Microwave-Activated Two-Qutrit SWAP Gate

• The paper introduces a microwave-activated two-qutrit SWAP gate, detailing synthesis routes via conditional swaps, coupler-assisted exchange, and indirect compilation using phase gates.
• It clarifies the operator structure by distinguishing the full SWAP from a conditional swap that acts only on the |1,2⟩ and |2,1⟩ subspace for selective state exchange.
• The work evaluates implementations in cavity-QED and superconducting circuits, addressing error mechanisms and calibration strategies to optimize gate fidelity.

Searching arXiv for the cited and closely related papers on microwave-activated qutrit/qutrit-swap mechanisms. A microwave-activated two-qutrit SWAP gate is a two-body unitary on the qutrit basis $\{|0\rangle,|1\rangle,|2\rangle\}$ that exchanges the local states of two three-level systems according to

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$

In the cited constructions, this object appears in three closely related forms: a conditional two-qutrit swap $U_{\mathrm{CSWAP}}$ acting only on the $\{|1,2\rangle,|2,1\rangle\}$ subspace; a full SWAP synthesized from microwave-activated exchange interactions across fixed excitation manifolds; and an indirect compilation from microwave-activated qutrit controlled-phase gates plus local Fourier transforms (Zheng, 2012, Shirai et al., 2023, Goss et al., 2022).

1. Operator structure and gate taxonomy

The full two-qutrit SWAP is the permutation operator

$\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$

which leaves $|00\rangle$, $|11\rangle$, and $|22\rangle$ invariant while exchanging the off-diagonal basis states. In the ordered basis $$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$, the data give the corresponding $9\times 9$ permutation matrix explicitly (Goss et al., 2022).

By contrast, the two-qutrit conditional swap introduced as an elementary primitive in the auxiliary-level construction is not the full SWAP. It acts nontrivially only on the two-dimensional subspace spanned by $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$0:

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$1

and

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$2

for all other basis states. Equivalently,

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$3

Its matrix is the identity except for the $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$4 block that swaps $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$5 and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$6 (Zheng, 2012).

These distinctions matter because the literature uses microwave activation in two different ways. One route engineers exchange-like dynamics directly in selected excitation manifolds. Another route keeps the interaction diagonal and reconstructs SWAP indirectly from entangling phase gates and local basis changes. A concise comparison is given below.

Route	Activated interaction	SWAP synthesis
Auxiliary-level conditional swap	$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$7 on $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$8	Full level-selected swaps by conjugation with single-qutrit permutations
Coupler-assisted exchange	Effective $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$9	Sequential iSWAPs in the $U_{\mathrm{CSWAP}}$0 manifolds
Dynamic cross-Kerr	Diagonal $U_{\mathrm{CSWAP}}$1 in the $U_{\mathrm{CSWAP}}$2 sector	Three SUM gates built from $U_{\mathrm{CSWAP}}$3 or $U_{\mathrm{CSWAP}}$4 and $U_{\mathrm{CSWAP}}$5

2. Auxiliary-level conditional swap and the expose-versus-hide method

The auxiliary-level construction identifies the qutrit state $U_{\mathrm{CSWAP}}$6 with the paper’s auxiliary level $U_{\mathrm{CSWAP}}$7 and uses it to “expose” a single amplitude to a controlled phase, rather than to “hide” amplitudes from conditional dynamics (Zheng, 2012). The associated qutrit–qubit controlled-phase gate is

$U_{\mathrm{CSWAP}}$8

so only the state $U_{\mathrm{CSWAP}}$9 acquires the phase.

Within the $\{|1,2\rangle,|2,1\rangle\}$0-qubit controlled-phase construction, the sequence is

$\{|1,2\rangle,|2,1\rangle\}$1

which in forward order applies $\{|1,2\rangle,|2,1\rangle\}$2, then $\{|1,2\rangle,|2,1\rangle\}$3, then $\{|1,2\rangle,|2,1\rangle\}$4, then the inverse swap chain, and finally $\{|1,2\rangle,|2,1\rangle\}$5. Here $\{|1,2\rangle,|2,1\rangle\}$6 maps $\{|1,2\rangle,|2,1\rangle\}$7 on qubit $\{|1,2\rangle,|2,1\rangle\}$8, and $\{|1,2\rangle,|2,1\rangle\}$9 maps $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$0. This realizes

$\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$1

using exactly $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$2 two-qutrit conditional swaps, one qutrit–qubit controlled phase, and two single-qutrit operations (Zheng, 2012).

