Phase-Modulated Entangling Gates Robust to Static and Time-Varying Errors

Entangling operations are among the most important primitive gates employed in quantum computing, and it is crucial to ensure high-fidelity implementations as systems are scaled up. We experimentally realize and characterize a simple scheme to minimize errors in entangling operations related to the residual excitation of mediating bosonic oscillator modes that both improves gate robustness and provides scaling benefits in larger systems. The technique employs discrete phase shifts in the control field driving the gate operation, determined either analytically or numerically, to ensure all modes are de-excited at arbitrary user-defined times. We demonstrate an average gate fidelity of 99.4(2)% across a wide range of parameters in a system of ${{}^{171}\mathrm{Yb}}^{+}$ trapped ion qubits, and observe a reduction of gate error in the presence of common experimental error sources. Our approach provides a unified framework to achieve robustness against both static and time-varying laser amplitude and frequency detuning errors. We verify these capabilities through system-identification experiments revealing improvements in error susceptibility achieved in phase-modulated gates.

Figure 1

Construction of phase-modulation sequences. (a) Schematic plot of ${\alpha }_k(t)$ for $0\le t\le 2\tau$. Shifting the coupling phase, initially ${\varphi }_0$, by an amount $-(\pi +{\delta }_k\tau$) at $t=\tau$ returns the oscillator trajectory to the origin at $t=2\tau$. (b) Construction of the $\mathrm{\Phi }\mathrm{M}$ sequence $R_{{\delta }_2}{R}_{{\delta }_1}{r}_0(t;{\tau }_s)$ from the base sequence $r_0(t;{\tau }_s)$. Each application of $R_{{\delta }_k}$ produces a new sequence consisting of the original sequence (black arrow) followed by the entire original sequence phase-shifted by $-(\pi +{\delta }_k\tau$) (gray arrow), where $\tau$ is the duration of the original sequence. The example sequence closes the trajectories of two modes $k=1,2$, shown in (c), with detunings ${\delta }_1$, ${\delta }_2$ in four time segments of length ${\tau }_s={\tau }_g/4$. Colors indicate the varying coupling phase ${\varphi }_n$ in each time segment $t\in [n{\tau }_s,(n+1){\tau }_s]$. (d) Example phase space trajectory (left) and schematic showing the construction (right) of a standard numerically optimized phase-modulation sequence targeting two modes with $S=8$ phase segments of ${\tau }_g/8$ duration each. (e) For the robust numerical sequence, the number of phase segments is doubled to $S=16$, with each ${\tau }_g/16$ in length. The time-averaged position of the phase space trajectory (red dot) lies at the origin. This condition, combined with the constraint that $\mathrm{\Delta }{\varphi }_{n+1}=\mathrm{\Delta }{\varphi }_{S-(n+1)}$, results in a trajectory symmetric about a line through the origin (dashed line).

Figure 2

Motional mode decoupling in the single-ion case. Population $P_1$ as a function of laser detuning scanned about two radial modes, $k=1$ and $k=2$ (dashed vertical lines). Solid lines are fits to the data, with each mode’s initial average phonon number after sideband cooling (${\overline{n}}_k$) a free parameter. (a) Mode frequencies split far apart. Arrows indicate detunings at which ${\delta }_1=\pm 2\pi /{\tau }_g$ and the inset shows corresponding phase space trajectories for each mode, with ${\overline{n}}_{1,2}=(0.2,0.4)$. (b) Mode frequencies closely spaced. The phase-modulation sequence $r_{{\delta }_2{\delta }_1}(t;{\tau }_g)$ (purple) achieves decoupling at an arbitrary detuning ${\delta }_1/2\pi =-8.5$ kHz (arrow), with phase segments ${\varphi }_{0,1,2,3}\approx (0,1.34,0.343,1.68)\pi$. The inset compares unmodulated and phase-modulated trajectories. Fits give ${\overline{n}}_{1,2}=(1.2,1.2)$ for the unmodulated and ${\overline{n}}_{1,2}=(2.5,1)$ for the phase-modulated data.

Figure 3

Maximum achievable gate fidelity as a function of detuning. Solid lines show theoretical predictions for initial phonon numbers ${\overline{n}}_{2,3}=0$ and the dashed line show predictions for ${\overline{n}}_{2,3}=0.2$. Different $\mathrm{\Phi }\mathrm{M}$ sequences are implemented over the detuning range, with $r_{{\delta }_2{\delta }_3}(t;{\tau }_g)$ used for ${\delta }_2/\mathrm{\Delta }\ge -0.5$ and $r_{{\delta }_3{\delta }_2}(t;{\tau }_g)$ for ${\delta }_2/\mathrm{\Delta }<-0.5$. The required Rabi frequency $\mathrm{\Omega }$ ranges from $2\pi \times (24–26)$ kHz for the phase-modulated gates and $2\pi \times (19–26)$ kHz for the unmodulated gates. Error bars are derived from quantum projection noise on the state population estimates and a fit of the parity contrast. The inset shows the underlying data for the phase-modulated gate at ${\delta }_2/\mathrm{\Delta }=-0.5$ (arrow), for which $\mathcal{F}=99.4(5)\mathrm{\%}$.

