Fig. 1 | npj Quantum Information

Fig. 1

From: Spectral quantum tomography

Fig. 1

Preliminary study of the numerical accuracy of the matrix-pencil method as a function of L, K and Nsamples. (Left) We use the matrix-pencil method with different Ls and Ks to estimate the eigenvalues of a random single-qubit channel, for Nsamples = 1000. On the vertical axis, we give the variance in the estimate of the eigenvalues: \(\Delta ^2 = \frac{1}{3}(\mathop {\sum}\nolimits_{j = 1}^{N = 3} | \lambda _j - \lambda _j^{{\mathrm{est}}}|^2)\). We see that, as long as the matrix-pencil parameter L is chosen away from 0 or K, the accuracy of the reconstructed signal is nearly independent of L. Furthermore, we see that higher K’s can achieve a lower Δ2. (Right) We generate a random single-qubit channel and set L = K/2. We plot Δ2 as a function of K for two different values of Nsamples = 1000 and Nsamples = 5000, showing how a larger Nsamples suppresses the total variance. We see that for constant Nsamples the accuracy of the method increases rapidly at first when K is increased, but it increases more slowly if K is already large. This can be explained by the fact that the signal decreases exponentially in K and so data points for large K have much lower signal-to-noise ratio. For both figures, random channels were generated using QuTip’s random TPCP map functionality, and measurement noise was approximated by additive Gaussian noise with standard deviation equal to \(1/\sqrt {N_{{\mathrm{samples}}}}\)

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