Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction
Abstract
We describe quantum circuits with only O ~ (N ) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (e.g., molecular) orbitals. With O (λ /ϵ ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1norm of Hamiltonian coefficients and ϵ is the target precision. This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonalbasis functions diagonalizing the Coloumb operator (e.g., the planewave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC factors. We then use qubitization to simulate the nonorthogonal THC Hamiltonian, in a fashion that avoids most complications of the nonorthogonal basis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling spacetime volume, compilation of our algorithm for challenging finitesized molecules such as FeMoCo reveals that our method requires the least faulttolerant resources of any known approach. By laying out and optimizing the surfacecode resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under 4 days of runtime, assuming 1μ s cycle times and physical gateerror rates no worse than 0.1 % .
 Publication:

PRX Quantum
 Pub Date:
 July 2021
 DOI:
 10.1103/PRXQuantum.2.030305
 arXiv:
 arXiv:2011.03494
 Bibcode:
 2021PRXQ....2c0305L
 Keywords:

 Quantum Physics;
 Physics  Chemical Physics
 EPrint:
 73 pages, fixed typos