---
title: Microwave-Activated Two-Qutrit SWAP Gate
url: https://www.emergentmind.com/topics/microwave-activated-two-qutrit-swap-gate
type: topic
---

# Microwave-Activated Two-Qutrit SWAP Gate

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 [1207.2882] [2302.06930] [2206.07216].

## 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 [2206.07216].

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 $\{|1,2\rangle,|2,1\rangle\}$:
$$
U_{\mathrm{CSWAP}}|1,2\rangle=|2,1\rangle,\qquad
U_{\mathrm{CSWAP}}|2,1\rangle=|1,2\rangle,
$$
and
$$
U_{\mathrm{CSWAP}}|i,j\rangle=|i,j\rangle
$$
for all other basis states. Equivalently,
$$
U_{\mathrm{CSWAP}}
= I-|1,2\rangle\langle1,2|-|2,1\rangle\langle2,1|
+|1,2\rangle\langle2,1|+|2,1\rangle\langle1,2|.
$$
Its matrix is the identity except for the $2\times2$ block that swaps $|12\rangle$ and $|21\rangle$ [1207.2882].

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 | $U_{\mathrm{CSWAP}}$ on $\{|12\rangle,|21\rangle\}$ | Full level-selected swaps by conjugation with single-qutrit permutations |
| Coupler-assisted exchange | Effective $a_1^\dagger a_2 + a_1 a_2^\dagger$ | Sequential iSWAPs in the $N=1,2,3$ manifolds |
| Dynamic cross-Kerr | Diagonal $H_{\mathrm{CK}}$ in the $\{|1\rangle,|2\rangle\}^{\otimes 2}$ sector | Three SUM gates built from $U_{\mathrm{CZ}}$ or $U_{\mathrm{CZ}}^\dagger$ and $F,F^{-1}$ |

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

The auxiliary-level construction identifies the qutrit state $|2\rangle$ with the paper’s auxiliary level $|a\rangle$ and uses it to “expose” a single amplitude to a controlled phase, rather than to “hide” amplitudes from conditional dynamics [1207.2882]. The associated qutrit–qubit controlled-phase gate is
$$
U_{\mathrm{CP}}^{(i,k)}(\phi)=I+\left(e^{i\phi}-1\right)|2_i,1_k\rangle\langle 2_i,1_k|,
$$
so only the state $|2_i,1_k\rangle$ acquires the phase.

Within the $n$-qubit controlled-phase construction, the sequence is
$$
U_{\mathrm{CP}}^{(n)}(\phi)
= M_1 \cdot \Big[\prod_{k=2}^{1} U_{k,k-1}\Big]\cdot V_{n-1,n}(\phi)\cdot
\Big[\prod_{k=1}^{n-2} U_{k,k+1}\Big]\cdot L_1,
$$
which in forward order applies $L_1$, then $U_{1,2},U_{2,3},\ldots,U_{n-2,n-1}$, then $V_{n-1,n}(\phi)$, then the inverse swap chain, and finally $M_1$. Here $L_1$ maps $|1\rangle\to|2\rangle$ on qubit $1$, and $M_1$ maps $|2\rangle\to|1\rangle$. This realizes
$$
U_{\mathrm{CP}}^{(n)}(\phi)=I+\left(e^{i\phi}-1\right)|11\ldots1\rangle\langle11\ldots1|
$$
using exactly $2n-4$ two-qutrit conditional swaps, one qutrit–qubit controlled phase, and two single-qutrit operations [1207.2882].

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 $\alpha$ of qutrit $j$ and $\beta$ of qutrit $k$ can be obtained by conjugating $U_{\mathrm{CSWAP}}$ with single-qutrit permutations $P_j$ and $Q_k$:
$$
U_{\mathrm{SWAP}}^{(\alpha\leftrightarrow\beta)}
=(P_j\otimes Q_k)\,U_{\mathrm{CSWAP}}\,(P_j\otimes Q_k)^\dagger.
$$
This makes the conditional primitive a building block for richer two-qutrit SWAP operations [1207.2882].

