Accelerated adiabatic quantum gates: Optimizing speed versus robustness

We develop protocols for high-fidelity single-qubit gates that exploit and extend theoretical ideas for accelerated adiabatic evolution. Our protocols are compatible with qubit architectures where direct transitions between logical states are either vanishingly small or nonexistent; in such systems traditional approaches cannot be implemented. Prime examples are superconducting fluxonium qubits, which have highly localized states, and AMO systems, where there are no dipole allowed transitions between the ground states encoding the logical states. By using an accelerated adiabatic protocol we can enforce the desired adiabatic evolution while having gate times that are comparable to the inverse adiabatic energy gap (a scale that is ultimately set by the amount of power used in the control pulses). By modeling the effects of decoherence, we explore the trade-off between speed and robustness that is inherent to shortcuts-to-adiabaticity approaches.

Figure 1

(a) Schematic of a tripod system. We choose $|0\rangle$ and $|1\rangle$ to encode the qubit states. The ground-state manifold couples to an excited state $|e\rangle$ via control fields denoted ${\mathrm{\Omega }}_{0e}\left(t\right), {\mathrm{\Omega }}_{1e}\left(t\right)$, and ${\mathrm{\Omega }}_{ae}\left(t\right)$. (b) Effective $\mathrm{\Lambda }$ system describing the evolution of the tripod system. The control pulses ${\mathrm{\Omega }}_{b_{1}e}\left(t\right)$ and ${\mathrm{\Omega }}_{ae}\left(t\right)$ are chosen to generate a cyclic evolution based on STIRAP. (c) Example of a control sequence generating a cyclic evolution. The relative phase of ${\mathrm{\Omega }}_{ae}\left(t\right)$ is changed instantaneously at $t=t_g/2$. This does not result in any discontinuity in the evolution since ${\mathrm{\Omega }}_{ae}(t_g/2)=0$. (d) Geometric representation of the evolution of the system on the Bloch sphere. The geometric phase accumulated by $|\tilde{1}\rangle$ is equal to the solid angle encapsulated during the evolution.

Figure 2

Error $\varepsilon =1-\overline{F}$ [see Eq. (22)] as a function of the gate time for the unitary operator ${\widehat{U}}_G$ acting on both the qubit and auxiliary subspace and its projection ${\widehat{U}}_q$ on the qubit subspace (${\varepsilon }_q$). Corrections to the ideal gate due to nonadiabatic transitions only affect the auxiliary subspace at leading order in ${\widehat{V}}_{\mathrm{err}}$, while the dynamics in the qubit subspace is only affected at third order. This is a consequence of our choice of $\theta \left(t\right)$, whose first and second derivatives vanish at $t=0, t_g/2$, and $t_g$.

Figure 3

Comparison of the average error ${\varepsilon }_{\mathrm{map}}=1-{\overline{F}}_{\mathrm{map}}$ between adiabatic (upper blue traces) and accelerated (lower orange traces) gates either without uncertainty on ${\mathrm{\Omega }}_0$ (dashed traces) or with $40\%$ ($k=0.2$) uncertainty on ${\mathrm{\Omega }}_0$ (solid traces) as a function of the gate time for $\alpha =\pi /4, \beta =0, {\gamma }_0=\pi , {\mathrm{\Gamma }}_{\phi ,e}={10}^{-2}({\mathrm{\Omega }}_0/2\pi )$, and ${\mathrm{\Gamma }}_{\phi ,j}=0$ for $j\in \{0,1,a\}$. Gate times shorter than the one indicated by the green arrow (first arrow from the right) result in a modified control sequence whose maximal amplitude is higher than ${\mathrm{\Omega }}_0$. The blue (second from the right) and red (third from the right) arrows indicate a modified control sequence that requires twice and three times the energy cost [see Eq. (48)] of the adiabatic sequence, respectively.

Figure 4

Comparison of the average error ${\varepsilon }_{\mathrm{map}}=1-{\overline{F}}_{\mathrm{map}}$ between adiabatic (upper blue traces) and accelerated (lower orange traces) gates either without uncertainty on ${\mathrm{\Omega }}_0$ (dashed traces) or with $40\%$ ($k=0.2$) uncertainty on ${\mathrm{\Omega }}_0$ (solid traces) as a function of the gate time for $\alpha =\pi /4, \beta =0, {\gamma }_0=\pi , {\mathrm{\Gamma }}_{\phi ,e}={10}^{-2}({\mathrm{\Omega }}_0/2\pi )$, and ${\mathrm{\Gamma }}_{\phi ,j}={10}^{-2}({\mathrm{\Omega }}_0/2\pi )$ for $j\in \{0,1,a\}$. Gate times shorter than the one indicated by the green arrow (first from the right) result in a modified control sequence whose maximal amplitude is higher than ${\mathrm{\Omega }}_0$. The blue (second from the right) and red (third from the right) arrows indicate modified control sequences that require twice and three times the energy cost [see Eq. (48)] of the adiabatic sequence, respectively.

Figure 5

Comparison of the smallest gate error between adiabatic and accelerated gates. For fixed values of the dephasing rates we look for the gate time that yields the smallest error. We consider an uncertainty on ${\mathrm{\Omega }}_0$ of $10\%$ ($k=0.05$) and ${\mathrm{\Gamma }}_{\phi ,0}={\mathrm{\Gamma }}_{\phi ,1}={\mathrm{\Gamma }}_{\phi ,a}={\mathrm{\Gamma }}_{\phi ,\mathrm{GS}}$. The accelerated gate reaches the same error as the adiabatic gate for dephasing rates that are roughly one order of magnitude larger.

Figure 6

Comparison of the average error ${\varepsilon }_{\mathrm{map}}=1-{\overline{F}}_{\mathrm{map}}$ calculated numerically (solid blue trace) vs using Eq. (49) (dashed orange trace). We used $\alpha =\pi /4, \beta =0, {\gamma }_0=\pi , {\mathrm{\Gamma }}_{\phi ,e}={\mathrm{\Omega }}_0/(2\pi \times 100)$.