Quantum State Learning via Single-Shot Measurements

A novel machine-learning algorithm based on single-shot measurements, named single-shot measurement learning, is demonstrated achieving the theoretical optimal accuracy. The method is at least as efficient as existing tomographic schemes and computationally much less demanding. The merits are attributed to the inclusion of weighted randomness in the learning rule governing the exploration of diverse learning routes. These advantages are explored experimentally by a linear-optical setup that is designed to draw the fullest potential of the proposed method. The experimental results show an unprecedented high level of accuracy for qubit-state learning and reproduction exhibiting (nearly) optimal infidelity scaling, $O(N^{-0.983})$, for the number $N$ of unknown state copies, down to $<{10}^{-5}$ without any compensation for experimental nonidealities. Extension to high dimensions is discussed with simulation results.

Figure 1

(a) Schematic of the SSML for unknown state identification. (b) Experimental setup involving linear optics. $\mathbf{P}$, $\mathbf{U}$, $\mathbf{M}$, and $\mathbf{F}$ denote state preparation, unitary operation, measurement, and feedback, respectively. The auxiliary beam is the fundamental laser light whose center wavelength (780 nm) is the same as the photons. BS: beam-splitter; SHG: second-harmonic generator; BBO: $\beta \text{−}\mathrm{Ba}{\mathrm{B}}_2{\mathrm{O}}_4$ crystal (thickness 1 mm); M: mirror; FM: flip mirror; IF: interference filter; FC: fiber coupler; SMF: single-mode fiber; FPC: fiber paddle polarization controller; GTP: Glan-Thompson polarizer; Q(H)WP: rotatable quarter(half)-wave plate; PM: polarimeter; WP: Wollaston prism; $H(V)$: horizontal (vertical) polarization; SPD: single-photon detector; $t$: trigger; $s$: success; $f$ failure; TDC: time-to-digital converter.

Figure 2

(a) Arrival time distribution of output photons relative to the trigger signal (bin size is about 81 ps, collection time: 10 h, Time-to-digital conversion by Qutools quTAU). Photons are initialized to horizontal polarization and the wave plates in $\mathbf{P}$ and $\mathbf{U}$ are omitted during this measurement. The curves are Gaussian fits to the measured data. The center peak of the $f$ counts shows the (nonideal) extinction ratio of polarizers. (b) Dependence of SNR of the measurement setup on the coincidence window. The window size (25 bins $\simeq 2\mathrm{ns}$) sets an SNR of $1.55(24)\times {10}^6$.

Figure 3

Accuracy of state estimation versus the amount of used resources. (a) Independently measured infidelity $\epsilon$. (b) Monitored infidelity $(1+M_S{)}^{-1}$. (c) Infidelity estimated by the rotation angles of the wave plates. The learning accuracy defined by the number of consecutive success detections matches reasonably well with the measured infidelity as shown in the inset of (b). The (yellow) shaded regions in (a) and (b) denote the measurement limit given by the finite SNR of the current setup. See the main texts for more details.

Figure 4

Numerical simulation results for higher dimensional SSML. (a) $d=3$, (b) $d=4$, (c) $d=5$, (d) $d=6$. All the fitted curves (solid lines) verify $O(N^{-1})$ scaling.