Happy new year everyone.

Subarea of Variational Quantum Algorithm(VQA) had developed drastically last year.

So, I introduce the variational methods of quantum chemistry using quantum computers as my first post of this year.

VQE:

Variational methods of quantum chemistry using quantum computers are all advanced methods of Variational Quantum Eigensolver (VQE) method.

VQE method is one of VQA used for quantum calculation. Second quantized hamiltonians can be expressed by Pauli operators. Hence, the eigenvaslues of quantum state can be calculated by quantum computers.

At first, second quantized hamiltonian that electrons are localized on sites, called Hubbard model is

$H=\sum_{i,j}t_{ij}c_i^\dagger c_j + \sum_{i,j,k,l}U_{ijkl}c_i^\dagger c_j^\dagger c_k c_l.$ -(1)

Then, $c_j^\dagger$ and $c_j$ are creation and annihilation operator of electron on j-th site.

i, j, k and l may be orbital in some cases. This hamiltonian is transformed into Pauli wards by Jordan-Wigner or Bravyi-Kitaev transformation[1]. Initial basis are also transformed in order to match Pauli words. Cluster is necessary too in order to derive the eigenvalues of trial states.

There are many types of cluster. For example, UCCSD[2] is as follows.

$T=\sum_{j\in ~occ., k\in ~vac.}u_{jk}(c_j^\dagger c_k + c_k^\dagger c_j)+\sum_{j,l\in ~occ.,k,m\in ~ vac.}u_{jklm}(c_j^\dagger c_l^\dagger c_k c_m + c_k^\dagger c_m^\dagger c_j c_l).$ -(2)

This cluster makes given superpositioned state by excitation operators of single and double electron and their conjugates. Cluster is the sum of operators. Hence, it must be translated to the operator products for sake quantum computer can process.

Clsuter must be decomposed by Suzuki-Trotter decomposition as follows.

$iT=\Pi_{j,k}exp(-i\theta_k/Nt_{jk}P_{jk})^N.$ -(3)

Then, $P_{jk}$ and $t_{jk}$ are translated operator of cluster and its coefficient, respectively.

N is the depth of cluster, number of repetition of cluster. $\theta_k$ is the variational variable.

By acting eq.(3) on initial state, trial state is expressed as follows.

$\mid \Phi_j \rangle = exp(iT)\mid \Phi_{ini.} \rangle.$ -(4)

Using this trial state, evaluation function of VQE is,

$F(\theta)= \langle \Phi_j \mid H \mid \Phi_j \rangle.$ -(5)

In case aimed state is excited state, deflation term,

$E_{def.}^j= \sum_{k<j}\mid \langle \Phi_k \mid \Phi_j \rangle \mid$ -(6)

is necessary for eq. (5). Next pert is the introduction of advanced VQE methods.

[1] Andrew Tranter, and et. al., Journal of Chemical Theory and Computation, 14(11), 5617-5630(2018)

[2] Panagiotis K.L. Barkoutsos, and et. al., Phys. Rev. A.98(2).022322(2018)

Review of Variational Quantum Eigensolver: part 1

Hikaru Wakaura