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arxiv: 2412.05202 · v4 · pith:KW7INAGUnew · submitted 2024-12-06 · 🪐 quant-ph

Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations

Vladyslav Bohun , Illia Lukin , Mykola Luhanko , Georgios Korpas , Philippe J.S. De Brouwer , Mykola Maksymenko , Maciej Koch-Janusz This is my paper

Pith reviewed 2026-05-23 07:37 UTC · model grok-4.3

classification 🪐 quant-ph
keywords matrix product statesentanglement scalingsmooth functionsquantum circuit approximationtensor cross interpolationLevy distributionsshallow circuits
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The pith

Entanglement across bonds in the MPS representation of a function decays asymptotically according to the function's smoothness class, whether real or complex.

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

The paper establishes rigorous asymptotic expansions that link the rate at which entanglement decays across successive bonds in a matrix-product-state encoding to the differentiability or analyticity order of the input function. Lower entanglement for smoother functions directly permits construction of shallower quantum circuits that still approximate the encoded distribution to high accuracy. The authors incorporate localization and support properties into the same scaling analysis. From these expansions they derive an improved MPS construction algorithm that employs tensor cross interpolation to build the required circuits in a memory-efficient manner. The resulting circuits are validated by sampling heavy-tailed distributions, including Levy distributions, on IBM quantum hardware with up to 156 qubits.

Core claim

For input functions that are smooth (real or complex), the bond entanglement in their matrix-product-state representation admits rigorous asymptotic expansions governed by the smoothness class; these expansions, together with localization and support considerations, yield an improved tensor-cross-interpolation algorithm that produces shallow, accurate quantum circuits for encoding the functions.

What carries the argument

Matrix product state representation of a function, whose bond entanglement scaling is controlled by the smoothness class and is used to construct shallow quantum circuits via tensor cross interpolation.

If this is right

  • The derived expansions give explicit decay rates for entanglement as a function of differentiability order or analyticity.
  • Tensor cross interpolation combined with the scaling results produces circuits whose gate count scales linearly with system size while maintaining accuracy.
  • Heavy-tailed distributions such as Levy distributions can be loaded and sampled on current quantum hardware up to at least 156 qubits.
  • The same scaling analysis applies to both real-valued and complex-valued input functions.

Where Pith is reading between the lines

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

  • The same smoothness-based scaling could be used to pre-select bond dimensions before circuit construction, reducing classical preprocessing cost.
  • If the asymptotic expansions hold for functions on higher-dimensional domains, the method may extend to multivariate probability distributions without exponential growth in circuit depth.
  • The approach offers a concrete route to test whether smoothness-induced entanglement reduction improves performance on other quantum algorithms that rely on function encoding.

Load-bearing premise

The entanglement decay rate in the MPS representation is governed by the smoothness class of the function in a manner that permits a rigorous asymptotic expansion.

What would settle it

A concrete counter-example would be a C^infty function whose measured bond entanglement in an MPS truncation fails to follow the predicted asymptotic decay rate (for instance, exponential rather than the expected polynomial or faster).

Figures

Figures reproduced from arXiv: 2412.05202 by Georgios Korpas, Illia Lukin, Maciej Koch-Janusz, Mykola Luhanko, Mykola Maksymenko, Philippe J.S. De Brouwer, Vladyslav Bohun.

Figure 1
Figure 1. Figure 1: a) A real or complex function f(x) on [0, 1] is discretized on a binary grid x = P i σi2 −i . Successive digits σi of the binary expansion describe decreasing spatial scales: parts of the interval [0, 1] sharing the same value of σi are shown in green and blue. b) Discretized function can be represented as a rank-N tensor fσ1σ2...σN . c) A matrix product state (MPS) can approximate fσ1σ2...σN , but the bon… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the analytical (see Theorem [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Entanglement decay for exponential and polynomial localization: [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An overview of our MPS based-circuit construction algorithm. First, the standard canonical form MPS is transformed [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The dependence of the infidelity on the number and [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: a) The log-log plot of infidelity of a 1-layer quantum circuit as a function of the size of the truncated support interval L. We fix the discretization step size, with the qubit number given by N = 10 + ⌈log2 L⌉. Here we consider the truncated normal distribution with σ = 2, show the numerically computed infidelity, and compare it to the analytical estimates. The label “Analytical” refers to Eq.(17) and “A… view at source ↗
Figure 7
Figure 7. Figure 7: The encoded L´evy distribution (c = 1) for different truncated support and discretization (number of qubits) sizes on an ideal noiseless simulator. Amplitudes of the quantum state created using a 2-layer encoding circuit are compared to the exact distribution. In a) for N = 6 qubits and L = 1; in b) for N = 10 qubits and L = 5. a general unitary synthesis requires, however, 3 CNOT gates and is not optimal.… view at source ↗
Figure 8
Figure 8. Figure 8: Encoding circuits execution on the ibm torino QPU. a) L´evy distribution with c = 50 truncated to L = 29 with N = 20 qubits b) L´evy distribution with c = 1310 on L = 214 with N = 25 qubits c) Log-normal distribution with on N = 25 qubits d) Gamma distribution with shape parameter k = 1 on N = 25 qubits. In each case 5000 shots were used. All executions pass the KS test with 200 samples. The better of the … view at source ↗
Figure 9
Figure 9. Figure 9: Tensor Cross Interpolation (TCI) constructed 2-layer encoding circuits for the [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (left) The amplitudes of the quantum state encoding the L´evy distribution on 40 qubits obtained using Tensor Cross Interpolation (TCI), compared to the exact Matrix Product State (MPS) amplitudes (right) The results of sampling from the 40-qubit quantum circuit constructed using TCI approach on an ideal simulator [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Tensor Cross Interpolation (TCI) constructed 2-layer encoding circuits for the L´evy distributions on 50 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The L´evy distribution encoded with 1- and [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
read the original abstract

