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arXiv:1401.3186 · The Journal of Chemical Physics

Adiabatic state preparation study of methylene

Libor Veis, Jiří Pittner

Published Jun 6, 2014• 89 citations
arXivHTMLPDF

Abstract

Quantum computers attract much attention as they promise to outperform their classical counterparts in solving certain type of problems. One of them with practical applications in quantum chemistry is simulation of complex quantum systems. An essential ingredient of efficient quantum simulation algorithms are initial guesses of the exact wave functions with high enough fidelity. As was proposed in Aspuru-Guzik et al. [Science 309, 1704 (2005)], the exact ground states can in principle be prepared by the adiabatic state preparation method. Here, we apply this approach to preparation of the lowest lying multireference singlet electronic state of methylene and numerically investigate preparation of this state at different molecular geometries. We then propose modifications that lead to speeding up the preparation process. Finally, we decompose the minimal adiabatic state preparation employing the direct mapping in terms of two-qubit interactions.

DOI
10.1063/1.4880755

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Quantum computers attract much attention as they promise to outperform their classical counterparts in solving certain type of problems. One of them with practical applications in quantum chemistry is simulation of complex quantum systems. An essential ingredient of efficient quantum simulation algorithms are initial guesses of the exact wave functions with high enough fidelity. As was proposed in Aspuru-Guzik et al. [Science 309, 1704 (2005)], the exact ground states can in principle be prepared by the adiabatic state preparation method. Here, we apply this approach to preparation of the lowest lying multireference singlet electronic state of methylene and numerically investigate preparation of this state at different molecular geometries. We then propose modifications that lead to speeding up the preparation process. Finally, we decompose the minimal adiabatic state preparation employing the direct mapping in terms of two-qubit interactions.
A variety of quantum computing algorithms exist for the preparation of approximate Hamiltonian ground states. A natural and important question is how these ground-state approximations can be further improved using adiabatic state preparation. Here, we present a heuristic method to carry out adiabatic state preparation starting from a generic initial wavefunction. Given a quantum circuit that prepares the initial wavefunction, and a target Hamiltonian for which one wishes to prepare the ground state, we present an algorithm to construct an adiabatic path between these two states. This method works by approximating a parent Hamiltonian for the initial wavefunction, and the quality of this approximation can be can be checked prior to running the ASP algorithm. We apply this technique to simulate the ground state of water and the lowest-lying multireference singlet state of methylene, using various initial wavefunctions as the starting point of the adiabatic path.
Quantum computing has the potential to transform simulations of quantum many-body problems at the heart of electronic structure theory. Efficient quantum algorithms to compute the eigenstates of fermionic Hamiltonians, such as quantum phase estimation, rely critically on high-accuracy initial state preparation. While several state preparation algorithms have been proposed for fermionic ground states, the preparation of excited states remains a major challenge, limiting the applicability of quantum algorithms to photochemistry and photophysics. In this contribution, we describe a physically motivated adiabatic state preparation technique for low-lying bright excited states using the explicit coupling between electrons and photons. Our approach systematically converges to the first bright excited state and can target different symmetry sectors by changing the photon polarization. We demonstrate the preparation of high-fidelity excited states for the Hubbard model and methylene molecule across a range of correlation regimes, and perform a successful hardware implementation for a model Hamiltonian.
Quantum computation promises to provide substantial speedups in many practical applications with a particularly exciting one being the simulation of quantum many-body systems. Adiabatic state preparation (ASP) is one way that quantum computers could recreate and simulate the ground state of a physical system. In this paper, we explore a novel approach for classically simulating the time dynamics of ASP with high accuracy and with only modest computational resources via an adaptive sampling configuration interaction scheme for truncating the Hilbert space to only the most important determinants. We verify that this truncation introduces negligible error and use this new approach to simulate ASP for sets of small molecular systems and Hubbard models. Furthermore, we examine two approaches to speeding up ASP when performed on quantum hardware: (i) using the complete active space configuration interaction (CASCI) wave function instead of the Hartree-Fock initial state and (ii) a nonlinear interpolation between the initial and target Hamiltonians. We find that starting with a CASCI wave function with a limited active space yields substantial speedups for many of the systems examined, while nonlinear interpolation does not. In additional, we observe interesting trends in the minimum gap location (based on the initial state) as well as how state preparation time can depend on certain molecular properties, such as the number of valence electrons. Importantly, we find that the required state preparation times do not show an immediate exponential wall that would preclude an efficient run of ASP on actual hardware.
Quantum computers represent a relatively new and promising development in computing technology. One of the applications of quantum computers is modelling quantum many-body systems, which describe interactions between particles on atomic and subatomic scales. Such systems are highly complex and challenging to model with classical computers due to the vast number of possible states and entanglement of the particles. In this dissertation, we examine the extent to which quantum computers can accelerate the modelling of quantum many-body systems compared to classical computers. Specifically, this dissertation presents research on adiabatic state preparation: a quantum algorithmic technique that uses the adiabatic principle from quantum mechanics to approximate eigenstates. We describe three new techniques that fall within this category and, in certain cases, offer advantages over standard methods. First, we consider cases of ground state preparation for fermionic many-body systems, where standard direct interpolation between the initial and final hamiltonian is hindered by level crossings due to discrete symmetries. As an alternative to direct interpolation, we propose adiabatic paths in a higher-dimensional space, which break the relevant symmetries. Next, we present an adiabatic echo verification protocol which mitigates both coherent and incoherent errors, arising from non-adiabatic transitions and hardware noise, respectively. We show that the estimator bias of the observable is reduced when compared to standard adiabatic preparation, achieving up to a quadratic improvement. Finally, we propose a general, fully gate-based and nonvariational quantum algorithm for counterdiabatic driving. We provide a rigorous quantum gate complexity upper bound in terms of the minimum gap Δ\DeltaΔ around this eigenstate. We find that, in the worst case, the algorithm can be run with at most O~(Δ−(3+o(1))ϵ−(1+o(1)))\tilde O(\Delta^{-(3 + o(1))} \epsilon^{-(1 + o(1))})O~(Δ−(3+o(1))ϵ−(1+o(1))) quantum gates such that a target state fidelity of at least 1−ϵ21 - \epsilon^21−ϵ2 is achieved. In certain cases, the gap dependence can be improved to quadratic.
Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH2_22​). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.