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arXiv:quant-ph/0308016 · Physical Review Letters

Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation

Péter Jaksch, Anargyros Papageorgiou

Published Dec 19, 2003• 43 citations
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Abstract

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.

DOI
10.1103/physrevlett.91.257902

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arXiv:1505.08126•Leibniz international proceedings in informatics•May 29, 2015•3 citations
We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.
The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va . The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the Va for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.
It has been shown that, starting from the state |0>, in the general case, an arbitrary quantum state |ψ> cannot be prepared with exponential precision in polynomial time. However, we show that for the important special case when |ψ> represents discrete values of some real, continuous function ψ(x), efficient preparation is possible by applying the eigenvalue estimation algorithm to a Hamiltonian which has ψ(x) as an eigenstate. We construct the required Hamiltonian explicitly and present an iterative algorithm for removing unwanted superpositions from the output state in order to reach |ψ> within exponential accuracy. The method works under very general conditions and can be used to provide the quantum simulation algorithm with very accurate and general starting states.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size nnn of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If MMM is sss sparse, then the complexity is O~(sn∥M∥max⁡κ2/ϵ2)\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)O(sn∥M∥max​κ2/ϵ2), where κ\kappaκ is the condition number of the matrix of eigenvectors.
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum computing. In this work, we propose a quantum algorithm that given a non-normal matrix, outputs an estimate of an eigenvalue to within additive error εεε with probability at least 1−pfail1-p_{\rm fail}1−pfail​. Our estimation strategy is to sample points on the complex plane and examine the distance between the sampled point and the eigenvalues. We show that the distance is related to the smallest singular value of the shifted matrix, hence reducing the problem to ground state energy estimation via Hermitianization. With the knowledge of an eigenvalue, we are able to prepare the associated eigenvector using ground state preparation. Our estimating scheme can also be modified to approximate the extreme eigenvalue, and in particular the spectral gap. The algorithm is implemented based on the block encoding input model and requires O(κ2ε−(2m−1)log⁡(1/pfail))O(κ^2ε^{-(2m-1)}\log(1/p_{\rm fail}))O(κ2ε−(2m−1)log(1/pfail​)) queries to the block encoding oracle. Our algorithm is the first general eigenvalue algorithm that achieves this scaling. We also perform numerical simulation to validate the algorithms.
Quantum phase estimation algorithm (PEA) has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. In this paper, we show that the principal eigenvalue of an irreducible non-negative operator can be determined by using an equal superposition initial state in the PEA. This removes the necessity of the existence of an initial good approximate eigenvector. Moreover, we show that the success probability of the algorithm is related to the closeness of the operator to a stochastic matrix. Therefore, we draw an estimate for the success probability by using the variance of the column sums of the operator. This provides a priori information which can be used to know the success probability of the algorithm beforehand for the non-negative matrices and apply the algorithm only in cases when the estimated probability is reasonably high. Finally, we discuss the possible applications and show the results for random symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal elements.
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in O(log⁡2n)O(\log^2 n)O(log2n) parallel time with a total cost of O(nω+1)O(n^{ω+1})O(nω+1) Boolean operations. This Boolean complexity matches the best known rigorous O(log⁡2n)O(\log^2 n)O(log2n) parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases (mod  (2π))(\mod (2π))(mod(2π)) is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in O(log⁡2n)O(\log^2 n)O(log2n) parallel time and O(nω)O(n^ω)O(nω) Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.