APS March Meeting 2023
YY03.00005

Geometric optimization based on first-quantized Hamiltonian
using imaginary-time evolution on a quantum computer
arXiv:2210.09883
Yu-Ichiro Matsushita1,2,3 , Hirofumi Nishi1,2, Taichi Kosugi1,2
1Tokyo Institute of Technology
2Quemix Inc
3National Institute for Quantum Science and Technology

1/12

Background
One of the important tasks: geometric structure optimization of molecules and crystals
Quantum computer → It is expected to accelerate the geometric structure optimization
Methodologies developed so far:Born-Oppenheimer surface

Force-field based optimization methodQPE: Kassal and Aspuru-Guzik, JCP 131, 224102 (2009).
VQE: Delgado et al., PRA 104, 052402 (2021).
Protein folding using Quantum annealer
Perdomo-Ortiz et al., Sci. Rep. 2, 571 (2012).

Propose a quantum algorithm of exhaustive search from candidate geometries on the BornOppenheimer surface

2/12

Hamiltonian for electrons and nuclei

Ignore nuclear kinetic part → Treat nuclei as classical particles(point charge)
We treat the target molecule as quantum mechanical electrons and classical nuclei.

By solving the Hamiltonian, we get the nuclear positions {Rν}ν so that the total energy is the lowest.
3/12

Encoding nuclear positions on a quantum computer
Dissociation curve of two atoms

Encode the geometric candidates by using nqn qubits.
e.g., nqn = 2 (2nqn=4 candidates)

Many-electron state

Nuclear positions

Total Energy

Sum over possible geometries

Displacement

Bond lengthe.g., nqn=2 (2nqn=4 candidates)

Find the most stable state bysolving the Hamiltonian

We perform an ITE calculation for this superposition of all candidate geometries.
4/12

Features of this method
ü We treat the target molecule as quantum mechanical electrons and classical nuclei.
ü We find the most stable geometry by solving the Hamiltonianü Our approach can be applied to any organic and inorganic materials
ü There is no need for knowing the energies of the candidate geometries for finding the optimaloneü We don’t need to calculate the gradients of the energies for the nuclei coordinates

5/12

Probabilistic Imaginary-Time Evolution (PITE) method
Kosugi, Nishiya, Nishi, and Matsushita, Phys. Rev. Res. 4, 033121 (2022)

Approximate PITE circuit within first order of imaginary-time step Δτ

|0i

ancilla bit

initial state| iH⌦n•W/URTE ( ⌧ )URTE (RzW†⌧)Feature: Express ITE operator with real-time evolution operators (URTE)
Input

ancilla bit

Outputsuccess state

failure state

The approximate PITE circuit uses the Hamiltonian only as black boxes, so which can be applied to
any types of Hamiltonian.

6/12

Circuit for PITE steps
Prepare initial state

Gate for giving initial weights to candidates(If no information, equal weights are appropriate.)

Gate for generating reference electronic states

After sufficiently many steps, we measure a geometry |J> on
nuclear register with the probability wJ.

After sufficient PITE steps, the state with highest probability is the most stable geometry
7/12

Numerical results(Example 1) Numerical simulations for a LiH model system
Structure optimization for nuclei coupled to electrons(Example 2) Classical nuclei-only system as a special case
Structure optimization for only nuclei(Example 3) Geometry optimization based on variational calculations
Structure optimization with variational method

8/12

(Example 1) Numerical simulations for a LiH model system
Setup

Numerical diagonalization

2 electrons and 2 atoms in 1d space,
ZH = ZLi = 1 (Li 1s electrons frozen)
All interactions are soft-Coulomb type:
Tempel et al, JCTC 5, 770 (2009)

Geometry optimization

nqe = 12 bits for ne = 2 electrons
nqn = 3 bits for nuclei

By using nqn = 3 qubits, we tried 2nqn = 8 candidate bond lengths for optimization: dJ = 0.55 + 0.5J (J = 0, …, 7)
Most stable geometry has the highest prob.

9/12

(Example 2) Classical nuclei-only system as a special caseInteraction energy

Ar atom above a benzene molecule
Treat only nuclei

Improved Lennard-Jones potential
Pirani et al, Chem. Phys. Lett. 394, 37 (2004)

Geometry optimization using PITE
nqn = 3 for each of x and z directions

Most stable geometry has the highest probability

Reduce the number of qubits by using force-field
10/12

(Example 3) Geometry optimization based on variational calculation
Our scheme can also be adapted to variational imaginary-time evolution (VITE).
Geometry optimization of H2+ model system
By using nqn = 3 qubits, we tried 2nqn = 8 candidate bond lengths for optimization: dJ = 0.5 + (7.5/8)J (J = 0, …, 7)

Ansatz circuit (d =12)

McLachlan, Mol. Phys. 8, 39 (1964)
Jones et al., Phys. Rev. A 99, 062304 (2019)

The weight of the most stablegeometry was enhanced

11/12

Conclusions
Summaryü Proposed a quantum algorithm for geometry optimization.
ü The most stable geometry is calculated from a superposition of all
candidate geometries.

This talk

Kosugi, Nishi, and Matsushita,
arXiv:2210.09883

Related studies
Detail of PITE

Kosugi, Nishiya, Nishi, and Matsushita,
Phys. Rev. Res. 4, 033121 (2022)

Under a magnetic field

With amplitude amplification

Kosugi, Nishi, and Matsushita,
arXiv:2212.13800

Nishi, Kosugi, Nishiya and Matsushita,
arXiv:2212.13816

12/12