# The lecture of Subspace-Search Variational Quantum Eigensolver method for beginners

I will introduce about Subspace-Search Variational Quantum Eigensolver (SSVQE) method[1], that is the most famous method to calculate the energies of excited states of quantum systems as Variational Quantum Deflation (VQD) method. Regardless of its simplicity, this algorithm is very witty. Moreover, this algorithm has the potential. In fact, the bow of this algorithm has applicated for Multiscale-Contracted VQE (MCVQE)[2] method and Variational Quantum State Eigensolver (VQSE)[3] method.

First of all, the space of quantum states include ground and excited states is orthonormal. The space of initial qubit states is also orthonormal.

Hence, the state in their space can be expressed by the sum of states. Transformation between them can be realized. The aim of this algorithm is to find the transformation matrix that transforms initial basis sets to states of quantum systems.

﻿$\mid \Phi_{ini.} \rangle = c_0\mid 1000 \rangle + c_1\mid 0110 \rangle + c_2\mid 1100 \rangle + c_3\mid 0010 \rangle$﻿

﻿$U\mid \Phi_{ini.} \rangle = c_0\mid \Phi_0 \rangle + c_1\mid \Phi_1 \rangle + c_2\mid \Phi_2 \rangle + c_3\mid \Phi_3 \rangle$﻿ -(1)

The expectation value of hamiltonian for eq.(1) can be assumed as the summation of each expectation value for each state. Left terms are,

﻿$F=\sum_{j=0}^{3}c_j\langle \Phi_j \mid H \mid \Phi_j \rangle$﻿ -(2).

Each term can be calculated by the circuit of ordinary VQE method. This algorithm is performed by optimizing eq.(2).

Then, coefficients have the relation ,

﻿$c_j>c_k~(k>j)$﻿ -(3),

in order to derive the energy levels in increasing order. Next is the result of calculation.

[1]Nakanishi, K. M., Phys. Rev. Research 1, 033062 (2019)

[2]Parrish, R. M. and et. al., arXiv quant-ph:1901.01234v2(2019)

[3]Cerezo, M., and et. al., arXiv quant-ph:2004.01372v1(2020)

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