Optimizing QAOA circuit transpilation with parity twine and SWAP network encodings

David E. Bernal Neira; J. A. Montanez-Barrera; Kristel Michielsen; Michael R. von Spakovsky; Yanjun Ji

REVIEW 3 major objections 6 minor 2 cited by

Optimizing QAOA circuit transpilation with parity twine and SWAP network encodings

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that parity twine chains and SWAP networks, when paired with simulated-annealing qubit ordering, outperform standard transpilers for QAOA circuits once problem edge density exceeds a threshold that shrinks as qubit count…

desk verdict Solid single-layer benchmarking with a useful SA heuristic, but the multi-layer claims rest on an unproved reverse-pass rule that needs to be fixed before the hardware results can be trusted. read the letter →

arxiv 2505.17944 v1 pith:3U4UUYNS submitted 2025-05-23 quant-ph

J. A. Montanez-Barrera , Yanjun Ji , Michael R. von Spakovsky , David E. Bernal Neira , Kristel Michielsen This is my paper

classification quant-ph

keywords paritytwinechainSWAPnetworkQAOAsimulatedannealingqubitmappingtranspilationweightedMaxCutLR-QAOA

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that parity twine chains (PTC) and SWAP networks, two encodings that map fully connected circuits onto a line of qubits, can be made competitive for sparsely connected QAOA problems by optimizing the initial qubit order with simulated annealing. Above an edge-density threshold that falls as the system grows, the optimized encodings beat standard transpilers in both two-qubit gate count and circuit depth, with an 85% depth reduction and 28% gate reduction at 120 qubits and 25% connectivity. The authors also demonstrate on the ibm_fez device that PTC finds optimal solutions for fully connected weighted MaxCut problems up to 20 qubits, versus 15 qubits with SWAP networks.

What carries the argument

Parity twine chains (PTC) are the central object: a CNOT-based encoding that folds the parities of several logical qubits onto a single physical qubit, so that a two-qubit ZZ interaction becomes a local single-qubit RZ rotation. A simulated-annealing step relabels the qubits' initial order so that the parities appearing at the end of the encoding correspond to edges missing from the problem graph and can be dropped; the same relabeling is applied to SWAP networks, where the dropped operations are SWAP gates belonging to missing edges. The paper's resource identities are $N_g = N_q^2 - 1$ and $d = 2N_q + 2$ for PTC per layer on a 1D chain, versus $N_g = \frac{3}{2}N_q^2 - \frac{5}{2}N_q + 1$ and $d = 3N_q - 2$ for SWAP networks.

What would settle it

Run a noiseless statevector simulation of a 6–8 qubit non-fully connected weighted MaxCut instance for $p=3$ and $p=4$ using the alternating forward/backward PTC or SWAP construction and compare the output distribution against the exact QAOA unitary evolution; any discrepancy would falsify the truncation rule.

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Extended reading notes

Core claim

The central claim is that PTC and SWAP encodings, when their initial qubit order is chosen by simulated annealing, outperform Qiskit's optimization-level-3 transpiler for QAOA instances once the problem graph's edge density exceeds a threshold that decreases with qubit count: roughly $E_d = 0.35$ at 20 qubits and $E_d = 0.13$ at 120 qubits. Above the threshold the PTC+SA construction yields both fewer two-qubit gates and smaller circuit depth; at 120 qubits with $E_d = 0.248$ it reports an 85% reduction in depth and a 28% reduction in two-qubit gates relative to Qiskit-T. The paper further claims this is the first demonstration of these encodings beating standard transpilers in reduced-connectivity scenarios, and validates PTC on real hardware, finding optimal solutions up to 20 qubits whereas SWAP networks stop at 15.

Load-bearing premise

The multi-layer hardware claims assume that, for a non-fully connected graph, truncating the first QAOA layer at $t=2$ and alternating forward/backward passes over subsequent layers reproduces $p$ independent cost-Hamiltonian evolutions without correction circuits, an assumption the paper states but does not derive.

