entanglement renormalization. Phys. Rev. Lett., 99:220405, Nov 2007

    work page 2007

  9. [9]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Continuous matrix product states for quantum fields. Phys. Rev. Lett. , 104:190405, May 2010

    work page 2010

  10. [10]

    Pervishko and Jacob Biamonte

    Anastasiia A. Pervishko and Jacob Biamonte. Pushing tensor networks to the limit. Physics, 12:59, 2019. FIG. 1. Tilloy and Cirac have extended the application of tensor networks from a 2D lattice case (left) to a continuous case (right) by replacing a sum over discrete indices with a functional integral. (APS/Alan Stonebraker)

    work page 2019