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International Journal of Theoretical Physics

Entanglement in Phase Estimation Algorithm and Quantum Counting Algorithm

Jianing Tan, Zhihao Liu, Hanwu Chen

Published Apr 13, 2020• 4 citations
Publisher
DOI
10.1007/s10773-019-04341-y

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The Kitaev's Phase Estimation Algorithm (also referred to as quantum eigenvalue estimation algorithm) is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. We explain the algorithm and indicate an example of the determination of the eigenvalue of the T gate. Then we introduce the quantum counting algorithm that is based on the quantum phase estimation algorithm and on the Grover's search algorithm. Quantum counting algorithm is a quantum algorithm for counting the number of solutions for a given search problem.
Quantum phase estimation algorithm (PEA) is one of the most important algorithms in early studies of quantum computation. However, we find that the PEA is not an unbiased estimation, which prevents the estimation error from achieving an arbitrarily small level. In this paper, we propose an unbiased phase estimation algorithm (UPEA) based on the original PEA. We also show that a maximum likelihood estimation (MLE) post-processing step applied on UPEA has a smaller mean absolute error than MLE applied on PEA. In the end, we apply UPEA to quantum counting, and use an additional correction step to make the quantum counting algorithm unbiased.
Implementations of the quantum phase estimation and quantum counting algorithms with qudits are proposed. The construction of the basic building blocks of the algorithms is described in detail and the fidelity of the quantum counting algorithm with qubits and qudits is simulated numerically. Along with the exponential increase of the size of the target register, the simulations with qudits demonstrate significantly higher probability of finding the number of solutions than qubits and more consistent performance for a different number of solutions.