Editorial extensions

If this is right

For QAOA on linear-chain hardware with non-trivial connectivity, PTC+SA and SWAP+SA become the preferred transpilation strategy above the density threshold, with the threshold shrinking as system size increases.
The PTC+SA construction reduces circuit depth relative to all tested transpilers across almost the entire edge-density range, even in regimes where it uses more two-qubit gates.
Under a depolarizing noise model, PTC improves success probability over SWAP networks by up to 5.65 percentage points at a moderate error strength, while approximation ratios remain comparable.
On the ibm_fez device, PTC extends the size at which optimal solutions to fully connected weighted MaxCut are found from 15 to 20 qubits, and extends distinguishability from a random sampler from 20 to 22 qubits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the alternating forward/backward layer rule holds, the annealed qubit ordering amortizes over all $p$ layers, meaning the reported per-layer resource counts understate the encodings' advantage for deep QAOA circuits.
The simulated-annealing cost function is a generic graph-relabeling heuristic that could transfer to other parity-based encodings or to T- and H-shaped hardware topologies, a direction the paper mentions but does not demonstrate.
The systematic fall of the crossing threshold with qubit count suggests a scaling law for when structured encodings beat general transpilers; fitting such a law to more sizes and hardware graphs would sharpen practical guidance.
A testable extension is to check whether the threshold depends only on edge density or also on graph structure, for instance by comparing random regular graphs with Erdős–Rényi graphs of the same density.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Single-layer resource comparisons are solid and the SA qubit-ordering heuristic is a real, if incremental, improvement. The thing to check before trusting the p>1 hardware and noise results is the unproved 'reverse direction' rule in Sec. II-C; if that rule is wrong, the multi-layer claims collapse, though the single-layer numbers stand independently.

The paper does a competent job benchmarking PTC and SWAP network encodings against Qiskit-T, Qiskit-P, Qiskit-AI, and TKET on random graphs with varying edge density, and it clearly shows the crossover thresholds where parity-based encodings beat the transpilers. The SA cost function tailored to trailing removable interactions is new relative to [20], and the systematic threshold map across Nq and Ed is useful. The first hardware run of PTC on ibm_fez is a legitimate data point. Code and data are linked, so the resource counts are reproducible.

Two soft spots. First, the reversal rule: Sec. II-C says subsequent layers proceed in the reverse direction after truncating at t=2, with no derivation. The CNOTs in a PTC layer conjugate the RZ rotations; reversing the pass changes which interactions are effectively implemented unless a specific angle relation holds. The paper doesn't provide that. Since the ibm_fez experiments with p=3..20 and the p=50 noise simulation both depend on it, this is load-bearing. It could be a correct identity, but it needs a proof or at least a numerical check on the full unitary. Second, Table I reports PTC+SA depth 66 for Nq=20, Ed=1.0, while the paper's own formula gives 2Nq+2=42; SWAP+SA depth 77 vs 58. Maybe the reported depths include something extra, but the discrepancy is unexplained and makes the exact numbers harder to trust. Minor: only three random graphs per point, so the error bars are thin, but the trends are consistent.

For whom: practitioners choosing a transpilation strategy for QAOA on fixed superconducting layouts, and people working on parity encodings. It deserves a serious referee; the single-layer results and the threshold map are citable regardless of the reversal-rule outcome. My recommendation: send it to review, but the referee should push for a derivation or validation of the reverse-pass rule before the multi-layer conclusions are accepted.

Referee Report

3 major / 6 minor

Summary. The paper proposes a simulated-annealing-based qubit-order optimization for two connectivity-aware encodings of QAOA circuits on a linear chain: the SWAP network and parity twine chains (PTC). It benchmarks PTC+SA and SWAP+SA against Qiskit-T, Qiskit-P, Qiskit-AI, and TKET on weighted MaxCut instances with varying edge density, reporting resource tables for 20, 60, and 120 qubits, a depolarizing-noise simulation for 8 qubits at p=50, and ibm_fez hardware experiments for up to 22–25 qubits. The central claim is that above a connectivity threshold, these encodings reduce two-qubit gate count and depth relative to Qiskit-T, e.g., an 85% depth reduction and 28% gate reduction for a 120-qubit instance at 25% connectivity, and that PTC extends LR-QAOA to larger qubit counts than SWAP networks on real hardware.

