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arxiv: 2606.23823 · v1 · pith:GMGZB5EDnew · submitted 2026-06-22 · 🪐 quant-ph · cond-mat.str-el· hep-th

Wavelet Matrix Product States for Quantum Fields

Molly Kaplan , Antoine Tilloy This is my paper

Pith reviewed 2026-06-26 07:48 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elhep-th
keywords wavelet matrix product statescontinuum quantum fieldstensor networksvariational methodsLieb-Liniger modelDaubechies waveletsmulti-resolution analysisFock space
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The pith

Wavelet matrix product states built on Daubechies scaling functions let standard tensor network algorithms optimize states for interacting continuum quantum field theories.

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

The paper introduces a variational method that embeds discrete matrix product states into the continuum using sufficiently regular Daubechies scaling functions. These wavelet matrix product states reside in the Fock space of quantum field theories, maintain finite energy density, and remain compatible with existing optimization routines even when the underlying model includes interactions. The multi-resolution structure of the wavelets further permits successive refinement of the same ansatz to finer length scales without restarting the calculation. The method is illustrated by computing energy densities and correlation functions in the Lieb-Liniger model.

Core claim

Wavelet matrix product states (wMPS) are matrix product states constructed on top of Daubechies scaling functions with N at least 6. These states live directly in the continuum field theory Fock space, possess finite energy density, and can be variationally optimized with standard algorithms without restriction to free theories. The multi-resolution analysis inherent to wavelets supplies a quantum-circuit description that permits iterative refinement to arbitrarily fine length scales.

What carries the argument

The wavelet matrix product state (wMPS), formed by placing a matrix product state on Daubechies scaling functions of sufficient regularity, which embeds the discrete tensor network into continuum Fock space while preserving finite energy.

If this is right

  • Standard matrix product state algorithms can be applied directly to interacting continuum models.
  • Energy density and correlation functions become accessible for the Lieb-Liniger model without discretization artifacts.
  • The same variational state can be refined iteratively to finer scales using the wavelet multi-resolution structure.
  • The approach works for both free and interacting theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may allow tensor-network studies of other one-dimensional continuum models whose critical behavior is sensitive to the ultraviolet cutoff.
  • Because the states remain in the continuum Fock space, the same ansatz could be compared directly with field-theoretic renormalization-group flows.
  • The quantum-circuit description of the multi-resolution step suggests a route to hybrid classical-quantum refinement protocols.

Load-bearing premise

Daubechies scaling functions with N at least 6 are regular enough that the resulting states have finite energy density when they represent interacting continuum fields.

What would settle it

An explicit computation in which the energy density of an optimized wMPS for the Lieb-Liniger model diverges or fails to converge under standard algorithms would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.23823 by Antoine Tilloy, Molly Kaplan.

Figure 1
Figure 1. Figure 1: FIG. 1. The variational manifold. From the full Hilbert space [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Relative error in the wMPS ground state energy [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Observables in the wMPS ground state with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the three steps of the refinement algorithm: (i) writing the inverse wavelet transform as a brick-wall [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Contraction of the original MPS with three layers of the IWT brick-wall circuit into an MPS with a larger unit cell [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparing the output of the refinement algorithm [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparing optimization routines with and without [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We introduce a variational method to solve continuum quantum models with discrete tensor network techniques. The method leverages wavelet matrix product states (wMPS): matrix product states built on top of sufficiently regular ($N\geq 6$) Daubechies scaling functions. These states live in the continuum field theory Fock space, have finite energy density, and can be optimized with standard algorithms, without restriction to free theories. Further, exploiting the multi-resolution analysis built into wavelets, and its quantum circuit description, we can iteratively refine wMPS to obtain accurate approximations at arbitrarily fine length-scales. We showcase the efficiency of the method on the Lieb-Liniger model, computing energy density and correlation functions.

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

2 major / 2 minor

Summary. The paper introduces wavelet matrix product states (wMPS) as a variational ansatz for continuum quantum field theories. These are MPS constructed on Daubechies scaling functions with regularity order N≥6; the states are asserted to lie in the continuum Fock space, possess finite energy density even for interacting Hamiltonians, admit optimization by standard discrete MPS algorithms without restriction to free theories, and support iterative multi-resolution refinement via the wavelet quantum circuit structure. The method is demonstrated on the Lieb-Liniger model by computing energy density and correlation functions.

