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arXiv:0904.3426 · Journal of Physics A Mathematical and Theoretical

Iterative phase estimation

Caleb J O'Loan

Published Dec 4, 2009• 18 citations
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Abstract

We give an iterative algorithm for phase estimation of a parameter θ, which is within a logarithmic factor of the Heisenberg limit. Unlike other methods, we do not need any entanglement or an extra rotation gate which can perform arbitrary rotations with almost perfect accuracy: only a single copy of the unitary channel and basic measurements are needed. Simulations show that the algorithm is successful. We also look at iterative phase estimation when depolarizing noise is present. It is seen that the algorithm is still successful provided the number of iterative stages is below a certain threshold.

DOI
10.1088/1751-8113/43/1/015301

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We give an iterative algorithm for phase estimation of a parameter θ, which is within a logarithmic factor of the Heisenberg limit. Unlike other methods, we do not need any entanglement or an extra rotation gate which can perform arbitrary rotations with almost perfect accuracy: only a single copy of the unitary channel and basic measurements are needed. Simulations show that the algorithm is successful. We also look at iterative phase estimation when depolarizing noise is present. It is seen that the algorithm is still successful provided the number of iterative stages is below a certain threshold.
In this work we consider Kitaev's algorithm for quantum phase estimation. We analyze the use of phase shifts that simplify the estimation of successive bits in the estimation of unknown phase φ, By using increasingly accurate shifts we reduce the number of measurements to the point where only a single measurement is needed for each additional bit. This results in an algorithm that can estimate φ to m + 2 bits of accuracy with probability at least 1- ϵ using Nϵ+ m measurements, where Nϵis a quantity that depends only on ϵ and the particular sampling algorithm. We present different sampling algorithms and study the exact number of measurements needed through careful numerical evaluation, and provide theoretical bounds and numerical values for Nϵ.
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Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision εεε in Heisenberg-limited time T=Θ(1/ε)T=Θ(1/ε)T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(1εlog⁡(δ−1))T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)T=O(ε1​log(δ−1)) in both the precision εεε and failure probability δδδ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.