Tunable Partial-SWAP in Quantum Systems

• Tunable partial-SWAP is a one-parameter gate that interpolates between the identity and full SWAP, enabling controlled exchange and entangling operations.
• It leverages parametrically activated interactions in superconducting circuits, fluxonium devices, and other qudit architectures to set the exchange angle.
• Its tunability allows for controlled amplitude damping and memory effects, impacting quantum computing, thermodynamics, and reservoir network applications.

Searching arXiv for recent and foundational papers on tunable partial-SWAP. arXiv search: tunable partial-SWAP, iSWAP, bSWAP, tunable coupler, quantum reservoir partial-SWAP. A tunable partial-SWAP is a one-parameter exchange operation that interpolates between no exchange and a full SWAP, while preserving a controlled structure on the relevant two-body subspace. In the most standard formulation, the unitary is written as $$U(\theta)=\exp(-i\theta E)=\cos\theta\,I-i\sin\theta\,E$$, where $E$ is the SWAP operator and $\theta$ is set by the interaction strength and duration (Sacchi, 2021). In contemporary hardware literature, the same underlying idea also appears as a parametrically activated exchange gate of arbitrary angle, such as $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$ in superconducting circuits (Roth et al., 2017), as a partial-SWAP generated by a tunable coupler in fluxonium devices (Zhang et al., 2023), and as a gate-level decomposition $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$ in quantum reservoir networks (Connerty et al., 12 May 2026). Across these settings, the common feature is continuous tunability of exchange angle, but the physical meaning of the control parameter depends on the platform and Hamiltonian.

1. Formal definitions and parameterizations

In the interaction-picture qudit construction, the two subsystems are coupled by the constant Hamiltonian $H_I=\kappa E$, where $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$ is the SWAP operator and $\kappa$ is a real coupling constant (Sacchi, 2021). Because $E^2=I$, one immediately has

$$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$

with the closed form

$E$0

The special cases are explicit: $E$1, $E$2, and intermediate $E$3 smoothly interpolate between identity and full swap (Sacchi, 2021).

In exchange-gate realizations, the same tunability is expressed through an $E$4 interaction. For two fixed-frequency transmons coupled via a parametrically driven tunable bus, the effective interaction is

$E$5

which generates

$E$6

Thus the swap-angle is tuned by the drive amplitude $E$7 and pulse duration $E$8 (Roth et al., 2017).

A gate-based qubit version uses the notation $E$9 and defines a partial-SWAP as a one-parameter interpolation between the identity $\theta$0 and SWAP $\theta$1:

$\theta$2

In that formulation,

$\theta$3

so the tuning parameter is directly mapped to an amplitude-damping strength under readout reset (Connerty et al., 12 May 2026).

A related but distinct construction is the Boolean-phase “p-SWAP gate,” whose unitary acts exactly like a SWAP on the amplitudes but attaches a tunable phase $\theta$4 on the $\theta$5 sector:

$\theta$6

This is tunable, but its defining degree of freedom is a phase label rather than the continuous exchange angle of the standard partial-SWAP family (Al-Bayaty et al., 2024).

2. Operator structure and subspace action

The SWAP operator $\theta$7 is both Hermitian and unitary, with $\theta$8 and $\theta$9, hence its eigenvalues are $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$0 (Sacchi, 2021). More precisely, the two-qudit Hilbert space decomposes into a symmetric subspace $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$1 of dimension $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$2 and an antisymmetric subspace $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$3 of dimension $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$4 (Sacchi, 2021). Functional calculus then gives

$$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$5

where $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$6 are the projectors onto the $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$7 eigenspaces.

On the computational basis $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$8, one has $$U(\phi)=\exp[-i(\phi/2)(\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y)]$$9, and therefore

$$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$0

For $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$1, the relevant two-dimensional block in the ordered basis $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$2 is

$$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$3

whereas for $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$4 the state is multiplied by $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$5 (Sacchi, 2021). This block structure is the algebraic reason why partial-SWAP naturally appears as a controllable rotation in an exchange subspace.

In superconducting-qubit language, the same subspace rotation is commonly written on $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$6 as

$$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$7

for an exchange Hamiltonian $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$8 (Sete et al., 2021). In the fluxonium realization, driving at the difference frequency yields resonant exchange in the $$U_{\mathrm{swap}}(\gamma)\equiv \mathrm{SWAP}^\gamma$$9 subspace,

$H_I=\kappa E$0

and $H_I=\kappa E$1 is identified with $H_I=\kappa E$2 when $H_I=\kappa E$3 (Zhang et al., 2023).

A common misconception is that all tunable partial-SWAPs are merely truncated versions of a full SWAP. The operator forms above indicate that the unitary can equally be understood as a controlled exchange rotation, an $H_I=\kappa E$4 entangler, or a gate that induces a specific effective channel after subsystem reset. This suggests that “partial-SWAP” is best treated as an operator class rather than a single circuit identity.

