Gate count estimates for performing quantum chemistry on small quantum computers (1312.1695v3)

Abstract: As quantum computing technology improves and quantum computers with a small but non-trivial number of N > 100 qubits appear feasible in the near future the question of possible applications of small quantum computers gains importance. One frequently mentioned application is Feynman's original proposal of simulating quantum systems, and in particular the electronic structure of molecules and materials. In this paper, we analyze the computational requirements for one of the standard algorithms to perform quantum chemistry on a quantum computer. We focus on the quantum resources required to find the ground state of a molecule twice as large as what current classical computers can solve exactly. We find that while such a problem requires about a ten-fold increase in the number of qubits over current technology, the required increase in the number of gates that can be coherently executed is many orders of magnitude larger. This suggests that for quantum computation to become useful for quantum chemistry problems, drastic algorithmic improvements will be needed.

• The paper demonstrates that achieving chemical accuracy for a water molecule simulation requires approximately six million Trotter steps.
• It employs quantum full configuration interaction and Trotterization schemes to quantitatively assess the scaling of gate counts with system size.
• The findings underscore the urgent need for novel algorithms and error mitigation techniques to overcome the prohibitive costs of simulating larger molecules.

Resource Constraints in Quantum Chemistry Simulations on Quantum Computers

The paper "Gate count estimates for performing quantum chemistry on small quantum computers" by Wecker et al. provides a quantitative analysis of the resource requirements necessary for conducting quantum chemistry calculations on quantum computing platforms. The authors explore the necessary computational resources to simulate molecular electronic structures using quantum computers, emphasizing the gap between current technology and what is required to surpass classical computing capabilities.

Overview

This research specifically explores simulating the ground state energy of molecules larger than those feasibly addressed by classical computers. The paper is motivated by Feynman's original proposal of harnessing quantum computers for quantum system simulations. While quantum computers hold the promise of solving classically intractable problems, the paper demonstrates that current hardware falls short, especially in terms of gate operations required, rather than by qubit count alone.

Quantum Resources and Algorithmic Challenges

The authors assess the quantum resources essential for quantum full configuration interaction (QFCI) algorithms: the number of qubits, gate count, and circuit depth. Their approach simulates the energy of a water molecule in a minimal basis and validates it against existing classical results. Importantly, they highlight that achieving chemical accuracy necessitates roughly six million Trotter steps due to the requisite time evolution in standard quantum dynamics simulation.

The gate count escalates quickly with system size, and while qubits might only need an order-of-magnitude increase to handle classically intractable problems, the gate operations must scale more significantly—potentially by orders of magnitude larger. The paper notes that algorithmic improvements are crucial, as hardware alone cannot close this gap efficiently within the foreseeable future.

Numerical Estimates and Scaling Analysis

Using a water molecule benchmark, they estimate the gate count and evaluate the influence of different trotterization schemes on the computational cost. The findings reveal that solving larger molecules' ground state energy, such as Fe₂S₂, presents even more substantial challenges—requiring upwards of 10¹⁸ gates, unattainable by foreseeable quantum hardware technology.

By examining the scaling laws, the research illustrates the demands increase with the number of spin orbitals in a manner largely attributable to quantum operations' inherent complexity. The scaling of quantum time steps to maintain chemical accuracy underscores the necessity for finer optimization beyond straightforward Trotter expansions or other decomposition methods.

Future Directions

1. Algorithmic Innovation: The paper suggests the requirement for new algorithms that dramatically reduce gate operations per computation step while maintaining, or improving, accuracy.
2. Error Mitigation Techniques: Exploration in adaptive trotterization and error-correction strategies presents theoretical opportunities in reducing computational costs, albeit not tackled exhaustively within this paper.
3. Hybrid Classical-Quantum Approaches: While not directly mentioned, future developments may also benefit from hybrid approaches that leverage classical pre-processing or post-processing to trim quantum requirements.

In conclusion, Wecker et al. provide critical insight into the quantum resource dilemma confronting quantum chemists. While the potential remains evident, the results of this paper act as a call for a concerted effort towards algorithmic and architectural innovations to reach the practical utility in quantum chemistry. Such strategic efforts may eventually render practical the exponential speedups quantum computers theoretically offer.