Machine Learning for Precise Quantum Measurement

Adaptive feedback schemes are promising for quantum-enhanced measurements yet are complicated to design. Machine learning can autonomously generate algorithms in a classical setting. Here we adapt machine learning for quantum information and use our framework to generate autonomous adaptive feedback schemes for quantum measurement. In particular, our approach replaces guesswork in quantum measurement by a logical, fully automatic, programable routine. We show that our method yields schemes that outperform the best known adaptive scheme for interferometric phase estimation.

Figure 1

Adaptive feedback scheme for interferometric phase estimation: Mach-Zehnder interferometer with an unknown phase difference $\phi$ between the two arms and an additional controllable phase shifter $\Phi$. The input state $|{\Psi }_N⟩$ is stored in a quantum memory and one qubit at a time is transformed into a photonic qubit and sent through the interferometer. The processing unit (PU) sets the value of the phase shifter $\Phi$ depending on the measurement outcome of the single photon detectors $c_0$ and $c_1$. Adaptive feedback at step $m=2$ is depicted. That is, two of the $N$ input photons (the lowest two circles) have been sent through the interferometer and measured in previous steps.

Figure 2

Decision tree representations of two adaptive feedback policies for $N=6$ photons. The PSO-generated policy is graphed in blue solid lines, the BWB policy in gray dotted lines. All $2^N$ possible experimental runs are represented by paths in the tree. The path corresponding to an experiment with detections $u_1{u}_2\dots u_6=100000$ is marked by red diamonds, the path corresponding to the detections $u_1{u}_2\dots u_6=011010$ is highlighted by green squares. For each path in the tree, the inner nodes represent the applied feedback phases ${\Phi }_m$ and the leaf shows the final phase estimate $\tilde{\phi }$.

Figure 3

(a) Holevo phase variance $V_{\phi }$ of the PSO-optimized policies in comparison to the BWB policy and globally optimal policies for varying operational cost $N$. The blue shaded area shows the domain of quantum-enhanced measurements. (b) Performance of the PSO policies with probability of photon loss $\eta$. All curves follow a power law for $N\ge 7$ indicated by solid lines.