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arxiv: 2606.23727 · v1 · pith:N5AZZONZnew · submitted 2026-06-19 · 🪐 quant-ph · cs.DS

Certifying Quantum Optimization and Circuit Cutting by Using Quantum-Classical Moment Duality

Ammar Daskin This is my paper

Pith reviewed 2026-06-26 13:53 UTC · model grok-4.3

classification 🪐 quant-ph cs.DS
keywords quantum optimizationGoemans-Williamson relaxationSum-of-Squarescircuit cuttingMax-Cutvariational quantum algorithmsmoment matrixPauli correlations
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The pith

The two-qubit Pauli-Z correlation matrix from any quantum state is always feasible for the classical Goemans-Williamson Max-Cut relaxation.

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

The paper shows that the matrix of two-qubit Pauli-Z correlations extracted from an arbitrary quantum state automatically satisfies the constraints of the classical degree-2 Sum-of-Squares semidefinite program. Because this matrix is therefore a valid input to the Goemans-Williamson relaxation, random-hyperplane rounding of the matrix produces a classical cut whose expected value is guaranteed to be at least the Goemans-Williamson factor times the quantum expectation value of the cut Hamiltonian. The same matrix also encodes the tensor-product structure of the underlying circuit, allowing a polynomial-time procedure that cuts the circuit into smaller pieces while supplying explicit error bounds on the reconstructed observables.

Core claim

Any quantum state ρ yields a two-qubit Pauli-Z moment matrix that lies inside the feasible set of the classical Goemans-Williamson semidefinite relaxation; rounding this matrix therefore certifies an expected cut value E[Cut] ≥ α_GW ⟨H⟩_ρ for every variational state, and the same matrix identifies tensor-product factorizations that permit rigorous circuit cutting.

What carries the argument

The two-qubit Pauli-Z correlation matrix extracted from a quantum state ρ, which is shown to be feasible for the classical Goemans-Williamson SDP relaxation.

If this is right

  • Every variational quantum optimization run on Max-Cut automatically receives a certified classical approximation ratio without requiring convergence to the ground state.
  • The moment matrix supplies a polynomial-time, correlation-based method to cut a quantum circuit into independent subcircuits together with explicit reconstruction error bounds.
  • The certification holds for any state produced by QAOA, VQPM, or any other variational procedure, independent of how close the state is to optimality.

Where Pith is reading between the lines

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

  • Classical SDP solvers could be seeded directly with quantum-estimated moment matrices to obtain hybrid certificates on larger instances.
  • When the quantum state is a product state the moment matrix reduces exactly to the classical relaxation, recovering the standard Goemans-Williamson guarantee as a special case.
  • The same correlation data might be used to certify other classical combinatorial problems whose SDP relaxations share the degree-2 Sum-of-Squares cone.

Load-bearing premise

The two-qubit Pauli-Z correlation matrix constructed from any quantum state lies inside the degree-2 Sum-of-Squares SDP cone.

What would settle it

A single quantum state whose measured two-qubit Pauli-Z correlations violate at least one constraint of the Goemans-Williamson SDP, or a variational run in which the rounded classical cut value is strictly less than α_GW times the observed quantum energy.

Figures

Figures reproduced from arXiv: 2606.23727 by Ammar Daskin.

Figure 1
Figure 1. Figure 1: Left: GW-rounded objective F(γ, β). Middle: QAOA energy EQAOA(γ, β). Right: optimal values. The GW-rounded bound is lower than the energy, yet respects the αGW guarantee. 0 5 10 15 20 25 30 35 Optimization Iterations 1.7 1.8 1.9 2.0 2.1 2.2 Expectation Value (Cut Size) QAOA State Trajectory & SoS Certificate Bounds on C4 GW-Rounded Bound F( , ) QAOA Energy E Analytical Floor GW E [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 2
Figure 2. Figure 2: Continuous optimization trajectory of p=1 QAOA on C4. The GW-rounded cut FGW (blue) stays above the certified lower bound αGWEρ (dashed) at all iterations, even when the raw energy is poor. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: VQPM trajectory for a 12-qubit instance. The GW-rounded cut [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Scatter plot of the actual 2-norm error against the mutual information [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

