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REVIEW 3 major objections 4 minor 59 references

A Cost-Effective Quantum Boolean-Phase SWAP Gate with Only Two CNOT Gates

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A swap gate built from two CNOT gates instead of three can carry a selectable phase and cut transpiled quantum cost by about 23 percent.

desk verdict A modest iSWAP variant whose Boolean-SWAP claim is refuted by its own phase table. read the letter →

arxiv 2507.17164 v1 pith:T4YLR3KK submitted 2025-07-23 quant-ph

Ali Al-Bayaty , Shanyan Chen , Steven A. Bleiler , Marek Perkowski This is my paper · ORCID
classification quant-ph
keywords p-SWAPgateSWAPCNOTcountquantumcostcircuitdepthBlochsphereapproachphaseoracletranspilation
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 proposes a two-qubit swap gate called the p-SWAP gate that uses only two CNOT gates instead of the three needed for a standard SWAP, while adding a selectable phase difference p between -pi and +pi radians. The authors argue that the gate is generic because it can serve Boolean oracles, where p is ignored, and phase oracles, where p is chosen so the gate votes for particular basis states. They build the circuit from the native sqrt(X), RZ, and CNOT gates of the target architecture using a Bloch sphere design method, avoiding the Hadamard gates that inflate the cost of the related iSWAP circuit. After transpilation on a 127-qubit superconducting processor, the p-SWAP circuit has about 23% lower quantum cost and 26% lower depth than the transpiled standard SWAP. If the claim holds, replacing standard SWAPs in large circuits would reduce both gate count and the time qubits spend exposed to decoherence.

What carries the argument

The load-bearing object is the four-stage p-SWAP circuit: an entanglement stage (sqrt(X) on the first qubit followed by a CNOT), a differentiation stage (two RZ(+pi/2) rotations chosen from the XY-plane of the Bloch sphere), an unentanglement stage (a CNOT and a sqrt(X)), and a phase selection stage consisting of two RZ gates, called v and w, which set p for the swapped outcomes. The Bloch sphere approach, a geometric design method that reads quantum rotations off intersections of planes with the sphere, fixes the RZ angles visually rather than by matrix multiplication. Replacing Hadamard gates with sqrt(X) gates keeps the circuit inside the native gate set, and Table I records which choices of v and w produce which phases.

What would settle it

Prepare the input $\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$, apply the Boolean p-SWAP (the first three stages), and compare with the standard three-CNOT SWAP on the same input by measuring in the $X$ basis. With $v=w=\text{None}$, Table I assigns phase $+\pi$ to the $|10\rangle$ outcome, so the Boolean circuit outputs $\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$ while the ideal SWAP outputs $\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$; a difference in the $X$-basis statistics would show that p cannot be ignored for superposition inputs.

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Extended reading notes

Core claim

The central claim is that the p-SWAP gate is a generic, cost-effective Boolean-phase swapping gate. It swaps the basis states of two qubits exactly as a SWAP does, but it also lets the designer choose a phase p for the swapped outcomes: with no phase-selection gates, p is 0 on three basis states and +pi on one, while with chosen RZ gates p can be set so the gate votes for a selected state, reproduces the iSWAP, or reproduces the inverse iSWAP. For Boolean use the paper treats p as ignorable; for phase use p is the feature. The gate's circuit, designed visually through the Bloch sphere approach, uses two CNOT gates and only native gates, so after transpilation the p-SWAP beats the three-CNOT SWAP by about 23% in cost and 26% in depth.

Load-bearing premise

The Boolean case assumes the chosen phase p can be ignored after the swap, even though Table I shows the gate places different relative phases on different basis states, so the phase is physically real whenever the input is a superposition.

Editorial extensions

If this is right

  • Replacing each standard SWAP with a p-SWAP in a transpiled circuit lowers the gate count by roughly 23% and the depth by roughly 26% on the targeted 127-qubit processor.
  • The same two-CNOT structure with different RZ choices implements phase oracles that vote for a single basis state, as well as the iSWAP and its inverse, as listed in Table I.
  • Because the circuit uses only native gates, it can be dropped into existing compilation flows without decomposing Hadamard gates, which is what makes the comparison with iSWAP favorable.
  • Setting p=+pi/2 on both swapped channels reproduces the iSWAP gate exactly, so the p-SWAP is a strict generalization of a known two-CNOT gate.
  • The Bloch sphere method is presented as a general visual design recipe, so the four-stage construction can be adapted to other hardware by choosing a different intersection plane.

