High-Fidelity Single-Shot Toffoli Gate via Quantum Control

A single-shot Toffoli, or controlled-controlled-not, gate is desirable for classical and quantum information processing. The Toffoli gate alone is universal for reversible computing and, accompanied by the Hadamard gate, forms a universal gate set for quantum computing. The Toffoli gate is also a key ingredient for (nontopological) quantum error correction. Currently Toffoli gates are achieved by decomposing into sequentially implemented single- and two-qubit gates, which require much longer times and yields lower overall fidelities compared to a single-shot implementation. We develop a quantum-control procedure to construct a single-shot Toffoli gate for three nearest-neighbor-coupled superconducting transmon systems such that the fidelity is 99.9% and is as fast as an entangling two-qubit gate under the same realistic conditions. The gate is achieved by a nongreedy quantum control procedure using our enhanced version of the differential evolution algorithm.

Figure 1

Optimal pulse shapes for the Toffoli gate given as frequency detunings ${\mathrm{\Delta }}_i$ (for the $i=1,2,3$) of the superconducting atoms, corresponding to the (a) piecewise-constant and (b) error-function-based pulse profiles, as a function of time $\tau$ with constant step-size time interval $\mathrm{\Delta }\tau =1$ ns and with $\mathcal{F}=0.9999$ and $g=30\mathrm{MHz}$. The black dots on both plots show the control parameters used to optimize the shape of the pulses.

Figure 2

(a) Intrinsic fidelity $\mathcal{F}$ vs total gate time $\mathrm{\Theta }$ for various coupling strengths $g$ and (b) $1/\mathrm{\Theta }$ vs $g$ for $\mathcal{F}=0.999$. The ◇, △, ∘, and □ denote the actual numerical computations using 1DSuSSADE, and solid lines depict cubic-fit curves.

Figure 3

The fidelity $\overline{\mathcal{F}}$ is plotted against the coherence times $T$. We assume $T=T_1=T_2$, with $T_1$ and $T_2$ the relaxation time and dephasing time of each transmon, respectively. This assumption is valid for tunable transmons. Each ◇ denotes an actual numerical result obtained from the decoherence calculation. The solid line depicts the cubic-fit curve.