Towards Quantum Chemistry on a Quantum Computer (0905.0887v3)

Abstract: The fundamental problem faced in quantum chemistry is the calculation of molecular properties, which are of practical importance in fields ranging from materials science to biochemistry. Within chemical precision, the total energy of a molecule as well as most other properties, can be calculated by solving the Schrodinger equation. However, the computational resources required to obtain exact solutions on a conventional computer generally increase exponentially with the number of atoms involved. This renders such calculations intractable for all but the smallest of systems. Recently, an efficient algorithm has been proposed enabling a quantum computer to overcome this problem by achieving only a polynomial resource scaling with system size. Such a tool would therefore provide an extremely powerful tool for new science and technology. Here we present a photonic implementation for the smallest problem: obtaining the energies of H2, the hydrogen molecule in a minimal basis. We perform a key algorithmic step - the iterative phase estimation algorithm - in full, achieving a high level of precision and robustness to error. We implement other algorithmic steps with assistance from a classical computer and explain how this non-scalable approach could be avoided. Finally, we provide new theoretical results which lay the foundations for the next generation of simulation experiments using quantum computers. We have made early experimental progress towards the long-term goal of exploiting quantum information to speed up quantum chemistry calculations.

• The paper introduces a photonic implementation of IPEA for calculating H₂ energies with 10⁻⁵ Eₕ precision.
• The study demonstrates a method to reduce computational scaling from exponential to polynomial in molecular simulations.
• The research underscores the need for efficient eigenstate preparation and error correction to enable scalable quantum simulations.

Towards Quantum Chemistry on a Quantum Computer

The paper, "Towards Quantum Chemistry on a Quantum Computer," presents significant advancements in addressing one of the longstanding and computationally demanding problems in quantum chemistry: accurate calculation of molecular properties, which traditionally requires solving the Schrödinger equation. The intrinsic complexity of solving this equation for multi-atom systems causes the computational resources required to scale exponentially with system size, rendering such calculations intractable for all but the smallest systems when using classical computers.

The authors propose a novel approach leveraging quantum computing to achieve a polynomial resource scaling with system size, thereby significantly improving the feasibility of these computations. They specifically demonstrate a photonic implementation for the smallest nontrivial problem: determining the energies of H₂, the hydrogen molecule, in a minimal basis set. Through the implementation of a key algorithmic step, the iterative phase estimation algorithm (IPEA), they achieve a high level of precision and robustness to experimental errors in their calculations.

Key Contributions and Results

The work progresses through a structured implementation involving:

1. Algorithmic Structure: The paper employs the iterative phase estimation algorithm to acquire the molecular energies. By encoding the molecular wavefunction into qubits, simulating its time evolution with quantum logic gates, and extracting the energy using phase estimation, the authors establish an efficient computational pathway that significantly challenges the limitations of classical full configuration interaction (FCI) methodologies.
2. Photonic Implementation: The experimental results demonstrate the successful execution of the IPEA on a photonic architecture. The paper achieved energy calculations and potential energy surface reconstruction for H₂ with a precision of approximately 10⁻⁵ Eₕ across various internuclear separations, representing the ground state energy with notable accuracy at the equilibrium bond length.
3. Scalability Pathways: The authors also address the scalability challenges in executing quantum algorithms that could overcome classical computational constraints. They highlight two essential tasks — eigenstate preparation and molecular evolution operator decomposition — that require solutions for scaling the quantum algorithm to larger systems efficiently.
4. Resource Counting and Error Management: A comprehensive resource count reveals that replicating a scalable simulation of H₂ within chemical precision would require 4 qubits and approximately 522 perfect gates. The approach underscores the need for error-correcting techniques and scalable quantum logic circuits to make larger-scale quantum simulations feasible.

Implications and Future Directions

This paper sets a foundation for future experimental and theoretical efforts aimed at quantum simulations for chemical systems:

• Experimental Control: The research highlights the necessity of improved experimental techniques for encoding eigenstates and non-trivial decomposition of molecular evolution operators, suggesting next steps for experimentalists in controlling quantum systems.
• Quantum Algorithm Expansion: The framework established by the authors can be extended into more sophisticated molecular systems, potentially providing evaluative pathways for larger, chemically significant molecules.
• Scalability and Error Correction: With advances in quantum error correction and fault-tolerant quantum computation, this approach can be expanded to accommodate larger molecules, crucial for numerous applications in chemistry and materials science.

The discourse in this paper thereby serves as a significant intersection of quantum computing and quantum chemistry, showcasing the practical utility of quantum algorithms in overcoming computation-bound problems that have traditionally limited the chemical sciences.

The paper advances toward the realization of quantum computational chemistry, setting a precedent for how quantum computing capabilities can be leveraged to transcend the traditional limits of molecular energy calculations, paving the way for novel explorations in material science and beyond.