A universal quantum circuit scheme for finding complex eigenvalues

Daskin, Ammar; Grama, Ananth; Kais, Sabre

doi:10.1007/s11128-013-0654-1

Cited by 26 publications

(24 citation statements)

References 35 publications

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“…In Ref. [42,43], a general circuit design consisting of 2log 2 N qubits and O(N 2 ) quantum gates is given for the simulation of a matrix of order N not necessarily Hermitian. In particular, it is shown that any matrix can be simulated on certain predetermined states by using log 2 N ancilla qubits.…”

Section: F Possible Applicationsmentioning

confidence: 99%

Obtaining a linear combination of the principal components of a matrix on quantum computers

Daskin

2016

Quantum Inf Process

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Principal component analysis is a multivariate statistical method frequently used in science and engineering to reduce the dimension of a problem or extract the most significant features from a dataset. In this paper, using a similar notion to the quantum counting, we show how to apply the amplitude amplification together with the phase estimation algorithm to an operator in order to procure the eigenvectors of the operator associated to the eigenvalues defined in the range [a, b], where a and b are real and 0 ≤ a ≤ b ≤ 1. This makes possible to obtain a combination of the eigenvectors associated to the largest eigenvalues and so can be used to do principal component analysis on quantum computers.

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Section: F Possible Applicationsmentioning

confidence: 99%

Obtaining a linear combination of the principal components of a matrix on quantum computers

Daskin

2016

Quantum Inf Process

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“…Some work has been done on this track to solve the resonance problem by quantum computers. By designing a general quantum circuit for non-unitary matrices, Daskin et al [36] explored the resonance states of a model non-Hermitian Hamiltonian. To be specific, he introduced a systematic way to estimate the complex eigenvalues of a general matrix using the standard iterative phase estimation algorithm with a programmable circuit design.…”

Section: Excited States and Resonancesmentioning

confidence: 99%

“…Then by complex-scaling method, the internal coordinates of the Hamiltonian is dilated by a complex factor η = αe −iθ such that H(x) → H(x/η) ≡ H η (x). We can solve the complex eigenvalue of H η (x) by the method D or using our previous method [36].…”

Section: Excited States and Resonancesmentioning

confidence: 99%

Quantum computing methods for electronic states of the water molecule

et al. 2019

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We compare recently proposed methods to compute the electronic state energies of the water molecule on a quantum computer. The methods include the phase estimation algorithm based on Trotter decomposition, the phase estimation algorithm based on the direct implementation of the Hamiltonian, direct measurement based on the implementation of the Hamiltonian and a specific variational quantum eigensolver, Pairwise VQE. After deriving the Hamiltonian using STO-3G basis, we first explain how each method works and then compare the simulation results in terms of gate complexity and the number of measurements for the ground state of the water molecule with different O-H bond lengths. Moreover, we present the analytical analyses of the error and the gate-complexity for each method. While the required number of qubits for each method is almost the same, the number of gates and the error vary a lot. In conclusion, among methods based on the phase estimation algorithm, the second order direct method provides the most efficient circuit implementations in terms of the gate complexity. With large scale quantum computation, the second order direct method seems to be better for large molecule systems. Moreover, Pairewise VQE serves the most practical method for near-term applications on the current available quantum computers. Finally the possibility of extending the calculation to excited states and resonances is discussed.

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“…where the first part of the matrix is equal to H with/without some normalization, and a • indicates another part of the matrix. As also mentioned in the introduction, these types of matrices are used in various contexts: either to design quantum circuits from matrix elements [11,12] or to be able to use it with the oblivious amplitude amplification [14], or to estimate unitary dynamic of a Hamiltonian through truncated Taylor series [13] and its generalized form [15].…”

Section: Proposed Eigenvalue Estimationmentioning

confidence: 99%

“…A quantum operator (or a dynamic of a system) can be implemented inside a larger system, where some reduced part of the system represents the action of the original system. This idea is used in different contexts such as designing circuits from matrix elements [11,12], and generating exponential of a matrix written as a sum of unitary matrices through Taylor series [13]. Moreover, in Ref.…”

Section: Introductionmentioning

confidence: 99%

Direct application of the phase estimation algorithm to find the eigenvalues of the Hamiltonians

Daskin

Kais

2018

Chemical Physics

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The eigenvalue of a Hamiltonian, H, can be estimated through the phase estimation algorithm given the matrix exponential of the Hamiltonian, exp(−iH). The difficulty of this exponentiation impedes the applications of the phase estimation algorithm particularly when H is composed of non-commuting terms. In this paper, we present a method to use the Hamiltonian matrix directly in the phase estimation algorithm by using an ancilla based framework: In this framework, we also show how to find the power of the Hamiltonian matrix-which is necessary in the phase estimation algorithm-through the successive applications. This may eliminate the necessity of matrix exponential for the phase estimation algorithm and therefore provide an efficient way to estimate the eigenvalues of particular Hamiltonians. The classical and quantum algorithmic complexities of the framework are analyzed for the Hamiltonians which can be written as a sum of simple unitary matrices and shown that a Hamiltonian of order 2 n written as a sum of L number of simple terms can be used in the phase estimation algorithm with (n + 1 + logL) number of qubits and O(2 a nL) number of quantum operations, where a is the number of iterations in the phase estimation. In addition, we use the Hamiltonian of the hydrogen molecule as an example system and present the simulation results for finding its ground state energy.

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A universal quantum circuit scheme for finding complex eigenvalues

Abstract: We present a general quantum circuit design for finding eigenvalues of non-unitary matrices on quantum computers using the iterative phase estimation algorithm. In addition, we show how the method can be used for the simulation of resonance states for quantum systems.

Cited by 26 publications

References 35 publications

Obtaining a linear combination of the principal components of a matrix on quantum computers

Obtaining a linear combination of the principal components of a matrix on quantum computers

Quantum computing methods for electronic states of the water molecule

Direct application of the phase estimation algorithm to find the eigenvalues of the Hamiltonians

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