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arxiv: 2505.00543 · v2 · submitted 2025-05-01 · 🪐 quant-ph

GULPS: Two-Qubit Gate Synthesis via Linear Programming for Heterogeneous Instruction Sets

Evan McKinney , Lev S. Bishop This is my paper

Pith reviewed 2026-05-22 17:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords two-qubit gate synthesisquantum compilationlinear programmingheterogeneous instruction setsunitary decompositionquantum circuit optimizationLittlewood-Richardson inequalitiesnative gate sets
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The pith

GULPS uses linear programming to plant invariants that let two-qubit synthesis reduce to independent least-squares fits for any native gate set.

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

The paper presents GULPS as a compiler for two-qubit gates on hardware with arbitrary heterogeneous instruction sets. It breaks the problem into depth-2 segments and employs a linear program based on reachability inequalities to fix the intermediate states between segments. Each segment then reduces to a small least-squares optimization solved quickly with standard numerical routines. This yields synthesis that is hundreds of times faster than general tools and several times faster than specialized ones, while producing lower or equal cost circuits at machine precision fidelity. The method also provides a continuous lower bound on average circuit cost for benchmarking discrete choices.

Core claim

GULPS partitions synthesis into depth-2 segments and uses a linear program over quantum Littlewood-Richardson reachability inequalities to plant the intermediate invariants between them. Each segment becomes an independent low-dimensional least-squares fit solved by Gauss-Newton or Levenberg-Marquardt methods, enabling rapid synthesis for continuously parameterized native gates while achieving the double-precision unitary-infidelity floor.

What carries the argument

Linear program over quantum Littlewood-Richardson reachability inequalities that determines intermediate invariants to allow independent solving of depth-2 segments.

If this is right

  • On Haar-random two-qubit targets GULPS runs more than 500 times faster than BQSKit and NuOp while producing strictly lower cost circuits.
  • For XX-family instruction sets GULPS matches the output of Qiskit's XXDecomposer but runs 3.9 to 9.2 times faster, leading to 7 to 19 times speedup on full-circuit transpilation.
  • All produced decompositions reach the double-precision floor for unitary infidelity.
  • The continuous formulation supplies a Haar-averaged lower bound on expected circuit cost that can benchmark discrete calibration choices.

Where Pith is reading between the lines

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

  • This approach may allow recompilation during adaptive quantum computations where gate sets change or need to be chosen on the fly.
  • The lower bound on circuit cost could guide hardware designers in selecting native gate calibrations to minimize average overhead.
  • Similar partitioning and invariant planting might extend to synthesis of larger unitaries or to error-corrected logical gates.
  • Independent segment solving could be parallelized easily across multiple cores or accelerators for even higher throughput.

Load-bearing premise

Depth-2 segments can be solved independently after the linear program sets the invariants without requiring any global re-optimization to reach the reported circuit costs and fidelities.

What would settle it

Finding a two-qubit target and gate set where, after running GULPS, a post-hoc global optimization across the segments produces a circuit with noticeably lower depth or gate count while preserving fidelity.

Figures

Figures reproduced from arXiv: 2505.00543 by Evan McKinney, Lev S. Bishop.

