The Chemical Master Equation (CME) provides a highly accurate yet extremely resource-intensive representation of stochastic chemical reaction networks and their kinetics due to the exponential scaling of its possible states with the number of reacting species. In this work, we investigate how quantum computing can be employed to model stochastic chemical kinetics as described by the CME using the Schlögl model of a trimolecular reaction network as an illustrative example. We analyze the mono- and bistable regimes of the Schlögl model, identifying the bistable regime as more suitable for quantum computation due to the availability of a Hermitian operator form that preserves eigenvalues and eigenvectors. Employing the Variational Quantum Deflation (VQD) algorithm, we compute the smallest-magnitude eigenvalues, λ0, and λ1. We use VQD and a combination of Quantum Phase Estimation (QPE) and Variational Quantum Singular Value Decomposition (VQSVD) to estimate the zeromode (non-equilibrium steady state) of the bistable case. Our results from noiseless quantum simulations (VQD) and quantum hardware (QPE + VQSVD) agree within a few percent with the classically computed eigenvalues and zeromodes of up to 4-qubit operators. We show that achieving an exact solution requires at least 5 qubits, which is within reach of near-term quantum computers.
The Chemical Master Equation (CME) provides a highly accurate yet extremely resource-intensive representation of stochastic chemical reaction networks and their kinetics due to the exponential scaling of its possible states with the number of reacting species. In this work, we investigate how quantum computing can be employed to model stochastic chemical kinetics as described by the CME using the Schlögl model of a trimolecular reaction network as an illustrative example. We analyze the mono- and bistable regimes of the Schlögl model, identifying the bistable regime as more suitable for quantum computation due to the availability of a Hermitian operator form that preserves eigenvalues and eigenvectors. Employing the Variational Quantum Deflation (VQD) algorithm, we compute the smallest-magnitude eigenvalues, λ0, and λ1. We use VQD and a combination of Quantum Phase Estimation (QPE) and Variational Quantum Singular Value Decomposition (VQSVD) to estimate the zeromode (non-equilibrium steady state) of the bistable case. Our results from noiseless quantum simulations (VQD) and quantum hardware (QPE + VQSVD) agree within a few percent with the classically computed eigenvalues and zeromodes of up to 4-qubit operators. We show that achieving an exact solution requires at least 5 qubits, which is within reach of near-term quantum computers.
One of the most important application areas of molecular quantum chemistry is the study and prediction of chemical reactivity. Large-scale, fully error-tolerant quantum computers could provide exact or near-exact solutions to the underlying electronic structure problem with exponentially less effort than a classical computer thus enabling highly accurate predictions for comparably large molecular systems. In the nearer future, however, only “noisy” devices with a limited number of qubits that are subject to decoherence will be available. For such near-term quantum computers the hybrid quantum-classical variational quantum eigensolver algorithm in combination with the unitary coupled-cluster ansatz (UCCSD-VQE) [Peruzzo et al. Nat. Commun. 2014, 5, 4213 and McClean et al. New J. Phys. 2016, 18, 023023] has become an intensively discussed approach that could provide accurate results before the dawn of error-tolerant quantum computing. In this work we present an implementation of UCCSD-VQE that allows for the first time to treat both open- and closed-shell molecules. We study the accuracy of the obtained energies for nine small molecular systems as well as for four exemplary chemical reactions by comparing to well-established electronic structure methods like (nonunitary) coupled-cluster and density functional theory. Finally, we roughly estimate the required quantum hardware resources to obtain “useful” results for practical purposes.
Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.
Modeling chemical dynamics is essential for understanding, predicting, and optimizing chemical processes. Because realistic chemical systems involve many interacting degrees of freedom, direct propagation of the many-body Schr"{o}dinger equation is intractable for all but the smallest systems. Quantum information platforms offer a fundamentally different representation: a d-dimensional state may be encoded using k=log2(d) qubits rather than storing each amplitude independently. To harness this advantage for chemistry, it is necessary to develop quantum algorithms and hardware architectures capable of simulating the open-system dynamics ubiquitous in condensed-phase and molecular environments. In the first chapter, we develop a theoretical framework for simulating a cavity-modified photoinduced charge-transfer reaction in the carotenoid–porphyrin–C60 (CPC60) molecular triad dissolved in tetrahydrofuran (THF). Nonequilibrium Fermi’s Golden Rule (NE-FGR) rate coefficients are obtained as a product of cavity-free and cavity-modified correlation functions. We further introduce an approximation based on Instantaneous Marcus Theory (IMT) that closely tracks the full NE-FGR result across all relevant time scales. Population transfer rates computed from the population-only off-diagonal quantum master equation (p-tcl-OD-QME) are shown to be consistent with the NE-FGR predictions. Our results demonstrate that nonequilibrium charge-transfer dynamics are significantly modified by coupling to an optical cavity. The second chapter addresses the integration of linear nonautonomous ordinary differential equations on digital quantum hardware. Rather than classically constructing a nonunitary propagator at each time step and embedding it into a larger unitary operation for execution on quantum hardware, we eliminate this classical–quantum partitioning. We derive and propagate differential equations for the singular value decomposition (SVD) factors of the propagator directly, enabling time evolution to be implemented within a quantum-compatible framework. This approach shifts the locus of time integration from the classical processor to the quantum device. In the third chapter, we investigate the feasibility of simulating chemical reaction dynamics on a parametrically driven bosonic superconducting Kerr-cat device. This analog platform enables direct control over parameters defining an effective double-well free-energy landscape, as well as over environmental factors such as temperature and system–bath coupling strength. We demonstrate that proton transfer dynamics in prototypical benchmark systems, including hydrogen-bonded dimers of malonaldehyde and DNA base pairs, can be accurately reproduced using experimentally accessible Kerr-cat architectures.
