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arxiv: 2606.17589 · v1 · pith:VIGMRDSRnew · submitted 2026-06-16 · 🪐 quant-ph

Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids

Chengzhuo Xu , Xiao Chen , Zhihao Liu , Zhigang Li This is my paper

Pith reviewed 2026-06-27 00:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords diagonal unitariesquantum compilationnearest neighbor gridsasymptotic depthGray codeQAOAHamiltonian simulationcircuit synthesis
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The pith

Gray-Path Framework synthesizes and compiles any diagonal unitary to optimal O(2^n/n) depth on 2D grids.

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

The paper introduces the Gray-Path Framework to synthesize any n-qubit diagonal unitary using asymptotically optimal depth O(2^n/n) with Rz and CNOT gates and no ancillas. It then proves that routing this fixed sequence of operations onto a two-dimensional nearest-neighbor grid adds only matching Θ(2^n/n) depth overhead, turning the problem into deterministic scheduling rather than heuristic search. This joint synthesis and compilation approach yields closed-form bounds for resource costs on grids and linear chains, with optional ancilla-assisted space-time tradeoffs that preserve the asymptotic optimality.

Core claim

Any n-qubit diagonal unitary can be realized in asymptotically optimal Rz and CNOT depth O(2^n/n) by the Gray-Path Framework. When this framework is compiled to a two-dimensional nearest-neighbor grid, the routing cost is also Θ(2^n/n) depth and Θ(2^n) gate count because the interaction structure is predetermined, allowing exact scheduling without search. The result holds with or without ancillas and on linear chains as well.

What carries the argument

The Gray-Path Framework, a construction that encodes the diagonal unitary into a fixed sequence of phase and entangling gates whose interaction graph can be routed deterministically on grid hardware.

Load-bearing premise

The Gray-Path Framework can fix the complete sequence of gates and interactions for synthesizing any given diagonal unitary in advance.

What would settle it

Finding an n-qubit diagonal unitary for which no circuit of depth o(2^n/n) exists on a 2D grid, or showing that GPF routing on the grid requires more than Θ(2^n/n) depth for some unitaries.

Figures

Figures reproduced from arXiv: 2606.17589 by Chengzhuo Xu, Xiao Chen, Zhigang Li, Zhihao Liu.

Figure 1
Figure 1. Figure 1: Two equivalent basic Gray Path circuits for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: CNOT skeletons of the two local mixing modules for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Recursive composition of the GPF and GPF∗ skele￾tons; both carry no phase gates, which are inserted into the slots of their GPS/GPS∗ modules (Fig. 2a). high subspace supplies the local parallel traversal, and the pure high-space modes recurse. The coefficient κ in the DGPF∗ CNOT line is the fold factor of Eq. (8): κ = 3 exactly for the BRGC, while a low-jump sequence gives κ ∈ (3, 4) with no closed form. T… view at source ↗
Figure 4
Figure 4. Figure 4: Complete GPF phase-insertion circuit for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Bipartite connectivity abstraction of GPS. Gray edges [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Normalized depth constants for GPF∗ . The phase￾depth ratio is Gray-code-independent; the CNOT-depth ratio carries the fold factor κ, converging to 3C/2 ≈ 5.10 for the BRGC (κ = 3) and slightly higher for the low-jump sequence (κ ∈ (3, 4); Appendix F). Hollow markers: n = 2λ . IV. FROM GPF STRUCTURE TO TWO-ROW PLACEMENT The synthesis half showed that GPF realizes any n-qubit diagonal unitary in depth O(2n/… view at source ↗
Figure 8
Figure 8. Figure 8: Two-row embedding for n = 2 × 4: the high half on the upper row, the low half on the lower row (sites Hi/Li , occupancy labels hi/li ; Table I). Each column’s vertical edge carries one high–low interaction; horizontal edges are intra￾row SWAPs. cording to their Gray-path schedules. In other words, GPS and GPS∗ share the same connectivity type at the synthesis level— multi-layer bipartite matching between t… view at source ↗
Figure 9
Figure 9. Figure 9: Unified recursion model. GPF(n) is (i) a set of top￾level GPS-2L mixings between the two halves on a 2 × nH grid, plus (ii) a high-half recursion GPF(nH) and (iii) a low-half recursion GPF∗ (nL), both on single-row segments. Two-row geometry occurs only at the top level; after the first split, all deeper blocks are single-row (with higher-level qubits acting as bystanders on the line). This figure is the m… view at source ↗
Figure 10
Figure 10. Figure 10: Base offset ring for nH = 4. Each cell shows one offset state together with the SWAP layer that advances it to the next state. Alternating SWAP-A/B cycles s0 → s1 → s2 → s3 → s0, reaching every offset state. one top-level GPS block. From sft to sft+1 requires DistnH (ft, ft+1) layers, so the internal state-switching depth of the full closed cycle is J = 2 nXH −1 t=0 DistnH [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 11
Figure 11. Figure 11: Routed top-level GPS circuit (n = 8). Boxes abbreviate the inserted Rz(φs) phase gates. is summarized below. It is not known to be globally optimal— the name is descriptive. Lemma 5 (low-jump attains the cost lower bound for small nH). For the low-jump transition sequence, • nH ≤ 5: µjump = 1.0, attaining the hard lower bound J = 2nH and proved globally optimal by iterative￾deepening exact search; • nH = … view at source ↗
Figure 13
Figure 13. Figure 13: The SWAP-A/B layers used by single-row GPS/GPS [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Routed single-row GPS module (n = 8). 4) Cost and Complexity: A GPS-1L node at level r of the GPF recursion tree (r = 0 is the top level) acts on Kr = n/2 r positions, with its high sub-group containing wr = n/2 r+1 qubits and its offset ring containing Jr = µjump 2 wr SWAP￾A/B layers. As r increases, the offset ring shortens while each brickwork layer deepens: bystander spacing right-shifts the distance … view at source ↗
Figure 15
Figure 15. Figure 15: Normalized routing-depth complexity of complete [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Normalized routing gate-count complexity of com [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Fan-out replication of GPF phase layers ( [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Complete expanded GPF(8) circuit, split into two consecutive panels. The phase labels use decimal Walsh-mode indices; the lower panel continues from the right end of the upper panel [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Complete routed GPF(8) circuit on the 2 × 4 grid (folded into six segments; low-jump sequence). Dark cross-row links are top-level GPS-2L matchings (Section V-A); light intra-row links are the recursive GPS-1L/GPS∗-1L relay (Section V-B). The whole circuit—278 CNOTs plus 350 SWAPs, all nearest-neighbor— realizes the original GPF(8) [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Logical top-level GPS to be routed (n = 8): the cross-block CNOT spans vary with the transition sequence, so most are non-nearest-neighbor—this is why routing is needed [PITH_FULL_IMAGE:figures/full_fig_p021_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Logical single-row GPS∗ to be routed (n = 8): the CNOT spans are highly irregular, and the long-range pairs are exactly what single-row routing must decompose to nearest-neighbor [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: 2.5D depth chart of the routed top-level GPS block ( [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Example 2: routed single-row GPS∗ module (n = 8); same folding and brickwork as [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Example 3: single-row GPS∗ on segment {1, 2, 5, 6} (n = 8): the remote SWAP(2, 5) is brickwork-decomposed through bystanders {3, 4} and restored—a deeper-recursion intra-group SWAP becoming remote. This realizes GPS∗ (4). B. Balanced Subsequence n = 2λ Here nH = nL = 2λ−1 , both powers of two. By the balanced-subsequence telescoping of Appendix C, DGPF∗ z (2λ ) = Y λ j=1 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
read the original abstract