3. Parametric activation in superconducting-qubit architectures

A major hardware route to tunable partial-SWAP gates is parametric activation through a tunable bus or tunable coupler. In the transmon-bus analysis, the lab-frame Hamiltonian is

$H_I=\kappa E$5

with

$H_I=\kappa E$6

and the external flux modulation

$H_I=\kappa E$7

A time-dependent Schrieffer–Wolff transformation yields an effective two-qubit Hamiltonian with exchange and two-photon sectors (Roth et al., 2017).

Specializing to harmonic drive at $H_I=\kappa E$8 gives the iSWAP band, with effective coupling

$H_I=\kappa E$9

To leading order in $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$0,

$$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$1

so the desired swap angle is set by $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$2 (Roth et al., 2017). The same framework analyzes a two-photon interaction that produces a bSWAP process, and the study notes that the bSWAP gate is generally slower than the more commonly used iSWAP gate, but features favorable scalability properties with less severe frequency crowding effects (Roth et al., 2017).

A different activation mechanism is parametric resonance with a tunable coupler. There, the exchange Hamiltonian

$$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$3

is obtained in the regime where $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$4, with

$$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$5

An arbitrary partial-SWAP of angle $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$6 is then implemented by

$$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$7

and the study reports iSWAP and CZ gates between two qubits coupled via a tunable coupler with average process fidelities as high as $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$8 and $$E(|\psi\rangle_A\otimes|\phi\rangle_B)=|\phi\rangle_A\otimes|\psi\rangle_B$$9, respectively (Sete et al., 2021).

The fluxonium realization provides a particularly explicit partial-SWAP instantiation. Its effective two-qubit Hamiltonian is

$\kappa$0

where $\kappa$1 and $\kappa$2 (Zhang et al., 2023). The coupler enables the qubits to have a large tuning range of $\kappa$3 coupling strengths ($\kappa$4 to $\kappa$5 MHz), while the $\kappa$6 coupling strength is $\kappa$7kHz across the entire coupler bias range, and $\kappa$8Hz at the coupler off-position (Zhang et al., 2023). By driving at the difference frequency of the two qubits, the device realizes a $\kappa$9 gate in $E^2=I$0ns with fidelity $E^2=I$1, and by driving at the sum frequency of the two qubits, it achieves a $E^2=I$2 gate in $E^2=I$3ns with fidelity $E^2=I$4; the latter gate is only 5 qubit Larmor periods in length (Zhang et al., 2023).

These results establish that tunable partial-SWAPs in superconducting hardware are not limited to a single exchange primitive. They include ordinary exchange in $E^2=I$5, two-photon exchange in $E^2=I$6, and parameterized fSim-like operations derived from tunable-coupler biasing and modulation. A plausible implication is that the notion of tunability is as much about spectral selectivity and crosstalk suppression as about dialling a target angle.

4. Fluxonium partial-SWAP protocols and calibration

In the fluxonium device, parametric activation is implemented through a time-dependent coupler flux bias

$E^2=I$7

Choosing $E^2=I$8 yields resonant exchange in $E^2=I$9 and produces

$$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$0

whereas choosing $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$1 yields resonant exchange in $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$2 and produces

$$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$3

The identification with $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$4 or $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$5 is made at $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$6 (Zhang et al., 2023).

The exchange angle is controlled by the drive parameters according to

$$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$7

For a Gaussian envelope $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$8, $$U(\tau_w)\equiv e^{-iH_I\tau_w}=e^{-i\kappa\tau_w E}\equiv U(\theta), \qquad \theta\equiv \kappa\tau_w,$$9, and one sets $E$00 so that $E$01 (Zhang et al., 2023). The reported pulse shapes are Gaussian envelope $E$02 on $E$03, with no occupation of higher levels. The $E$04 gate has $E$05 ns, and the $E$06 gate has $E$07 ns (Zhang et al., 2023).

The calibration procedure is stated explicitly:

1. Calibrate $E$08 such that repeated back-to-back applications of the gate drive population to the target state $E$09 or $E$10 with maximal contrast.
2. Calibrate pulse phase $E$11 and virtual Z-corrections on each qubit to cancel accumulated single-qubit and two-qubit phases.
3. Interleave small Z rotations on $E$12 and adjust drive-phase so that $E$13 or $E$14 matches the ideal unitary up to $E$15 phase error (Zhang et al., 2023).

The same device reports static ZZ crosstalk suppression at the dedicated off-point, where Ramsey-based measurement yields $E$16 Hz, and cross-entropy benchmarking over extended runs with $E$17 drift $E$18 and $E$19 drift $E$20 (Zhang et al., 2023). These details matter because tunable partial-SWAPs are often limited less by the existence of an exchange mechanism than by the ability to turn it off cleanly and maintain phase stability over long calibration windows.

