APS March meeting 2023

Imaginary-time evolution with a single ancilla:
first-quantized eigensolver for electronic structure
calculation in quantum chemistry
Taichi Kosugi1,2, Yusuke Nishiya1,2,Hirofumi Nishi1,2, Yu-ichiro Matsushita1,2,3
1 Tokyo Institute of Technology
2 Quemix Inc.
3 National Institutes for Quantum Science and Technology

Purpose
To obtain the ground state of a given Hamiltonian …
Imaginary-time evolution (ITE)

Nonunitary

Initial state

The weights of excited states decay rapidly.
No need for an appropriate ansatz
Nonunitarity : cannot be implemented directly in a quantum circuit

We propose a new framework using ITE for fault-tolerant quantum computers.

Probabilistic Imaginary-Time Evolution (PITE)
How to realize ITE operation in quantum circuits
Probabilistic imaginary-time evolution (PITE)Space expansion & Measurement

Ancillary qubit

Success state

By measuring the ancillary qubit, the Success state is obtained with probability

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

PITE circuit

Taylorexpansion

|0i
| iH
⌦n/

•WURTE†URTERz ( 2✓)W†

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

PITE for quantum chemistryFirst-quantized Hamiltonian of ne electrons
cf. Kassal et al., PNAS 105, 18681 (2008)
for real-time dynamics

Encoding of wave functions
Discretize wave function by using 3ndne qubits: Position eigenstate of an electron at (kxex+kyey+kzez)Δx

Implementation of RTE
1st-order Suzuki-Trotter :Numerical simulation
Model : Single particle in 1D space
Harmonic potential

Simulation result

Approaching the ground state as step proceeds(PITE step)

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

First-quantized Eigensolver (FQE)
Circuit : Interacting many-electron system in 3D space

Antisymmetrization
Initial trial state preparation

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

Scaling
Comparison with 2nd-quantized formalism

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

1st-quantized2nd-quantized# of qubits fora many-electron stateO(ne log ne)O(ne)# of oprs at each PITE stepO(ne2(log ne)p’)O(ne4)

p’: degree of polynomial fore-e interactions

Better scaling of 1st-quantized formalism !

Optimization of molecular geometry
Quantum mechanical electrons & classical nuclei

Kosugi, Nishi, and Matsushita,
arXiv:2210.09883

Input register

Nuclear positions
Many-electron state
Sum over possible geomatries

Most stable structure in the candidates( the ground state of the Hamiltonian above)

In detail …
March 22, 10:00~@Virtual room 3

Conclusions
Summary- Generic construction of PITE circuit with a single ancillaand RTE oprs.
- FQE : a new framework for quantum chemistryThis talk

Kosugi, Nishiya, Nishi, and Matsushita,
Phys. Rev. Res. 4, 033121 (2022)- FQE for an electronic system exhibits better scaling than
2nd-quantized formalism.
Related studies
Structural search with PITE

Under a magnetic field

With amplitude amplification

Kosugi, Nishi, and Matsushita,
arXiv:2210.09883

Kosugi, Nishi, and Matsushita,
arXiv:2212.13800

Nishi, Kosugi, Nishiya and Matsushita,
arXiv:2212.13816