Diagonal unitaries are a fundamental but resource-intensive class of quantum operations, arising as the phase separators of QAOA and the time-evolution blocks of Hamiltonian simulation. Under all-to-all connectivity their optimal depth is established, but on nearest-neighbor hardware general-purpose compilers fall back on heuristic search, which yields no analyzable cost bound and becomes intractable at the very sizes where depth is the bottleneck. We address synthesis and compilation jointly. On the synthesis side, we develop a Gray-Path Framework (GPF) that realizes any $n$-qubit diagonal unitary in asymptotically optimal $R_z$ and CNOT depth $O(2^n/n)$ without ancillas. Our main result is that compiling GPF onto a two-dimensional nearest-neighbor grid preserves this optimality: routing adds depth $\Theta(2^n/n)$ and gate count $\Theta(2^n)$. Because GPF fixes its entire interaction structure in advance, routing reduces to scheduling a known sequence, with no heuristic search. We give the construction both with and without ancillas: the ancilla-free, cost-optimized layout is a two-row grid, and a $2k$-row layout introduces a space--time tradeoff that cuts depth by $1/k$ while remaining asymptotically optimal for the enlarged register; both are deterministic and analyzed in closed form. The same complexity is also attained on a linear nearest-neighbor chain, so the preservation is topology-independent, holding on any architecture that contains such a chain. All routing bounds are closed-form, giving the concrete resource estimates that heuristic compilers cannot provide at scale.

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 / 2 minor

Summary. The paper introduces the Gray-Path Framework (GPF) for synthesizing arbitrary n-qubit diagonal unitaries in asymptotically optimal O(2^n/n) depth using Rz and CNOT gates without ancillas under all-to-all connectivity. Its central result is that the fixed, predetermined interaction sequence of GPF permits deterministic routing onto a 2D nearest-neighbor grid (or linear chain) whose overhead remains heta(2^n/n) in depth and heta(2^n) in gate count, thereby preserving optimality; closed-form analyses are given for both an ancilla-free two-row layout and a 2k-row space-time tradeoff.

Significance. If the constructions and closed-form bounds hold, the work supplies the first rigorous, analyzable compilation strategy for diagonal unitaries on nearest-neighbor hardware that matches the known all-to-all optimum up to constants. The topology independence within any architecture containing a chain, together with the elimination of heuristic search, constitutes a concrete advance over existing compilers.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'all routing bounds are closed-form' would be strengthened by an explicit forward reference to the theorem or proposition that states the leading-term expressions for the 2D and chain cases.
  2. The manuscript would benefit from one additional sentence in the introduction clarifying how the GPF interaction sequence is generated from the target diagonal phases, to make the 'fixed in advance' property immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the provided report, so we have no specific points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on explicit constructions of the Gray-Path Framework (GPF) for O(2^n/n) depth synthesis of diagonal unitaries under all-to-all connectivity, followed by closed-form deterministic scheduling arguments that bound routing overhead on 2D grids or chains by the same leading term. No equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing premises depend on self-citations, and the fixed-sequence property converts routing to a scheduling task whose cost is analyzed independently of the target asymptotic result. The paper is self-contained against external benchmarks with no self-referential definitions or ansatzes smuggled via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard quantum circuit model with Rz and CNOT as elementary gates and the assumption of nearest-neighbor connectivity on grids or chains; no free parameters, ad-hoc constants, or new entities are introduced.

axioms (2)
  • standard math Quantum circuit depth is measured in Rz and CNOT gates under the standard gate set
    Invoked when stating O(2^n/n) depth for synthesis.
  • domain assumption Hardware supports only nearest-neighbor interactions on a 2D grid or linear chain
    Central to the compilation and routing analysis.

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discussion (0)

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