Boolean-Phase Swapping Gate (p-SWAP)

• Boolean-Phase Swapping Gate is a circuit primitive that swaps two binary inputs while applying a controllable phase shift, thereby generalizing the classic swap operation.
• The p-SWAP gate uses a geometric Bloch sphere method to achieve a cost-effective design, reducing quantum cost by using only two CNOT gates and minimizing circuit depth.
• Its versatility enables applications in error-resilient classical logic and resource-efficient quantum algorithms, bridging conventional and quantum computing paradigms.

A Boolean-Phase Swapping Gate (BPSG) is a logic or quantum circuit primitive that exchanges two binary variables while also manipulating their phase information—realizing a transformation that generalizes conventional swap gates by introducing conditional or programmable phase shifts between the swapped states. These gates play a foundational role in both classical and quantum computational architectures, with applications ranging from error-resilient digital logic to resource-efficient quantum circuit compilation. BPSGs are essential in systems where phase encoding carries information or is exploited for algorithmic cost minimization, as in quantum or unconventional logic substrates.

1. Principles of Boolean-Phase Swapping and Gate Generality

A Boolean-Phase Swapping Gate, denoted as a “p-SWAP” gate, operates by swapping two input variables (classical or quantum) and, critically, introducing a controllable phase factor $p$ (typically in radians, $-\pi \leq p \leq +\pi$) to the swapped outputs. In classical systems leveraging phase encoding, like self-sustaining nonlinear oscillators or spin-wave interferometers, the “phase” refers to signal phase relative to a reference, switching between $0$ and $\pi$ to represent Boolean values. In quantum systems, phase is interpreted as the relative phase of computational basis states or as a factor in the off-diagonal elements of the unitary transformation.

This class of gates is defined such that for inputs $|a, b\rangle$, the output is

$$|a, b\rangle \longrightarrow e^{i p \, f(a,b)} |b, a\rangle$$

where the function $f(a,b)$ is determined by the physical implementation, but often yields the conditional or global phase factor on the swapped result. Setting $p=0$ recovers the standard SWAP operation; $p=\pi/2$ yields the iSWAP transformation, central in quantum information.

2. Circuit Constructions and the Bloch Sphere Approach

The p-SWAP gate introduced in recent literature (Al-Bayaty et al., 22 Oct 2024, Al-Bayaty et al., 23 Jul 2025) is notable for its explicit cost-effectiveness and configurable phase, employing only two CNOT gates, compared to three for a standard SWAP. The construction, validated on IBM Quantum’s “ibm_brisbane” native gate set, is achieved via a geometric approach grounded in the Bloch sphere formalism rather than explicit matrix algebra.

The circuit design is structured as follows:

Stage	Operation	Gate Usage
Entanglement	Prepare one target in superposition (VX gate), entangle via CNOT	VX, CNOT
Differentiation	Apply RZ rotations to both qubits, tuning phase by $+\pi/2$	RZ
Unentanglement	Second CNOT reverses entanglement, final VX brings back to Z-axis	CNOT, VX
Phase Selection	Apply two RZ gates to select desired phase $p$	RZ

The phase selection is achieved by assigning the rotation angles in the RZ gates according to

$v = RZ(\pm A \cdot \pi/B), \quad @ = RZ(D)$

with $A, B, D$ being integer cofactors related to quadrant or octant segmentation on the Bloch sphere’s XY-plane (Al-Bayaty et al., 22 Oct 2024, Al-Bayaty et al., 23 Jul 2025). For $p = \pi/2$, the p-SWAP behaves as an iSWAP gate.

The geometric approach enables intuitive visualization and modular expansion, where each differentiation and selection step corresponds to a specific rotation (quadrant shift) in Bloch coordinates. The design forgoes explicit matrix decomposition, relying instead on path planning for phase manipulation.

3. Quantum Cost, Depth Minimization and Experimental Advantages

A central metric for practical adoption is the transpilation quantum cost (TQC), which sums the number of 1-bit gates, 2-bit gates (notably ECR or CNOT), and overall circuit depth $D$. For the p-SWAP gate, post-transpilation evaluations on a 127-qubit system yield:

• Quantum cost reduction: $\sim$23% lower than the standard SWAP (3 CNOTs), due to using only 2 CNOTs.
• Circuit depth reduction: $\sim$26% lower, resulting in decreased exposure to decoherence and operational errors.