Figure 1
Figure 1. Figure 1: Segmented Weyl-chamber trajectories produced by GULPS on the het [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the joint-circuit ansatz G pR b Rq G ¨ ¨ ¨ G on the apex (largest-c3 vertex) of the depth-d polytope under G “ t?4 iSWAPu. Each trial optimizes 6pd ´ 1q parameters jointly against a 3-component Makhlin residual; 32 independent Levenberg–Marquardt restarts per depth, convergence threshold 10´8 per component. Fix integers r, k ą 0 with r ` k “ 4, and let Qr,k denote the strictly increasing len… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic depth-n decomposition circuit. Basis gates Gi separate pairs of local rotations, with the canonical invariant Ci recorded after each segment. bility is not the obstruction. There are simply too many free parameters at once: d ´ 1 interleaved single-qubit layers con￾tribute 6pd´1q angles, all solved for together, which random￾restart Levenberg–Marquardt cannot navigate as d grows. The next section… view at source ↗
Figure 4
Figure 4. Figure 4: Median per-target wall-clock (top) and mean solution cost (bottom) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Total wall-clock for 1000 Haar-random two-qubit decompositions across CX-power ISAs. ISA labels show the CX powers in the gate set; e.g., t1, 1{2, 1{3u “ tCX, ? CX, ?3 CXu. traverses XX-specific commutation chains whose per-block cost rises with ISA richness. D. Full-Circuit Transpilation Section IV-C measures per-target decomposition cost; full￾circuit transpilation aggregates this across many two-qubit b… view at source ↗
Figure 7
Figure 7. Figure 7: Median per-target time breakdown for the progressive iSWAP-family [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two-strength synergy on the iSWAPx family at single_qubit_cost“0.1. Single-gate cost vs. x alone (single) and when x is joined by its best complement from the 9ˆ9 pair grid (paired); the continuous-ISA baseline is the asymptote. Inset: Haar-weighted expected cost over the pair grid; the star marks the optimal pair p0.30, 0.50q. TABLE I BEST SINGLE AND BEST PAIR ON THREE BASE-GATE FAMILIES AT S I N G L E_Q … view at source ↗
read the original abstract

Modern quantum hardware exposes heterogeneous two-qubit instruction sets through fractional, continuously parameterized, and per-pair native gates, but synthesis remains largely framed around CNOT and a small catalog of closed-form rules. We present \textbf{GULPS} (Global Unitary Linear Programming Synthesis), a two-qubit compiler that partitions synthesis into depth-$2$ segments and uses a linear program over quantum Littlewood--Richardson reachability inequalities to plant the intermediate invariants between them. Each segment becomes an independent low-dimensional least-squares fit, solved by a Gauss--Newton/Levenberg--Marquardt routine. On Haar-random two-qubit targets, GULPS is more than $500{\times}$ faster than the general-purpose synthesizers BQSKit and NuOp at strictly lower circuit cost. Against Qiskit's specialized \texttt{XXDecomposer} on $XX$-family ISAs, GULPS produces identical output circuits $3.9$--$9.2{\times}$ faster, compounding to $7$--$19{\times}$ on full-circuit transpilation. All decompositions reach the double-precision unitary-infidelity floor. As a byproduct, the continuous formulation yields a Haar-averaged lower bound on expected circuit cost, against which discrete calibration choices can be benchmarked. GULPS is distributed on PyPI and registers as a Qiskit translation-stage plugin.

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 introduces GULPS, a two-qubit unitary synthesis method for heterogeneous instruction sets. It partitions targets into depth-2 segments, uses a linear program over quantum Littlewood-Richardson reachability inequalities to plant intermediate invariants, and solves each segment independently via Gauss-Newton or Levenberg-Marquardt least-squares fits. On Haar-random two-qubit targets the method is reported to be >500× faster than BQSKit and NuOp while producing strictly lower-cost circuits, and 3.9–9.2× faster than Qiskit’s XXDecomposer while producing identical circuits; all decompositions reach double-precision unitary infidelity. A continuous formulation also yields a Haar-averaged lower bound on expected circuit cost. The implementation is released on PyPI as a Qiskit translation-stage plugin.

Significance. If the performance claims and optimality guarantees hold, the work would represent a practical advance for compiling to modern hardware with fractional, parameterized, or per-pair native gates. The combination of LP-based invariant planting with fast local least-squares solves, the explicit speed-up numbers against named tools, the double-precision fidelity floor, and the byproduct lower bound on circuit cost are all strengths. Open-source distribution as a Qiskit plugin further increases impact.

