Fault-Tolerant Quantum Simulations of Chemistry in First Quantization

Abstract

Quantum simulations of chemistry in first quantization offer important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside the Born-Oppenheimer approximation. However, as all prior work on quantum simulation in first quantization has been limited to asymptotic analysis, it has been impossible to compare the resources required for these approaches to those for more commonly studied algorithms in second quantization. Here, we analyze and optimize the resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al that realize block encodings for the qubitization and interaction picture frameworks of Low et al. The two algorithms we study enable simulation with gate complexities $\tilde{\mathcal{O}}({\eta }^{8/3}{N}^{1/3}t+{\eta }^{4/3}{N}^{2/3}t)$ and $\tilde{\mathcal{O}}({\eta }^{8/3}{N}^{1/3}t)$ where $\eta$ is the number of electrons, $N$ is the number of plane wave basis functions, and $t$ is the duration of time-evolution ($t$ is 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 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. 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 qubitized algorithm often requires much less surface code spacetime volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals.

96 pages, 9 figures, 8 tables

References in corpus (13)

• Surface codes: Towards practical large-scale quantum computation
• Simulated Quantum Computation of Molecular Energies
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Polynomial-time quantum algorithm for the simulation of chemical dynamics
• Truncation of Periodic Image Interactions for Confined Systems
• Efficient Distributed Quantum Computing
• Tapering off qubits to simulate fermionic Hamiltonians
• Ideal regularization of the Coulomb singularity in exact exchange by Wigner-Seitz truncated interactions: towards chemical accuracy in non-trivial systems
• Quantum computing and the entanglement frontier
• Simulating chemistry efficiently on fault-tolerant quantum computers
• Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules
• Downfolding of many-body Hamiltonians using active-space models: extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms
• Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian

Cited by in corpus (30)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules
• Exact and efficient Lanczos method on a quantum computer
• Towards near-term quantum simulation of materials
• Simulating key properties of lithium-ion batteries with a fault-tolerant quantum computer
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory
• Efficient quantum computation of molecular forces and other energy gradients
• Hybridized Methods for Quantum Simulation in the Interaction Picture
• Grid-based methods for chemistry simulations on a quantum computer
• Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation
• T-count and T-depth of any multi-qubit unitary
• Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods
• Quantum Computation for Periodic Solids in Second Quantization
• On the complexity of implementing Trotter steps
• Quantum simulation of real-space dynamics
• Quantum algorithm for time-dependent differential equations using Dyson series
• The Basics of Quantum Computing for Chemists
• Composite Quantum Simulations
• Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian
• Importance sampling for stochastic quantum simulations
• Quantum computation of molecular structure using data from challenging-to-classically-simulate nuclear magnetic resonance experiments
• Success of digital adiabatic simulation with large Trotter step
• Quantifying $T$-gate-count improvements for ground-state-energy estimation with near-optimal state preparation
• Selection and improvement of product formulae for best performance of quantum simulation
• TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry
• Differentiable quantum computational chemistry with PennyLane
• Near-term quantum algorithm for computing molecular and materials properties based on recursive variational series methods
• Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics
• Quantum Computing in Pharma: A Multilayer Embedding Approach for Near Future Applications
• Some Error Analysis for the Quantum Phase Estimation Algorithms