A 12-CNOT Double Qubit Excitation Gate

Irfansha Shaik

REVIEW 4 major objections 4 minor 22 references

A 12-CNOT Double Qubit Excitation Gate

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A 12-CNOT circuit implements the double qubit excitation operator, the fewest CNOTs reported to date.

desk verdict A concrete one-CNOT improvement with the best reported depth numbers, but the central circuit is unverifiable from the text and needs a machine-checked companion. read the letter →

arxiv 2608.11733 v1 pith:QRYGC7UT submitted 2026-08-12 quant-ph cs.AI

Irfansha Shaik This is my paper · ORCID

classification quant-phcs.AI MSC 81P68 PACS 03.67.Lx

keywords doublequbitexcitationoperatorCNOTcountcircuitdepthquantumdecompositionunitarycoupledclusterchemistrytriple-controlledrotation

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

This paper presents a new explicit quantum circuit for the double qubit excitation operator, a four-qubit gate that rotates between the computational basis states $|0011\rangle$ and $|1100\rangle$ and is a building block of quantum chemistry ansätze and Hamiltonian simulation. The author claims the circuit uses 12 CNOT gates, one fewer than the best previously published implementations, and that it also achieves the lowest CNOT depth (10) and total circuit depth (16) among the compared circuits. If the claimed identity is correct, this is a concrete efficiency improvement: fewer two-qubit gates means less error accumulation on near-term hardware, and a shallower circuit means less exposure to decoherence. The construction starts from the standard $L$--$C^3R_y(2\theta)$--$R$ sandwich and optimizes the CNOT placement using circuit synthesis software, adding only two one-qubit gates compared to the most gate-efficient prior circuit.

What carries the argument

The load-bearing object is the $L$--$R$ CNOT sandwich around a triple-controlled $R_y$ rotation: the double-excitation operator is written as $L\, C^3R_y(2\theta)\, R$, with $R = L^{-1}$, where $L$ is a CNOT network sending $|0011\rangle$ and $|1100\rangle$ to states whose three control qubits are in $|1\rangle$ and all other basis states to some $|x'\rangle$ that does not trigger the rotation. The new 12-CNOT circuit is a specific optimized realization of this sandwich, in which CNOTs from the controlled-rotation decomposition are commuted, cancelled, or shared with the flanking networks. The whole argument depends on the previously established identity (cited to [21]) that such a sandwich exactly equals the eight-term Pauli exponential in Eq. (1); this identity supplies the correctness of the high-level circuit, and the paper's contribution is the concrete 12-CNOT layout of that circuit.

What would settle it

Compute the $16 \times 16$ unitary of the Fig. 5 circuit symbolically and compare it entry-by-entry with the matrix exponential of Eq. (1) at a generic angle such as $\theta = \pi/7$; any nonzero difference or a relative phase on the $|0011\rangle \leftrightarrow |1100\rangle$ block would falsify the claimed 12-CNOT implementation.

Watch

Extended reading notes

Core claim

The central claim is that the double qubit excitation operator $U(\theta)$ of Eq. (1), which exponentiates a sum of eight Pauli strings, can be implemented with only 12 CNOT gates, improving on all known 13-CNOT implementations. The circuit shown in Fig. 5 realizes this with CNOT depth 10, total depth 16, and 13 one-qubit gates; in the paper's comparison table these are the lowest values among the listed circuits for all three metrics. The construction is an explicit four-qubit circuit obtained by decomposing the operator as $L\,C^3R_y(2\theta)\,R$ with $R=L^{-1}$, where $L$ maps the two coupled basis states $|0011\rangle$ and $|1100\rangle$ to states that activate the triple-controlled rotation, and then optimizing the CNOT structure of the whole sandwich. The author reports that no previously published decomposition with fewer than 13 CNOTs was found in the literature.

Load-bearing premise

The construction assumes that the sandwich of the $L$ CNOT network, the triple-controlled $R_y(2\theta)$ rotation, and $R=L^{-1}$ reproduces exactly the eight-term Pauli exponential in Eq. (1) with no spurious phase or sign; the paper takes this high-level identity from an earlier reference rather than proving it, so any mismatch between that identity and the angle layout of Fig. 5 would mean the circuit does not realize the intended gate.

