Readout rebalancing for near-term quantum computers

Readout errors are a significant source of noise for near-term intermediate-scale quantum computers. Mismeasuring a qubit as $|1\rangle$ when it should be $|0\rangle$ occurs much less often than mismeasuring a qubit as $|0\rangle$ when it should have been $|1\rangle$. We make the simple observation that one can improve the readout fidelity of quantum computers by applying targeted $X$ gates prior to performing a measurement. These $X$ gates are placed so that the expected number of qubits in the $|1\rangle$ state is minimized. Classical postprocessing can undo the effect of the $X$ gates so that the expectation value of any observable is unchanged. This protocol is designed to have no effect when readout errors are symmetric. We show that the statistical uncertainty following readout error corrections is smaller when using readout rebalancing. The statistical advantage is circuit and computer dependent and is demonstrated for the $W$ state, a Grover search, and for a Gaussian state. The benefit in statistical precision is most pronounced (and nearly a factor of 2 in some cases) when states with many qubits in the excited state have high probability.

Figure 1

An illustration of the readout rebalancing protocol. Here $U_{\mathrm{circuit}}$ represents the state preparation that must happen for each measurement and the readout error mitigation represented by an inverted response matrix $R^{-1}$ (in practice, a more sophisticated readout error mitigation scheme may be used) is performed on an ensemble of measured states. From the measured values of the qubits of the first circuit which uses a small fraction of the total number of runs one then determines which qubits have $\langle q_i\rangle >0.5$ and should, therefore, be flipped (the first and fifth in our example). One then runs the remaining large fraction of the runs on the modified second circuit.

Figure 2

Top: An example response matrix for the IBM Q Tokyo machine with five qubits. Bottom: the probability to measure the correct string (diagonal elements of the top plot) as a function of the number of 0's in the bitstring.

Figure 3

The inverted $W$ state (top) and Grover state with the 11111 oracle (bottom) for the truth counts ($t$), the raw counts ($m$), and the readout corrected counts ($\widehat{t}$).

Figure 4

Top: the true counts for the truncated Gaussian states with fixed variance and shifted means. Middle (right): the average value (standard deviation) of the state when converting the bitstring to base 10 as a function of the Gaussian mean. $\langle O\rangle$ is computed over a particular set of measurements, and then this procedure is repeated 1000 times to compute the mean and standard deviation. Error bars represent statistical uncertainties.

Figure 5

The probability that the rebalancing map $m$ is optimal $=(1,1,1,1,1)$ as a function of the number of initial measurements reserved for determining the rebalancing map. The probability is determined by averaging over 20 runs.

Figure 6

The average rebalancing map value for each of the five qubits and for $\mu =-0.75$ (top) and $\mu =0.75$ (bottom). A value of 0 means that the qubit is untouched and a value of 1 represents a qubit that is flipped. The probability is determined by averaging over 20 runs, and error bands represent the standard error.