Abstract
Quantum simulations of chemistry in first quantization offer some important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside of the Born-Oppenheimer approximation. However, since all prior work on quantum simulation of chemistry in first quantization has been limited to asymptotic analysis, it has been impossible to directly compare the resources required for these approaches to those required for the more commonly studied algorithms in second quantization. Here, we compile, optimize, and analyze the finite resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al. [Npj Quantum Inf. 5, 92 (2019)] that realize block encodings for the qubitization and interaction-picture frameworks of Low et al. [Quantum 3, 163 (2019), arXiv:1805.00675 (2018)]. The two algorithms we study enable simulation with gate complexities of and where is the number of electrons, is the number of plane-wave basis functions, and is the duration of time evolution ( is linearly inverse to target precision when the goal is to estimate energies). In addition to providing the first explicit circuits and constant factors for any first quantized simulation, and then introducing improvements, which reduce circuit complexity by about a thousandfold over naive implementations for modest sized systems, we also describe new algorithms that asymptotically achieve the same scaling in a real-space representation. Finally, we assess the resources required to simulate various molecules and materials and conclude that the qubitized algorithm will often be more practical than the interaction-picture algorithm. We demonstrate that our qubitization algorithm often requires much less surface-code space-time volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals.
2 More- Received 31 May 2021
- Accepted 4 October 2021
- Corrected 21 September 2022
DOI:https://doi.org/10.1103/PRXQuantum.2.040332
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
21 September 2022
Correction: Headings for Sec. II C and Sec. II D contained typographical errors introduced during the production process and have been set right.
Popular Summary
The simulation of molecular and material systems is one of the most anticipated applications of quantum computers, fulfilling Feynman’s original vision for the field. As such, it is important to come up with the best algorithms for this application in order to assess what sort of quantum computers would be required to tackle impactful problems. Here, we perform the first complete analysis of the cost required to implement a rarely studied approach to the problem that represents molecular systems in “first quantization.”
Unlike the more commonly used second quantized representation, in first quantization one encodes the symmetry of fermions and bosons in the wave function as opposed to in the operators. This leads to conceptually more complex algorithms but with some important advantages such as the ability to use enormously large basis sets that can represent systems in the continuum limit and enable explicit nuclear quantum dynamics.
By compiling and optimizing first quantized algorithms that have only been described asymptotically in prior work, we show that these approaches are unrivaled in terms of efficiency when it comes to simulating solid-state materials. Even more surprisingly, these methods substantially outperform all prior methods even for small molecules, with an advantage that increases with system size. As a result, our work opens an exciting new paradigm for quantum-computing chemistry, suggesting that the field’s future is first quantized!