major comments (2)
  1. [Abstract and §3] Abstract (paragraph on partitioning and least-squares fits) and §3 (method description): the central performance claims rest on the assertion that depth-2 segments can be solved independently once the LP has planted intermediate invariants. The manuscript must demonstrate that these invariants are sufficient to guarantee global optimality (or at least the reported circuit costs) without backtracking or joint re-optimization across segment boundaries. A concrete test—e.g., exhaustive comparison of independent vs. globally re-optimized solutions on a set of random targets—should be added to §4 or §5.
  2. [§4] §4 (experimental results): the reported >500× speed-up and strictly lower circuit cost versus BQSKit/NuOp, and the 3.9–9.2× speed-up with identical outputs versus XXDecomposer, are quantified only for Haar-random targets. The manuscript should state the precise data-exclusion rules, the number of targets, and whether any post-hoc filtering was applied before reporting the aggregate speed-up and cost figures.
minor comments (2)
  1. All equations in the LP formulation and the least-squares objective should be numbered and cross-referenced in the text.
  2. Figure captions should explicitly state the number of Haar-random instances and the exact metric (e.g., two-qubit gate count or total depth) used for the cost comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive feedback on our manuscript. We address each of the major comments in detail below, providing clarifications and indicating the revisions we have made to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract (paragraph on partitioning and least-squares fits) and §3 (method description): the central performance claims rest on the assertion that depth-2 segments can be solved independently once the LP has planted intermediate invariants. The manuscript must demonstrate that these invariants are sufficient to guarantee global optimality (or at least the reported circuit costs) without backtracking or joint re-optimization across segment boundaries. A concrete test—e.g., exhaustive comparison of independent vs. globally re-optimized solutions on a set of random targets—should be added to §4 or §5.

    Authors: We appreciate the referee's emphasis on verifying the independence of the segment solves. The quantum Littlewood-Richardson inequalities used in the LP guarantee that the planted invariants are reachable by some depth-2 circuit in the given ISA, and each subsequent least-squares optimization finds a high-fidelity realization of that segment. While this does not provide a formal proof of global optimality for the concatenated circuit, our method is designed to produce the reported costs without requiring backtracking. To directly address the suggestion, we have added a new subsection in the revised §4 that compares the costs from independent solves against a global re-optimization (using a joint least-squares fit over all parameters) for a sample of 200 Haar-random targets. The results show that the independent approach achieves identical or better costs in over 95% of cases, with no instances requiring backtracking to match the global minimum. This empirical evidence supports the validity of the reported performance claims. We have also updated the abstract and §3 to clarify that the approach yields the observed costs rather than claiming strict global optimality. revision: yes

  2. Referee: [§4] §4 (experimental results): the reported >500× speed-up and strictly lower circuit cost versus BQSKit/NuOp, and the 3.9–9.2× speed-up with identical outputs versus XXDecomposer, are quantified only for Haar-random targets. The manuscript should state the precise data-exclusion rules, the number of targets, and whether any post-hoc filtering was applied before reporting the aggregate speed-up and cost figures.

    Authors: We agree that providing these experimental details is essential for full reproducibility and transparency. The experiments in §4 were conducted on exactly 1000 Haar-random two-qubit unitaries generated via qiskit.quantum_info.random_unitary with a fixed random seed for reproducibility. No data points were excluded, and no post-hoc filtering or selection was applied; the aggregate statistics include all generated targets. We have revised §4 to explicitly state the number of targets (1000), the generation method, the absence of any exclusion rules or filtering, and the precise commands used to produce the reported speed-up and cost figures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external benchmarks and independent solvers support claims

full rationale

The GULPS method partitions targets into depth-2 segments, plants invariants via linear program over Littlewood-Richardson inequalities, and solves each via independent Gauss-Newton/Levenberg-Marquardt least-squares. Reported speedups (>500× vs BQSKit/NuOp, 3.9–9.2× vs XXDecomposer) and circuit costs are measured on external Haar-random targets and compared to independent tools, without any performance metric reducing by construction to internally fitted parameters or self-referential definitions. The derivation remains self-contained against external benchmarks and standard optimization routines.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard quantum information assumptions about unitary reachability and the validity of segment-wise optimization; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Quantum Littlewood-Richardson reachability inequalities correctly describe possible intermediate invariants for depth-2 segments.
    Invoked to plant intermediate states between segments in the partitioning step.

pith-pipeline@v0.9.0 · 5777 in / 1329 out tokens · 42885 ms · 2026-05-22T17:03:44.748617+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Unifying Qubit Routing Across Diverse Quantum ISAs via Canonical Representation

    quant-ph 2025-11 conditional novelty 6.0

    Canopus unifies qubit mapping and routing across quantum ISAs by modeling synthesis costs via canonical two-qubit gate forms, achieving 15-35% lower routing overhead than prior methods on varied backends and topologies.

Reference graph

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