Reading between the lines

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

  • The paper leaves implicit that the Boolean identity only holds for computational-basis inputs; for superpositions, the state-dependent phases in Table I change the output state, so the Boolean use should be scoped to inputs where those phases are harmless or absorbed into the oracle.
  • The same entangle-rotate-unentangle-phase-select recipe could be applied to other two-qubit operations, potentially yielding cost reductions for gates other than swaps on machines with different native gate sets.
  • The reported 23%/26% savings are an empirical outcome of one transpiler and one processor; the ratio could shift with qubit routing, transpiler version, or hardware native gates, so the advantage should be re-measured when porting to a new device.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces a two-qubit gate called the p-SWAP gate, claimed to swap two qubits while optionally applying a selected relative phase p, and to use only two CNOT gates compared to the standard three-CNOT SWAP. The gate is designed visually using the authors' Bloch-sphere approach, and the authors claim it is a generic cost-effective Boolean-Phase swapping gate: for Boolean applications 'the value of p is ignored,' while for Phase applications p is chosen via two RZ gates v and w. The paper reports transpiled-circuit results on ibm_brisbane showing approximately 23% lower quantum cost and 26% lower depth than the transpiled standard SWAP. Table I lists six phase selections said to implement voting on basis states, the iSWAP, and the inverse iSWAP.

Significance. If the central claim were correct, a two-CNOT swap-type gate with tunable phases would be a useful building block for phase oracles and could reduce transpiled circuit costs on IBM hardware. The paper is honest in explicitly admitting that two Bloch spheres cannot represent entangled two-qubit states, and it provides concrete transpilation counts on a real backend. However, the Boolean-use claim is mathematically incompatible with the gate's own phase table: no row of Table I gives a common phase on all basis states, so the gate is not a SWAP up to global phase on superposition inputs. Because Boolean oracles necessarily process superpositions, the phase 'ignored' by the authors is physically observable. The absence of any unitary-matrix verification means the correctness of even the Phase variants is not established. These issues undermine the paper's central contribution as stated.

major comments (3)
  1. [Section II.C and Table I] The Boolean-use claim fails on superposition inputs. The paper states that 'when the value of p is ignored, i.e., p can be arbitrarily set to any angular value, such a p-SWAP gate is a generic cost-effective Boolean swapping gate.' But Table I assigns different relative phases to different swapped basis states in every row. For example, with v=w=None the output has phase +π on |10> and phase 0 on |00>, |01>, and |11>. For the input (|01>+|10>)/√2, the two swapped branches acquire different phases, producing a state orthogonal to the intended SWAP output (|10>+|01>)/√2. Relative phases are physical in any superposition, so they cannot be dismissed when p is 'ignored.' This contradicts the known result that an exact SWAP requires three CNOT gates, and it invalidates the central claim that the gate is a generic Boolean swapping gate.
  2. [Section II.C and Fig. 4] The paper provides no unitary matrix or algebraic verification of the two-CNOT circuit. The correctness argument relies on the Bloch-sphere visual method, and the text itself concedes that 'two Bloch spheres could not visualize the maximally entangled states of two qubits.' Since the circuit passes through entangled states in the differentiation stage, the visual tracking in Fig. 4 cannot certify the unitary. A direct matrix computation should be provided for the full circuit (including the RZ(θ1), RZ(θ2), √X, and CNOT/ECR gates) and should reproduce the phase entries in Table I. Without this, the correctness of even the Phase variants is asserted, not demonstrated.
  3. [Section III, Table II] The claimed 23% cost reduction and 26% depth reduction are based on a single transpilation target (ibm_brisbane) and on the paper's own TQC metric, which adds depth into the cost. The conclusion that the p-SWAP gate 'always has a lower quantum cost and depth' is not supported: transpilation outcomes depend on the qubit topology, the initial layout, and the transpiler optimization level. At minimum, the results should be reported for several layouts and optimization levels, and the claim should be restricted to the specific metric and backend used.
minor comments (4)
  1. [Abstract] The word 'trans piled' in the abstract is a typo for 'transpiled.'
  2. [Section I] The paper uses 'n-bit' for 'n-qubit' gates repeatedly; this is nonstandard and may confuse readers, though it is consistent with the authors' previous work.
  3. [Section II.C] The phrase 'Boolean oracles' is not formally defined. Since the Boolean/Phase distinction is central to the claimed generality, a precise definition of how the relative phase is treated in each kind of oracle would clarify the argument.
  4. [Fig. 3 and Fig. 5] The text says that on 127-qubit IBM systems the native CNOT is replaced by ECR, but Fig. 3 and Fig. 5 still show CNOT symbols. It would be helpful to state explicitly which level of abstraction each figure refers to, and to show the ECR-level decomposition rather than only the CNOT-level circuit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the p-SWAP construction and transpilation cost comparison are self-contained; the Boolean-use caveat is a correctness risk, not a circular derivation.

