Randomized benchmarking of quantum gates

A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized $\pi /2$ pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.

Figure 1

Fidelity as a function of the number of steps for each randomized sequence. The fidelity (given by 1 minus the probability of error) is plotted on a logarithmic scale. The fidelity for the final state is measured for each randomized sequence. There are 32 points for each number of steps, corresponding to $N_P=8$ randomizations of each of $N_G=4$ different computational sequences. We used $N_e=8160$ for these experiments. Different symbols are used for the data for each computational sequence. The standard error of each point is between 0.001 (near fidelities of 1) and 0.006 (for the smaller fidelities). The scatter greatly exceeds the standard error, suggesting that coherent errors contribute significantly to the loss of fidelity.

Figure 2

Average fidelity as a function of the number of steps for each computational sequence. The points show the average randomized fidelity for four different computational gate sequences (indicated by the different symbols) as a function of the length. The average fidelity is plotted on a logarithmic scale. The middle line shows the fitted exponential decay. The upper and lower line show the boundaries of the 68% confidence interval for the fit. The standard deviation of each point due to measurement noise ranges from 0.0004 for values near 1 to 0.002 for the lower values, smaller than the size of the symbols. The empirical standard deviation based on the scatter in the points shown in Fig. 1 ranges from 0.0011 to 0.014. The slope implies an error probability of 0.00482(17) per randomized computational gate. The data is consistent with the gate’s errors not depending on position in the sequence (${\chi }^2=17.72$, 15 degrees of freedom, significance $p=0.28$).

Figure 3

Measurement of phase decoherence with refocusing. We measured the probability of $|1⟩$ as a function of time for the standard refocused decoherence measurement. The pulse sequence consisted of a $\pi /2$ pulse at phase 0 followed by a delay of $T/2$, a $\pi$ pulse at phase $\pi$, another delay of $T/2$ and a final $\pi /2$ pulse at phase $\pi$. The straight line shows the fit for exponential decay on the interval from 1 to $200\mu \text{s}$. Its extrapolation to larger times is shown dashed. The deviation from an exponential decay at larger times can be attributed to slow phase drifts that are no longer refocused by the single $\pi$ pulse in the pulse sequence. From the fit, the contribution of unrefocusable phase decoherence to the error probability per step is 0.0037(1). The standard deviation of the plotted points ranges from 0.002 for values near 1 to 0.008 for the smallest values, similar to the apparent scatter of the plotted points.

Figure 4

Measurement of phase decoherence without refocusing. The randomized benchmark does not systematically refocus changes in frequency. To estimate the contribution to error from decoherence including refocusable decoherence, we performed the experiment of Fig. 3 without the refocusing pulse. This is essentially an on-resonance Ramsey experiment. It was not experimentally possible to eliminate the oscillatory shape of the curve by calibrating the frequency indicating that the oscillation was not simply caused by detuning from the resonant frequency. However, the shape is similar to what one would expect from a roughly periodic change in frequency that is not synchronized with the experiment. Such changes could come from magnetic field fluctuations and phase noise due to air currents in the paths of the two Raman beams. To estimate the contribution to the probability of error per step, we fitted an exponentially decaying $\text{cos}\left(t\right)$ curve to the points with time coordinates less than $220\mu \text{s}$. The extrapolation of the fitted curve (dashed) clearly deviates from the data. Note that for sinusoidal phase noise, the curve should be related to a decaying Bessel function. Fits to such a function also deviate from the experimental data, consistent with the phase fluctuations not being sinusoidal. Since the contribution to the probability of error is derived from the short-time behavior, the effect of the different models on the inferred probability of error per step is small. For the fit shown, the inferred contribution to the probability of error per step is 0.0090(7), larger than the error per step derived from Fig. 2. This is likely due to the fact that in the randomized sequences, the centering of the explicit $\pi$ pulses in their intervals reduces this contribution by refocusing.

Figure 5

Contribution of spontaneous emission to phase decoherence. To experimentally determine the contribution of spontaneous emission to decoherence, we applied the two Raman beams separately for half the time of each arm of the refocused decoherence measurement and compared the resulting data to that of Fig. 3 [32]. The points shown here were obtained by dividing the probabilities measured by the corresponding probabilities of Fig. 3, interpolating between the nearest points to match the time coordinates. The straight line shows the fitted exponential decay. The fit was weighted and used linear approximation to determine standard deviations of the points. The standard deviations used range from 0.003 to 0.015, which is substantially less than the apparent scatter of the plotted points. The inferred contribution to the error probability per step is 0.00038(3). This contribution can be estimated theoretically [32], which for the relevant configuration gives a value of approximately 0.0003.

Figure 6

Rabi flopping experiment. To determine the contribution to the probability of error per step due to pulse area error and associated decoherence, we performed a Rabi flopping experiment. We fitted the points to a decaying cosine curve with a possible phase offset and both linear and quadratic decay. Again, we restricted the fit to an initial segment of the data (black curve). The extrapolation (dashed curve) shows significant deviations. The random uncertainty in the points ranges from 0.002 to 0.007, less than the symbol size of the plotted points. The apparent scatter in the points near the end of the curve is likely due to slow fluctuations in pulse amplitude. The contribution to the probability of error per step as detected in this experiment is 0.006(3) if the calibration were based on this experiment. Automatically calibrated pulse times fluctuated by around $0.02\mu \text{s}$. For pulse times differing by this amount, the contribution to the error per step is 0.007(3).