Significance. If the finite-energy-density claim for interacting models is rigorously established, the construction would provide a concrete bridge between discrete tensor-network algorithms and continuum QFT, enabling controlled variational calculations at arbitrary resolution without ad-hoc cutoffs. The use of a standard benchmark (Lieb-Liniger) and the built-in multi-resolution property are concrete strengths that would be of interest to the tensor-network and quantum-simulation communities.

major comments (2)
  1. [§3] §3 (or the section deriving finite energy density): the Sobolev regularity index for N=6 Daubechies scaling functions is only marginally above 1, which bounds the kinetic term but does not automatically guarantee that the interaction term (e.g., the delta-function contact interaction in Lieb-Liniger) remains finite when the variational parameters are optimized over the discrete MPS manifold. An explicit domain-of-Hamiltonian argument or UV convergence bound for the full interacting operator is required to close this gap.
  2. [§4] §4 (Lieb-Liniger application): the quantitative accuracy claims for energy density and correlations rest on the premise that the wMPS manifold is dense enough in the interacting continuum Hilbert space; without a convergence theorem or explicit error bound relating bond dimension, wavelet order N, and resolution level to the continuum limit, it is unclear whether the reported results are controlled or merely consistent with known values.
minor comments (2)
  1. Notation for the wavelet basis and the embedding into Fock space should be made fully explicit (e.g., the precise definition of the scaling-function coefficients and their relation to the field operators).
  2. Figure captions and axis labels in the Lieb-Liniger plots would benefit from explicit mention of the wavelet order N and bond dimension used for each data point.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight important points regarding rigor in the finite-energy claim and the controlled nature of the numerical results. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (or the section deriving finite energy density): the Sobolev regularity index for N=6 Daubechies scaling functions is only marginally above 1, which bounds the kinetic term but does not automatically guarantee that the interaction term (e.g., the delta-function contact interaction in Lieb-Liniger) remains finite when the variational parameters are optimized over the discrete MPS manifold. An explicit domain-of-Hamiltonian argument or UV convergence bound for the full interacting operator is required to close this gap.

    Authors: The manuscript grounds the finite-energy-density claim in the Sobolev regularity (H^s with s>1) of N≥6 Daubechies scaling functions, which places the wMPS in the domain of the kinetic operator. For the 1D delta-function interaction, the local regularity and compact support of the scaling functions ensure that the quadratic form remains finite; the interaction is a relatively mild perturbation in one dimension. We agree that an expanded discussion would strengthen the presentation and will add an explicit paragraph in the revised §3 deriving the domain of the full Hamiltonian, citing standard results on 1D delta interactions in Sobolev spaces. This clarification does not alter the core construction but makes the argument more self-contained. revision: yes

  2. Referee: [§4] §4 (Lieb-Liniger application): the quantitative accuracy claims for energy density and correlations rest on the premise that the wMPS manifold is dense enough in the interacting continuum Hilbert space; without a convergence theorem or explicit error bound relating bond dimension, wavelet order N, and resolution level to the continuum limit, it is unclear whether the reported results are controlled or merely consistent with known values.

    Authors: Section 4 presents numerical benchmarks on the Lieb-Liniger model to demonstrate that standard MPS algorithms can be applied directly to an interacting continuum theory and that the multi-resolution structure permits iterative refinement. The manuscript does not assert a rigorous density or convergence theorem; such a result would constitute a separate mathematical contribution. The reported agreement with known values is therefore consistency evidence rather than a controlled error analysis. We will revise the text in §4 and the conclusions to state this distinction more explicitly and to emphasize that practical convergence can be assessed by increasing resolution level and bond dimension, as enabled by the wavelet circuit. No new theorem will be added. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces wMPS as a variational ansatz on Daubechies scaling functions (N≥6) and applies standard MPS optimization plus multi-resolution refinement to the Lieb-Liniger model. No quoted step reduces a central claim to a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The finite-energy-density statement is presented as following from the regularity premise rather than being derived circularly from the method itself; the benchmark application uses an independent external model. This is the normal case of a self-contained new ansatz with external validation.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The method rests on standard properties of Daubechies wavelets and introduces the wMPS ansatz as a new variational object; free parameters include wavelet order, bond dimension, and truncation thresholds typical of MPS.

free parameters (2)
  • wavelet regularity order N
    Chosen >=6 to ensure finite energy density; value is a modeling choice rather than fitted to data.
  • MPS bond dimension
    Variational truncation parameter controlling approximation accuracy.
axioms (1)
  • domain assumption Daubechies scaling functions of order N>=6 possess sufficient smoothness and vanishing moments to represent continuum field configurations with finite energy density.
    Invoked to justify the basis choice for the continuum Fock space representation.
invented entities (1)
  • wavelet matrix product state (wMPS) no independent evidence
    purpose: Variational ansatz combining MPS with wavelet basis for continuum quantum fields.
    New state class introduced by the paper; no independent evidence supplied beyond the construction itself.

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discussion (0)

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