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 1-norm 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 orthogonal-basis functions diagonalizing the Coloumb operator (e.g., the plane-wave 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 space-time volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach. By laying out and optimizing the surface-code 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 1-μs cycle times and physical gate-error rates no worse than 0.1%.13 MoreReceived 12 December 2020Revised 7 April 2021Accepted 24 May 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030305Published 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 SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum computationQuantum simulationPhysical SystemsStrongly correlated systemsTechniquesElectronic structureCondensed Matter, Materials & Applied PhysicsQuantum InformationAtomic, Molecular & Optical
Highlights
The primary contribution of this paper is to introduce a method for simulating electronic Hamiltonians in arbitrary basis on a quantum computer
The method we introduce uses a number of standard techniques for quantum simulation, including phase estimation of quantum walks, qubitization, QROM, coherent alias sampling, and unary iteration
Our key innovation is to adapt this set of tools to the tensor hypercontraction representation of quantum chemistry, which had previously gone unexplored in the context of quantum computing
Summary
The quantum computation of quantum chemistry is commonly regarded as one of the most promising applications of quantum computers [1,2,3]. Some algorithms achieved a lower scaling by performing simulation in a basis [15,19,20] that diagonalizes the Coulomb operator V such that Vpqrs = 0 unless p = q and r = s Those representations typically require a significantly larger N in order to model molecular systems within target accuracy of the continuum limit. [38] and optimized for use in our context [9] (where it is referred to as “QROAM”—a portmanteau of QROM and QRAM) Using these tools one can often construct algorithms to realize qubitized q√uantum walks with gate √complexity that scales as O( ) where one requires O( ) ancilla, and is the amount of information required to specify the Hamiltonian within a particular tensor factorization. It is critical to consider how the algorithmic choices that one makes will affect the value of λ
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