Self-Guided Quantum Tomography

We introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that Self-guided quantum tomography is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of postprocessed state estimation.

Figure 1

Estimating the state of a qubit via SGQT and the fidelity metric. On the left, each fidelity estimate was generated with $N={10}^2$ single shot experiments—on the right, $N={10}^4$. Each line represents one of three runs with $k={10}^3$ iterations.

Figure 2

The infidelity vs the number of iterations $k$ achieved by SGQT. Each line is the median performance of SGQT over 100 randomly (according to Haar measure) chosen pure states. The shaded regions represent the interquartile range of infidelities. The inset shows the performance as function $Nk$ by simply shifting the original lines by their corresponding $N$. The SGQT parameters chosen for these simulations where $a=3$, $A=0$, and $b=0.1$.

Figure 3

The infidelity vs the number of iterations $k$ achieved by SGQT for states of increasing qubit number (shown on the right). The number of experiments per fidelity estimate for all cases was $N={10}^4$. Each line is the median performance of SGQT over 100 randomly (according to Haar measure) chosen pure states, where the initial guess was a random perturbation of 0.01 standard deviation in each dimension of the parametrized space (this was done to more efficiently extract the asymptotic scaling). The shaded regions represent the interquartile range of infidelities. The inset shows the performance as a function of the number of qubits for fixed values of $k$ as marked. The SGQT parameters used for these simulations (and all further simulations) were $a=0.3$, $A=1000$, and $b=0.1$.

Figure 4

The rescaled infidelity (difference between the minimum achievable infidelity and actual infidelity) vs the number of iterations $k$ achieved by SGQT for $W$ states of increasing qubit number (shown on the right). Parameters are as in Fig. 3.

Figure 5

The infidelity vs the number of iterations $k$ achieved by SGQT for $W$ states of increasing qubit number. For each fidelity estimation measurement, an unknown zero-mean, 0.1 standard deviation, Gaussian random perturbation was applied to the target state. All other parameters are as in Fig. 3.