Wavelet Matrix Product States (wMPS)

• Wavelet Matrix Product States (wMPS) are a continuum variational ansatz that integrates Daubechies wavelet discretization with matrix product states to represent quantum fields.
• They use compactly supported, multi-resolution scaling functions to construct a local tensor-product Fock space, enabling standard MPS optimization for interacting theories.
• The multi-resolution refinement via wavelet transforms ensures a systematic and exact variational embedding of continuum Hamiltonians into discrete tensor networks.

Searching arXiv for the specified paper and closely related context. Wavelet Matrix Product States (wMPS) are a variational ansatz for continuum quantum field theory states built by combining a wavelet-based discretization of the field with a standard matrix product state (MPS) representation on the resulting local tensor-product Hilbert space. In the formulation introduced in "Wavelet Matrix Product States for Quantum Fields" (Kaplan et al., 22 Jun 2026), the construction uses sufficiently regular Daubechies scaling functions, specifically with order $N \ge 6$, so that the resulting 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. A central conceptual feature is that wMPS are not lattice approximations in the usual sense; they are bona fide states in the continuum QFT Fock space, expressed in a discrete orthonormal basis adapted to locality and multi-resolution refinement.

1. Definition and continuum embedding

For a continuum bosonic field $\hat\psi(x)$, wMPS are built from coarse-grained mode operators obtained by projecting the field onto Daubechies scaling functions $s_n^r(x)$,

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$

At fixed resolution $r$, these operators generate a local tensor-product Fock space

$\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$

which is a subspace of the continuum Fock space $\mathcal{F}(L^2(\mathbb{R}))$ (Kaplan et al., 22 Jun 2026).

A wMPS is then an MPS living in this truncated, resolution-$r$ Fock space. In finite systems it is written as an MPS over local occupation-number states associated with the resolution-$r$ modes, while in the translation-invariant thermodynamic limit it is an infinite, translation-invariant MPS on the resolution-$r$ local Fock spaces. The terminology "wavelet-MPS" or "wMPS" refers precisely to this continuum-embedded MPS construction.

The continuum setting is explicit at the level of the bosonic Fock expansion,

$\hat\psi(x)$0

Because the scaling functions are compactly supported and orthonormal, the induced Hilbert space at fixed resolution has a natural tensor-product structure. This is the structural input that allows standard MPS techniques to act directly inside the continuum Hilbert space rather than on an external lattice discretization.

2. Wavelet basis, regularity, and the role of Daubechies functions

The construction starts from a single compactly supported scaling function $\hat\psi(x)$1 on $\hat\psi(x)$2, together with dyadic dilations and translations,

$\hat\psi(x)$3

so that

$\hat\psi(x)$4

These functions satisfy the refinement relation

$\hat\psi(x)$5

with filter coefficients $\hat\psi(x)$6, or equivalently

$\hat\psi(x)$7

for the mother scaling function (Kaplan et al., 22 Jun 2026).

The basis selection is governed by three properties identified as necessary for a variational continuum-QFT basis. First, locality requires compact support so that the Hamiltonian remains local in the discrete description. Second, regularity is needed because the non-relativistic kinetic energy contains $\hat\psi(x)$8, and the basis must therefore be differentiable to ensure finite energy. Third, inclusion across scales is required so that the basis at resolution $\hat\psi(x)$9 embeds into that at resolution $s_n^r(x)$0, allowing systematic refinement.

Daubechies scaling functions satisfy these conditions. The family is selected by additionally requiring exact representation of polynomials up to degree $s_n^r(x)$1,

$s_n^r(x)$2

For $s_n^r(x)$3, the scaling function has a continuous first derivative. In the wMPS construction, this regularity threshold is crucial because it makes the kinetic energy finite. A common misconception is to treat the wavelet basis merely as a numerical grid; here its regularity and nested-scale structure are constitutive parts of the variational ansatz.

3. Exact projected Hamiltonians and strict variationality

A central claim of the framework is exact variationality at fixed resolution. For states $s_n^r(x)$4 in the resolution-$s_n^r(x)$5 subspace, the projected Hamiltonian satisfies

$s_n^r(x)$6

with

$s_n^r(x)$7

The coefficients are overlap integrals of scaling functions,

$s_n^r(x)$8

$s_n^r(x)$9

so the continuum Hamiltonian is rewritten exactly in the local discrete basis (Kaplan et al., 22 Jun 2026).

Because the scaling functions are compactly supported, these couplings vanish beyond finite separation. The projected Hamiltonian $\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$0 is therefore local and can be written exactly as a matrix product operator (MPO). This locality is the technical reason that standard MPS optimization applies. The interaction term $\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$1 is retained explicitly, rather than being absorbed into a Gaussian or free-field reduction, so the method is not restricted to free theories.