Significance. If the central claims hold, the work is a practically valuable contribution to QAOA transpilation. The SA-based ordering heuristic is simple and scales to 120 qubits, and the paper provides a public repository with instances and results, which aids reproducibility. The claim that PTC and SWAP encodings can outperform a general-purpose transpiler above density thresholds is useful for practitioners. However, the validation is incomplete: the multi-layer extension relies on an unproved alternating-pass rule, and the depth metric in Table I is inconsistent with the paper's own formulas. The single-layer resource-reduction results are credible enough to warrant revision rather than rejection.

major comments (3)

[II-C, Fig. 3(b), Figs. 7–8] The paper states that for a non-fully connected graph the PTC circuit can be truncated at t=2 for the first QAOA layer and that 'subsequent layers can then proceed in the reverse direction.' This alternating-pass rule is the only stated mechanism for extending the single-layer encoding to the p>1 circuits used in the depolarizing-noise simulation (Fig. 7, p=50) and the ibm_fez experiments (Fig. 8, p=3..20). No derivation, circuit identity, or numerical validation is provided to show that a reversed truncated pass, with interleaved RZ rotations and mixers, implements another independent cost-Hamiltonian layer; CNOT conjugation of the diagonal RZ gates generally changes the effective two-qubit interactions. Please supply a formal argument or a statevector-level check for p=2 and state explicitly how the mixer and the reversed RZ angles are scheduled. Without this, the multi-layer hardware and noise claims are unsupported, although the single-layer resource comparisons could still stand.
[II-A, II-B, Table I] The depth formulas in Sec. II state d=2Nq+2 for PTC and d=3Nq-2 for SWAP networks, and Ng=Nq^2-1 for PTC. Table I at Ed=1.0 reports substantially larger depths for the SA-optimized encodings (e.g., Nq=20: PTC+SA depth 66 vs 42, SWAP+SA depth 77 vs 58; Nq=120: 366 vs 242 and 477 vs 358). Please clarify whether Table I counts native-gate depth, includes final uncompute/measurement-order restoration, or includes single-qubit rotations, and correct either the table or the formulas. Since the abstract's depth-reduction percentages are computed from Table I, this inconsistency is load-bearing and must be resolved.
[III-A, Fig. 6, Table I] The quantitative thresholds quoted in the abstract and conclusions (e.g., Ed≈0.35 for Nq=20 and Ed≈0.13 for Nq=120) are based on only three randomly generated graphs per data point. The error bars in Fig. 6 are standard deviations over these three instances, and no confidence intervals or significance tests are reported. Given the visible variance, the crossing points should be presented as ranges or supported by more instances, and the text should avoid presenting them as precise thresholds. The qualitative trend is not in question, but the specific threshold values are not statistically established.

minor comments (6)

[II-C] The stopping temperature T_s is mentioned in the SA description but never given a value; only T0, delta, and max_iter are specified.
[II-D] 'Qiskit-P allows implemention' should read 'implementation.'
[II-F, Eq. (8)] The expression U_{ij}^{2Q}ρU_{ij}^{2Q} should presumably be U_{ij}^{2Q}ρU_{ij}^{2Q†} unless the gate is assumed Hermitian.
[III-A, Fig. 4] The 'minimum possible depth' d_min used for the ratios in Fig. 4 is not defined in the text; please define it explicitly.
[III-C, Fig. 8(b)] The formula for the effective approximation ratio r_eff appears only in the inset and is not defined in the text; please add a definition.
[II-E] The paragraphs introducing r and p_gs do not specify the sample count used for the noiseless simulation in Fig. 8(a), which would help interpret the comparison with the random sampler.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: SA-optimized gate counts are empirical outputs, not fitted inputs; the reverse-direction multilayer rule is an unproved correctness risk, not a circular reduction.

full rationale

We find no input-output reduction of the kind that would constitute circularity. The SA method optimizes an explicit cost function (number of removable terminal two-qubit interactions) and then reports the actual gate counts and depths of the circuits produced; the threshold values E_d=0.35, 0.13, etc. are empirical crossings against external baseline transpilers (Qiskit-T, Qiskit-P, Qiskit-AI, TKET), not parameters fitted to the claim. The PTC and SWAP encodings are introduced from prior literature [6]–[13], and the SA contribution is benchmarked independently. The paper does reuse the authors' own LR-QAOA protocol [17], [18] in Secs. II-E and III-B/C; this self-citation is real but not load-bearing for the central transpilation comparison, since resource counts are schedule-independent and PTC/SWAP hardware results use identical protocols. The one notable unsupported assertion is Sec. II-C's rule that after truncating the first QAOA layer at t=2, 'subsequent layers can then proceed in the reverse direction, effectively alternating forward and backward passes as the QAOA p increases.' No circuit identity or derivation is given, and all p>1 noise/hardware results inherit this risk. That is a missing-proof/correctness concern, not a circularity, because the rule is not defined in terms of the results it is used to produce.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on prior encoding schedules, a hand-tuned SA heuristic, and an unproved multi-layer truncation rule. No new physical entities are introduced. The SA hyperparameters are not fitted to the reported resource counts, so circularity burden is low.

