Noise analysis for high-fidelity quantum entangling gates in an anharmonic linear Paul trap

The realization of high-fidelity quantum gates in a multiqubit system, with a typical target set at $99.9\%$, is a critical requirement for the implementation of fault-tolerant quantum computation. To reach this level of fidelity, one needs to carefully analyze the noises and imperfections in the experimental system and optimize the gate operations to mitigate their effects. Here, we consider one of the leading experimental systems for the fault-tolerant quantum computation, ions in an anharmonic linear Paul trap, and optimize entangling quantum gates using segmented laser pulses with the assistance of all the collective transverse phonon modes of the ion crystal. We present detailed analyses of the effects of various kinds of intrinsic experimental noises as well as errors from imperfect experimental controls. Through explicit calculations, we find the requirements on these relevant noise levels and control precisions to achieve the targeted high fidelity of $99.9\%$ for the entangling quantum gates in a multi-ion crystal.

Figure 1

Schematic diagram of the three levels and the two Raman beams.

Figure 2

(a) Schematic experimental setup. Three beams are shined on the ion, with one beam in the direction of ${\mathit{k}}_1$ and the other two red- and blue-detuned beams in the direction of ${\mathit{k}}_2$. (b) Schematic energy levels and the two pairs of Raman transitions.

Figure 3

A 17-qubit surface code layout. The open circles represent the data qubits and the filled circles represent the syndrome qubits. Labels 1–17 correspond to the real order of qubits in the 1D chain.

Figure 4

Infidelity for an entangling gate between ion 1 and ion 4 as a function of the detuning $\mu$. Here gate time $\tau =300\mu \mathrm{s}$ and three segment numbers $n_{\mathrm{seg}}=10,12,14$ are used. The vertical dash-dotted lines give the range of the spectrum of the transverse normal modes.

Figure 5

Optimized effective Rabi frequency sequence ${\mathbf{\Omega }}_0$ on ion 5 and ion 6 for $n_{\mathrm{seg}}=10$, detuning ${\mu }_0=0.995{\omega }_x$, and gate time $\tau =80.4\mu \mathrm{s}$. Here we allow the Rabi frequency to take negative values by adding a phase shift of $\pi$. If such a phase shift is not available, we can look for other solutions where all the Rabi frequencies are positive. Some examples are shown in Ref. [26].

Figure 6

Parameter sensitivity for the entangling gate between ion 5 and ion 6. (a) Infidelity as a function of shift in detuning. (b) Infidelity as a function of relative shift in laser intensity. Here we assume that the frequency of the noise is low so that the laser intensities of all the segments are shifted by the same percentage. (c) Infidelity as a function of shift in gate time $\tau$. (d) Dependence on the motional phase ${\phi }^{\left(m\right)}$. Here we consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal infidelity for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in Rabi frequency, and $0.4\mu \mathrm{s}$ change in total gate time, respectively.

Figure 7

Optimized laser sequence ${\mathbf{\Omega }}_0$ on ion 1 and ion 4 for $n_{\mathrm{seg}}=17$, detuning ${\mu }_0=0.997{\omega }_x$, and gate time $\tau =250\mu \mathrm{s}$.

Figure 8

Parameter sensitivity for the entangling gate between ion 1 and ion 4. (a) Infidelity as a function of shift in detuning. (b) Infidelity as a function of relative shift in Rabi frequency. We assume the laser intensities of all the segments are shifted by the same percentage. (c) Infidelity as a function of shift in gate time $\tau$. (d) Consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal infidelity for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in intensity, and $0.4\mu \mathrm{s}$ change in total gate time, respectively.

Figure 9

Optimized laser sequence ${\mathbf{\Omega }}_0$ on ion 9 and ion 14 for $n_{\mathrm{seg}}=24$, detuning ${\mu }_0=0.997{\omega }_x$ and gate time $\tau =482\mu \mathrm{s}$.

Figure 10

Parameter sensitivity for the entangling gate between ion 9 and ion 14. (a) Infidelity as a function of shift in detuning. (b) Infidelity as a function of relative shift in Rabi frequency. (c) Infidelity as a function of shift in gate time $\tau$. (d) Consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal infidelity for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in intensity, and $0.4\mu \mathrm{s}$ change in total gate time, respectively.

Figure 11

Parameter sensitivity for the entangling gate between ions 5 and 6. (a) Diamond norm as a function of shift in detuning. (b) Diamond norm as a function of relative shift in Rabi frequency. (c) Diamond norm as a function of shift in gate time $\tau$. (d) Consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal diamond norm for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in intensity, and $0.4\mu \mathrm{s}$ change in total gate time, respectively. For panels (a), (b), and (c), the diamond norms below ${10}^{-8}$ are not shown, since they are subject to numerical errors.

Figure 12

Parameter sensitivity for the entangling gate between ions 1 and 4. (a) Diamond norm as a function of shift in detuning. (b) Diamond norm as a function of relative shift in Rabi frequency. (c) Diamond norm as a function of shift in gate time $\tau$. (d) Consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal diamond norm for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in intensity, and $0.4\mu \mathrm{s}$ change in total gate time, respectively.

Figure 13

Parameter sensitivity for the entangling gate between ions 9 and 14. (a) Diamond norm as a function of shift in detuning. (b) Diamond norm as a function of relative shift in Rabi frequency. (c) Diamond norm as a function of shift in gate time $\tau$. (d) Consider ${\phi }_i^{\left(m\right)}={\phi }_j^{\left(m\right)}$ between 0 and $2\pi$. Solid blue, dashed red, and dotted green curves are the maximal diamond norm for a shift of $1\mathrm{kHz}$ in detuning $\mu$, a 1% change in intensity, and $0.4\mu \mathrm{s}$ change in total gate time, respectively.

Figure 14

(a) Equilibrium positions for an anharmonic trap potential with ${\gamma }_4=4.3$ and $l_0=40$ (arbitrary unit). (b) Equilibrium positions for a harmonic trap potential that can produce the same average ion spacing.

Figure 15

Infidelity due to residual coupling to the phonon modes vs motional phase ${\phi }^{\left(m\right)}$. The pulse sequence is optimized for ${\phi }^{\left(m\right)}=0$, but the infidelity is almost independent of ${\phi }^{\left(m\right)}$.

Figure 16

$XX$ rotation angle ${\mathrm{\Theta }}_{ij}$ vs motional phase ${\phi }^{\left(m\right)}$. The pulse sequence is chosen such that ${\mathrm{\Theta }}_{ij}=\pm \pi /4$ at ${\phi }^{\left(m\right)}=0$.

Figure 17

Total gate design infidelity vs number of repeated gates $m$ between ions 9 and 14. Here we consider a special case where there is no interval between two adjacent gates.