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arXiv:quant-ph/0610214 · Physical Review A

Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark

Miroslav Dobšíček, Göran Johansson, Vitaly S. Shumeiko, Goran P. Wendin

Published Sep 19, 2007• 208 citations
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Abstract

We discuss the implementation of an iterative quantum phase estimation algorithm with a single ancillary qubit. We suggest using this algorithm as a benchmark for multiqubit implementations. Furthermore, we describe in detail the smallest possible realization, using only two qubits, and exemplify with a superconducting circuit. We discuss the robustness of the algorithm in the presence of gate errors, and show that seven bits of precision is obtainable, even with very limited gate accuracies.

DOI
10.1103/physreva.76.030306

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We discuss the implementation of an iterative quantum phase estimation algorithm with a single ancillary qubit. We suggest using this algorithm as a benchmark for multiqubit implementations. Furthermore, we describe in detail the smallest possible realization, using only two qubits, and exemplify with a superconducting circuit. We discuss the robustness of the algorithm in the presence of gate errors, and show that seven bits of precision is obtainable, even with very limited gate accuracies.
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We report the first experimental demonstration of a nuclear magnetic resonance (NMR) realization of iterative phase estimation algorithms. Using feedback and iterations, we experimentally obtain the phase with 6 bits of precision on a two-qubit NMR quantum computer. Furthermore, we experimentally demonstrate the effect of gate noise on the iterative phase estimation algorithm. Our experimental results show that errors of measurements of the phase depend strongly on the precision of coupling gates. This experiment can be used as a benchmark for multi-qubit realizations of quantum information processing and precision measurements.
Abstract An elementary algorithm of quantum phase estimation based on the modified Kitaev algorithm is implemented on two qubits of an IBM quantum processor. This algorithm includes adiabatic preparation of the initial state, controlled phase shift with allowance for the results of previous measurements of qubit states, and single measurement of the qubit quantum state in order to obtain each significant bit of the measured phase. Classical error correction is applied to determine the correct sequence of bits, which makes it possible to eliminate the influence of limited accuracy of two-qubit gates.
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.
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.