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arxiv: 2605.00259 · v2 · pith:NMGZ5QDMnew · submitted 2026-04-30 · 🪐 quant-ph

Entanglement Distance of Two- and Multi-Qubit Variational States and Its Quantification with Quantum Computing

Kh. P. Gnatenko , R. O. Hredil , V. Y. Pinchuk , M. Z. Seniak , Y. T. Shevchuk This is my paper

Pith reviewed 2026-05-09 19:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglement distancevariational quantum circuitsRY gatesCZ gatesPauli observablesquantum correlationsrecurrence relationsmulti-qubit systems
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The pith

Recurrence relations on Pauli observables yield exact entanglement distances for RY-CZ variational circuits on qubit chains.

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

This paper derives analytical tools to compute the entanglement distance for states prepared by variational quantum circuits consisting of RY rotation gates and CZ entangling gates. Starting with two-qubit systems, recurrence relations are obtained for the expectation values of Pauli operators, enabling closed-form expressions for quantum correlators and thus the entanglement distance as a function of circuit parameters and depth. The framework is then applied to a closed chain of N qubits, demonstrating that increasing the number of layers causes the influence on any qubit to encompass more distant qubits, corresponding to the propagation of quantum correlations. Explicit analytical formulas are provided specifically for circuits with two layers, and these are validated against numerical simulations using quantum computing tools.

Core claim

By applying successive RY and CZ operations to the Pauli operators and deriving recurrence relations for their expectation values, the entanglement distance of the resulting variational states is obtained analytically for both two-qubit systems and larger closed chains, with the two-layer case yielding explicit expressions that show the depth-dependent spreading of correlations.

What carries the argument

Recurrence relations for the expectation values of Pauli observables under successive layers of RY rotations and CZ entangling gates.

Load-bearing premise

The recurrence relations derived from the specific layered structure of RY and CZ gates fully capture the expectation values of all relevant Pauli observables without additional approximations or truncation.

What would settle it

Exact simulation of a three-layer RY-CZ circuit on a four-qubit closed chain producing an entanglement distance that deviates from the value obtained by applying the two-layer recurrence formulas to the three-layer case.

Figures

Figures reproduced from arXiv: 2605.00259 by Kh. P. Gnatenko, M. Z. Seniak, R. O. Hredil, V. Y. Pinchuk, Y. T. Shevchuk.

Figure 1
Figure 1. Figure 1: Periodic layered architecture of the two-qubit parameterized quantum circuit ansatz. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Surface plot of the entanglement distance [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Surface plot of the entanglement distance [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Surface plot of the entanglement distance [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Quantum circuit diagram for a 5-qubit closed 1D chain ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Surface plot of the entanglement distance [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Entanglement distance surface showing the interplay between the target qubit’s local [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

We study the entanglement distance of variational quantum states for two-qubit and multi-qubit systems. These states are constructed using variational quantum circuits with $R_Y$ rotations and entangling $CZ$ gates. For the two-qubit case, we analytically derive recurrence relations for expectation values of Pauli observables. This approach allows us to calculate quantum correlators and evaluate the entanglement distance as a function of the circuit parameters and depth. The analysis was extended to a closed one-dimensional chain of $N$ qubits. An explicit analytical expression for the entanglement is derived for the case of two layers. We conclude that the entanglement of a qubit with the rest of the system depends on the parameters of the gates acting on first- and second-nearest neighbors in the chain topology of the entangling layers. We also quantify the entanglement of the variational quantum states using quantum computing on the AerSimulator. The corresponding quantum protocols are constructed, and the dependence of the entanglement on the parameters of the variational quantum states is studied. The results of the quantum programming are in good agreement with the theoretical predictions.

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 manuscript studies the entanglement distance of variational quantum states for two-qubit and multi-qubit systems constructed from RY rotation gates and CZ entangling gates. For the two-qubit case, recurrence relations are derived for the expectation values of Pauli observables, allowing analytical computation of correlators and the entanglement distance as functions of circuit parameters and depth. The analysis is extended to a closed one-dimensional chain of N qubits, demonstrating that increasing circuit depth causes more qubits to influence a given site (reflecting the spread of quantum correlations). Explicit analytical expressions are derived for the two-layer case on the closed chain, and these are compared to numerical simulations performed with quantum programming tools, with reported agreement.