## 3. Microwave activation in dispersive cavity and circuit QED

In the cavity-QED realization, the system consists of $n$ identical atoms in a cavity, each with one excited state $|r\rangle$ and three long-lived ground states $|0\rangle$, $|1\rangle$, and $|2\rangle$. The cavity mode couples $|1\rangle\leftrightarrow|r\rangle$ with coupling $g$, and a classical field drives $|2\rangle\leftrightarrow|r\rangle$ with Rabi frequency $\Omega_j$ and phase $\phi_j$. The cavity and classical field are detuned by $\Delta_2$ and $\Delta_1$, respectively, giving the interaction Hamiltonian
$$
H = \sum_{m=1}^n g\left(a e^{i\Delta_2 t}|r_m\rangle\langle1_m| + a^\dagger e^{-i\Delta_2 t}|1_m\rangle\langle r_m|\right)
+\sum_j\left(\Omega_j e^{i(\Delta_1 t-\phi_j)}|r_j\rangle\langle2_j|
+\Omega_j e^{-i(\Delta_1 t-\phi_j)}|2_j\rangle\langle r_j|\right).
$$
Under the dispersive conditions $|\Delta_1|,|\Delta_2|\gg \Omega_j,g$, adiabatic elimination of $|r\rangle$ yields the effective Hamiltonian
$$
H_{\mathrm{eff}}=
- M_j |2_j\rangle\langle2_j|
- M_{j+1} |2_{j+1}\rangle\langle2_{j+1}|
+ S\left(e^{i\varphi} S_j^+ S_{j+1}^- + e^{-i\varphi} S_j^- S_{j+1}^+\right),
$$
with $\varphi=\phi_j-\phi_{j+1}$ and $S_j^+=|2_j\rangle\langle1_j|$, $S_j^- = |1_j\rangle\langle2_j|$ [1207.2882].

Restricting to the subspace $\{|2_j,1_{j+1}\rangle,|1_j,2_{j+1}\rangle\}$ produces conditional-swap dynamics. For symmetric Stark shifts $M_j=M_{j+1}$, one has $\Delta=0$ and $\eta=S$. Choosing
$$
S t_{\mathrm{swap}}=\frac{\pi}{2},\qquad \beta t_{\mathrm{swap}}=-\frac{\pi}{2}\ \text{(or }+\frac{\pi}{2}\text{)},
$$
and then applying single-qutrit $Z$ phases to remove the residual phase implements the conditional swap $U_{j,j+1}$. The paper gives the gate time as
$$
t_{\mathrm{swap}}=\frac{\pi}{2S},
$$
while the controlled phase uses $t_{\mathrm{CP}}=\pi/\eta$ [1207.2882].

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 $\Omega_j(t)$. In that translation, the effective Hamiltonian retains the same form, the dispersive conditions remain $|\Delta_1|,|\Delta_2|\gg \Omega_j,g$, and the phases $\phi_j$ control the sign of the exchange term through $\varphi=\phi_j-\phi_{j+1}$. This suggests a direct microwave-activated route to conditional two-qutrit exchange in circuit QED [1207.2882].

## 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 [2302.06930]. The bare Hamiltonian is modeled with weakly anharmonic Duffing oscillators,
$$
H_0=\sum_{k\in\{1,2,c\}}\left[\omega_k a_k^\dagger a_k + \frac{\alpha_k}{2} a_k^\dagger a_k^\dagger a_k a_k\right],
$$
with capacitive exchange
$$
H_{\mathrm{int}}=\sum_{k\in\{1,2\}} g_{kc}(a_k^\dagger a_c + a_k a_c^\dagger),
$$
and drive
$$
H_d(t)=\epsilon(t)(a_c+a_c^\dagger)\cos(\omega_d t).
$$
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
$$
H_{\mathrm{eff}}\approx J_{\mathrm{eff}}(\epsilon,\omega_d,\Delta,\alpha)(a_1^\dagger a_2 + a_1 a_2^\dagger) + \zeta ZZ + \ldots
$$
[2302.06930].

For the demonstrated qubit manifolds, the data give blue and red coupler-assisted swap rates
$$
\Omega_b = \frac{2 g_{1c} g_{2c} \alpha_c \Omega_d}
{\Delta_{12} (\omega_c-\omega_1+\alpha_c)(\omega_c-\omega_2)},
\qquad
J_{\mathrm{eff}}^{(b)}=\frac{\Omega_b}{2},
$$
and
$$
\Omega_r = \frac{-2 g_{1c} g_{2c} \alpha_c \Omega_d}
{\Delta_{12} (\omega_c-\omega_2+\alpha_c)(\omega_c-\omega_1)},
\qquad
J_{\mathrm{eff}}^{(r)}=\frac{\Omega_r}{2}.
$$
The paper states that the negative anharmonicity $\alpha_c$ 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 [2302.06930].