Encoding classical data in a quantum state is a key prerequisite of many quantum algorithms. Recently matrix product state (MPS) methods emerged as the most promising approach for constructing shallow quantum circuits approximating input functions, including probability distributions, with only linear number of gates. We derive rigorous asymptotic expansions for the decay of entanglement across bonds in the MPS representation depending on the smoothness of the input function, real or complex. We also consider the dependence of the entanglement on localization properties and function support. Based on these analytical results we construct an improved MPS-based algorithm yielding shallow and accurate encoding quantum circuits. By using Tensor Cross Interpolation we are able to construct utility-scale quantum circuits in a compute- and memory-efficient way. We validate our methods by loading heavy-tailed distributions, including Levy, important in finance, but they apply to any smooth function inputs. We test the performance of the resulting quantum circuits by executing and sampling from them on IBM quantum devices, for up to 156 qubits.

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 / 3 minor

Summary. The manuscript derives rigorous asymptotic expansions relating the decay of entanglement across bonds in the MPS representation of input functions (real or complex) to the smoothness class of the function, including dependence on localization and support properties. These analytic results are used to construct an improved MPS-based algorithm for shallow quantum circuit approximations of smooth functions, implemented efficiently via Tensor Cross Interpolation. The approach is validated by encoding heavy-tailed distributions (e.g., Lévy) and executing the resulting circuits on IBM quantum hardware for systems up to 156 qubits.

Significance. If the claimed expansions are rigorously established, the work supplies a parameter-free theoretical link between function smoothness and MPS entanglement scaling that directly informs circuit depth and accuracy for quantum data encoding. The hardware demonstration on utility-scale systems and the focus on finance-relevant distributions (Lévy) add practical value; the combination of analytic derivations with reproducible numerical validation on real devices is a clear strength.

minor comments (3)
  1. §3 (or wherever the expansions are stated): the asymptotic forms should be accompanied by explicit remainder bounds or convergence rates to make the 'rigorous' claim fully verifiable without external references.
  2. Figure 4 (hardware sampling results): error bars or shot statistics are not visible in the caption; adding them would clarify the statistical significance of the reported fidelities for 156-qubit circuits.
  3. Notation: the definition of the smoothness class (e.g., C^k vs. analytic) and the precise bond index for the entanglement cut should be restated once in the main text before the first expansion to avoid ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of our manuscript, including the accurate summary of our analytic results on entanglement scaling and the practical demonstration on IBM hardware up to 156 qubits. The recommendation for minor revision is noted; however, the report contains no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central claim consists of analytic derivations of asymptotic expansions relating entanglement decay rates in MPS representations directly to the smoothness class (analyticity or differentiability order) of the input function. These steps are presented as independent mathematical results from function class definitions, with no reduction to fitted parameters, self-definitional constructions, or load-bearing self-citations that would make the expansions equivalent to their inputs by construction. Numerical tests and circuit constructions are downstream applications and do not retroactively define the claimed expansions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the existence of an MPS representation whose entanglement is controlled by function smoothness; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

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    Proof of Theorem 2 Let us consider the reduced density matrix of the first k qubits ρk and compute its particular element: ρk(uk, u′ k) = 1 2k Z 1 0 f(uk + v/2k)f ∗(u′ k + v/2k)dv, (A16) where uk, u′ k are the combined indices of the reduced density matrix. Expanding the functions f, f ∗ in v/2k we can write the reduced density matrix as ρk = 1 2k X m,n≥0...

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    To the leading order for large enough k we have: Λk,2 ≈ s ⟨f(2)(u)|P1|f(2)(u)⟩ 720 × 16k = r g2(f) 720 × 16k , (B1) where we defined g2(f) = ⟨f(2)(u)|P1|f(2)(u)⟩

    The third eigenvalue and their sum For the error analysis we compute the expression for the third entanglement coefficient Λ k,2. To the leading order for large enough k we have: Λk,2 ≈ s ⟨f(2)(u)|P1|f(2)(u)⟩ 720 × 16k = r g2(f) 720 × 16k , (B1) where we defined g2(f) = ⟨f(2)(u)|P1|f(2)(u)⟩. The sum of the remaining eigenvalues appear in com- puting fidel...

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    The leading term 21L2/8c2 defines the characteristic scale k = log L/c, where the entanglement spectra become small

    Exact functional coefficients for probability distributions Here we give explicit expressions for the g1(f) func- tions for the symmetrical normal ( µ = L/2), log-normal and L´ evy distributions, obtained by performing the inte- grals in Eq.A13 for these examples: g1( p p(x)normal) (B3) = L2 4σ2  1 − r L2 2σ2 exp − L2 8σ2 (Γ(1/2) − Γ(1/2, L2/8σ2))   g...

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  59. [60]

    − Γ( 1 2 , L2 8σ2 ) − − L6 32σ6 exp h − L2 4σ2 i [Γ( 1

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  60. [61]

    function values

    − Γ( 1 2 , L2 8σ2 )]2 (B6) If we drop the finitness of the interval, that is, consider the large L limit, the expression simplifies to: g2( p p(x)normal) = 1 8σ4 . (B7) Appendix C: Entanglement and function localization In Sec.III A we showed that the entanglement on the smallest scales exhibits universal exponential decay. We can similarly ask how the en...

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