Householder methods for quantum circuit design

Urı́as, Jesús; Quiñones, Diego A.

doi:10.1139/cjp-2015-0490

Cited by 6 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also note that U f is a special case of the general Householder transformation (or reflection). Householder transformations are used as ingredients to simulate arbitrary unitary matrices [28][29][30][31] and hence can be efficiently simulatable on quantum computers.…”

Section: Numerical Examplementioning

confidence: 99%

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

Daskin

2016

Quantum Inf Process

View full text Add to dashboard Cite

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.

show abstract

Section: Numerical Examplementioning

confidence: 99%

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

Daskin

2016

Quantum Inf Process

View full text Add to dashboard Cite

show abstract

“…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%

Quantum computing methods for electronic states of the water molecule

et al. 2019

View full text Add to dashboard Cite

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.

show abstract

“…For a normalized L-dimensional column-vector |x j ∈ C ⊗l and the identity matrix I, R j = I−2 |x j x j | is an Householder transformation describing a reflection operator around the vector |x j . On quantum computers, a Householder transformation [22] can be implemented by using O(2 l ) total number of two-and one-qubit quantum gates [23][24][25] ( In Ref. [23], an implementation with the same complexity is also presented for a general version of the Householder transformation: i.e., I − (e iϕ − 1) |x j x j |).…”

Section: B Circuit Implementation Of Bmentioning

confidence: 99%

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

Daskin

Kais

2018

Chemical Physics

View full text Add to dashboard Cite

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.

show abstract

Householder methods for quantum circuit design

Cited by 6 publications

References 22 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

Contact Info

Product

Resources

About