Study of Adaptative Derivative-Assemble Pseudo-Trotter Ansatzes in VQE through qiskit API Max Alteg Baptiste Chevalier Octave MestoudjianJohan-Luca Rossi

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Study of Adaptative Derivative-Assemble

Pseudo-Trotter Ansatzes in VQE through qiskit API

Max Alteg, Baptiste Chevalier, Octave Mestoudjian,Johan-Luca Rossi

April 2022

arXiv:2210.15438v1 [quant-ph] 25 Oct 2022

Contents

1 Theoretical Background 5

1.1 QuantumChemistry .............................. 5

1.1.1 Hamiltonian and Born-Oppenheimer approximation . . . . . . . . . 5

1.1.2 Spin-orbital and Slater determinant . . . . . . . . . . . . . . . . . . 6

1.1.3 First and Second Quantization of the Hamiltonian . . . . . . . . . . 6

1.1.4 Mapping................................. 7

1.2 Ansatzes..................................... 8

1.2.1 Introduction : Variational Method . . . . . . . . . . . . . . . . . . 8

1.2.2 Hardware Eﬃcient Ansatz . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Unitary-Coupled Cluster Ansatz . . . . . . . . . . . . . . . . . . . . 11

1.2.4 ADAPT-VQE.............................. 14

1.3 OtheraspectsofVQE ............................. 16

1.3.1 Classical Optimisers . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 BarrenPlateau ............................. 18

2 Implementation 19

2.1 Qiskit general framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Hamiltonian Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.2 Operators representation . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.3 Aersimulator .............................. 21

2.1.4 VQE................................... 21

2.2 Pre-existingAnsatz............................... 21

2.2.1 HardwareEﬃcient ........................... 21

2.2.2 UCCSD ................................. 21

2.3 ADAPT-VQE development . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Fermionic-ADAPT-VQE . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Qubit-ADAPT-VQE . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3 QEB-ADAPT-VQE........................... 23

3 Results and discussion 24

3.1 Comparison of various HEA . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Evaluation of the accuracy of UCCSD . . . . . . . . . . . . . . . . . . . . . 26

3.3 Comparison of ADAPT-VQE Methods . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Results for H2.............................. 28

3.3.2 Results for LiH ............................. 29

Introduction

The solution of the time-independent Schrodinger equation for molecular systems allows

for the prediction of chemical properties. Quantum Computers are expected to make

possible the simulation of quantum systems more eﬃciently and accurately than classical

ones, including the simulation of molecules. This is due to the exponential size of the

Hilbert Space.

The ﬁrst algorithm which has been proposed to solve the problem of simulating elec-

tronic structure on a Quantum Computer is well-known as the Phase Estimation Algo-

rithm (PEA). Despite that PEA gives the exact ground state for a given problem, the long

circuit and the complex gates used in this algorithm make it impractical on any current

or near-term Quantum Hardware referred as NISQ (Noisy intermediate Scale Quantum)

devices.

In order to answer the problem of NISQ era and allows one to outperform classical comput-

ers, Variational Quantum Algorithms (VQAs) were designed. VQAs are hybrid classical-

quantum algorithms due to the fact that there are using a quantum hardware for com-

putation while using a classical optimizer for optimization. In this way, VQA algorithms

allows the creation of shallower circuits and are thus much more noise resilient. The VQA

algorithm we are interested in is the so-called Variational Quantum Eigensolver (VQE)

algorithm and was originally designed to simulate electronic structures and to compute

the ground state of a given molecule.

VQE is made of two main components. First, an ansatz which has several tunable param-

eters and his associated quantum circuit. The ansatz is run on the quantum device and

aim to simulate the wavefunction, the parameters of the ansatz will be optimize until the

expectation value is minimum meaning that one has found the ground state. The second

component is then the classical optimizer, it is run on a classical computer and is the often

the same as the one used in Machine Learning. At this point, it is clear that if there is a

limiting factor for VQE it can only be the ansatz.

The very ﬁrst ansatz that has originally been used is called UCCSD and it is based on

Coupled Cluster Theory truncated at single and double excitations. One other way that

has been investigated alongside is the Hardware eﬃcient ansatz (HEA) which was de-

signed to be eﬃciently implemented on NISQ devices but is not as eﬃcient as UCCSD.

The main issue considering UCCSD is the large amount of parameters to optimize and

this leads us to the introduction of Adaptive Derivative-Assembled Pseudo-Trotter ansatz

VQE (ADAPT-VQE) which determines a quasi-optimal ansatz with a minimal number of

parameters.

Unlike ansatzes introduced above, the key point of ADAPT-VQE is to grow the ansatz at

every step, by adding operators chosen from a pre-determined pool of operators one-at-a-

time, assuring that the maximal amount of correlation energy is recovered at each step.

There exists diﬀerent kind of ADAPT-VQE depending on the starting pool of operators as

the fermionic-ADAPT, the qubit-ADAPT or even the qubit excitation based (QEB).

The goal of this project is to implement the diﬀerent types of ADAPT-VQE mentioned

before. After being sure to understand the theoretical background under all of these

concepts, we will implement each algorithm using quiskit. We will also compare all of

these algorithms on diﬀerent criterions such as the number of parameters, the accuracy

or the number of CNOT gate used on H2and LiH molecules. Then we will have a small

discussion about the results we obtained.

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StudyofAdaptativeDerivative-AssemblePseudo-TrotterAnsatzesinVQEthroughqiskitAPIMaxAlteg,BaptisteChevalier,OctaveMestoudjian,Johan-LucaRossiApril2022Contents1TheoreticalBackground51.1QuantumChemistry..............................51.1.1HamiltonianandBorn-Oppenheimerapproximation.........51.1.2Spin-orb...

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Study of Adaptative Derivative-Assemble Pseudo-Trotter Ansatzes in VQE through qiskit API Max Alteg Baptiste Chevalier Octave MestoudjianJohan-Luca Rossi

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