Key cost relationships:

$\text{TQC (p-SWAP)} = N_1 + N_2 + D$

with $N_1$ the number of single-qubit operations, $N_2$ the number of two-qubit (CNOT/ECR) gates.

This reduction is especially significant for large-scale circuits where high-depth, high-cost swapping operations introduce randomization, hardware error, and increased latency.

4. Applications in Boolean, Quantum, and Phase Oracle Computing

The BPSG concept’s generality is reflected in its utility for both Boolean (classical truth-preserving) and phase-sensitive (quantum oracle or phase estimation) contexts (Al-Bayaty et al., 22 Oct 2024, Al-Bayaty et al., 23 Jul 2025, Amy et al., 2021):

• Phase oracles: The configurable phase parameter $p$ enables the encoding of algorithmically relevant global or conditional phase shifts in state transfer and query oracles for quantum algorithms.
• Boolean logic and state transfer: When $p$ is ignored, the gate reduces to an efficient SWAP suitable for Boolean circuits and reversible classical computation.
• Resource optimization: Integration of the p-SWAP in transpiler libraries (e.g., GALA-n, CALA-n) and quantum compilers leads to resource-optimal implementations for quantum algorithms on hardware platforms with constrained gate sets and error budgets.

5. Comparative Analysis: Physical Implementations and Theoretical Constructs

Boolean-Phase Swapping concepts generalize across hardware domains:

• Statistical mechanics of noisy Boolean gates: Phase swapping arises as phase transitions in order parameters ($m(\ell)$, $C(\ell)$) in noisy formulas under specific noise thresholds (0908.3981).
• NMR and spin-wave systems: Swapping is realized via rotation operator manipulation (e.g., pulse phase and flip angle), mapping physical phase changes directly onto logical swaps (1109.0918, Mahmoud et al., 2021).
• Quantum optical and atomic ensemble devices: Swapping protocols exploit spatial motion of excitations (e.g., swapping collective Rydberg excitations) and precise phase shifts via stored-light or single-photon dressed states (1102.3266, Khazali et al., 2014, Hizhnyakov, 9 Jan 2025).
• Superconducting and trapped ion circuits: Hardware-efficient controlled-swap operations introduce phase as a natural consequence of eigenenergy resonance, conditional excitation, or shuttling-induced phase accumulation (Kaufmann et al., 2016, Rasmussen et al., 2018, Rasmussen et al., 2020, Heya et al., 2021).
• Algebraic designs via Boolean polynomials: Swap operations are mapped to multivariate quadratic polynomials mod 8, with phase encoded in the polynomial spectrum—bridging combinatorial and quantum computing (Gangopadhyay et al., 2019).

6. Implications, Challenges, and Future Research

The p-SWAP gate’s approach to Boolean-phase manipulation sets a precedent for hardware-agnostic gate design incorporating:

• Phase/customization flexibility for algorithmic or error-minimization applications.
• Quantitative resource savings relevant in noisy intermediate-scale quantum (NISQ) devices or dense classical logic fabrics.
• Geometric and visual design tools, such as the Bloch Sphere Approach, for scalable, intuitive circuit engineering.

Challenges for widespread adoption include generalizing phase selection procedures over arbitrary ranges, extending the framework to multi-valued logic or higher-dimensional swaps, and integrating with scalable transpiler and error correction infrastructures. Continued work may optimize phase assignment in cofactor selection, adapt BPSG to hybrid classical-quantum systems, and test performance on emergent architectures. A plausible implication is broad deployment of p-SWAP-type gates as standard primitives in future compilation stacks, especially where circuit cost and phase precision are critical.

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Gate type	#CNOTs (or equiv.)	Phase control	Application domain(s)
Standard SWAP	3	None	Boolean, quantum
iSWAP	2	$p = \pi/2$	Quantum, XY models
p-SWAP	2	$p$ selectable	Quantum, phase, Boolean

In sum, the Boolean-Phase Swapping Gate—exemplified by the p-SWAP—provides a generic, cost-effective, and phase-configurable mechanism at the intersection of classical and quantum logic, optimizing essential circuit primitives for both resource efficiency and algorithmic versatility (Al-Bayaty et al., 22 Oct 2024, Al-Bayaty et al., 23 Jul 2025).