Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.

Aug 31, 20260 citations

Effects of decoherence and imperfections for quantum algorithms

A. A. Pomeransky, O. V. Zhirov, Dima L. Shepelyansky

We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength $ε$ . The critical border $ε_c$ between two phases drops polynomially with the number of qubits $n_q$ as $ε_c \sim n_q^{-3/2}$ . In the regular phase $(ε< ε_c)$ the algorithm remains robust against imperfections showing the efficiency gain $ε_c / ε$ for $ε> 2^{-n_q/2}$ . In the chaotic phase $(ε> ε_c)$ the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength $ε$. The critical border $ε_c$ between two phases drops polynomially with the number of qubits $n_q$ as $ε_c \sim n_q^{-3/2}$. In the regular phase $(ε< ε_c)$ the algorithm remains robust against imperfections showing the efficiency gain $ε_c / ε$ for $ε> 2^{-n_q/2}$. In the chaotic phase $(ε> ε_c)$ the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.

arXiv:quant-ph/0407264Jul 30, 20040 citations

Phase diagram for the Grover algorithm with static imperfections

Pomeransky, A A, Zhirov, O V, Dima L. Shepelyansky

We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength $\\epsilon$ . The critical border $\\epsilon_c$ between two phases drops polynomially with the number of qubits $n_q$ as $\\epsilon_c \\sim n_q^{-3/2}$ . In the regular phase $(\\epsilon < \\epsilon_c)$ the algorithm remains robust against imperfections showing the efficiency gain $\\epsilon_c / \\epsilon$ for $\\epsilon \\gtrsim 2^{-n_q/2}$ . In the chaotic phase $(\\epsilon > \\epsilon_c)$ the algorithm is completely destroyed.We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength $\\epsilon$. The critical border $\\epsilon_c$ between two phases drops polynomially with the number of qubits $n_q$ as $\\epsilon_c \\sim n_q^{-3/2}$. In the regular phase $(\\epsilon < \\epsilon_c)$ the algorithm remains robust against imperfections showing the efficiency gain $\\epsilon_c / \\epsilon$ for $\\epsilon \\gtrsim 2^{-n_q/2}$. In the chaotic phase $(\\epsilon > \\epsilon_c)$ the algorithm is completely destroyed.

Jan 1, 200410 citations

ε

. The critical border

ε_c

between two phases drops polynomially with the number of qubits

n_q

ε_c \sim n_q^{-3/2}

. In the regular phase

(ε< ε_c)

the algorithm remains robust against imperfections showing the efficiency gain

ε_c / ε

for

ε> 2^{-n_q/2}

. In the chaotic phase

(ε> ε_c)

the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.

\\epsilon

. The critical border

\\epsilon_c

between two phases drops polynomially with the number of qubits

n_q

\\epsilon_c \\sim n_q^{-3/2}

. In the regular phase

(\\epsilon < \\epsilon_c)

the algorithm remains robust against imperfections showing the efficiency gain

\\epsilon_c / \\epsilon

for

\\epsilon \\gtrsim 2^{-n_q/2}

. In the chaotic phase

(\\epsilon > \\epsilon_c)

the algorithm is completely destroyed.