Digital quantum computers provide a computational framework for solving the Schrodinger equation for a variety of many-particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self-contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented.

Emerging quantum computing algorithms for quantum chemistry

Abstract Variational quantum eigensolvers (VQE) are among the most promising approaches for solving electronic structure problems on near-term quantum computers. A critical challenge for VQE in practice is that one needs to strike a balance between the expressivity of the VQE ansatz versus the number of quantum gates required to implement the ansatz, given the reality of noisy quantum operations on near-term quantum computers. In this work, we consider an orbital-optimized pair-correlated approximation to the unitary coupled cluster with singles and doubles (uCCSD) ansatz and report a highly efficient quantum circuit implementation for trapped-ion architectures. We show that orbital optimization can recover significant additional electron correlation energy without sacrificing efficiency through measurements of low-order reduced density matrices (RDMs). In the dissociation of small molecules, the method gives qualitatively accurate predictions in the strongly-correlated regime when running on noise-free quantum simulators. On IonQ’s Harmony and Aria trapped-ion quantum computers, we run end-to-end VQE algorithms with up to 12 qubits and 72 variational parameters—the largest full VQE simulation with a correlated wave function on quantum hardware. We find that even without error mitigation techniques, the predicted relative energies across different molecular geometries are in excellent agreement with noise-free simulators. 

/pdf/orbital-optimized-pair-correlated-electron-simulations-on-3vwdcyoi.pdf

Orbital-optimized pair-correlated electron simulations on trapped-ion quantum computers

The exact evaluation of the molecular ground state in quantum chemistry requires an exponentially increasing computational cost. Quantum computation is a promising way to overcome the exponential problem using polynomial-time quantum algorithms. A quantum-classical hybrid optimization scheme known as the variational quantum eigensolver is preferred for noisy intermediate-scale quantum devices. However, the circuit depth becomes one of the bottlenecks of its application to large molecules of more than 20 qubits. In this work, we employ point-group symmetry to reduce the number of operators in constructing ansatz so as to achieve a more compact quantum circuit. We illustrate this methodology with a series of molecules ranging from LiH (12 qubits) to ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$ (28 qubits). A significant reduction of up to 82% of the operator numbers is reached on ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$. This also sheds light onto further work in this direction to construct even shallower ansatz with enough expressive power and simulate even larger scale systems. 

Progress toward larger molecular simulation on a quantum computer: Simulating a system with up to 28 qubits accelerated by point-group symmetry

Molecular science is governed by the dynamics of electrons, atomic nuclei, and their interaction with electromagnetic fields. A reliable physicochemical understanding of these processes is crucial for the design and synthesis of chemicals and materials of economic value. Although some problems in this field are adequately addressed by classical mechanics, many require an explicit quantum mechanical description. Such quantum problems represented by exponentially large wave function should naturally benefit from quantum computation on a number of logical qubits that scales only linearly with system size. In this perspective, we focus on the potential of quantum computing for solving relevant problems in the molecular sciences -- molecular physics, chemistry, biochemistry, and materials science.

pdf/prospects-of-quantum-computing-for-molecular-sciences-4k8dwk6jk4.pdf

Prospects of Quantum Computing for Molecular Sciences

Quantum computers hold promise to improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties that are more complex than tractable at present on classical architectures. Here, we discuss computational frameworks to carry out electronic structure calculations of solids on noisy intermediate-scale quantum computers using embedding theories, and we give examples for a specific class of materials, that is, solid materials hosting spin defects. These are promising systems to build future quantum technologies, such as quantum computers, quantum sensors and quantum communication devices. Although quantum simulations on quantum architectures are in their infancy, promising results for realistic systems appear to be within reach. Quantum embedding theory promises the simulation of realistic materials in quantum computers. In this Perspective, challenges and opportunities of applying different embedding frameworks to calculate solid materials properties are discussed, with a focus on electronic structures of spin defects. 

/pdf/quantum-embedding-theories-to-simulate-condensed-systems-on-ng9nt7o4.pdf

Quantum embedding theories to simulate condensed systems on quantum computers

Quantum computers have the potential to advance material design and drug discovery by performing costly electronic structure calculations. A critical aspect of this application requires optimizing the limited resources of the quantum hardware. Here, we experimentally demonstrate an end-to-end pipeline that focuses on minimizing quantum resources while maintaining accuracy. Using density matrix embedding theory as a problem decomposition technique, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms without freezing any electrons. The originally 20-qubit system is decomposed into 10 two-qubit problems, making it amenable to currently available hardware. Combining this decomposition with a qubit coupled cluster circuit ansatz, circuit optimization, and density matrix purification, we accurately reproduce the potential energy curve in agreement with the full configuration interaction energy in the minimal basis set. Our experimental results are an early demonstration of the potential for problem decomposition to accurately simulate large molecules on quantum hardware. Problem decomposition methods may help to overcome the size limitations of quantum hardware and allow largescale electronic structure simulations. Here, a method to simulate a ten-atom Hydrogen ring by decomposing it into smaller fragments that are amenable to a currently available trapped ion quantum computer is demonstrated experimentally.

Optimizing electronic structure simulations on a trapped-ion quantum computer using problem decomposition

Quantum computers have the potential to perform accurate and efficient electronic structure calculations, enabling the simulation of properties of materials. However, today's noisy, intermediate-scale quantum (NISQ) devices have a limited number of qubits and gate operations due to the presence of errors. Here, we propose a systematically improvable end-to-end pipeline to alleviate these limitations. Our proposed resource-efficient pipeline combines problem decomposition techniques for compact molecular representations, circuit optimization methods for compilation, solving the eigenvalue problem on advanced quantum hardware, and error-mitigation techniques in post-processing the results. Using the density matrix embedding theory for compact representation, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms taking into account all electrons equally and explicitly in the electronic structure calculation. In our experiment, we simulated the largest molecular system on a quantum computer within chemical accuracy with respect to total molecular energy calculated by the full CI method. Our methods reduce the number of qubits required for high-accuracy quantum simulations by an order of magnitude in the present work, enabling the simulation of larger, more industrially relevant molecules using NISQ devices. They are further systematically improvable as devices' computational capacity continues to grow.

Efficient and Accurate Electronic Structure Simulation Demonstrated on a Trapped-Ion Quantum Computer

Quantum computers have the potential to advance material design and drug discovery by performing costly electronic structure calculations. A critical aspect of this application requires optimizing the limited resources of the quantum hardware. Here, we experimentally demonstrate an end-to-end pipeline that focuses on minimizing quantum resources while maintaining accuracy. Using density matrix embedding theory as a problem decomposition technique, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms without freezing any electrons. The originally 20-qubit system is decomposed into 10 two-qubit problems, making it amenable to currently available hardware. Combining this decomposition with a qubit coupled cluster circuit ansatz, circuit optimization, and density matrix purification, we accurately reproduce the potential energy curve in agreement with the full configuration interaction energy in the minimal basis set. Our experimental results are an early demonstration of the potential for problem decomposition to accurately simulate large molecules on quantum hardware.

Optimizing Electronic Structure Simulations on a Trapped-ion Quantum Computer using Problem Decomposition

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Optimizing electronic structure simulations on a trapped-ion quantum computer using problem decomposition

Efficient and Accurate Electronic Structure Simulation Demonstrated on a Trapped-Ion Quantum Computer

Optimizing Electronic Structure Simulations on a Trapped-ion Quantum Computer using Problem Decomposition