There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena such as high temperature superconductivity, solid-state physics, transition metal catalysis, and certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable these problems to be tackled with fewer quantum resources should be considered important. Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry. This review provides a comprehensive introduction to both computational chemistry and quantum computing, bridging the current knowledge gap. Major developments in this area are reviewed, with a particular focus on near-term quantum computation. Illustrations of key methods are provided, explicitly demonstrating how to map chemical problems onto a quantum computer, and how to solve them. The review concludes with an outlook on this nascent field.

Quantum computational chemistry

Quantum computers hold promise to enable efficient simulations of the properties of molecules and materials; however, at present they only permit ab initio calculations of a few atoms, due to a limited number of qubits. In order to harness the power of near-term quantum computers for simulations of larger systems, it is desirable to develop hybrid quantum-classical methods where the quantum computation is restricted to a small portion of the system. This is of particular relevance for molecules and solids where an active region requires a higher level of theoretical accuracy than its environment. Here we present a quantum embedding theory for the calculation of strongly-correlated electronic states of active regions, with the rest of the system described within density functional theory. We demonstrate the accuracy and effectiveness of the approach by investigating several defect quantum bits in semiconductors that are of great interest for quantum information technologies. We perform calculations on quantum computers and show that they yield results in agreement with those obtained with exact diagonalization on classical architectures, paving the way to simulations of realistic materials on near-term quantum computers.

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Quantum simulations of materials on near-term quantum computers

The Variational Quantum Eigensolver: A review of methods and best practices

The variational quantum eigensolver (or VQE) uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits. The VQE may be used to model complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims to provide an overview of the progress that has been made on the different parts of the algorithm. All the different components of the algorithm are reviewed in detail including representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process, the post-processing mitigation of errors, and best practices are suggested. We identify four main areas of future research:(1) optimal measurement schemes for reduction of circuit repetitions; (2) large scale parallelization across many quantum computers;(3) ways to overcome the potential appearance of vanishing gradients in the optimization process, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.

The Variational Quantum Eigensolver: a review of methods and best practices

With the aim of establishing a framework to efficiently perform the practical application of quantum chemistry simulation on near-term quantum devices, we envision a hybrid quantum--classical framework for leveraging problem decomposition (PD) techniques in quantum chemistry. Specifically, we use PD techniques to decompose a target molecular system into smaller subsystems requiring fewer computational resources. In our framework, there are two levels of hybridization. At the first level, we use a classical algorithm to decompose a target molecule into subsystems, and utilize a quantum algorithm to simulate the quantum nature of the subsystems. The second level is in the quantum algorithm. We consider the quantum--classical variational algorithm that iterates between an expectation estimation using a quantum device and a parameter optimization using a classical device. We investigate three popular PD techniques for our hybrid approach: the fragment molecular-orbital (FMO) method, the divide-and-conquer (DC) technique, and the density matrix embedding theory (DMET). We examine the efficacy of these techniques in correctly differentiating conformations of simple alkane molecules. In particular, we consider the ratio between the number of qubits for PD and that of the full system; the mean absolute deviation; and the Pearson correlation coefficient and Spearman's rank correlation coefficient. Sampling error is introduced when expectation values are measured on the quantum device. Therefore, we study how this error affects the predictive performance of PD techniques. The present study is our first step to opening up the possibility of using quantum chemistry simulations at a scale close to the size of molecules relevant to industry on near-term quantum hardware.

Towards the Practical Application of Near-Term Quantum Computers in Quantum Chemistry Simulations: A Problem Decomposition Approach.

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

The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum (NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation energies for the case of small molecules (i.e., BeH2, CH4, NH3, H2O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex catalyst that is utilized in industrial settings for the polymerization of α-olefins. We show that the MI-FNO approach provides a significant reduction in the quantum bit (qubit) requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices.

Scaling up electronic structure calculations on quantum computers: The frozen natural orbital based method of increments.

The accelerated progress in manufacturing noisy intermediate-scale quantum (NISQ) computing hardware has opened the possibility of exploring its application in transforming approaches to solving computationally challenging problems. The important limitations common among all NISQ computing technologies are the absence of error correction and the short coherence time, which limit the computational power of these systems. Shortening the required time of a single run of a quantum algorithm is essential for reducing environment-induced errors and for the efficiency of the computation. We have investigated the ability of a variational version of adiabatic quantum computation (AQC) to generate an accurate state more efficiently compared to existing adiabatic methods. The standard AQC method uses a time-dependent Hamiltonian, connecting the initial Hamiltonian with the final Hamiltonian. In the current approach, a navigator Hamiltonian is introduced which has a non-zero amplitude only in the middle of the annealing process. Both the initial and navigator Hamiltonians are determined using variational methods. A hermitian cluster operator, inspired by coupled-cluster theory and truncated to single and double excitations/de-excitations, is used as a navigator Hamiltonian. A comparative study of our variational algorithm (VanQver) with that of standard AQC, starting with a Hartree--Fock Hamiltonian, is presented. The results indicate that the introduction of the navigator Hamiltonian significantly improves the annealing time required to achieve chemical accuracy by two to three orders of magnitude. The efficiency of the method is demonstrated in the ground-state energy estimation of molecular systems, namely, H$_2$, P4, and LiH.

VanQver: the variational and adiabatically navigated quantum eigensolver

The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum~(NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation energies for the case of small molecules (i.e., BeH$_2$, CH$_4$, NH$_3$, H$_2$O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex (CGC) catalyst that is utilized in industrial settings for the polymerization of $\alpha$-olefins. We show that the MI-FNO approach provides a significant reduction in the qubit requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices.

Takeshi Yamazaki

Papers

Towards the Practical Application of Near-Term Quantum Computers in Quantum Chemistry Simulations: A Problem Decomposition Approach.

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

Scaling up electronic structure calculations on quantum computers: The frozen natural orbital based method of increments.

VanQver: the variational and adiabatically navigated quantum eigensolver

Scaling Up Electronic Structure Calculations on Quantum Computers: The Frozen Natural Orbital Based Method of Increments