Editorial extensions

If this is right

Quantum chemistry circuits that use the double excitation gate, such as UCCSD variants and adaptive VQE ansätze, can be compiled with one fewer CNOT than before, reducing the two-qubit error budget on current processors.
The new circuit's CNOT depth of 10 and total depth of 16 make it the shallowest reported implementation, a favourable property for devices where decoherence limits the number of sequential operations.
The paper's Table 1 provides a new benchmark: any future compiler or synthesis tool targeting the double excitation operator now has a 12-CNOT reference point to beat.
Because the circuit adds only two one-qubit gates relative to the minimum-known 11, the improvement in CNOT count does not come at a large single-qubit cost.

Reading between the lines

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

The same $L$--controlled-rotation--$R$ template, optimized with the same style of CNOT minimization, may yield one-gate reductions for single, triple, and other fermionic excitation operators used in quantum chemistry.
A formal lower-bound proof would be the natural next step: the author reports no circuit with fewer than 12 CNOTs, but the paper does not attempt to show 12 is optimal.
On fully connected hardware, the fan-out CNOTs in Fig. 5 could be parallelized further, potentially reducing the CNOT depth below 10 without altering the circuit.
Real-device benchmarking against the 13-CNOT circuits would test whether the lower depth translates into higher end-to-end fidelity, given that the one-qubit gate count rises from 11 to 13.

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Signed reviews

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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

Short version: this is a real, incremental result—a 12-CNOT decomposition of the double qubit excitation operator, one fewer CNOT than the previous best, with the best reported CNOT depth and total depth. The comparison table is clear, the novelty claim is properly qualified, and the author is honest about having corrected a prior circuit. No red flags in the approach: the L/R sandwich identity is standard, and the claim is specific, not hand-wavy.

The soft spot is exactly what the reader flagged: the paper gives no way to verify the central artifact. Figure 5 is drawn with compact fan-out notation, the CNOT placements are not listed in the text, and there is no OpenQASM, no Qiskit code, no numerical unitary check, and no proof that the circuit equals Eq. (1). The acknowledgment that a published 13-CNOT circuit needed two added gates shows how easy it is to slip up. I cannot reconstruct the circuit from the text myself, so the central claim is credible but unconfirmed.

That said, the logic of the construction is not circular, and the burden is not on the author's identity—it's just missing evidence. This is the kind of paper that should have a one-line verification script as a supplementary file. Without it, the claim is essentially "trust the figure."

Is the result worth a referee's time? Yes. A one-CNOT reduction in a widely used building block for UCCSD/VQE and Hamiltonian simulation is a modest but real contribution to circuit compilation. The paper does not need a desk reject; it needs a referee who will actually run a unitary equivalence check. That is standard for this subfield.

My recommendation: send it to peer review, but make the acceptance conditional on shipping a machine-readable circuit (QASM or Qiskit) and a verification script that checks the unitary against Eq. (1). Also ask the author to clarify the synthesis method—saying "we used Q-Synth, Qiskit, tket" without details on how the 12-CNOT circuit was found or why it is optimal is thin. For a smaller point, the table is useful but would be stronger with an explicit statement of how single-qubit gates are merged into u3 counts, which the caption mentions.

Who benefits: people who need concrete resource counts for excitation operators in compiled circuits. I would not cite it myself until the circuit is independently verified, but I would bring it to a reading group working on quantum compilation.

Referee Report

4 major / 4 minor

Summary. The paper proposes a new quantum circuit for the double qubit excitation operator U(θ) of Eq. (1), claiming to be the first 12-CNOT decomposition, improving over previously reported 13-CNOT circuits. The construction is based on the standard L/R CNOT sandwich around a triple-controlled Ry rotation, with the new circuit shown in Figure 5. The paper reports resource counts of 12 CNOTs, CNOT depth 10, total circuit depth 16, and 13 single-qubit gates, and compares these in Table 1 with prior circuits from Yordanov et al., Nam et al., and Wang et al. No equivalence proof, numerical verification, or machine-readable circuit is provided; the central claim rests entirely on the correctness of Figure 5.

