Designing High-Fidelity Single-Shot Three-Qubit Gates: A Machine-Learning Approach

Three-qubit quantum gates are key ingredients for quantum error correction and quantum-information processing. We generate quantum-control procedures to design three types of three-qubit gates, namely Toffoli, controlled-not-not, and Fredkin gates. The design procedures are applicable to a system comprising three nearest-neighbor-coupled superconducting artificial atoms. For each three-qubit gate, the numerical simulation of the proposed scheme achieves 99.9% fidelity, which is an accepted threshold fidelity for fault-tolerant quantum computing. We test our procedure in the presence of decoherence-induced noise and show its robustness against random external noise generated by the control electronics. The three-qubit gates are designed via the machine-learning algorithm called subspace-selective self-adaptive differential evolution.

Figure 1

The energy ($E$) spectrum of the system Hamiltonian of two capacitively coupled transmons. The frequency of the first transmon is fixed at 6.5 GHz. The frequency ${ϵ}_2(t)$ of the second transmon varies from 7.5 to 6.5 GHz.

Figure 2

The energy ($E$) spectrum of three nearest-neighbor-coupled transmons. The first and third transmon frequencies are fixed at 4.8 and 6.8 GHz, respectively. The frequency of the second transmon varies from 4.5 to 7.5 GHz.

Figure 3

The quantum-circuit representation of Fredkin (controlled-swap) gate. The horizontal solid back line is the circuit wire, the filled circle denotes the control qubit, and the large cross sign shows the swap gate which acts on the target qubits (the second and third qubits).

Figure 4

The quantum-circuit representation of the cxx gate (left panel), which is equivalent to the czz gate up to local Hadamard gates (right panel). The horizontal solid black lines are circuit wires, the filled circle represents the control qubit, and the bun symbol denotes the Pauli-$X$ operator acting on the target qubit. The boxes with $Z$ and $H$ denote the Pauli-$Z$ and Hadamard operations, respectively.

Figure 5

Intrinsic fidelity of the Toffoli gate $\mathcal{F}$ as a function of $\delta ϵ$ for the ccz gate. The vertical red dotted line denotes the threshold, such that $\mathcal{F}>0.9999$ on the left of the dotted line.

Figure 6

Optimal pulses for designing a Fredkin gate with a resultant fidelity better than 0.999 and a gate-operation time of 26 ns. System frequencies, ${ϵ}_i$, are varied from $-2.5$ to 2.5 GHz, which are within the experimental requirements of transmon implementation. The black dots denote the learning parameters for SUSSADE. (a) The piecewise-constant pulses for each transmon frequency. (b) The piecewise-error-function pulses for each transmon frequency.

Figure 7

(a) The dependence of intrinsic fidelity of the Fredkin gate on the evolution time $\tau$ of the system for various values of $g$. The discretized values show the actual numerical data with the triangles, squares, circles, and diamonds corresponding to the following values of $g$: 20, 30, 40, and 50 MHz, respectively. A cubic interpolation fits the curves to the data. (b) The relation between the inverse of the gate-operation time and the coupling strength between transmons, where the dots denote the actual numerical results for various values of $g$: $g\in \{20,30,40,50\}$. A linear fit interpolates the points to the actual data.

Figure 8

Fidelity vs coherence time for the Fredkin gate. The dots denote the actual numerical data and the red solid line shows a cubic-fit interpolation on the actual data.

Figure 9

Intrinsic fidelity $\mathcal{F}$ vs random noise applied to the optimal pulse of the Fredkin gate. The vertical red dotted line denotes the threshold, such that on the left side of the line $\mathcal{F}>0.9999$.

Figure 10

Optimal pulses for designing a czz gate with a resultant fidelity better than 0.999 and a gate-operation time of 31 ns. System frequencies, ${ϵ}_i$, vary from $-2.5$ to 2.5 GHz, which are within the experimental constraints of transmon implementation. The black dots denote the learning parameter for SUSSADE. (a) The piecewise-constant pulses for each transmon frequency. (b) The piecewise-error-function pulses for each transmon frequency.

Figure 11

(a) The dependence of the intrinsic fidelity of the czz gate on the evolution time $\tau$ of the system for various values of $g$. The discretized values show the actual numerical data, with the diamonds, triangles, circles, and squares corresponding to the following values of $g$: 20, 30, 40, and 50 MHz, respectively. A cubic interpolation fits the curves to the data. (b) The relation between the inverse of the gate-operation time and the coupling strength between transmons, where the dots denote the actual numerical results for various values of $g$: $g\in \{20,30,40,50\}$. A linear fit interpolates the points to the actual data.

Figure 12

Fidelity vs coherence time for the czz gate. The dots denote the actual numerical data and the blue solid line shows a cubic-fit interpolation on the actual data.

Figure 13

Intrinsic fidelity $\mathcal{F}$ vs $\delta ϵ$ for the czz gate. The vertical red dotted line denotes the threshold, such that on the left side of this line $\mathcal{F}>0.9999$.