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APS March Meeting 2023
Y72.00002

Improving success probability ofimaginary-time evolution on a quantum computer
arXiv:2212.13816
Hirofumi Nishi1,2, Taichi Kosugi1,2, Yusuke Nishiya1,2, Yu-Ichiro Matsushita1,2,3
1Tokyo Institute of Technology
2Quemix Inc
3National Institute for Quantum Science and Technology

Background

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One of the motivations: ground state calculation of quantum chemistry
Imaginary-time evolution (ITE) method on a quantum computer
Difficulty：Implementation of nonunitary is not straightforward
Methodologies developed so far：ü Variational imaginary-time evolution: VQE-based[X. Yuan et al., Quantum 3, 191 (2019)]

NISQ

ü Quantum imaginary-time evolution[Motta et al., Nat. Phys. 16(2) 205 (2020)], [Nishi et al. npj QI 7, 85 (2021).]

ü Probabilistic imaginary-time evolution (PITE)[Kosugi, Nishi, et. al., Phys. Rev. Res. 4, 033121 (2022)]

Solving the difficulty by introducing ancilla bit and measurement
Advangage: reduction of number of measurements

FTQC

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Probabilistic Imaginary-Time Evolution (PITE)

Y72.00007 9:12am-9:24am: Kosugi, Nishi, et. al., Phys. Rev. Res. 4, 033121 (2022)

Approximate PITE circuit within first order of imaginary-time step Δτ
ancilla|0i| iH⌦n•W/URTE ( ⌧ )URTE (RzW†⌧)

Feature: Express ITE operator with real-time evolution operators (URTE)
Input

Outputancilla bit

success state probabilistically obtained4/10

Goal of this study

Y72.00007 9:12am-9:24am: Kosugi, Nishi, et. al., Phys. Rev. Res. 4, 033121 (2022)

Problems of PITE method：
Success probability ≠ 100%

2.

Exponential decay of total success probability according to ITE steps
increasing

Input

Outputancilla bit

success state probabilistically obtained

Total success probability

1.

Imaginary-time step

Goal：Improve success probability of PITE method combined with QAAQAA (Quantum Amplitude Amplification)

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Quantum Amplitude Amplification (QAA)
Success state(Target state)

Initial state
1st step QAA

success state

failure state

Reflection

m times
QAA steps

Initial state
Failure state(Orthogonal state)oracleaa

Amplified state after m steps of QAA

Brassard, Gilles, et al. "Quantum amplitude amplification and estimation." Contemporary Mathematics 305, 53 (2002)

Quantum Amplitude Amplification (QAA)

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Amplified state after m steps of QAA

Success probability

desirable

Determine QAA steps ms.t. success prob. is maximizedn : integer
Estimate θa by QAEQAA steps m

Optimized mopt depends on θa,which is derived from initial state,
Smaller value of optimized mopt ispreferable

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Deterministic Imaginary-Time Evolution
In order to reduce the optimized m, look back again PITE

Focus: PITE has a freely choosing parameter γ (in the range 0<γ<1, γ≠1/√2)
Initial success prob.:
Success prob. is almost one

θa depends on parameter γ
γ is an degree of freedom to tune theinitial success prob., leading to smaller
value of the optimized mopt.
Importantly,
by using QAA, make success prob. = 1
Realize deterministic ITE

Success probability

even when m=1γγγγγ

QAA steps mDeterministic Imaginary-Time Evolution

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By adopting the optimized γopt and mopt, the success state is always realized(deterministic PITE)
No need to observe the ancilla bitanymore

1st step|0i| iemQUref

CPITE

Second step

Can repeat this procedure|0iCPITE

We can continue subsequent
2nd step of PITE withoutmeasurement

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Numerical Results
Setup:
Max-cut problem (Ising Hamiltonian)with QAA
Success prob. pn

Total success prob.

w/o QAAü Dramatically improve the total success probability

Fidelity

Summary

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ü Improved success probability of the PITE method by QAA
ü Realized deterministic ITE by optimizing an parameter γ included in the PITE
Nishi,et al., arXiv:2212.13816

See Also
1. For details of PITE
Y72.00007 9:12am-9:24am
2. Geometric optimization using PITE method
TK, Nishi, et al., arXiv:2210.09883, YY03.00005 Wed. March 22, 2023 (Virtual)
3. Electron states under a uniform magnetic field using PITE method,TK, Nishi, et. al., arXiv:2212.13800.