User-specified random sampling of quantum channels and its applications

Random samples of quantum channels have many applications in quantum information processing tasks. Due to the Choi-Jamiołkowski isomorphism, there is a well-known correspondence between channels and states, and one can imagine adapting state sampling methods to sample quantum channels. Here, we discuss such an adaptation, using the Hamiltonian Monte Carlo method, a well-known classical method capable of producing high-quality samples from arbitrary, user-specified distributions. Its implementation requires an exact parametrization of the space of quantum channels, with no superfluous parameters and no constraints. We construct such a parametrization, and demonstrate its use in three common channel sampling applications.

Figure 1

Size $s_{\lambda }$ and credibility $c_{\lambda }$ of the BLRs ${\mathcal{R}}_{\lambda }$, plotted against ${log}_{10}\lambda$, for (a) the qubit amplitude-damping channel, and (b) the qutrit amplitude-damping channel. The red vertical dashed lines mark the respective critical $\lambda$ values ${\lambda }_{\text{crit}}=0.0073$ for (a), and ${\lambda }_{\text{crit}}=3.5598\times {10}^{-20}$ for (b). These identify the plausible regions.

Figure 2

The marginal likelihood $L\left(D\right|F_{\mathrm{avg}})$, computed using the iterative procedure of Ref. [25] and HMC with our channel parametrization.

Figure 3

(a) Size (blue) and credibility (green) curves for the bounded likelihood intervals for $F_{\mathrm{avg}}$. The red vertical dashed line marks the critical value of $\lambda$, at ${\lambda }_{\text{crit}}=0.3819$. (b) SCI for $F_{\mathrm{avg}}$. The blue curve indicates the boundaries of the SCIs for different credibility values. The black horizontal line marks the plausible interval and the arrow indicates the true value of $F_{\mathrm{avg}}=0.7333$.

Figure 4

The marginal likelihood $L\left(D\right|F_{\min })$, computed using the iterative procedure of Ref. [25] and HMC with our channel parametrization.

Figure 5

The hierarchy of the five candidate models for the example in Sec. 4c.

Figure 6

Average gate fidelity. (a) The green dots depict the MC values of $P_{r,0}\left(F_{\mathrm{avg}}\right)$, and the black curve is fitted to them; the inset table reports the fitting parameters. The values of ${\tilde{P}}_{r,0}\left(F_{\mathrm{avg}}\right)$ are traced out by the blue dots. (b) The blue dots show the MC values of ${\tilde{P}}_{r,0}\left(F_{\mathrm{avg}}\right)$ after subtracting the straight line $\frac{3}{2}(F_{\mathrm{avg}}-\frac{1}{3})$. The blue curve, a truncated Fourier series, is fitted to the dots. (c) Fourier amplitudes for ${\tilde{P}}_{r,0}\left(F_{\mathrm{avg}}\right)$. The high-frequency noise is removed from the fit in (b) by discarding the red amplitudes whose magnitude is less than $2\%$ of that of largest amplitude.

Figure 7

Minimum gate fidelity. (a) The green dots depict the MC values of $P_{r,0}\left(F_{\min }\right)$, and the black curve is fitted to them; the inset table reports the fitting parameters. The values of ${\tilde{P}}_{r,0}\left(F_{\min }\right)$ are traced out by the blue dots. (b) The blue dots show the MC values of ${\tilde{P}}_{r,0}\left(F_{\min }\right)$ after subtracting the straight line $F_{\min }$. The blue curve, a truncated Fourier series, is fitted to the dots. (c) Fourier amplitudes for ${\tilde{P}}_{r,0}\left(F_{\min }\right)$. The high-frequency noise is removed from the fit in (b) by discarding the red amplitudes whose magnitude is less than $2\%$ of that of largest amplitude.