Significance. If the recurrence relations are rigorously derived and the numerical agreement holds quantitatively for the periodic boundary conditions, the work provides useful closed-form expressions for tracking entanglement spreading in shallow variational circuits. This could support analysis of variational quantum algorithms by quantifying how correlations propagate layer by layer without requiring full state-vector simulation. The explicit two-layer formulas for the closed chain represent a concrete advance if they correctly capture the periodic wrapping.

major comments (2)
  1. [multi-qubit closed-chain analysis] The central derivation of recurrence relations for Pauli expectations (abstract and multi-qubit section) propagates through successive RY and CZ layers. For the closed 1D chain, the final CZ layer introduces a wrap-around from qubit N to qubit 1. The manuscript must explicitly show whether an additional consistency condition is imposed on the wrap-around correlators; without it, the expressions for the number of influencing qubits may be inexact for N>2.
  2. [numerical comparison] The claim that 'the results agree with the theoretical predictions' (abstract) is load-bearing for the validation of the two-layer closed-chain expressions. The manuscript should report quantitative agreement metrics (e.g., maximum absolute deviation on correlators or entanglement distance) and confirm that the numerics used the exact periodic boundary conditions rather than open-chain proxies.
minor comments (2)
  1. [abstract] Abstract contains a grammatical error: 'The analysis were extended' should read 'The analysis was extended'.
  2. [two-qubit section] Notation for the entanglement distance and the specific Pauli observables whose expectations enter the recurrence relations should be defined at first use with a clear equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us improve the clarity of the derivations and the validation of the results. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [multi-qubit closed-chain analysis] The central derivation of recurrence relations for Pauli expectations (abstract and multi-qubit section) propagates through successive RY and CZ layers. For the closed 1D chain, the final CZ layer introduces a wrap-around from qubit N to qubit 1. The manuscript must explicitly show whether an additional consistency condition is imposed on the wrap-around correlators; without it, the expressions for the number of influencing qubits may be inexact for N>2.

    Authors: We appreciate the referee pointing out the need for explicit treatment of the periodic boundary. The recurrence relations are derived by applying the unitary layers sequentially to the Pauli operators, with the closed-chain circuit including the CZ gate between qubits N and 1 as part of the final entangling layer. Because the chain is defined with periodic boundaries from the start, the same recurrence rule is used for the wrap-around correlator as for all other nearest-neighbor pairs; no separate consistency condition is imposed. The number of influencing qubits is obtained by counting the sites that acquire non-zero correlators after each layer, which automatically incorporates the wrap-around. To address the concern directly, we will insert a short paragraph in the multi-qubit section that walks through the application of the recurrence to the N-to-1 correlator and confirms that the resulting expressions remain exact for any N>2. revision: yes

  2. Referee: [numerical comparison] The claim that 'the results agree with the theoretical predictions' (abstract) is load-bearing for the validation of the two-layer closed-chain expressions. The manuscript should report quantitative agreement metrics (e.g., maximum absolute deviation on correlators or entanglement distance) and confirm that the numerics used the exact periodic boundary conditions rather than open-chain proxies.

    Authors: We agree that quantitative metrics and an explicit statement on boundary conditions will strengthen the validation. In the revised manuscript we will add a table (or inline values) reporting the maximum absolute deviation between the analytical two-layer expressions and the numerical results for the relevant Pauli correlators and the entanglement distance. We will also state in the numerical-methods paragraph that the simulations were performed with the exact closed-chain circuit, i.e., the final CZ layer explicitly connects qubit N to qubit 1, matching the periodic boundary conditions used in the analytical derivation. revision: yes

Circularity Check

0 steps flagged

Recurrence relations derived directly from RY/CZ gate structure; analytical expressions for closed chain are independent of fitted inputs.

full rationale

The paper derives recurrence relations for Pauli expectation values by propagating through successive layers of RY rotations and CZ entangling gates. For the two-qubit case and the two-layer closed 1D chain, these relations are obtained by direct application of the gate operators to the initial state, yielding closed-form expressions for correlators and entanglement distance as functions of circuit parameters and depth. These expressions are then compared against independent numerical simulations on quantum programming tools. No parameters are fitted to data and then relabeled as predictions; no self-citations are invoked as load-bearing uniqueness theorems; the closed-chain wrap-around is handled explicitly within the recurrence for N qubits. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum mechanics, the algebraic properties of Pauli operators, and the assumption that the chosen variational ansatz generates the states of interest. No new entities are postulated.

free parameters (1)
  • rotation angles in RY gates
    Entanglement distance is stated to depend on these circuit parameters; their specific values are chosen per instance but not fitted globally in the abstract.
axioms (2)
  • standard math Pauli operators satisfy the standard commutation and anticommutation relations used to derive recurrence relations for expectation values.
    Invoked implicitly when constructing recurrence relations from the circuit gates.
  • domain assumption The variational circuit is composed of alternating RY rotations and CZ entangling layers in a fixed pattern.
    The recurrence and spreading results are derived specifically for this architecture.

pith-pipeline@v0.9.0 · 5475 in / 1322 out tokens · 33370 ms · 2026-05-09T19:46:21.959676+00:00 · methodology

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