Significance. If the Figure 5 circuit is correct, the result is a modest but concrete improvement over the prior 13-CNOT state of the art, lowering CNOT count, CNOT depth, and total depth, at the cost of two additional single-qubit gates relative to the best prior one-qubit count. Such resource reductions are relevant for near-term ansatz circuits and Hamiltonian simulation. The paper makes falsifiable and precise metric claims in Table 1 and correctly attributes prior constructions. However, the result is not backed by any machine-checked proof, simulation, or code artifact, and the figure is not independently reconstructible from the text, so the central claim is currently unverified.

major comments (4)

[Section 2, Figure 5] The central claim that the circuit in Figure 5 realizes U(θ) of Eq. (1) with 12 CNOTs is not supported by any equivalence proof, numerical simulation, or automated verification. The figure uses compact fan-out notation and does not list the explicit CNOT control/target pairs, so a reader cannot reconstruct the circuit from the text. This matters because the paper itself notes in the acknowledgements and Section 1.2 that corrections were needed to a previously published Yordanov circuit, demonstrating that this class of construction is error-prone. Please provide an explicit gate-by-gate circuit, a unitary equality check, or a rigorous derivation showing that Figure 5 equals Eq. (1).
[Section 2, synthesis-tool description] The description in Section 2 that various synthesis and optimization tools, including Q-Synth, Qiskit, and tket, were used to explore decompositions is not accompanied by any reproducible details: no synthesis scripts, parameter settings, output certificates, or machine-readable circuits are given. Since the paper reports a record resource count obtained from undisclosed synthesis runs, the central result is not reproducible as written. Please include a circuit file (e.g., OpenQASM), a verification script, or a synthesis certificate.
[Table 1 and Section 2, metric definitions] The reported metrics, especially CNOT depth 10 and total circuit depth 16, cannot be checked from the paper as printed. The caption of Figure 5 says fan-out CNOTs are expanded when computing depth, but the expanded circuit is never shown, and the single-qubit gate counting rule based on merging maximal runs into u3 gates is not applied explicitly to the displayed gates. Please provide the expanded circuit and a transparent depth computation to substantiate the Table 1 entries.
[Section 1.1, Eq. (3)] The L transformation is stated in Eq. (3) but no explicit CNOT circuit realizing L is given in the paper; the text refers to [21] for the sandwich construction. Since the new 12-CNOT circuit is presented as a standalone result and not shown to be derived from that identity, the paper should either prove the L/R decomposition used in Figure 5 or cite the specific circuit from [21] with a clear mapping to the new figure.

minor comments (4)

[Section 1, Eq. (1) and Eq. (2)] There are missing spaces in 'Usingthecomputational-basisordering' and several other OCR-like spacing errors throughout the text; these should be corrected in a revision.
[Section 1, references] Equation (1) is attributed to 'Eq. 20 of [21]'; please give the exact equation number in the cited paper and verify that the sign convention matches the present Eq. (1).
[Figures 3 and 4] The figures for the previous 13-CNOT circuits are difficult to parse because control/target positions and timing are not always visually aligned; adding explicit circuit coordinates or an OpenQASM listing would improve comparability.
[Section 2, Figure 5 caption] The notation '√X†' is used without defining the phase convention; since Rz angles and one-qubit gate counts depend on the convention, please specify it explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 12-CNOT circuit is a concrete synthesis result, not a re-labeled input, and the cited prior work is background rather than load-bearing.

full rationale

The central claim is that Fig. 5 is a 12-CNOT implementation of the double-excitation operator defined in Eq. (1). This is a constructive synthesis claim: the circuit is produced by exploration with Q-Synth, Qiskit, and tket, and the counted resource metrics are evaluated on the displayed circuit, not fitted from the metric being predicted. The high-level L-C3Ry-R sandwich and the L/R map of Eq. (3) are taken from Yordanov et al. [21], but the 12-CNOT result is not derived from those references; the references supply a baseline (the 14-CNOT Gray-code circuit and previous 13-CNOT circuits) against which the new circuit is compared. The author's self-citations ([11], [12], [13], [14]) are cited only as synthesis tools used in the search, not as the source of the claimed equality, so they are not load-bearing. No parameter is fitted from the claimed output, no known result is merely renamed, and no uniqueness theorem is invoked. Verification concerns about the unshown CNOT layout or unstated synthesis details are correctness/verifiability risks, not circularity. Thus the derivation is self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The decomposition leans on standard circuit identities from the literature (Gray-code expansion, Pauli-string decompositions, the L/R high-level construction from [21]). No parameters are fitted to data and no new entities are introduced. The correctness of the final Figure 5 circuit is asserted rather than derived, which is the key gap.