For the present topic, the crucial point is that the conditional swap is a selective transport mechanism in Hilbert space. It propagates the auxiliary label through the register without disturbing other computational amplitudes. The paper also states that a full two-qutrit swap on selected levels $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$3 of qutrit $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$4 and $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$5 of qutrit $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$6 can be obtained by conjugating $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$7 with single-qutrit permutations $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$8 and $\mathrm{SWAP}=\sum_{i,j}|ji\rangle\langle ij|,$9:

$|00\rangle$0

This makes the conditional primitive a building block for richer two-qutrit SWAP operations (Zheng, 2012).

3. Microwave activation in dispersive cavity and circuit QED

In the cavity-QED realization, the system consists of $|00\rangle$1 identical atoms in a cavity, each with one excited state $|00\rangle$2 and three long-lived ground states $|00\rangle$3, $|00\rangle$4, and $|00\rangle$5. The cavity mode couples $|00\rangle$6 with coupling $|00\rangle$7, and a classical field drives $|00\rangle$8 with Rabi frequency $|00\rangle$9 and phase $|11\rangle$0. The cavity and classical field are detuned by $|11\rangle$1 and $|11\rangle$2, respectively, giving the interaction Hamiltonian

$|11\rangle$3

Under the dispersive conditions $|11\rangle$4, adiabatic elimination of $|11\rangle$5 yields the effective Hamiltonian

$|11\rangle$6

with $|11\rangle$7 and $|11\rangle$8, $|11\rangle$9 (Zheng, 2012).

Restricting to the subspace $|22\rangle$0 produces conditional-swap dynamics. For symmetric Stark shifts $|22\rangle$1, one has $|22\rangle$2 and $|22\rangle$3. Choosing

$|22\rangle$4

and then applying single-qutrit $|22\rangle$5 phases to remove the residual phase implements the conditional swap $|22\rangle$6. The paper gives the gate time as

$|22\rangle$7

while the controlled phase uses $|22\rangle$8 (Zheng, 2012).

The same data explicitly translate this mechanism to superconducting circuits by replacing the atoms with weakly anharmonic superconducting qutrits, such as transmons or fluxonium, replacing the optical cavity with a microwave resonator, and using classical microwave drives $|22\rangle$9. In that translation, the effective Hamiltonian retains the same form, the dispersive conditions remain $$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$0, and the phases $$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$1 control the sign of the exchange term through $$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$2. This suggests a direct microwave-activated route to conditional two-qutrit exchange in circuit QED (Zheng, 2012).

4. Coupler-assisted microwave exchange and direct full-SWAP synthesis

The all-microwave fixed-frequency transmon-coupler architecture consists of two data transmons and one fixed-frequency transmon coupler driven by a microwave tone (Shirai et al., 2023). The bare Hamiltonian is modeled with weakly anharmonic Duffing oscillators,

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$3

with capacitive exchange

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$4

and drive

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$5

Under a Schrieffer–Wolff expansion, the coupler’s third-order nonlinearity generates a drive-enabled effective exchange between the data transmons. In the rotating frame and under RWA, the effective interaction is

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$6

(Shirai et al., 2023).

For the demonstrated qubit manifolds, the data give blue and red coupler-assisted swap rates

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$7

and

$$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$8

The paper states that the negative anharmonicity $$\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\}$$9 pushes disturbing transitions away for the blue CAS, making it cleaner than red CAS, and that high drive efficiency with small residual interaction persists over a wide range of detuning between the data transmons (Shirai et al., 2023).

The qutrit truncation of the same exchange Hamiltonian suggests a direct two-qutrit SWAP synthesis. In the qutrit basis,

$9\times 9$0

so a resonantly activated exchange can address multiple excitation manifolds. The data specify the resonances

$9\times 9$1

and the additional $9\times 9$2 links

$9\times 9$3

This yields a sequential construction: an iSWAP in the $9\times 9$4 manifold $9\times 9$5, two linked iSWAPs in the $9\times 9$6 manifold $9\times 9$7, and an iSWAP in the $9\times 9$8 manifold $9\times 9$9, together with virtual-$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$00 corrections. The corresponding segment times are

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$01

and, because the relevant matrix elements carry $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$02 factors,

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$03

The data state that sequential drives are generally preferable to simultaneous multi-tone driving because the latter risks undesired interference and off-resonant couplings (Shirai et al., 2023).

