Quantum Algorithm for Optimal Molecular Geometries
Based on Probabilistic Imaginary-Time Evolution
Taichi Kosugi
Quemix Inc. and Tokyo Institute ofTechnology

Various ITE approaches on quantum computers
Variational ITE (VITE)
Circuit parameters θ(τ) for unitaries are updated by solving
McLachlan, Mol. Phys. 8, 39 (1964)
Jones et al., Phys. Rev. A 99, 062304 (2019)

Quantum ITE (QITE)
Determine coeffs in Hermitian oprs

for approximate unitaries.
Motta et al., Nat. Phys. 16, 205 (2020)
cf. Nishi, Kosugi, and Matsushita, npj Quant. Inf. 7, 85 (2021)

Probabilistic ITE (PITE)
Implements the nonunitary opr

by using real-time evolution (RTE) gates:
Kosugi, Nishiya, Nishi, and Matsushita,
Phys. Rev. Res. 4, 033121 (2022)

Success state≈ M|ψ>

Failure state😀Every impl of RTE can be transferred to PITE.
Finite-temperature calc is also possible.
Copyright @Quemix Inc. ALL RIGHTS RESERVEDPITE for first-quantized systems

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First-quantized formalism for many-electron systems
Hamiltonian
Hamiltonian of ne electrons

cf. Kassal et al., PNAS 105, 18681 (2008)
for real-time dynamics

Encoding of wave functions
Many-electron wave function in pos repr|k>: Position eigenstate of an electron
Implementation of RTE
1st-order Suzuki-Trotter
For example, from phase gate for 2 electrons
we can construct e-e part of RTE opr:

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N ≡ 2nqΔx ≡ L/N: grid step
L: cell size

First-quantized eigensolver (FQE) based on PITE
Nonvariational energy minimization for a molecular system

Simulation for one particle in 1dIntroduction of an ext. magnetic field is possible.

Kosugi, Nishi, and Matsushita,
Jpn. J. Appl. Phys. 62, 062004 (2023)

from measurements
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First-quantized eigensolver (FQE) based on PITE
Comparison with 2nd-quantized formalism
Typical simulation cells satisfy L3 ∝ ne, we have nq ∝ log ne for each electron.
Classical and quantum cost for performing a PITE calculation is estimated as follows:
1st-quantized2nd-quantizedClassical# of oprs for obtaining MOs

None# of oprs forconstructing HamiltonianNone# of qubits fora many-electron stateO(ne log ne)

Quantum# of oprs at each PITE stepO(ne2(log ne)p’)
p’: degree of polynomial fore-e interactions

O(ne4) for two-electron integrals(and O(ne3) for diagonalization)O(ne5)(Uabcd for AOs -> Upqrs for MOs)O(ne)O(ne4)(cp†cq†crcs terms)

😀 Better scaling of 1st-quantized formalism both for classical and quantum oprs
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Geometry optimization based on PITE

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Hamiltonian for electrons and nuclei
We treat the target molecule as quantum mechanical electrons and classical nuclei.

Kosugi, Nishi, and Matsushita,
arXiv:2210.09883

What we have to do is to determine the nuclear positions {Rν}ν so that the total energy is the lowest.
There is no need for knowing the energies of the candidate geometries for finding the optimal one.
We should know only which geometry has the unknown lowest energy.
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Many candidates for optimal nuclear positions
We encode the candidates by using nqn qubits per direction for each nucleus.
Nuclear-position opr:Original position

Displacement

We use the nuclear qubits not for wave functions of nuclei, but for classical data of nuclear positions.
Each geometry is specified by 3nqnnnucl integers.
The qubits in our approach encode a state looking like:

Summation over possible geoms

Many-electron state

Nuclear positions

We perform an ITE calculation for this superposition of all candidate geometries.
This approach finds the optimal geometry, for which the total energy is the lowest.
Copyright @Quemix Inc. ALL RIGHTS RESERVEDCircuit for PITE steps

For giving initial weights to candidates

For energy minimization

For generating reference electronic states
Resultant state😀Geoms having lower energies will be observed with higher probs.
Computational time for finding the lowest-energy state among Ncand candidates is
PITE with quantum amplitude amplification (QAA) -> quadratic speedup
Nishi et al. in poster session
See also Nishi et al., arXiv:2212.13816
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Numerical simulations for a LiH model system
Setup
2 electrons and 2 atoms in 1d space, ZH = ZLi = 1 (Li 1s electrons frozen)
All interactions are soft-Coulomb type:
nqe = 6 for spatial discretization
Geometry optimization
By using nqn = 3 qubits, we tried 2nqn = 8 candidate bond lengths for optimization: dJ = 0.55 + 0.5J (J = 0, …, 7)
Starting from a symmetric state (spin triplet)

Most stable geometry has the highest prob.

Xm: bond center

Starting from an antisymmetric state (spin singlet)

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No energy minimum within this range

Classical ions-only system as a special caseSetup
Ar atom above a benzenemolecule
Improved Lennard-Jones potential
Pirani et al, Chem. Phys. Lett. 394, 37 (2004)

Geometry optimization using PITEnqn = 3 for each of x and z directions -> 64 candidates

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Most stable geometry has the highest prob.

Geometry optimization based on variational calculations
McLachlan, Mol. Phys. 8, 39 (1964)

Our scheme can also be adapted to vatiational imaginary-time evolution (VITE). Jones et al., Phys. Rev. A 99, 062304 (2019)
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)
Most stable geometry

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Summary- PITE performs nonvariational energy minimization for a generic system using
RTE oprs.
- When combined with first-quantized formalism, PITE for an electronic system
exhibits better scaling than second-quantized formalism.
- By using ITE for a superposition of candidate geometries, exhaustive search
for finding the optimal one is possible.
Kosugi, Nishiya, Nishi, and Matsushita,Phys. Rev. Res. 4, 033121 (2022)

Kosugi, Nishi, and Matsushita,
arXiv:2210.09883

Thank you.
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Implementation of RTE for a molecular system
Position-dependent part in RTE

Efficient impl of e-e part for 4 electrons

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