The qutrit truncation of the same exchange Hamiltonian suggests a direct two-qutrit SWAP synthesis. In the qutrit basis,
$$
a_k \approx |0\rangle\langle1| + \sqrt{2}\,|1\rangle\langle2|,
$$
so a resonantly activated exchange can address multiple excitation manifolds. The data specify the resonances
$$
\omega_d^{(01)} \approx \omega_c + (\omega_{1,01}-\omega_{2,01}),
\qquad
\omega_d^{(12)} \approx \omega_c + (\omega_{1,12}-\omega_{2,12}),
$$
and the additional $N=2$ links
$$
\omega_d^{(12\leftrightarrow01)} \approx \omega_c + (\omega_{1,12}-\omega_{2,01}),
\qquad
\omega_d^{(01\leftrightarrow12)} \approx \omega_c + (\omega_{1,01}-\omega_{2,12}).
$$
This yields a sequential construction: an iSWAP in the $N=1$ manifold $\{|10\rangle,|01\rangle\}$, two linked iSWAPs in the $N=2$ manifold $\{|20\rangle,|11\rangle,|02\rangle\}$, and an iSWAP in the $N=3$ manifold $\{|21\rangle,|12\rangle\}$, together with virtual-$Z$ corrections. The corresponding segment times are
$$
t_{01}=\frac{\pi}{2J_{01}},\qquad
t_{12}=\frac{\pi}{2J_{12}},
$$
and, because the relevant matrix elements carry $\sqrt{2}$ factors,
$$
t_{20\leftrightarrow11}=\frac{\pi}{2\sqrt{2}\,J_{12\leftrightarrow01}},\qquad
t_{11\leftrightarrow02}=\frac{\pi}{2\sqrt{2}\,J_{01\leftrightarrow12}}.
$$
The data state that sequential drives are generally preferable to simultaneous multi-tone driving because the latter risks undesired interference and off-resonant couplings [2302.06930].

## 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 $J$ [2206.07216]. In this method the entangling Hamiltonian is
$$
H_{\mathrm{CK}}=
\alpha_{11}|11\rangle\langle11|
+\alpha_{12}|12\rangle\langle12|
+\alpha_{21}|21\rangle\langle21|
+\alpha_{22}|22\rangle\langle22|,
$$
with
$$
\alpha_{ij}=A_{ij}(\omega_d)\,\Omega_a\Omega_b\cos(\phi_a-\phi_b).
$$
Including single-qutrit Stark shifts,
$$
H_{\mathrm{eff}}
=\sum_{i=0,1,2}\delta_{a,i}|i\rangle\langle i|\otimes I
+\sum_{j=0,1,2}\delta_{b,j}I\otimes|j\rangle\langle j|
+H_{\mathrm{CK}}.
$$
The paper uses this mechanism to realize qutrit $CZ^\dagger$ and $CZ$ gates with estimated process fidelities of $97.3(1)\%$ and $95.2(3)\%$, and total gate times of $580$ ns and $783$ ns, respectively [2206.07216].

The interaction is diagonal rather than exchange-like, so the SWAP is obtained indirectly. The qutrit Fourier transform is
$$
F=\frac{1}{\sqrt{3}}\sum_{m,n=0}^{2}\omega^{mn}|m\rangle\langle n|,
\qquad \omega=e^{2\pi i/3},
$$
with matrix
$$
F=\frac{1}{\sqrt{3}}
\begin{bmatrix}
1&1&1\\
1&\omega&\omega^2\\
1&\omega^2&\omega
\end{bmatrix}.
$$
Using $U_{\mathrm{CZ}}$ or $U_{\mathrm{CZ}}^\dagger$, one constructs the SUM gate,
$$
\mathrm{SUM}_{a\to b}=(I\otimes F^{-1})U_{\mathrm{CZ}}(I\otimes F)
=(I\otimes F)\,U_{\mathrm{CZ}}^\dagger\,(I\otimes F^{-1}),
$$
and the SWAP identity is
$$
\mathrm{SWAP}_{a,b}
=\mathrm{SUM}_{b\to a}\,\mathrm{SUM}_{a\to b}\,\mathrm{SUM}_{b\to a}.
$$
The data explicitly recommend the indirect route on this platform, because exchange-like $XY$ interactions are not the operating point of the dynamic cross-Kerr method [2206.07216].

