Efficient Algorithm for Optimizing Adaptive Quantum Metrology Processes

Quantum-enhanced metrology infers an unknown quantity with accuracy beyond the standard quantum limit (SQL). Feedback-based metrological techniques are promising for beating the SQL but devising the feedback procedures is difficult and inefficient. Here we introduce an efficient self-learning swarm-intelligence algorithm for devising feedback-based quantum metrological procedures. Our algorithm can be trained with simulated or real-world trials and accommodates experimental imperfections, losses, and decoherence.

Figure 1

Adaptive-feedback scheme for estimating an interferometric phase $\phi$. The input state $|{\Psi }_N⟩$ is fed into the unital quantum channel $\mathcal{C}$ one qubit at a time and the output qubit is measured or lost. The processing unit (PU) shifts the interferometric phase by $\Phi$ after each successful measurement prior to processing the next qubit.

Figure 2

(a) Decision tree representation of a GLS policy for $N=2$ (solid) and $N=3$ (entire tree). For each path in the tree, the inner nodes represent the applied feedback phases ${\Phi }_m$ and the leaf shows the final phase estimate $\tilde{\phi }$. At depth $m$, a measurement $u_{m+1}=0$ directs the path to the left and $u_{m+1}=1$ to the right. (b) Embedding the best policy ${\mathit{\varrho }}_N^{\prime }\in {\mathcal{P}}_N$ in the policy space ${\mathcal{P}}_{N+1}$, shown for $N=2$. From the best two-qubit policy ${\mathit{\varrho }}_2^{\prime }$, the policy ${\mathit{\varrho }}_3^{\prime }\in {\mathcal{P}}_3$ is generated as a guideline. The initial candidate policies for three input qubits are chosen according to probability density (4), indicated by the shaded area around ${\mathit{\varrho }}_3^{\prime }$. [For clarity, the $N=2$ case is depicted, although only candidate policies for $N>10$ are chosen according to (4).]

Figure 3

Holevo phase variance $V_H$ of PSO-optimized policies compared to other schemes vs the number of input qubits $N$ for (a) $|{\Psi }_N⟩$ and (b) $|0\cdots 00⟩$ as input states, respectively. The dashed line shows the SQL. Because of limited computational resources, some simulations are carried out only to $N\le 25$. (Loss rate $\eta$ is in percent; ${\sigma }_{{\mathit{n}}_x}={\sigma }_{{\mathit{n}}_y}={\sigma }_{{\mathit{n}}_z}=0.2{\sigma }_{\theta }$.)