Usefulness of an enhanced Kitaev phase-estimation algorithm in quantum metrology and computation

We analyze the performance of a generalized Kitaev's phase-estimation algorithm where $N$ phase gates, acting on $M$ qubits prepared in a product state, may be distributed in an arbitrary way. Unlike the standard algorithm, where the mean square error scales as $1/N$, the optimal generalizations offer the Heisenberg $1/N^2$ error scaling and we show that they are in fact very close to the fundamental Bayesian estimation bound. We also demonstrate that the optimality of the algorithm breaks down when losses are taken into account, in which case the performance is inferior to the optimal entanglement-based estimation strategies. Finally, we show that when an alternative resource quantification is adopted, which describes the phase estimation in Shor's algorithm more accurately, the standard Kitaev's procedure is indeed optimal and there is no need to consider its generalized version.

Figure 1

The circuit of a generalized Kitaev's algorithm. The $i$-th qubit receives the phase shift $m_i\phi$, while the total number of used gates is $\sum m_i=N$. A general POVM measurement is applied and the results are used to construct the phase estimator.

Figure 2

Plot showing the estimation costs with respect to the number of used resources. The black line connects the best numerically found solutions. Filled blue (red) circles show the results for ${\mathbf{m}}_2 \left({\mathbf{m}}_3\right)$, while lines are the corresponding analytical upper bounds. The dark-gray ribbon depicts the area between the Bayesian estimation optimum and the shot-noise limit given by ${\mathbf{m}}_{\otimes }$. Inset: Cost ratios of the best found parameter $\mathbf{m}$ to the Bayesian optimum. Ratios correspond to the costs presented in the main plot.

Figure 3

Plot showing the approximate performance of the enhanced Kitaev's algorithm for two lossy setups. Filled circles reveal the numerically computed cost values of the algorithm, solid lines are the cost limits for entanglement-free strategies, and dashed lines depict the ultimate bounds for the lossy setup where any entangled start states are permitted.