full rationale

The paper's central construction is presented directly: the p-SWAP circuit in Fig. 3(c) and the phase table in Table I are explicit, not fitted to the cost data. The claimed 23% TQC and 26% depth reductions are measurements of transpiled circuits on ibm_brisbane (Table II), not predictions derived from a fitted parameter. The self-citations to the authors' Bloch sphere approach and TQC metric are descriptive rather than load-bearing: the circuit and the metric definition (N1 + N2 + D) are given in the paper itself, and the cost comparison is an experimental report. The only questionable step is the Section II.C assertion that when "the value of p is ignored" the gate is a "generic cost-effective Boolean swapping gate." Table I shows state-dependent relative phases (e.g., v = w = None gives +π on |10> but 0 on the other basis states), so ignoring p is not justified for superposition inputs such as (|01>+|10>)/√2. That is an unproven physical assumption and a correctness risk, not a circular reduction: the gate's unitary action is tabulated independently rather than being derived from the Boolean-use claim. No self-citation chain is required to obtain the cost or phase results, so no circularity is present.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

The central design rests on two hand-chosen angle sets (theta1=theta2=pi/2 and the v/w phase-selection angles), a visual proof method that cannot represent entangled states, and a self-defined cost metric. No fitted parameters appear, but the claimed arbitrary-p generality and Boolean applicability are supported only by the paper's own tables and figures.

free parameters (2)
  • Differentiation-stage RZ angles theta1 and theta2 = +pi/2 each
    Chosen by hand in Section II.C so that the unentanglement stage restores the first target to the Z-axis; no proof is given that this choice is necessary or that it works for all requested phase values p.
  • Phase-selection RZ angles v and w = None, +pi, -pi, +pi/2, -pi/2 per Table I
    Selected ad hoc for each application row in Table I; the paper gives no general formula mapping an arbitrary requested phase p to concrete v and w values, so the claimed arbitrary-p generality is not demonstrated.
assumptions (3)
  • ad hoc to paper The relative phase p can be ignored for Boolean oracles, so the state-dependent phases introduced by the p-SWAP gate do not affect the computation.
    Section II.C states 'the value of p is ignored ... for Boolean oracles', but Table I shows the phases differ across basis states (e.g., 0 and +pi in the same row), so this is only valid for basis-state inputs, not superpositions.
  • ad hoc to paper The two-Bloch-sphere visual tracking in the Bloch sphere approach is a valid proof of the two-qubit unitary's correctness.
    Section II.C uses separated XY-plane diagrams for each target, while the text admits 'two Bloch spheres could not visualize the maximally entangled states of two qubits' [52-54], leaving the proof method unsupported.
  • domain assumption TQC, defined as the sum of native single-qubit gates, ECR gates, and depth, is an appropriate measure of cost for the claimed improvement.
    Section III defines TQC from the authors' prior work [38,56]; physical error rates, runtimes, and other device metrics are explicitly excluded, so the 23%/26% reductions are in abstract gate counts only.
invented entities (1)
  • p-SWAP gate
    purpose: A two-qubit gate that swaps computational basis states while adding a customizable phase difference, built from two CNOT gates plus native rotations.
    The paper provides no external benchmark or independent verification of the gate's unitary action; the only support is the authors' own Bloch sphere figures and the TQC table.

