Trigonometric Continuous-Variable Quantum Gates: Realization with Trapped Ions and Nonperturbative Wigner Negativity

arXiv:2607.14085 · arXiv: 2607.14085

License: CC BY 4.0.

Abstract

We experimentally realize trigonometric continuous-variable gates on a trapped-ion processor, including a two-mode generalization, and characterize the resulting parity selection rules, open-system dynamics, and Wigner negativity.

Figures

38 panels with data across 10 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (1): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 1$ and $\theta = 0$. data.csv
• panel (2): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 1$ and $\theta = 0.8$. Black lines are the $W_0(x,p) = 0$ contours. data.csv
• panel (3): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 1$ and $\theta = 3$. Black lines are the $W_0(x,p) = 0$ contours. data.csv
• panel (4): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0$. data.csv
• panel (5): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0.8$. Black lines are the $W_0(x,p) = 0$ contours. data.csv
• panel (6): Wigner function $W_0(x,p)$ of the Fock vacuum $|0\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 3$. Black lines are the $W_0(x,p) = 0$ contours. data.csv

Fig. 2

• panel (1): Wigner function $W_1(x,p)$ of the Fock state $|1\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0$. Black lines are the $W_1(x,p) = 0$ contours. data.csv
• panel (2): Wigner function $W_1(x,p)$ of the Fock state $|1\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0.8$. Black lines are the $W_1(x,p) = 0$ contours. data.csv
• panel (3): Wigner function $W_1(x,p)$ of the Fock state $|1\rangle$ after the single-qumode cosine gate, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 3$. Black lines are the $W_1(x,p) = 0$ contours. data.csv

Fig. 3

• panel (1): Left printed panel, weak-angle regimes: mana $\mathcal{M}(W_0)$ of the Fock vacuum under the cosine gate against $\theta$, for $c = 0.5$, 1, 2, 3, 5 and 10. The dashed curve is the first-order bound. The inset of the printed panel is a separate panel, 1-inset. data.csv
• panel (1-inset): Inset of the left printed panel, nonperturbative onset: $\log_{10}\mathcal{M}$ against $\theta$ on a logarithmic axis, for $c = 0.5$, 1, 2, 3, 5 and 10, with the dashed first-order bound. data.csv
• panel (2): Right printed panel, stationary-phase growth: mana $\mathcal{M}(W_0)$ against $\theta$ on a logarithmic axis, for $c = 0.5$, 1, 2, 3, 5 and 10. Dash-dotted lines are the asymptotes, and triangles mark the onset for $c = 0.5$ and 1. data.csv

Fig. 4

Illustrative figure, no extractable data. Shown in the paper PDF. Page 13 of the PDF.

Fig. 5

• panel (1): Detection probability of each qubit, $q_{-1}$, $q_{0}$ and $q_{1}$, against lower-manifold radial-mode frequency (MHz) for the three-ion chain. The mode labels and arrows of the print are not shown. data.csv
• panel (2): Simulated Lamb-Dicke factor $\eta_{j,1}$ of the tilt mode, $k = 1$, for each qubit $q_j$ of the three-ion chain, with the probe and gate qubits marked in colour. data.csv
• panel (3): Detection probability of each qubit, $q_{-1}$, $q_{0}$, $q_{1}$ and $q_{2}$, against lower-manifold radial-mode frequency (MHz) for the four-ion chain. The mode labels and arrows of the print are not shown. data.csv
• panel (4): Simulated Lamb-Dicke factor $\eta_{j,3}$ of the zig-zag mode, $k = 3$, for each qubit $q_j$ of the four-ion chain, with the probe and gate qubits marked in colour. data.csv
• panel (5): Simulated Lamb-Dicke factor $\eta_{j,2}$ of the drum mode, $k = 2$, for each qubit $q_j$ of the four-ion chain, with the probe and gate qubits marked in colour. data.csv

Fig. 6

• panel (1): Wigner function $W_0^{(1)}(x,p)$ of the Fock vacuum after one Trotter step of the circuit, with the qubit traced out, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0.2$. Black lines are the zero contours. data.csv
• panel (2): Wigner function $W_0^{(1)}(x,p)$ of the Fock vacuum after one Trotter step of the circuit, with the qubit traced out, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 0.8$. Black lines are the zero contours. data.csv
• panel (3): Wigner function $W_0^{(1)}(x,p)$ of the Fock vacuum after one Trotter step of the circuit, with the qubit traced out, against position $x$ and momentum $p$, at $c = 2$ and $\theta = 3$. Black lines are the zero contours. data.csv