However, accurately simulating molecular interactions constitutes the central, but open question of computer chemistry. This is a core task for efforts in drug development, catalyst design, and materials science. Most classical algorithms perform well on smaller systems, but incumbent exponential scaling penalties are unavoidable as models grow larger and more complex to describe quantum behaviour. An entirely new paradigm for this problem is now introduced by quantum computing: superposition and entanglement resources natively encode quantum states and model their electronic structures. The paper will thus take a high-level view of the leading quantum algorithms, with particular regard to those called Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE), while evaluating their performance on today's Noisy Intermediate Scale Quantum (NISQ) devices. Next will follow discussions on algorithm design and current hardware capabilities, before which open challenges on scalability, error mitigation, and algorithm design will be outlined. Emphasis shall also be placed on the ethics and accessibility issues as well as the timelines with which quantum computing might gain a practical advantage. Linking together the recent advances and limitations, this paper posits how quantum computer implementation may change the face of chemical modelling and scientific discovery.
Quantum chemical calculations have attracted much attention as a practical application of quantum computing. Quantum computers can prepare superpositions of electronic states with various numbers of electrons on qubits. This special feature could be used to construct an efficient method for analyzing the structural variations of molecules and chemical reactions involving changes in molecular charge. The present work demonstrates a variational quantum eigensolver (VQE) algorithm based on a cost function (Lcost) having the same form as the grand potential of the grand canonical ensemble of electrons. The chemical potential of the electrons (w) is used as an input to these VQE calculations, whereas the molecular charge is not specified in advance but rather is a physical quantity that results from the calculations. Calculations involving model systems are carried out to show the viability of this new approach. Calculations for typical electron-donating and electron-accepting molecules using this technique yielded cationic, neutral or anionic species depending on the value of w. Models representing the adsorption of water or ammonia on copper-based catalysts predicted that oxidation would be associated with such adsorption. The molecular structures in which such reactions occurred were found to be dependent on the catalyst model, the adsorbed molecular species, and the value of w. These results arise because the electronic state that gives the lowest Lcost value depends on the value of w and the molecular structure. This behaviour was successfully simulated by the present VQE calculations.
The advancements in classical computational components such as CPUs and GPUs have had a reverberating impact on the democratization of computational power available, with the field of computational chemistry reaping significant benefits from the propulsion of the limits of systems studied. The accurate prediction of chemical properties and reaction dynamics through computational simulation is now a centerpiece of modern chemical research, significantly accelerating discoveries research areas of materials science and drug development, as well as industrially relevant processes such as the emission of aldehydes from soft foams, flame retardants and pyrolysis and the carbonylation of epoxides. These achievements of computational chemistry still are overshadowed in the face of fundamental bottlenecks where classical methods remain constrained by the exponential scaling when treating large and/or strongly correlated quantum many-body systems. This innate computational complexity often constitutes chemically accurate calculations based on first principles impractical, or plain impossible, for systems beyond modest sizes. Quantum computing offers a potential pathway around these limitations, originally formally formulated by David Deutsch as a tool for simulating quantum systems, with the first quantum algorithms indicating speed-ups compared to classical counterparts arising from Peter Shor and Lov Grover for the prime factorization of an integer number and unstructured search respectively. Relying on fundamental theorems by Claude Shannon, it has been demonstrated that through the use of error-correcting codes fault-tolerant quantum computers are able to safeguard information against noise and decoherence, strongly indicating towards exponential speedups in solving central computational chemistry problems, such as determining ground state energies or reaction pathways of chemical systems. Substantial strides in experimental quantum computing over the past decade, including proof-of-principle demonstrations of quantum error correction, mean to bring full-scale fault-tolerant quantum computing within the grasp of humanity. The breakthroughs needed to attain scalable, error-corrected quantum computation with capabilities exceeding those of classical methods in real-world scenarios is likely still some years away. While striving to address the engineering challenges standing in the way of achieving fault tolerant quantum computing, substantial effort has been focused toward developing and benchmarking hybrid algorithms tailored for the Noisy Intermediate- Scale Quantum (NISQ) devices, available now to researchers. These hybrid quantum-classical algorithms, with the most notable being the Variational Quantum Eigensolver (VQE), allow the probing of quantum computing’s utility in computational chemistry even in the face of significant hardware noise and limited coherence times. The work presented in this thesis introduces several key additions the VQE framework, geared especially towards to NISQ hardware, where the mindful use of available resources is imperative. First, the implementation of the Conjugate Model Search optimizer is described, an optimization method that integrates simple models that are surrogate to the original function, combined with conjugate gradient optimization to navigate effectively parameter landscapes typical of variational algorithms, prone to noise. Second, this thesis contains the proposal and extensive study of a hyperparameter tuning procedure employing genetic algorithms, allowing for comprehensive simulation benchmarks that augment the robustness and efficiency of VQE implementations with respect to convergence time through real wall-time models of quantum processors. Continuing, a bridge is presented between quantum simulation and quantum chemistry, through qubit-to-fermion mappings of existing quantum gates, further enabling the direct and accurate modeling of chemical systems on trapped atom quantum hardware with the added arsenal provided by techniques designed for qubit architectures. Additionally, experimental data contained in this thesis include detailed benchmarking data collected from state-of-the-art trapped ion quantum processors available at the Johannes Gutenberg Universität Mainz, demonstrating the actual viability and performance of various methodologies for extracting results. Lastly, the state of VQE is examined, as it served as a springboard for a plethora of advancements in the development of quantum computing hardware, as well as the direction the field of simulation of computational quantum chemistry via quantum devices is headed towards in the coming years.