Adaptive Bayesian quantum algorithm for phase estimation

Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary $N_{\mathrm{tot}}$ times, the variance of the estimates can scale as $O(1/N_{\mathrm{tot}}^2)$, compared to the best “classical” strategy with $O(1/N_{\mathrm{tot}})$. The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling $O(1/N_{\mathrm{tot}}^2)$. Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps.

Figure 1

The quantum circuit $(n,\varphi )$.

Figure 2

Unnormalized posterior distribution generated from 20 applications of $\widehat{U}\left(\theta \right)$ in the circuits (a) (1,0), (b) $(1,\pi /2),$ and (c) (2,0), with $\theta =3\pi /4$ [vertical line]. (d) By combining the results from (a)–(c), erroneous peaks can be reduced in size.

Figure 3

The circuits executed at each iteration step of our algorithm and the posterior distribution at the end of the step. The initial conditions were $\theta =\frac{2\pi }{3}$ (vertical line), $N_{\text{tot}}=300$ and $\beta =0.9$ ($n_{\text{opt}}=5$). The regions ${\mathrm{\Theta }}_i$ are shaded.

Figure 4

Simulated values of the MAE and MSE against total number of unitary applications for different phase-estimation algorithms with $\alpha =1$ and (a), (b) $\beta =1$; (c), (e) $\beta =0.99$; (d), (f) $\beta =0.9$. Error bars are plotted between the minimum and maximum average losses achieved after 1000 samples.