Figure 4

Robustness of phase-modulated gates to detuning offsets. (a) Motional excitation quantified by $P_1$ at the conclusion of a two-qubit entangling gate as a function of detuning offset magnitude. Solid lines are theory with ${\overline{n}}_4=0.1$. The gate time is fixed at ${\tau }_g=500\mu \mathrm{s}$, with a target detuning of $-2$ kHz from mode 4. The unmodulated gate (gray) is compared to the $\mathrm{\Phi }\mathrm{M}$ sequences $r_{{\delta }_4{\delta }_4}(t;{\tau }_g)$ (blue) and $r_{{\delta }_4{\delta }_4{\delta }_4{\delta }_3{\delta }_2}(t;{\tau }_g)$ (purple), which provide second- and third-order noise suppression for mode 4, respectively. Rabi frequency $\mathrm{\Omega }$ is scaled to enact a maximally entangling gate at zero detuning error and lies in the range $2\pi \times (18–39)$ kHz. The scaling required for the third-order sequence also necessitates the decoupling of modes 2 and 3. (b) Phase space trajectories for mode 4 with a detuning error of $+500$ Hz, marked via the dashed line in (a). All plots are of equal scale. (c) $P_1$ at the conclusion of a two-qubit entangling gate for numerically optimized phase-modulation sequences, constructed to sufficiently decouple from all four modes. Solid lines are theory with ${\overline{n}}_4=0.05$. The gate time is fixed at ${\tau }_g=400\mu \mathrm{s}$ with a target detuning of $-4$ kHz from mode 4. The standard numerically optimized gate (gray) with 16 phase segments is compared to a robust solution (purple) with 32 phase segments. Rabi frequencies are $2\pi \times 23$ kHz and $2\pi \times 28$ kHz for the standard and robust gates, respectively. (d) Corresponding phase space trajectories for mode 4 with a detuning error of +500 Hz, marked via the dashed line in (c).

Figure 5

Suppression of time-dependent noise. (a),(b) Noise-averaged measurements of $P_1$ for a single ion plotted against applied (a) detuning and (b) amplitude noise frequency. Overlaid are the analytic filter function predictions with ${\overline{n}}_{1,2}=0.9$ (dashed lines) and the predictions with an added frequency-independent offset (solid lines), determined by the average of the six lowest-frequency data points. For (a), the added offsets are $5.1\times {10}^{-3}$ (unmodulated) and $3.3\times {10}^{-3}$ (phase-modulated). For (b), the added offsets are $4.7\times {10}^{-3}$ (unmodulated) and $1.7\times {10}^{-3}$ (phase-modulated). For (a) the modulation depth is $\beta =0.1$ with ${\tau }_g=300\mu \mathrm{s}$ and for (b) $\beta =0.29$ with ${\tau }_g=250\mu \mathrm{s}$. The Rabi frequencies lie in the range $2\pi \times (14–34)$ kHz, and $\mathrm{\Omega }$ is scaled in the phase-modulated operations to enclose the same phase space area as the equivalent unmodulated operation. Shading indicates the measurement floor. (c),(d) Two-qubit gate fidelities with engineered (c) detuning and (d) amplitude noise. For both noise types, ${\tau }_g=500\mu \mathrm{s}$, and $\mathrm{\Omega }$ lies in the range $2\pi \times (18–36)$ kHz. Here, the detuning noise is engineered by directly modulating the frequency of the motional modes via the application of a sinusoidally oscillating voltage to the dc trap electrodes. Measured gate fidelities for (c) are 86%, 91% and 96%, respectively, under $\beta =0.25$. For (d), the measured gate fidelities are 82% and 87%, under $\beta =0.2$. Due to limitations on the experimentally achievable Rabi frequency in our setup ($\mathrm{\Omega }=2\pi \times 40$ kHz), only a second-order $\mathrm{\Phi }\mathrm{M}$ sequence is performed for the amplitude noise case shown in (d).

Figure 6

Scaling to larger qubit systems. (a) Shortest achievable gate time (${\tau }_B$) as a function of the number of qubits, decoupling from all $M=2N$ radial modes. The number of phase shifts for each $\mathrm{\Phi }\mathrm{M}$ approach is fixed at $S=2^M$ (analytic), $S=4M$ (standard numerical), and $S=8M$ (robust numerical). The mode spectrum is calculated using COM trapping frequencies of ${\omega }_{x,y,z}/2\pi =(1.62,1.54,0.15)$ MHz with the maximum Rabi frequency limited to $\mathrm{\Omega }=2\pi \times 100$ kHz. (b) Shortest achievable gate time (${\tau }_B$) represented as a color scale for standard numerically optimized entangling gates between different target pairs in an $N=10$ ion chain, decoupling from the radial modes. Black represents the median gate time, red indicates slower gates, and blue indicates faster. (c) Schematic depiction of the equilibrium positions for an $N=10$ ion chain and corresponding radial mode spectrum. Vertical bars positioned at the frequency of each mode $k$ indicate the value of $|\overline{{\eta }_k^{(\mu )}{\eta }_k^{(\nu )}}|$, which is $|{\eta }_k^{(\mu )}{\eta }_k^{(\nu )}|$ normalized to the maximum value of $|{\eta }_k^{(\mu )}{\eta }_k^{(\nu )}|$ for all $k$. Arrows indicate the detunings corresponding to ${\tau }_B$ for two target pairs: 1,4 (red) and 5,6 (blue).