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arxiv: 2606.05397 · v1 · pith:7F2FSB5Ynew · submitted 2026-06-03 · 🪐 quant-ph

Multi-Qubit Dyadic Phase Fixing for Fault-Tolerant Quantum Compilation

Justin Kalloor , Mathias Weiden , Ed Younis , John Kubiatowicz , Costin Iancu This is my paper

Pith reviewed 2026-06-28 05:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum compilationClifford+T synthesisT-count reductionphase kickbackfault-tolerant quantum computingsurface codedyadic angles
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The pith

Dyadic Phase Fixing extracts dyadic-angle rotations from arbitrary circuits to cut T-count by up to 70 percent.

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

The paper introduces Dyadic Phase Fixing to extend phase kickback beyond structured circuits in Clifford+T compilation. Numerical unitary synthesis greedily pulls out dyadic rotations from any input, paired with an automatic register-sizing matrix. On benchmarks this produces large T-count drops versus gridsynth and repeat-until-success methods. Mapping the results to surface codes also lowers space-time volume, though the two metrics sometimes diverge.

Core claim

Dyadic Phase Fixing is a multi-qubit synthesis tool that uses numerical unitary synthesis to greedily extract dyadic-angle rotations from general quantum circuits and a decision matrix to size the final phase gradient register, delivering up to 70 percent T-count reduction and up to 60 percent space-time volume reduction on surface-code mappings.

What carries the argument

Numerical unitary synthesis that greedily extracts dyadic-angle rotations, combined with a decision matrix for phase gradient register sizing.

Load-bearing premise

Numerical unitary synthesis can reliably extract dyadic-angle rotations from arbitrary circuits without introducing errors or extra overhead that cancels the savings.

What would settle it

A circuit on which the workflow produces a higher T-count than gridsynth or repeat-until-success synthesis.

Figures

Figures reproduced from arXiv: 2606.05397 by Costin Iancu, Ed Younis, John Kubiatowicz, Justin Kalloor, Mathias Weiden.

Figure 1
Figure 1. Figure 1: T-gate counts for the Quantum Fourier Transform circuit, comparing Rz decomposition via Phase Kickback against standard rotation synthesis (gridsynth). Phase kickback achieves signifi￾cantly lower T-counts for angles of the form 2π 2k , with the advantage growing as the approximation error threshold decreases. However, this efficiency does not extend to arbitrary rotation angles. For a generic angle θ, the… view at source ↗
Figure 2
Figure 2. Figure 2: Standard bus architecture for surface code qubit patches ( [16], [21], [28]). The routing qubits (blank squares) allow for logical all-to-all connectivity. Pauli Products (red, green, and purple regions) in the same cycle can be scheduled in parallel if there are non￾overlapping paths through the routing qubits. The perimeter contains magic state cultivation qubits (yellow) which generate resource states a… view at source ↗
Figure 3
Figure 3. Figure 3: High-level overview of our end-to-end compilation workflow. We accept a logical circuit in any gate set along with an acceptable approximation error tolerance. The circuit is partitioned into blocks and each block is simplified with BQSKit-FT [33]. Each block is then passed to our Dyadic Phase Fixing routine, which greedily extracts dyadic Rz rotations from the circuit. The resulting circuit is decomposed … view at source ↗
Figure 4
Figure 4. Figure 4: Decision matrix comparing T-gate counts between Phase Kickback and gridsynth as a function of approximation error threshold and number of fixed Rz gates, shown for k = 10 (left) and k = 20 (right). Green indicates configurations where phase kickback achieves fewer T gates; Blue indicate where standard gridsynth is preferable. phase kickback model uses the adder T-count of [12] together with the phase gradi… view at source ↗
Figure 5
Figure 5. Figure 5: Normalized T-gate counts across all benchmarks, comparing Baseline compilation, Naive Phase Kickback (Naive PK), and Dyadic Phase Fixing Synthesis (DPF Synth) at error thresholds of 10−3 and 10−5 . evaluate two techniques to translate our circuits to a sequence of Pauli Products that can be scheduled on chip: 1) Lightweight Pauli Basis Computation (LPBC): Single-qubit Clifford gates are conjugated rightwar… view at source ↗
Figure 6
Figure 6. Figure 6: The blue distribution shows the dyadic angles found [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of dyadic Rz angles across benchmark circuits that are multiples of 2π 2k , k ≤ 10. The blue distribution shows the angles found in the original circuit; the green distribution shows the angles after Phase Kickback Synthesis. Gates that were already T gates in the input circuit are excluded from both distributions, which is why the blue distribution contains no Rz( π 4 ) gates. B. Parallelizing … view at source ↗
Figure 8
Figure 8. Figure 8: Decision matrix comparing Phase Kickback and Repeat-Until￾Success (RUS) [2] as a function of approximation error threshold and number of fixed Rz gates. Green indicates configurations where Phase Kickback achieves fewer T gates; Purple indicate where RUS is preferable. RUS requires a considerably larger number of fixed Rz gates and higher precision to merit the use of Phase Kickback. Because the rest of ou… view at source ↗
Figure 7
Figure 7. Figure 7: Impact of the number of phase gradient states on the logical circuit depth and volume of the QAE-81q circuit [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: T-gate counts comparing our workflow (DPF Synth) and RUS-based rotation synthesis across benchmark circuits. DPF Synth still achieves significant reductions in T-gate count over the RUS baseline, demonstrating that the technique generalizes beyond gridsynth-based synthesis. Beyond rotation synthesis, our compiler can also operate under strict ancilla constraints. By specifying a maximum ancilla budget kmax… view at source ↗
read the original abstract

