The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry (1406.4920v1)

Abstract: The simulation of molecules is a widely anticipated application of quantum computers. However, recent studies \cite{WBCH13a,HWBT14a} have cast a shadow on this hope by revealing that the complexity in gate count of such simulations increases with the number of spin orbitals $N$ as $N^8$, which becomes prohibitive even for molecules of modest size $N\sim 100$. This study was partly based on a scaling analysis of the Trotter step required for an ensemble of random artificial molecules. Here, we revisit this analysis and find instead that the scaling is closer to $N^6$ in worst case for real model molecules we have studied, indicating that the random ensemble fails to accurately capture the statistical properties of real-world molecules. Actual scaling may be significantly better than this due to averaging effects. We then present an alternative simulation scheme and show that it can sometimes outperform existing schemes, but that this possibility depends crucially on the details of the simulated molecule. We obtain further improvements using a version of the coalescing scheme of \cite{WBCH13a}; this scheme is based on using different Trotter steps for different terms. The method we use to bound the complexity of simulating a given molecule is efficient, in contrast to the approach of \cite{WBCH13a,HWBT14a} which relied on exponentially costly classical exact simulation.

The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry

In the paper "The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry," the authors, David Poulin and colleagues, provide a detailed analysis of the computational complexity involved in simulating quantum chemical systems using quantum computers. The focus is on improving and understanding the gate count requirements for simulating molecules, which is a pivotal application of quantum computing. This work revisits scaling analyses and proposes alternative simulation schemes to refine our understanding of the necessary resources.

The anticipated quantum simulation of molecules is critically evaluated, particularly in terms of its gate complexity for molecules represented by a given number of spin orbitals, denoted by $N$. Past studies have shown prohibitive scaling laws—increasing as $N^8$—which make practical applications challenging. This paper reassesses and refines these scaling estimates to show that, for real-world molecules, the scaling could be as favorable as $N^6$ in the worst case scenario.

The main contribution of this paper is threefold:

1. Refinement of Scaling Analysis: By revisiting previous analyses that relied on an ensemble of random molecules, this paper suggests that such ensembles do not accurately reflect the properties of real molecules. New scaling results are derived, showing complexity scaling between $N^{1.5}$ to $N^{2.5}$, which challenges the previously accepted growth derived from artificial models.
2. Alternative Decomposition Approach: The authors propose a decomposition of the Hamiltonian into a sum of squared terms of free Hamiltonians, reducing the effective number of interaction terms. This method is shown to achieve significant improvements in the number of trotter steps required, particularly for certain ensembles of molecules, highlighting a better-than-expected performance due to more efficient error bounds.
3. Coalescing Scheme: Building on prior work, the paper discusses a refined coalescing approach whereby different Hamiltonian terms are allowed different Trotter step sizes. This enables the possibility of optimizing the simulation by executing certain terms less frequently, depending on their significance, thus reducing the total computational cost in terms of gate counts.

The implications of this research are significant. By suggesting that real molecules are easier to simulate than previously anticipated, it opens the door to more efficient quantum simulations that are computationally feasible with smaller quantum processors. Moreover, the approach used to bound the complexity is computationally efficient and avoids the exponentially costly exact simulations utilized in earlier studies.

The theoretical developments laid out in this paper also hint towards future possibilities where more refined error analyses and Hamiltonian decompositions could further bridge the gap toward practical quantum computational chemistry. As understanding and technology improve, these advancements could culminate in breakthroughs within both theoretical realms and practical applications, such as drug design and material science.

While the paper stops short of offering practical algorithms ready for immediate industrial application, it provides essential groundwork and several promising strategies for future exploration. Researchers in the field of quantum computation and quantum chemistry stand to gain from these insights, especially those developing quantum algorithms where optimizing gate counts and minimizing errors are paramount to success.