Practical Experimental Certification of Computational Quantum Gates Using a Twirling Procedure

Because of the technical difficulty of building large quantum computers, it is important to be able to estimate how faithful a given implementation is to an ideal quantum computer. The common approach of completely characterizing the computation process via quantum process tomography requires an exponential amount of resources, and thus is not practical even for relatively small devices. We solve this problem by demonstrating that twirling experiments previously used to characterize the average fidelity of quantum memories efficiently can be easily adapted to estimate the average fidelity of the experimental implementation of important quantum computation processes, such as unitaries in the Clifford group, in a practical and efficient manner with applicability in current quantum devices. Using this procedure, we demonstrate state-of-the-art coherent control of an ensemble of magnetic moments of nuclear spins in a single crystal solid by implementing the encoding operation for a 3-qubit code with only a 1% degradation in average fidelity discounting preparation and measurement errors. We also highlight one of the advances that was instrumental in achieving such high fidelity control.

Figure 1

The relationship between (a) comparing a physical process $\mathcal{E}$ to the identity, and (c) comparing a physical process superoperator $\tilde{\mathcal{U}}=\mathcal{U}\circ \mathcal{E}$ to a unitary superoperator $\mathcal{U}$ can be seen by (b) the appropriate insertion of the identity ${\mathcal{U}}^{†}\circ \mathcal{U}$. When $\mathcal{U}$ and all ${\mathcal{C}}_i$ are elements of the Clifford group, and ${\widehat{M}}_j$ is a Pauli operator, $f({\widehat{M}}_j,{\mathcal{C}}_i,\mathcal{U})$ is also a Pauli operator.

Figure 2

Portion of a typical pulse shape showing the designed target shape (solid blue line), the initial attempt at implementing the pulse including systematic imperfections due to nonlinearities in pulse generation and amplification as well as finite bandwidth effects from the probe’s resonant circuit (red dots), and the corrected shape after the feedback protocol (green circles). Full power corresponds to nutation frequency of 80 kHz.

Figure 3

Randomized benchmarking of single-qubit $\frac{\pi }{2}$ rotations required for state preparation and measurement—shown is the average fidelity decay of randomized sequences of $\frac{\pi }{2}$ pulses on each of the 3 qubits. Each data point is the average fidelity of 24 sequences. Fitting the data to [12] $\log (F-\frac{1}{2})=\log A_0+m\log p$, we extract an average error per gate of $1.6\pm 0.4\times {10}^{-3}$ for ${\mathrm{C}}_1$ (blue diamonds), $3.8\pm 0.7\times {10}^{-3}$ for ${\mathrm{C}}_2$ (red squares), and $4.4\pm 0.6\times {10}^{-3}$ for ${\mathrm{C}}_3$ (${\mathrm{C}}_m$) (green circles).

Figure 4

Summary of the experimental parameters and results for the three sets of certification experiments—the Target column shows the quantum circuit representation of the ideal process; the Experiment column represents the experimental setup to certify the corresponding implementation, including state preparation and measurement using local readout pulses as described in the text; $k_w$ and ${\lambda }_w$ are, respectively, the number of performed experiments, and the average surviving polarization, partitioned by the Pauli weight, $w$, of the input preparation [8, 27]. Shown also are the Bayesian estimated probability density functions for the probability of no error in the experimental implementation of the target gate, as well as the estimated average fidelity. The three sets of experiments are (a) state preparation and measurement compared to the identity operation—this can be thought of as a calibration for the certification procedure; (b) the target is the encoding operation for the 3-qubit phase quantum error correcting code, and the experimental implementation is a numerically designed pulse using gradient ascent pulse engineering; and (c) is the same as (b) but the pulse is corrected for implementation errors using the feedback procedure described in the text.