5. Indirect microwave SWAP from dynamic cross-Kerr qutrit entanglers

A distinct microwave-activated route uses the differential AC Stark shift to generate a diagonal two-qutrit interaction between two fixed-frequency transmon qutrits statically coupled with exchange $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$04 (Goss et al., 2022). In this method the entangling Hamiltonian is

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$05

with

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$06

Including single-qutrit Stark shifts,

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$07

The paper uses this mechanism to realize qutrit $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$08 and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$09 gates with estimated process fidelities of $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$10 and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$11, and total gate times of $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$12 ns and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$13 ns, respectively (Goss et al., 2022).

The interaction is diagonal rather than exchange-like, so the SWAP is obtained indirectly. The qutrit Fourier transform is

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$14

with matrix

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$15

Using $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$16 or $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$17, one constructs the SUM gate,

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$18

and the SWAP identity is

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$19

The data explicitly recommend the indirect route on this platform, because exchange-like $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$20 interactions are not the operating point of the dynamic cross-Kerr method (Goss et al., 2022).

For timing, each SUM uses one controlled-phase gate plus two local Fourier transforms. Hence three SUMs require three $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$21 or three $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$22 gates. The data give the corresponding Stark-driven times as approximately $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$23 using $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$24 and approximately $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$25 using $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$26, excluding the local transforms. Under the approximate multiplicative model

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$27

and assuming $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$28, the projected SWAP range is $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$29–$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$30 using $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$31 and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$32–$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$33 using $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$34 (Goss et al., 2022).

6. Error mechanisms, calibration practice, and conceptual clarifications

The dominant error channels depend on the microwave-activation mechanism. In the dispersive conditional-swap construction, detuning mismatch $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$35 gives $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$36, and the data state that the swap infidelity scales as approximately $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$37 to leading order. Residual cavity excitation and decay contribute dephasing of order $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$38. Qutrit decoherence in the ground manifold contributes through transient occupation of $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$39, with infidelity terms of the form

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$40

while inhomogeneous couplings can be compensated by calibrating $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$41 to equalize effective Stark shifts and exchange strengths (Zheng, 2012).

In the coupler-assisted exchange approach, the main issues are leakage, residual $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$42, and frequency crowding. For an unwanted transition detuned by $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$43, the instantaneous leakage probability is stated to scale approximately as

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$44

The residual static and ac-tunable $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$45 components are

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$46

and under an off-resonant blue CAS tone,

$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$47

The data recommend Gaussian edges, DRAG-like shaping, and virtual-$$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$48 compensation, and note that blue CAS relaxes frequency crowding relative to red CAS (Shirai et al., 2023).

In the dynamic cross-Kerr method, the dominant errors are single-qutrit phase errors, low-frequency drift, dephasing during relatively long Stark pulses, amplitude and phase miscalibration of the Stark drive, small leakage to $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$49 and above at higher drive amplitudes, classical cross-talk, and residual always-on $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$50. The paper’s experimental protocol calibrates the conditional phases by full two-qutrit state tomography after simultaneous ternary Hadamards and removes residual local phases with virtual $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$51 gates in both the $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$52 and $S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$53 subspaces (Goss et al., 2022).

Two recurring points of confusion are explicitly resolved by the data. First, the conditional swap of (Zheng, 2012) is not the full two-qutrit SWAP; it swaps only $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$54 and $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$55 and acts as identity elsewhere. Second, a microwave-activated SWAP need not be a direct exchange gate: in the dynamic cross-Kerr setting the recommended synthesis is indirect, through three SUM gates, precisely because the operating interaction is diagonal rather than $$S_3\,|i\rangle_1|j\rangle_2 = |j\rangle_1|i\rangle_2,\qquad i,j\in\{0,1,2\}.$$56-like (Zheng, 2012, Goss et al., 2022).

The broader significance is architectural rather than terminological. The auxiliary-level method minimizes single-qutrit overhead and exposes a single amplitude to phase accumulation; the coupler-assisted exchange method furnishes a tunable, efficient exchange with small residual interaction over a wide detuning window; and the dynamic cross-Kerr method provides a flexible microwave-only entangling primitive from which SWAP can be compiled. Taken together, these constructions define the main research-level meanings of a microwave-activated two-qutrit SWAP gate in cavity-QED and superconducting-circuit settings (Zheng, 2012, Shirai et al., 2023, Goss et al., 2022).