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Cite this review

Pith. "Pith review of A Cost-Effective Quantum Boolean-Phase SWAP Gate with Only Two CNOT Gates." pith.science (2026). https://pith.science/paper/T4YLR3KK

@misc{pith2026250717164,
  author       = {Pith},
  title        = {Pith review of: A Cost-Effective Quantum Boolean-Phase SWAP Gate with Only Two CNOT Gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4YLR3KK}},
  note         = {Machine review of arXiv:2507.17164}
}
abstract

A new Boolean-Phase swapping gate is presented with improved quantum generality and cost-effectiveness. Our swapping gate is termed the "p-SWAP gate", where p is the phase difference selected for a set of swapped qubits. The phase p is expressed in radians and $-{\pi}~{\leq}$ p ${\leq}~+{\pi}$. The generality of p-SWAP gate is demonstrated for Phase applications for selected values of p and for Boolean applications when the value of p is ignored. The cost-effectiveness of p-SWAP gate follows from the lower quantum cost and depth of its final realized (transpiled) quantum circuit, when compared to the standard SWAP gate composed of three Feynman (CNOT) gates. Specifically, our presented p-SWAP gate utilizes only two CNOT gates. The quantum circuit of the p-SWAP gate is visually designed using our previously developed Bloch sphere approach. Experimentally, after transpilation for an IBM quantum computer, the transpiled p-SWAP gate shows an approximate 23% quantum cost reduction and an approximate 26% depth minimization compared to the transpiled standard SWAP gate.

Figures

Figures reproduced from arXiv: 2507.17164 by the authors.

Figure 1
Figure 1. , where + θ represents a counterclockwise rotation and – θ represents a clockwise rotation for – π ≤ θ ≤ + π. For instance, the following Z gates rotate the states of a qubit by the Z-axis for the defined θ angles: • The Z gate performs the quantum rotation of + π radians. • The √Z 2 gate performs the quantum rotation of + π/2 radians. The √Z 2 gate is the so-called “S gate” [1–3, 6]. • The √Z 4 gate performs the qu… view at source ↗
Figure 2
Figure 2. Coordinate and parallel planes (P) perpendicular to various coordinate axes intersected with the Bloch sphere: (a) the XY-plane perpendicular to the Z-axis intersected with the Bloch sphere (as P3 in d), (b) the XZ-plane perpendicular to the Y-axis intersected with the Bloch sphere(as P3 in e), (c) the YZ-plane perpendicular to the X-axis intersected with the Bloch sphere (as P3 in f), (d) m parallel planes perpendi… view at source ↗
Figure 3
Figure 3. Decomposed quantum circuits of SWAP, iSWAP, and p-SWAP gates: (a) the standard SWAP gate, (b) the iSWAP gate [36], and (c) the p-SWAP gate consisting of IBM native gates only, four stages, and two 1-bit phase-based gates (v and w) for selecting the values of p, where |q0⟩ is the initial state of the first target, |q1⟩ is the initial state of the second target, and – π ≤ p ≤ + π in radians [PITH_FULL_IMAGE:figures/f… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Representations of the quantum transitions (the yellow arrows) for the swapped outcomes of a p-SWAP gate using two XY-planes intersecting two Bloch spheres, for all permutative states |q1q0⟩: (a) |00⟩ → |00⟩, (b) |01⟩ → |10⟩, (c) |10⟩ → |01⟩, and (d) |11⟩ → |11⟩, where…
Figure 5
Figure 5. Figure 5: Transpiled quantum circuits using ibm_brisbane quantum computer: (a) one Feynman (CNOT) gate, (b) the standard SWAP gate consisting of three CNOT gates, (c) the iSWAP gate consisting of two CNOT gates, and (d) our generic cost-effective p-SWAP gate consisting of two CN…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.