Error models in quantum computation: An application of model selection

Threshold theorems for fault-tolerant quantum computing assume that errors are of certain types. But how would one detect whether errors of the “wrong” type occur in one's experiment, especially if one does not even know what type of error to look for? The problem is that for many qubits a full state description is almost impossible to analyze due to the exponentially large state space, and a full process description requires even more resources. As a result, one simply cannot detect all types of errors. Here we show through a quantum state estimation example (on up to 25 qubits) how to attack this problem using model selection. We use, in particular, the Akaike information criterion. The example indicates that the number of measurements that one has to perform before noticing errors of the wrong type scales polynomially both with the number of qubits and with the error size.

Figure 1

The differences in AIC values $\Delta \text{AIC}$ for a state of $N=5$ qubits plotted against the total number of measurements. There are 21 measurements settings in this case, and the PI model contains 55 parameters. The simulation was run 100 times and the average $\Delta \text{AIC}$ is plotted. Error bars refer to the spread of $\Delta \text{AIC}$ over the 100 runs.

Figure 2

Plots for $q=0.02$. (a) The differences in AIC for several different numbers of qubits, plotted against the total number of measurements. Note that a single measurement on $N$ qubits yields $N$ binary outcomes. For very small numbers of measurements, $\Delta \text{AIC}$ approaches twice the difference in the number of parameters of the two models ($\approx N^3/3$). (b) The average number of measurements required to reach the point where both models are rated equally. Small even and odd numbers of qubits behave slightly differently.

Figure 3

(a) $\Delta \text{AIC}$ as a function of the number of measurements performed, for four different values of $q$ and $N=5$ qubits. (b) The minimum number of measurements $M$ needed to detect a perturbation of strength $q$, for both $N=5$ and 10 qubits.