5. Channel interpretation, memory control, and repeated application

A distinctive perspective emerges when the tunable partial-SWAP is applied to a memory qubit $E$21 and a readout qubit $E$22 initialized in $E$23, followed by tracing out and resetting $E$24. In that setting, the memory qubit undergoes exactly an amplitude-damping channel with damping probability

$E$25

Starting from $E$26 and $E$27, one finds after $E$28 and $E$29:

$E$30

with Kraus operators

$E$31

Under repeated application for $E$32 time-steps, the excited-state population decays as $E$33 and off-diagonals as $E$34, driving $E$35 (Connerty et al., 12 May 2026).

This construction is used to control fading memory in quantum reservoir networks. Because the partial-SWAP acts as amplitude damping with $E$36, the reservoir’s fading memory is directly tunable by $E$37 (Connerty et al., 12 May 2026). If $E$38 is small, $E$39 is small and the memory qubit retains nearly all previous amplitude, producing very slow leak; if $E$40, $E$41 and the memory is completely swapped or reset each step, producing no memory beyond one step (Connerty et al., 12 May 2026). The paper states that there is an intermediate “sweet-spot” $E$42 where past inputs fade at just the right rate to maximize memory capacity (Connerty et al., 12 May 2026).

Representative results are reported for a randomized short-term memory capacity recall benchmark and NARMA-5. For $E$43, $E$44, the minimum RMSE occurs at $E$45, while a hardware demonstration on an IBM QPU used $E$46, $E$47, $E$48, circuit depth $E$49, and $E$50 shots on ibm_boston, with Aer simulator RMSE $E$51 and IBM QPU RMSE $E$52 (Connerty et al., 12 May 2026).

The channel picture clarifies an often-overlooked point: a tunable partial-SWAP can be studied either as a unitary exchange gate or as an induced non-unitary memory-leak mechanism after ancillary reset. This suggests that the same operator family naturally links coherent gate synthesis and controlled dissipation.

6. Thermodynamic, bosonic, and related variants

In multilevel quantum thermodynamic swap engines, the work stroke is operated by a partial-swap unitary interaction between two qudits (Sacchi, 2021). For local Gibbs initial states with mean occupations $E$53, the average work per cycle is

$E$54

The three regimes of operation—heat engine, refrigerator, and thermal accelerator—are unchanged for partial swaps, while the factor $E$55 scales the magnitude of energy exchange (Sacchi, 2021). The Otto-efficiency $E$56 is independent of $E$57 and $E$58, so partial swaps only scale the amount of work but do not alter the efficiency (Sacchi, 2021). Here, tunability controls thermodynamic throughput rather than computational gate time.

A bosonic variant appears in a Rabi-driven qubit–cavity protocol. In a dressed frame, the effective Hamiltonian becomes

$E$59

which produces

$E$60

in the single-excitation manifold $E$61 (Karaev et al., 8 Apr 2026). The study reports single-photon SWAP in approximately $E$62 microseconds and identifies $E$63 with a $E$64-type operation (Karaev et al., 8 Apr 2026). This is again a tunable partial-SWAP, but now between a transmon and a high-Q cavity mode.

Not every “swap with a parameter” is an exchange-angle partial-SWAP. The p-SWAP construction uses two CNOTs plus two single-qubit $E$65 gates and attaches a tunable phase $E$66 to the $E$67 subspace (Al-Bayaty et al., 2024). For Boolean applications with $E$68, it becomes exactly a SWAP up to a global phase, and the reported transpiled circuit on ibm_brisbane uses $E$69 one-qubit gates, $E$70 ECR gates, depth $E$71, and TQC $E$72, compared with $E$73 one-qubit gates, $E$74 ECR gates, depth $E$75, and TQC $E$76 for a standard SWAP (Al-Bayaty et al., 2024). The relation to partial-SWAP is therefore taxonomic rather than identical: both are tunable swap-family gates, but they tune different invariants.

A further extension outside gate synthesis is “partial swappability” in swap-acceleration studies of glassy dynamics, where only a fraction $E$77 of particles are ever allowed to initiate swaps (Gopinath et al., 2021). In that setting, varying $E$78 moves continuously from the familiar full-swap limit to an effectively unswappable glass, and the diffusion coefficients obey $E$79 and $E$80 for small $E$81 (Gopinath et al., 2021). Although this usage is outside quantum information, it preserves the central idea that swap-like dynamics can be made continuously tunable through a single control parameter.

Taken together, these formulations show that tunable partial-SWAP is a cross-domain concept: an exchange-angle unitary on qudits, a parametrically activated entangling primitive in superconducting hardware, a controlled amplitude-damping mechanism in quantum reservoir networks, a work-stroke control in quantum thermodynamics, and, in related but nonidentical forms, a broader family of tunable swap operations. The recurring technical theme is that one continuously controls the degree of exchange while preserving a mathematically simple action on a low-dimensional invariant subspace.