Fault-tolerant quantum computing requires translating application-level quantum circuits into the Clifford+$T$ gate set, where the $T$ gate is the dominant resource cost. Phase kickback is an ancilla-based technique that can dramatically reduce $T$-count for rotations with dyadic angles, but has previously been limited to highly structured circuit families. We present Dyadic Phase Fixing (DPF), a general multi-qubit synthesis tool that extends phase kickback to general quantum circuits. DPF uses numerical unitary synthesis to greedily extract dyadic angle rotations from any input circuit. Combined with a decision matrix to automatically size the final phase gradient register, our end-to-end workflow achieves up to 70% reduction in $T$-count compared to \texttt{gridsynth} and up to 60% compared to Repeat-Until-Success synthesis on a diverse set of benchmarks. We map these compiled circuits to a surface-code architecture to evaluate space-time volume, demonstrating up to a 60\% reduction in this metric as well. However, for some circuits and mapping strategies the two metrics diverge significantly, demonstrating that $T$-count alone is a useful but incomplete proxy for fault-tolerant program costs.

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 Dyadic Phase Fixing (DPF), a general multi-qubit synthesis method that uses numerical unitary synthesis to greedily extract dyadic-angle rotations from arbitrary input circuits, thereby extending phase kickback beyond structured families. Combined with an automatic decision matrix for phase-gradient register sizing, the end-to-end workflow is reported to achieve up to 70% T-count reduction versus gridsynth and 60% versus Repeat-Until-Success synthesis on diverse benchmarks, with up to 60% space-time volume reduction when mapped to surface-code architectures; the authors also note that T-count and space-time metrics can diverge substantially.

Significance. If the numerical extraction step reliably identifies exact dyadic rotations without introducing approximation errors or net overhead, the technique would meaningfully lower the dominant T-gate cost for fault-tolerant compilation of general circuits and provide a concrete demonstration that T-count alone is an incomplete resource proxy. The generalization of phase kickback and the surface-code volume results would be a practical contribution to compilation toolchains.

major comments (2)
  1. [numerical synthesis subsection] The description of the numerical unitary synthesis procedure (greedy dyadic-angle extraction) supplies no explicit error bounds, floating-point tolerance, convergence criteria, or post-extraction fidelity/unitarity verification. This is load-bearing for the central claim: without these, it is impossible to confirm that the extracted rotations remain exact for arbitrary multi-qubit inputs or that the reported T-count savings are not offset by corrective overhead.
  2. [benchmark evaluation section] Benchmark results (T-count and space-time volume reductions) are stated without error bars, raw circuit lists, or explicit verification steps against post-hoc selection. This undermines assessment of whether the up-to-70% and up-to-60% figures hold across the full benchmark set or only selected instances.
minor comments (2)
  1. The abstract and methods mention a 'decision matrix' for register sizing but provide no pseudocode, construction details, or validation against manual sizing.
  2. Figure captions and tables would benefit from explicit column definitions for 'original T-count', 'DPF T-count', and 'space-time volume' to allow direct comparison with gridsynth and RUS baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript accordingly to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [numerical synthesis subsection] The description of the numerical unitary synthesis procedure (greedy dyadic-angle extraction) supplies no explicit error bounds, floating-point tolerance, convergence criteria, or post-extraction fidelity/unitarity verification. This is load-bearing for the central claim: without these, it is impossible to confirm that the extracted rotations remain exact for arbitrary multi-qubit inputs or that the reported T-count savings are not offset by corrective overhead.

    Authors: We agree that the numerical synthesis subsection requires additional technical detail to substantiate the exactness of extracted dyadic rotations. In the revised manuscript we will add an explicit subsection on numerical parameters, including: floating-point tolerance of 1e-12 for angle matching and unitarity residual; convergence criterion defined as residual norm below tolerance after at most 100 iterations; and post-extraction verification that computes the operator fidelity against the original unitary using exact dyadic arithmetic where possible. Any rotation whose angle deviates beyond tolerance is rejected, ensuring no corrective overhead is introduced and that reported T-count reductions reflect exact dyadic extractions within the stated bound. revision: yes

  2. Referee: [benchmark evaluation section] Benchmark results (T-count and space-time volume reductions) are stated without error bars, raw circuit lists, or explicit verification steps against post-hoc selection. This undermines assessment of whether the up-to-70% and up-to-60% figures hold across the full benchmark set or only selected instances.

    Authors: The synthesis procedure is fully deterministic, so statistical error bars are not applicable. We will nevertheless strengthen the benchmark section by (i) providing the complete list of benchmark circuits and per-circuit T-count and volume numbers in a supplementary data file, (ii) stating that all circuits in the reported set were processed without post-hoc filtering, and (iii) adding a short verification paragraph confirming that each output circuit was checked for unitary equivalence to the input within the numerical tolerance. Average as well as maximum reductions will also be reported to give a fuller picture of performance across the set. revision: yes

Circularity Check

0 steps flagged

No significant circularity; workflow is self-contained and benchmark-driven

full rationale

The paper describes an algorithmic workflow (numerical unitary synthesis + greedy dyadic extraction + decision matrix for register sizing) whose claimed T-count and space-time reductions are obtained by direct empirical comparison to external baselines (gridsynth, RUS) on a diverse benchmark set. No equations, fitted parameters, or self-citations are shown that would make any reported reduction equivalent to its own inputs by construction. The central claims rest on the correctness and efficiency of the described synthesis procedure rather than on any definitional loop or imported uniqueness result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the decision matrix and numerical synthesis procedure are described at high level without disclosed constants or assumptions.

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discussion (0)

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Reference graph

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