Benchmarking a High-Fidelity Mixed-Species Entangling Gate

We implement a two-qubit logic gate between a ${{}^{43}\mathrm{Ca}}^{+}$ hyperfine qubit and a ${{}^{88}\mathrm{Sr}}^{+}$ Zeeman qubit. For this pair of ion species, the S–P optical transitions are close enough that a single laser of wavelength 402 nm can be used to drive the gate but sufficiently well separated to give good spectral isolation and low photon scattering errors. We characterize the gate by full randomized benchmarking, gate set tomography, and Bell state analysis. The latter method gives a fidelity of 99.8(1)%, comparable to that of the best same-species gates and consistent with known sources of error.

Figure 1

(a) A single laser with wavelength $\lambda =c/f_L=402\mathrm{nm}$, detuned by ${\mathrm{\Delta }}_{\mathrm{Sr}}=+11.2\mathrm{THz}$, ${\mathrm{\Delta }}_{\mathrm{Ca}}=-9.0\mathrm{THz}$ from S–P transitions in ${{}^{88}\mathrm{Sr}}^{+}$ and ${{}^{43}\mathrm{Ca}}^{+}$, provides qubit-state-dependent forces on both ion species. The qubit states are ($|\downarrow ⟩,|\uparrow ⟩)=(5{\mathrm{S}}_{1/2}^{-1/2},5{\mathrm{S}}_{1/2}^{+1/2})$ in ${{}^{88}\mathrm{Sr}}^{+}$ and ($|\Downarrow ⟩,|\Uparrow ⟩)=(4{\mathrm{S}}_{1/2}^{4,+4},4{\mathrm{S}}_{1/2}^{3+3})$ in ${{}^{43}\mathrm{Ca}}^{+}$. A static magnetic field $B=146\mathrm{G}$ gives qubit frequencies $f_{\updownarrow }=409\mathrm{MHz}$ and $f_{\Updownarrow }=2.874\mathrm{GHz}$. (b) Both Raman beams are derived from a single frequency-doubled Ti:S laser. The beatnote ($f_{\mathrm{ax},\mathrm{oop}}+{\delta }_g$) between the two beams is created by acousto-optic modulators. The differential wave vector $\mathrm{\Delta }\mathit{k}$ of the Raman beams is parallel to the trap axis $\mathit{z}$, suppressing coupling to the radial modes of motion. (c) Values of (ax)ial and (rad)ial in-phase (ip) and out-of-phase (oop) normal mode frequencies.

Figure 2

(a) Schematic sequence of the ${\sigma }_z\otimes {\sigma }_z$ gate. The gate $G$ characterized in RBM and GST measurements consists of two Raman pulses $\mathrm{g}(t_{\mathrm{g}}/2)$, separated by a “spin-echo” $\pi$-pulse on each qubit. For PST, $G$ is bracketed by two $\pi /2$-pulses, and followed by an additional $(\pi /2{)}_{\varphi }$ analysis pulse (not shown) with phase $\varphi =45°$ or 135° to determine the parity contrast. (b) Gate dynamics measured with the sequence in (a). To improve sensitivity during calibration, the initial phase of the second gate pulse matches that of the first pulse. The asymmetry of the forces on ${{}^{43}\mathrm{Ca}}^{+}$ and ${{}^{88}\mathrm{Sr}}^{+}$ leads to asymmetric evolution of the populations $p_{\Downarrow \uparrow }$, $p_{\Uparrow \downarrow }$.

Figure 3

Interleaved RBM: (a) Gate sequences. Operations $C_1\dots C_L$ are randomly chosen from the 11 520 elements of the two-qubit Clifford group, generated by the single-qubit $\pi /2$-rotations $\{\pm X,\pm Y,\pm Z\}$, and $G$. In interleaved sequences, the entangling gate under test $G$ is applied after each $C_i$. To finish in an eigenstate of the measurement basis, each sequence terminates with the inverse operation $C^{-1}$ ($C_G^{-1}$) of all combined $C_i$ ($C_i$ and $G$) operations. The final eigenstate is randomized by single-qubit $\pi$-rotations $P$. (b) Combined datasets of 660 randomized sequences with $L=1\dots 60$. Each randomization is repeated 100 times to determine the sequence fidelity. Each data point gives the mean fidelity for all sequences of length $L$. (c) Error per interleaved $G$ gate, inferred from fitting the fidelity decays in (b), as a function of the maximum sequence length used in the fit. The error per gate increases with $L$ as certain error sources become more pronounced after longer durations. Error bars are from parametric bootstrapping.

Figure 4

Gate set tomography: (a) Structure of GST sequences. For each target sequence length $L=1,2,4,\dots ,64$, each germ sequence $g_k$ is repeated $l$ times such that the maximum number of operations is $L$ and is surrounded by combinations of preparation and measurement fiducials $p_i$ and $m_j$. (b) Final maximum-likelihood estimate for the entangling gate under CPTP constraints, given as a superoperator $\mathcal{G}$ in the Pauli-product basis. (c) Difference of the fit of the measured gate $\mathcal{G}$ to the ideal gate ${\mathcal{G}}_0$. Displayed is the error generator $\mathbb{L}$ such that $\mathcal{G}={\mathrm{e}}^{\mathbb{L}}{\mathcal{G}}_0$. (d) Goodness of fit of the final estimate for data up to circuit length $L$ in units of standard deviations of the expected ${\chi }^2$ distribution. A significant amount of non-Markovianity is present, presumably due to duty-cycle effects and motional heating.