We address the problem of estimating the phase $\ensuremath{\phi}$ given $N$ copies of the phase-rotation gate ${u}_{\ensuremath{\phi}}$. We consider, for the first time, the optimization of the general case where the circuit consists of an arbitrary input state, followed by any arrangement of the $N$ phase rotations interspersed with arbitrary quantum operations, and ending with a general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate $\stackrel{\texttildelow{}}{\ensuremath{\phi}}$ for $\ensuremath{\phi}$ depends only on the difference $\stackrel{\texttildelow{}}{\ensuremath{\phi}}\ensuremath{-}\ensuremath{\phi}$, the optimal scheme has a very simple fixed form. This implies that an optimal general phase estimation procedure can be found by just optimizing the amplitudes of the initial state.We address the problem of estimating the phase $\ensuremath{\phi}$ given $N$ copies of the phase-rotation gate ${u}_{\ensuremath{\phi}}$. We consider, for the first time, the optimization of the general case where the circuit consists of an arbitrary input state, followed by any arrangement of the $N$ phase rotations interspersed with arbitrary quantum operations, and ending with a general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate $\stackrel{\texttildelow{}}{\ensuremath{\phi}}$ for $\ensuremath{\phi}$ depends only on the difference $\stackrel{\texttildelow{}}{\ensuremath{\phi}}\ensuremath{-}\ensuremath{\phi}$, the optimal scheme has a very simple fixed form. This implies that an optimal general phase estimation procedure can be found by just optimizing the amplitudes of the initial state.

arXiv:quant-ph/0609160Physical Review LettersMar 1, 200794 citations

Phase Estimation and Its Applications

Weng-Long Chang, Athanasios V. Vasilakos

Studies in big dataDec 21, 20200 citations

Phase estimation Under energy conservation

Sen Han, Xueyuan Hu

Quantum Information ProcessingApr 26, 20230 citations

Fourier-transform quantum phase estimation with quantum phase noise

François Chapeau‐Blondeau, Étienne Belin

For the fundamental task of estimating a phase on an arbitrary quantum process, a variant of Fourier-based quantum phase estimation is devised, which uses a probing signal of multiple entangled qubits. For simple practical implementation, each probing qubit can be applied and measured separately. When the qubits are optimally entangled, the Heisenberg enhanced scaling of the estimation efficiency is obtained. The phase estimation protocol can be applied equally in the presence of quantum phase noise. This enables us to investigate the impact of a generic quantum phase noise on the performance of the Fourier-based phase estimation. Especially it reveals that the strategy found optimal with no noise, gradually loses its optimality as the noise increases. Also, in contrast to the noise-free situation, with noise the presence of entanglement is no longer uniformly beneficial to estimation; there exists an optimal amount of entanglement to maximize the efficiency and above which it becomes detrimental. The results contribute to better knowledge of quantum noise and entanglement for quantum signal and information processing.For the fundamental task of estimating a phase on an arbitrary quantum process, a variant of Fourier-based quantum phase estimation is devised, which uses a probing signal of multiple entangled qubits. For simple practical implementation, each probing qubit can be applied and measured separately. When the qubits are optimally entangled, the Heisenberg enhanced scaling of the estimation efficiency is obtained. The phase estimation protocol can be applied equally in the presence of quantum phase noise. This enables us to investigate the impact of a generic quantum phase noise on the performance of the Fourier-based phase estimation. Especially it reveals that the strategy found optimal with no noise, gradually loses its optimality as the noise increases. Also, in contrast to the noise-free situation, with noise the presence of entanglement is no longer uniformly beneficial to estimation; there exists an optimal amount of entanglement to maximize the efficiency and above which it becomes detrimental. The results contribute to better knowledge of quantum noise and entanglement for quantum signal and information processing.

Signal ProcessingDec 18, 201923 citations

Heisenberg-limited Bayesian phase estimation with low-depth digital quantum circuits

Su Direkci, Ran Finkelstein, Manuel Endres +1 more

Abstract Optimal phase estimation protocols require complex state preparation and readout schemes, generally unavailable or unscalable in many quantum platforms. We develop a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits. This is done by approximating the optimal initial states with products of Greenberger-Horne-Zeilinger states for Gaussian prior phase distributions with arbitrary widths. We study various protocols that employ this class of states with different levels of measurement and post-processing complexities, and obtain improvement compared to previously proposed schemes. We then use our scheme to address phase slip errors and laser noise, which impose a major limitation in Bayesian phase estimation and atomic clocks. Based on our scheme, we develop an efficient protocol to suppress this noise that outperforms existing methods.Abstract Optimal phase estimation protocols require complex state preparation and readout schemes, generally unavailable or unscalable in many quantum platforms. We develop a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits. This is done by approximating the optimal initial states with products of Greenberger-Horne-Zeilinger states for Gaussian prior phase distributions with arbitrary widths. We study various protocols that employ this class of states with different levels of measurement and post-processing complexities, and obtain improvement compared to previously proposed schemes. We then use our scheme to address phase slip errors and laser noise, which impose a major limitation in Bayesian phase estimation and atomic clocks. Based on our scheme, we develop an efficient protocol to suppress this noise that outperforms existing methods.

arXiv:2407.06006npj Quantum InformationJan 15, 20264 citations

We address the problem of estimating the phase $\ensuremath{\phi}$ given

N

copies of the phase-rotation gate ${u}_{\ensuremath{\phi}}$. We consider, for the first time, the optimization of the general case where the circuit consists of an arbitrary input state, followed by any arrangement of the

N

phase rotations interspersed with arbitrary quantum operations, and ending with a general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate $\stackrel{\texttildelow{}}{\ensuremath{\phi}}$ for $\ensuremath{\phi}$ depends only on the difference $\stackrel{\texttildelow{}}{\ensuremath{\phi}}\ensuremath{-}\ensuremath{\phi}$, the optimal scheme has a very simple fixed form. This implies that an optimal general phase estimation procedure can be found by just optimizing the amplitudes of the initial state.