We establish a direct quantum-classical duality based on the degree-$2$ Sum-of-Squares (SoS) semidefinite programming cone: the matrix of two-qubit Pauli-$Z$ correlation functions obtained from \emph{any} quantum state $\rho$ is automatically a feasible point of the classical Goemans-Williamson (GW) relaxation. This observation provides a universal ``safety net'' for quantum optimization algorithms: applying GW random hyperplane rounding to the quantum-driven moment matrix yields a certified expected cut value $\mathbb{E}[\mathrm{Cut}] \ge \alpha_{\mathrm{GW}}\langle\mathcal{H}\rangle_\rho$, valid for every state produced by variational algorithms such as QAOA or the Variational Quantum Power Method (VQPM), regardless of convergence quality. We further show that the same moment matrix reveals the tensor-product structure of the underlying unitary circuit, enabling a polynomial-time, correlation-based circuit cutting procedure with rigorous error bounds. The framework is validated numerically on Max-Cut instances for variational quantum algorithms and on random states for circuit cutting, demonstrating that the cheap two-point correlation data are sufficient to locate near-optimal bipartitions and that the theoretical error bounds hold in practice.

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 claims a quantum-classical duality based on the degree-2 Sum-of-Squares cone: for any quantum state ρ the two-qubit Pauli-Z correlation matrix C with entries C_ij = Tr(ρ Z_i Z_j) (C_ii=1) is feasible for the Goemans-Williamson SDP relaxation of Max-Cut. Consequently, GW random-hyperplane rounding applied to this matrix certifies E[Cut] ≥ α_GW ⟨H⟩_ρ for every variational state produced by QAOA or VQPM. The same matrix is further asserted to reveal tensor-product structure, enabling a polynomial-time correlation-based circuit-cutting procedure with rigorous error bounds. Numerical experiments on Max-Cut instances and random states are reported to confirm that the two-point data suffice to locate near-optimal bipartitions and that the error bounds hold in practice.

Significance. If the central duality holds, the result supplies a parameter-free, assumption-free certification mechanism that converts any quantum expectation value into a classical approximation guarantee via an off-the-shelf SDP rounding procedure. This is a direct, elementary link between quantum moment matrices and classical SDP relaxations that requires no additional fitting or convergence assumptions. The circuit-cutting application, if rigorously derived from the same matrix, would likewise be a useful byproduct for scalable quantum computation. The elementary positivity argument (x^T C x = Tr(ρ (∑ x_k Z_k)^2) ≥ 0) is a strength that makes the claim robust.

minor comments (3)
  1. [§2] §2 (or wherever the duality is stated): the explicit construction of the moment matrix from the Pauli-Z operators and the verification that it lies in the degree-2 SoS cone should be written out with the quadratic-form identity shown, even if elementary, to make the argument self-contained for readers unfamiliar with SoS.
  2. [circuit cutting section] The circuit-cutting section: the precise mapping from the correlation matrix to a tensor-product decomposition (or bipartition) and the derivation of the polynomial-time error bounds need an explicit algorithm box or pseudocode, together with the dependence of the error on the number of cuts.
  3. [numerical validation] Numerical section: the Max-Cut instances and random-state ensembles should report the distribution of the ratio E[Cut]/⟨H⟩_ρ across instances, not only selected examples, to substantiate the claim that the bounds hold in practice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting the elementary positivity argument, and for recommending minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim is that any two-qubit Pauli-Z correlation matrix C_ij = Tr(ρ Z_i Z_j) (C_ii=1) from an arbitrary state ρ lies in the degree-2 SoS cone because x^T C x = Tr(ρ (∑ x_k Z_k)^2) ≥ 0 for all real x, with SDP objective exactly ⟨H⟩_ρ. This is an elementary positivity fact that directly implies the GW rounding guarantee E[Cut] ≥ α_GW ⟨H⟩_ρ with no fitted parameters, self-definitions, or load-bearing self-citations required. The abstract and skeptic analysis present the construction and duality explicitly as a direct consequence of the SoS membership; the derivation is self-contained against external mathematical benchmarks and exhibits none of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on the standard properties of the degree-2 SoS cone and the definition of the two-qubit Pauli-Z moment matrix.

axioms (1)
  • standard math Degree-2 Sum-of-Squares SDP cone membership for the Pauli-Z moment matrix
    Invoked in the first sentence of the abstract as the basis for the duality.

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discussion (0)

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