I&#39;ll introduce about Multiscale-Contracted Variational Quantum Eigensolver (MCVQE)[1].This algorithm had been famous temporary in 2019.This algorithm is expected that it can derive the excited states of large molecules in high accuracy.Transformation of hamiltonian and basis is same as other VQE methods. Though, final step of calculation is unique.This algorithm diagonalize the Configuration Interaction Single state after convergence.This is the superpositioned state among ground and single electron excited states. CIS states are expressed as follows, ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo><msub><mi mathvariant="normal">Φ</mi><mrow><mi>C</mi><mi>I</mi><mi>S</mi></mrow></msub><mo stretchy="false">⟩</mo><mo>=</mo><msub><mo>∑</mo><mi>θ</mi></msub><mo>∣</mo><msub><mi mathvariant="normal">Φ</mi><mi>θ</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mid \Phi_{CIS} \rangle = \sum_\theta \mid \Phi_\theta \rangle</annotation></semantics></math>∣ΦCIS​⟩=∑θ​∣Φθ​⟩﻿ . Then, CIS hamiltonian is as follows, ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mi>V</mi><mo>=</mo><mi>E</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">HV=EV</annotation></semantics></math>HV=EV﻿ ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mrow><mi>θ</mi><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi mathvariant="normal">Φ</mi><mi>θ</mi></msub><mo>∣</mo><mi>H</mi><mo>∣</mo><msub><mi mathvariant="normal">Φ</mi><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">H_{\theta \theta&#39;}=\langle \Phi_\theta \mid H \mid \Phi_{\theta &#39;} \rangle</annotation></semantics></math>Hθθ′​=⟨Φθ​∣H∣Φθ′​⟩﻿ . By solving eq.(2), aimed states and eigenenergies are derived.Detail of calculation process are in my Slideshare.  <a href="https://www.slideshare.net/hikaruwakaura/multiscalecontracted-variational-quantum-eigensolver" rel="noopener noreferrer" target="_blank" style="background-color: rgb(255, 255, 255);">Multiscale-Contracted Variational Quantum Eigensolver (slideshare.net)</a> Then, deriving process of each state is performed by Quantum Subspace Expansion or Subspace Search VQE[1][2].After publishing this paper, QCWare published new papers about deriving excited states and transition matrices using MCVQE[3].Besides, some applications have been published.Next is numerical simulation. [1] Parrish, R. M. and et. al., arXiv quant-ph:1901.01234v2(2019) [2] Nakanishi, K. M., and et. al., arXiv quant-ph:1810.09434v2(2018) [3] Parrish, R. M., and et. al., arXiv quant-ph:1906.08728v1(2019)

Multiscale-Contracted Variational Quantum Eigensolver method part 1

Hikaru Wakaura