assumptions (2)

standard math Gray-code expansion decomposes a triple-controlled Ry rotation into 8 CNOTs and 8 Ry rotations (Section 1.1, cited [6,15]).
The baseline 14-CNOT circuit relies on this equality; if it failed, the baseline construction would not be valid. It is accepted from the literature.
domain assumption The L and R CNOT circuits in Figure 1 implement the state mapping in Eq (3), so the controlled rotation triggers only on |0011> and |1100>.
This is a central modeling assumption for the high-level construction; the paper refers to [21] for the explanation rather than proving it.

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Cite this review

Pith. "Pith review of A 12-CNOT Double Qubit Excitation Gate." pith.science (2026). https://pith.science/paper/QRYGC7UT

@misc{pith2026260811733,
  author       = {Pith},
  title        = {Pith review of: A 12-CNOT Double Qubit Excitation Gate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRYGC7UT}},
  note         = {Machine review of arXiv:2608.11733}
}

read the original abstract

Effective implementation of high-level quantum gates is essential for practical quantum computing. To the best of our knowledge, we present the first reported 12-CNOT decomposition of the double qubit excitation operator, improving upon state-of-the-art (SOTA) implementations with 13 CNOTs. Our new circuit has the lowest CNOT count (12), lowest CNOT depth (10), and lowest total circuit depth (16) among all the previous SOTA circuits. Further, we only added 2 extra one-qubit gates compared to the lowest one-qubit gate count (11) among the previous SOTA circuits.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [21]

Quantum Science and Technology , volume =

Sivarajah, Seyon and Dilkes, Silas and Cowtan, Alexander and Simmons, Will and Edgington, Alec and Duncan, Ross , title =. Quantum Science and Technology , volume =. 2021 , doi =. 2003.10611 , archivePrefix =

arXiv 2021

[2] [1]

and Chuang, Isaac L

Nielsen, Michael A. and Chuang, Isaac L. , title =. 2010 , edition =

work page 2010

[3] [2]

Efficient quantum circuits for quantum computational chemistry

Yordanov, Yordan S. and Arvidsson-Shukur, David R. M. and Barnes, Crispin H. W. , title =. Physical Review A , volume =. 2020 , doi =. 2005.14475 , archivePrefix =

work page Pith review arXiv 2020

[4] [3]

and Armaos, V

Yordanov, Yordan S. and Armaos, V. and Barnes, Crispin H. W. and Arvidsson-Shukur, David R. M. , title =. Communications Physics , volume =. 2021 , doi =

work page 2021

[5] [4]

and Barnes, Crispin H

Yordanov, Yordan S. and Barnes, Crispin H. W. and Arvidsson-Shukur, David R. M. , title =. Physical Review A , volume =. 2022 , doi =

work page 2022

[6] [5]

and Wright, Kenneth and Delaney, Conor and Maslov, Dmitri and Brown, Kenneth R

Nam, Yunseong and Chen, Jwo-Sy and Pisenti, Neal C. and Wright, Kenneth and Delaney, Conor and Maslov, Dmitri and Brown, Kenneth R. and Allen, Stewart and Amini, Jason M. and Apisdorf, Joel and Beck, Kristin M. and Blinov, Aleksey and Chaplin, Vandiver and Chmielewski, Mika and Collins, Coleman and Debnath, Shantanu and Hudek, Kai M. and Ducore, Andrew M....

arXiv 2020

[7] [6]

Quantum , volume =

Wang, Qingfeng and Li, Ming and Monroe, Christopher and Nam, Yunseong , title =. Quantum , volume =. 2021 , doi =. 2004.04151 , archivePrefix =

arXiv 2021

[8] [7]

Quantum circuits for general multiqubit gates , journal =

M. Quantum circuits for general multiqubit gates , journal =. 2004 , doi =. quant-ph/0404089 , archivePrefix =

arXiv 2004