Simulation of Electronic Structure Hamiltonians Using Quantum Computers (1001.3855v3)

Abstract: Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, to obtain the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule, of how a chemical Hamiltonian can be simulated using a quantum computer.

• The paper demonstrates how quantum phase estimation and quantum circuits can efficiently simulate electronic structure Hamiltonians derived from molecular integrals.
• It illustrates the mapping of second-quantized Hamiltonians into quantum circuits via the Jordan-Wigner transformation using a hydrogen molecule example.
• The study highlights the potential for exponential speedups, paving the way for advancements in quantum error correction and large-scale chemical simulations.

Simulation of Electronic Structure Hamiltonians Using Quantum Computers

The paper under discussion explores the potential for quantum computing to revolutionize the simulation of electronic structure Hamiltonians, a fundamental aspect of quantum chemistry. The authors, Whitfield, Biamonte, and Aspuru-Guzik, conduct a thorough investigation into how quantum computers can efficiently simulate chemical systems, overcoming limitations faced by classical computational techniques due to exponential scaling issues when solving Schrödinger's equation.

Key Focus and Methodology

The objective of the paper is to demonstrate how quantum computers can simulate the electronic structure of molecules using quantum circuits, specifically leveraging the quantum phase estimation (QPE) algorithm. The authors provide a detailed account of generating quantum circuits from pre-computed molecular integrals. These circuits can evaluate molecular energies for systems with fixed nuclear geometries.

The paper further discusses several extensions to established methodologies, particularly the adiabatic state preparation procedure critical for setting up initial states for QPE. The authors offer a practical example using the hydrogen molecule, illustrating the complete process from the creation of a quantum circuit to obtaining energy estimates through quantum simulation.

Theoretical Inceptions and Practical Implications

The discussion is steeped in consistent formalism, rooted in advancements from theoretical and computational chemistry. The paper presents a transformation from second-quantized molecular Hamiltonians into quantum circuits employing the Jordan-Wigner transformation. This is crucial as it allows simulations of molecular electronic structures directly on quantum platforms.

The implications of this research are substantial. Quantum computers hold promise for more efficiently finding solutions to electronic structure problems that besiege classical computational models due to resource constraints on larger molecules and complex systems. The quantum approach potentially offers exponential speedups by replacing simulations carried out through classical algorithms with quantum algorithms providing precise approximations of molecular observables.

Experimental Context and Quantum Error Considerations

Experimentally, this work sets a baseline for feasibly testing realistic quantum simulations in devices presently limited by decoherence and gate fidelity. The paper does not deeply delve into quantum error corrections. However, it touches upon resource estimations for q-bit redundancy in quantum error correction, emphasizing the prospective feasibility of quantum simulations without them in the near-term quantum devices.

Numerical Results and Future Prospects

The authors provide numerical demonstrations to validate their approach, notably the quantum simulation's applicability to the hydrogen molecule. Though illustrative, these simulations highlight a stepping stone towards applying this framework to more complex molecular systems.

Anticipated future developments in this domain include refining quantum error correction techniques to enhance practicability in near-term quantum devices. Additionally, optimization in state preparation methods and algorithmic advancements can further aid accurate simulations.

Conclusion

Overall, the paper offers an insightful exposition on the capacity of emerging quantum technologies to transcend the bottlenecks of classical modeling in quantum chemistry. While grounded in the simulation of simple molecules currently feasible with small quantum devices, the foundational methodology suggests a clear pathway towards broader applications in chemistry and materials science as quantum computational resources scale. As the field progresses, integrating robust quantum algorithms with expanding quantum hardware capabilities will be pivotal in realizing the full spectrum of possibilities that quantum simulations of electronic structure hold.