This exact projection differentiates wMPS from ad hoc discretizations that require an external continuum extrapolation. A plausible implication is that the resolution parameter $\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$2 should be understood as a variational scale parameter within the continuum Hilbert space, not as a literal lattice spacing in an independent lattice model.

4. Optimization algorithms and multi-resolution refinement

At fixed resolution, optimization proceeds with standard tensor-network methods. The implementation described for wMPS uses VUMPS for translation-invariant uniform MPS, followed near convergence by gradient descent on the Grassmann manifold (Kaplan et al., 22 Jun 2026). Since the projected Hamiltonian is an interacting local MPO, the optimization problem is the usual one for interacting MPS Hamiltonians.

The multi-resolution structure is inherited from the nested sequence of scaling spaces,

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$3

which lifts to

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$4

This inclusion property enables iterative refinement: a coarse-scale optimized state can be embedded into a finer-resolution Hilbert space instead of being reinitialized.

The refinement mechanism uses the wavelet complement $\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$5 with basis functions $\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$6,

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$7

and the inverse wavelet transform

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$8

On Fock space one defines wavelet modes

$\hat a_n^r := \int d x \, s_n^r(x)\,\hat\psi(x).$9

so that the field operator decomposes exactly as

$r$0

The inverse wavelet transform admits a circuit interpretation as a local brick-wall quantum circuit. For $r$1, the mode-space transform has depth $r$2. The procedure for refining a wMPS $r$3 is given in three steps:

1. Apply the inverse wavelet transform with wavelet modes in vacuum.
2. Contract the resulting tensor network using 3 steps of ITEBD, producing a two-site-unit-cell MPS at resolution $r$4.
3. Approximate projection back to a one-site translation-invariant MPS, yielding a new wMPS $r$5.

This refinement strategy is what allows optimization at successively smaller length scales without restarting from random states.

5. Lieb-Liniger benchmark and continuum observables

The primary application presented for wMPS is the one-dimensional Lieb-Liniger model,

$r$6

For $r$7, $r$8, and local cutoff $r$9, energy densities obtained with wMPS are compared against the exact Bethe-Ansatz solution (Kaplan et al., 22 Jun 2026).

At fixed bond dimension $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$0, the energy-density error decreases roughly exponentially in resolution $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$1, that is, linearly in lattice spacing $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$2. For larger $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$3, optimization without refinement becomes unstable. With iterative refinement, the calculation can be pushed to much larger $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$4, until the error is dominated by finite bond dimension rather than finite scale. The resulting plateau at fixed $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$5 matches the corresponding CMPS energy density, and the paper therefore conjectures that the $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$6 limit of wMPS coincides with CMPS at the same bond dimension.

The formalism also computes correlation functions directly in the continuum. For the one-body correlator,

$\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$7

compact support reduces the evaluation to a finite sum,

$\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$8

Density-density correlators are treated similarly. In the Tonks-Girardeau regime $\mathcal{H}^r = \bigotimes_n \mathcal{F}(f_n^r),$9, the reported results show the expected Friedel oscillations and approach the exact strongly interacting limit as $\mathcal{F}(L^2(\mathbb{R}))$0 increases.

6. Conceptual significance, relations, and limitations

The physical and practical significance attributed to wMPS is that they provide a strictly variational continuum tensor-network method that lives directly in QFT Fock space, uses exact local MPO representations of continuum Hamiltonians, works for interacting theories, can be optimized with standard MPS algorithms, and can be refined systematically to finer length scales via wavelet multi-resolution analysis (Kaplan et al., 22 Jun 2026). In this sense, the construction combines variational continuum structure with discrete tensor-network machinery.

Two clarifications are central to the interpretation of the method. First, wMPS are not ordinary lattice approximations. Their discrete data structure arises from an orthonormal basis inside the continuum Hilbert space, not from replacing the continuum theory with a separate lattice model. Second, the method does not rely on Gaussianity or free-field structure; the quartic interaction is preserved explicitly in the MPO representation.

The comparison with CMPS is suggestive but remains a conjectural point in the formulation presented for the Lieb-Liniger benchmark. The observed energy-density plateau at fixed bond dimension motivates the conjecture that $\mathcal{F}(L^2(\mathbb{R}))$1 wMPS coincide with CMPS at the same bond dimension. This suggests a possible unification between wavelet-based multi-resolution tensor networks and established continuum MPS constructions, with the refinement circuit providing a concrete route to reach fine scales while maintaining numerical stability.