free parameters (4)

SA initial temperature T0 = 0.01
Hand-chosen hyperparameter; no sensitivity analysis reported. Affects qubit-order quality and thus the reported gate and depth reductions.
SA cooling rate delta = 0.999
Hand-chosen; fixed for all runs. Not fitted to the target resource counts.
SA max_iter = 50000
Hand-chosen stopping criterion; no convergence guarantee.
LR-QAOA schedule amplitude Delta_beta_gamma = 0.63 for Nq<=15, 0.3 for Nq>15
Taken from the authors' prior benchmark [18]; used for hardware results, not for the transpilation resource counts.

assumptions (4)

domain assumption The PTC and SWAP schedules from prior work [6]-[13] generate all pair interactions on a 1D chain.
The resource formulas and SA cost function assume these schedules; they are not re-derived in this paper.
ad hoc to paper Trailing interactions can be removed and subsequent layers can alternate forward and backward without changing the QAOA evolution.
Asserted in Sec II-C; no proof supplied. This is the main correctness risk for multi-layer experiments.
domain assumption A depolarizing channel with uniform epsilon_g on all two-qubit gates captures the dominant hardware noise.
Stated in Sec II-F; used to compare PTC and SWAP noise resilience.
ad hoc to paper Simulated annealing converges to a near-optimal qubit order for the chosen hyperparameters.
No convergence guarantee; results are empirical for specific random graphs.

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Cite this review

Pith. "Pith review of Optimizing QAOA circuit transpilation with parity twine and SWAP network encodings." pith.science (2026). https://pith.science/paper/3U4UUYNS

@misc{pith2026250517944,
  author       = {Pith},
  title        = {Pith review of: Optimizing QAOA circuit transpilation with parity twine and SWAP network encodings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3U4UUYNS}},
  note         = {Machine review of arXiv:2505.17944}
}

read the original abstract

Mapping quantum approximate optimization algorithm (QAOA) circuits with non-trivial connectivity in fixed-layout quantum platforms such as superconducting-based quantum processing units (QPUs) requires a process of transpilation to match the quantum circuit on the given layout. This step is critical for reducing error rates when running on noisy QPUs. Two methodologies that improve the resource required to do such transpilation are the SWAP network and parity twine chains (PTC). These approaches reduce the two-qubit gate count and depth needed to represent fully connected circuits. In this work, a simulated annealing-based method is introduced that reduces the PTC and SWAP network encoding requirements in QAOA circuits with non-fully connected two-qubit gates. This method is benchmarked against various transpilers and demonstrates that, beyond specific connectivity thresholds, it achieves significant reductions in both two-qubit gate count and circuit depth, surpassing the performance of Qiskit transpiler at its highest optimization level. For example, for a 120-qubit QAOA instance with 25% connectivity, our method achieves an 85% reduction in depth and a 28% reduction in two-qubit gates. Finally, the practical impact of PTC encoding is validated by benchmarking QAOA on the ibm_fez device, showing improved performance up to 20 qubits, compared to a 15-qubit limit when using SWAP networks.

Figures

Figures reproduced from arXiv: 2505.17944 by the authors.

**Figure 2.** Flowchart of the simulated annealing algorithm for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** QAOA problem encoding with a reduced depth for [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Resources needed to implement a layer of QAOA using the Qiskit transpilation method level 3 on a heavy-hex layout [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: (a) Time used vs. Ed for the different transpilers to find a circuit that encodes a layer of QAOA for 120-qubit random problems. The error bars in all three plots represent the mean and standard deviation computed over three different randomly generated graphs. (b) Red…

**Figure 6.** Figure 6: Graph density vs. number of qubits. The blue region [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Depolarizing noise simulation of an 8-qubit WMC using LR-QAOA with [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: LR-QAOA experiments on ibm fez for WMC problems ranging from 5 to 22 qubits using PTC and SWAP encoding and comparing it against a random sampler and a noiseless simulation. (a) Best pgs vs. the number of qubits for the SWAP, PTC, a noiseless simulation, and a random s…

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Computationally efficient zero-noise extrapolation for quantum-gate-error mitigation,

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Performance of the quantum approximate optimization algorithm on the maximum cut problem,

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Quantum simulation of electronic structure with linear depth and connectivity,

I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, “Quantum simulation of electronic structure with linear depth and connectivity,”Physical review letters, vol. 120, no. 11, p. 110501, 2018

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