A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase accurate to 2 −n with the probability greater than 8/π 2 . By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.
A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase accurate to 2 −n with the probability greater than 8/π 2 . By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.
Abstract This paper is concerned with the phase estimation algorithm in quantum computing, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is approximated by Trotter or Taylor expansion methods; (3) random approximations are used for the unitary operator. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with error less or equal to 2 − n and probability at least 1 − ϵ , the required number of qubits is t ⩾ n + log 2 + δ 2 2 ϵ Δ E 2 . The parameter δ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and Δ E characterizes the spectral gap, i.e., the separation from the rest of the phase values. This analysis generalizes the standard result (Cleve et al 1998 Phys. Rev X 11 011020; Nielsen and Chuang 2002 Quantum Computation and Quantum Information ) by including these effects. More importantly, it shows that when δ < Δ E , the complexity remains the same. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.
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.
A quantum computation problem is discussed in this paper.Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence effect, which depends on the phase of quantum coherent state.Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced.And one of its most important applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details.The flow of phase estimation algorithm and the quantum circuit model are shown.And the error of the output phase value, as well as the probability of measurement, is analysed.The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.The researchis focused on how to determine the optimum number of qubit which satisfy the accuracy of phase estimation.
A quantum algorithm for the computation the order of an integer, which uses the quantum Fourier transform, is discussed. The cases of the exact and approximate Fourier transform are considered, and estimates of the probability of the successful solution of the problem that significantly improve the available results are obtained. The quantum algorithm for order finding is simulated using the QuantumCircuit package written in the computer algebra system Wolfram Mathematica , and its efficiency in combination with the approximate Fourier transform is demonstrated.
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop an algorithmic toolbox, quantum phase processing, that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigeninformation of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase related. As a notable application, we propose a quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropy estimation, demonstrating improvements or optimality for almost all cases.
Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. The standard quantum eigenvalue (phase) estimation algorithm accepts a Hermitian (unitary) matrix and a state in an unknown superposition of its eigenstates as input, and coherently records the estimates for real eigenvalues (eigenphases) in an ancillary register. Extension of quantum eigenvalue and phase estimation algorithms to the case of non-normal input matrices is obstructed by several factors such as non-orthogonality of eigenvectors, existence of generalized eigenvectors and the fact that eigenvalues may lie anywhere on the complex plane. In this work, we propose a novel approach for estimating the eigenvalues of non-normal matrices based on preparation of a state that we call the "resolvent state". We construct the first efficient algorithm for estimating the phases of the unimodular eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix, achieving complexities that match the best known results while operating under significantly relaxed assumptions on the non-real part of the spectrum. The resolvent-based approach that we introduce also extends to estimating eigenvalues that lie on a parametrized complex curve, subject to explicitly stated conditions, thereby paving the way for a new paradigm of parametric eigenvalue estimation.