For timing, each SUM uses one controlled-phase gate plus two local Fourier transforms. Hence three SUMs require three $U_{\mathrm{CZ}}$ or three $U_{\mathrm{CZ}}^\dagger$ gates. The data give the corresponding Stark-driven times as approximately $2.35\,\mu\text{s}$ using $U_{\mathrm{CZ}}$ and approximately $1.74\,\mu\text{s}$ using $U_{\mathrm{CZ}}^\dagger$, excluding the local transforms. Under the approximate multiplicative model
$$
F_{\mathrm{SUM}}\approx F_{\mathrm{CP}}(F_1)^2,\qquad
F_{\mathrm{SWAP}}\approx (F_{\mathrm{SUM}})^3\approx (F_{\mathrm{CP}})^3(F_1)^6,
$$
and assuming $F_1\in[0.995,0.999]$, the projected SWAP range is $0.838$–$0.857$ using $U_{\mathrm{CZ}}$ and $0.892$–$0.915$ using $U_{\mathrm{CZ}}^\dagger$ [2206.07216].

## 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 $M_j\neq M_{j+1}$ gives $\Delta\neq0$, and the data state that the swap infidelity scales as approximately $\Delta^2/(4S^2)$ to leading order. Residual cavity excitation and decay contribute dephasing of order $(g/\Delta_2)^2\kappa t_{\mathrm{gate}}$. Qutrit decoherence in the ground manifold contributes through transient occupation of $|2\rangle$, with infidelity terms of the form
$$
\int dt \left[\frac{p_2(t)}{T_{1,2}} + \frac{p_2(t)}{T_{\phi,2}}\right],
$$
while inhomogeneous couplings can be compensated by calibrating $\Omega_j$ to equalize effective Stark shifts and exchange strengths [1207.2882].

In the coupler-assisted exchange approach, the main issues are leakage, residual $ZZ$, and frequency crowding. For an unwanted transition detuned by $\Delta_{\mathrm{off}}$, the instantaneous leakage probability is stated to scale approximately as
$$
P_{\mathrm{leak}}\lesssim \left(\frac{\Omega_{\mathrm{eff}}}{\Delta_{\mathrm{off}}}\right)^2.
$$
The residual static and ac-tunable $ZZ$ components are
$$
\xi_0=\frac{2 g_{\mathrm{eff}}^2(\alpha_1+\alpha_2)}
{(\Delta_{12}+\alpha_1)(\alpha_2-\Delta_{12})},
\qquad
g_{\mathrm{eff}}=\frac{g_{1c}g_{2c}}{2}\left(\frac{1}{\Delta_{1c}}+\frac{1}{\Delta_{2c}}\right)+g_{12},
$$
and under an off-resonant blue CAS tone,
$$
\xi_{\mathrm{ac}}=-\frac{\Omega_b^2}{8\delta},
\qquad
\xi_{\mathrm{ZZ}}(\omega_d,\Omega_d)=\xi_0-\frac{\Omega_b^2}{8(\omega_d-\omega_b)}.
$$
The data recommend Gaussian edges, DRAG-like shaping, and virtual-$Z$ compensation, and note that blue CAS relaxes frequency crowding relative to red CAS [2302.06930].

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 $|3\rangle$ and above at higher drive amplitudes, classical cross-talk, and residual always-on $ZZ$. 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 $Z$ gates in both the $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|2\rangle\}$ subspaces [2206.07216].

Two recurring points of confusion are explicitly resolved by the data. First, the conditional swap of [1207.2882] is not the full two-qutrit SWAP; it swaps only $|1,2\rangle$ and $|2,1\rangle$ 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 $XY$-like [1207.2882] [2206.07216].

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 [1207.2882] [2302.06930] [2206.07216].

Source: https://www.emergentmind.com/topics/microwave-activated-two-qutrit-swap-gate