Fig. 7

• panel (1): Wigner function $W_0^{(N)}(x,p)$ of the Fock vacuum after $N = 2$ Trotter steps, against position $x$ and momentum $p$, at $\theta = 3$ and $c = 2$. Black lines are the zero contours. data.csv
• panel (2): Wigner function $W_0^{(N)}(x,p)$ of the Fock vacuum after $N = 4$ Trotter steps, against position $x$ and momentum $p$, at $\theta = 3$ and $c = 2$. Black lines are the zero contours. data.csv
• panel (3): Wigner function $W_0^{(N)}(x,p)$ of the Fock vacuum after $N = 8$ Trotter steps, against position $x$ and momentum $p$, at $\theta = 3$ and $c = 2$. Black lines are the zero contours. data.csv

Fig. 8

Illustrative figure, no extractable data. Shown in the paper PDF. Page 15 of the PDF.

Fig. 9

Illustrative figure, no extractable data. Shown in the paper PDF. Page 16 of the PDF.

Fig. 10

• panel (1): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 1$ circuit with 1 Trotter step, $n = 0$ to 4: data points with error bars and the optimized simulations as lines with uncertainty bands. The parameters printed atop the plot are not shown. data.csv
• panel (2): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 1$ circuit with 2 Trotter steps, $n = 0$ to 4: data points with error bars and the optimized simulations as lines with uncertainty bands. The parameters printed atop the plot are not shown. data.csv
• panel (3): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 1$ circuit with 4 Trotter steps, $n = 0$ to 4: data points with error bars and the optimized simulations as lines with uncertainty bands. The parameters printed atop the plot are not shown. data.csv
• panel (4): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 1$ circuit with 8 Trotter steps, $n = 0$ to 4: data points with error bars and the optimized simulations as lines with uncertainty bands. The parameters printed atop the plot are not shown. data.csv

Fig. 11

• panel (1): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 2$ circuit with one Trotter step, $n = 0$ to 4: data points with error bars and the simulations as lines. data.csv
• panel (2): Postselected transition probabilities $P_n(\theta)$ against $\theta$ for the $c = 3$ circuit with one Trotter step, $n = 0$ to 4: data points with error bars and the simulations as lines. data.csv

Fig. 12

• panel (1): Left printed panel, ideal simulation: real part $\mathrm{Re}[\chi(\beta)]$ of the characteristic function of the postselected two-step state at $\theta = 2$ and $c = 2$, against $\mathrm{Re}(\beta)$ and $\mathrm{Im}(\beta)$. data.csv
• panel (2): Middle printed panel, simulation including qumode dephasing, displacement miscalibration and thermal initialization: real part $\mathrm{Re}[\chi(\beta)]$ of the characteristic function of the postselected two-step state at $\theta = 2$ and $c = 2$, against $\mathrm{Re}(\beta)$ and $\mathrm{Im}(\beta)$. data.csv
• panel (3): Right printed panel, experimental reconstruction: real part $\mathrm{Re}[\chi(\beta)]$ of the characteristic function of the postselected two-step state at $\theta = 2$ and $c = 2$, against $\mathrm{Re}(\beta)$ and $\mathrm{Im}(\beta)$. data.csv

Fig. 13

• panel (1): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 2 (left column of the print) with one Trotter step, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. The dephasing and phonon-occupation labels atop the print are not shown. data.csv
• panel (2): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 3 (right column of the print) with one Trotter step, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. The dephasing and phonon-occupation labels atop the print are not shown. data.csv
• panel (3): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 2 (left column of the print) with two Trotter steps, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. data.csv
• panel (4): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 3 (right column of the print) with two Trotter steps, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. data.csv
• panel (5): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 2 (left column of the print) with four Trotter steps, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. data.csv
• panel (6): Transition probabilities $P_n(\theta)$ against $\theta$ of the two-mode circuit for qumode 3 (right column of the print) with four Trotter steps, $n = 0$ to 4: data points with error bars and simulations as lines with uncertainty bands. data.csv

Cite

Tommaso Rainaldi et al. (11 authors). Trigonometric Continuous-Variable Quantum Gates: Realization with Trapped Ions and Nonperturbative Wigner Negativity. arXiv:2607.14085

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