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

Daskin, Ammar; Kais, Sabre

doi:10.1016/j.chemphys.2018.01.002

Cited by 15 publications

(20 citation statements)

References 27 publications

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“…It was proposed [17] that given a Hamiltonian H and large κ we can construct an approximated unitary operator U such that:…”

Section: B Direct Implementation Of Hamiltonian In First Order (Dirementioning

confidence: 99%

“…where |u a = B |0 a − |0 a . The complexity of constructing this gate has been analyzed before [17,[45][46][47]. Since Givens rotation G L−2,L−1 (θ L−1 ) can nullify B 0,L−1 , it can also nullify all B j,L−1 for j = L − 1 and update B L−1,L−1 to 1 due to B's special form.…”

Section: C2 Direct Pea (First Order)mentioning

confidence: 99%

“…Babbush et al [15] further shows that it is possible to reduce the gate depth of the circuit to O(n) by using plane wave orbitals. Recently, a direct circuit implementation of the Hamiltonian within the phase estimation (Direct-PEA) is presented by authors of paper [16][17][18]: the circuit designs are provided to the time evolution operator by using the truncated series such as U = I − iH κ and U = tH + i(I − t 2 H 2 2 ), in which κ and t are parameters to restrict truncation error. These unitary operators are much simpler to implement than those of a Trotter decomposition, and can be also used to calculate ground state energies of molecular Hamiltonians.…”

Section: Introductionmentioning

confidence: 99%

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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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“…It was proposed [17] that given a Hamiltonian H and large κ we can construct an approximated unitary operator U such that:…”

Section: B Direct Implementation Of Hamiltonian In First Order (Dirementioning

confidence: 99%

Section: C2 Direct Pea (First Order)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum computing methods for electronic states of the water molecule

et al. 2019

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“…The construction of V is achieved through a permutation matrix Π (also used in Ref. [17]) which realigns matrix elements to obtain H 2 on the diagonal and is defined as:…”

Section: A Circuit Design Forū (T)mentioning

confidence: 99%

“…In Ref. [17], successive applications of I + iH/µ with µ ≥ 10||H|| are used to obtain the eigenvalue of H from the sine value of the phase in the phase estimation algorithm. Here, note that these approaches are assumed that the Hamiltonian is given as a sum of simple unitary matrices.…”

Section: Introductionmentioning

confidence: 99%

A generalized circuit for the Hamiltonian dynamics through the truncated series

Daskin

Kais

2018

Quantum Inf Process

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In this paper, we present a method for the Hamiltonian simulation in the context of eigenvalue estimation problems which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, H(t), through the truncated Taylor series method described by Berry et al. [1]. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable t in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires O(Ln) number of quantum gates for the simulation of a molecular Hamiltonian. Here, n is the number of states of a spin orbital, and L is the number of terms in the molecular Hamiltonian and generally bounded by O(n 4 ). We also discuss how to use the circuit in adaptive processes and eigenvalue related problems along with a slight modified version of the iterative phase estimation algorithm. In addition, a simple divide and conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by O(poly(n)) for a matrix with poly(n)−sparsity.

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Context-aware quantum simulation of a matrix stored in quantum memory

Daskin

Bian

Xia

et al. 2019

Quantum Inf Process

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In this paper a storage method and a context-aware circuit simulation idea are presented for the sum of block diagonal matrices. Using the design technique for a generalized circuit for the Hamiltonian dynamics through the truncated series, we generalize the idea to (0-1) matrices and discuss the generalization for the real matrices. The presented circuit requires O(n) number of quantum gates and yields the correct output with the success probability depending on the number of elements: for matrices with poly(n), the success probability is 1/poly(n). Since the operations on the circuit are controlled by the data itself, the circuit can be considered as a context aware computing gadget. In addition, it can be used in variational quantum eigensolver and in the simulation of molecular Hamiltonians.

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

Cited by 15 publications

References 27 publications

Quantum computing methods for electronic states of the water molecule

Quantum computing methods for electronic states of the water molecule

A generalized circuit for the Hamiltonian dynamics through the truncated series

Context-aware quantum simulation of a matrix stored in quantum memory

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