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arXiv:1807.09802 · npj Quantum Information

Quantum simulation of chemistry with sublinear scaling in basis size

Ryan Babbush, Dominic William Berry, Jarrod R. McClean, Hartmut Neven

Published Nov 1, 2019• 104 citations
arXivHTMLPDF

Abstract

Abstract We present a quantum algorithm for simulating quantum chemistry with gate complexity $$\tilde {\cal{O}}(N^{1/3}\eta ^{8/3})$$ O ̃ ( N 1 ∕ 3 η 8 ∕ 3 ) where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity $$\tilde {\cal{O}}(N^{8/3}{\mathrm{/}}\eta ^{2/3})$$ O ̃ ( N 8 ∕ 3 ∕ η 2 ∕ 3 ) . We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η , as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.

DOI
10.1038/s41534-019-0199-y

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Abstract We present a quantum algorithm for simulating quantum chemistry with gate complexity O~(N1/3η8/3)\tilde {\cal{O}}(N^{1/3}\eta ^{8/3})O~(N1/3η8/3) O ̃ ( N 1 ∕ 3 η 8 ∕ 3 ) where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity O~(N8/3/η2/3)\tilde {\cal{O}}(N^{8/3}{\mathrm{/}}\eta ^{2/3})O~(N8/3/η2/3) O ̃ ( N 8 ∕ 3 ∕ η 2 ∕ 3 ) . We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η , as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.
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Discretizing electronic degrees of freedom into atomic orbitals is the default choice to accurately model electrons in molecules in a compact and flexible way. We consider the task of simulating such electronic structure models on a quantum computer for NNN orbitals, keeping the number of orbitals per atom constant. Atomic orbitals give rise to Hamiltonians with potentially up to O(N4)\mathcal{O}(N^4)O(N4) terms and little structure, which made it challenging to match the complexity of methods based on plane waves and real-space grids. In this work, we show that a Trotter step for atomic orbitals can be implemented in depth O(polylog(N))\mathcal{O}(\mathrm{polylog}(N))O(polylog(N)) without ancillas by combining a rigorous treatment of orbital localization, a hierarchical Hamiltonian decomposition based on the Fast Multipole Method, and shallow quantum Fourier arithmetic circuits. By bounding the Trotter error, we obtain sublinear simulation depth, and our total gate count N5/3+o(1)N^{5/3 + o(1)}N5/3+o(1) matches the best scaling known for any basis.
Here we describe an approach for simulating electronic structure on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantized representation of the molecular Hamiltonian, which we propagate using high-order product formulas. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with O ˜ ( η ) complexity for a simulation with η particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with t ( η 4 / 3 N 1 / 3 + η 1 / 3 N 2 / 3 ) ( η N t / ϵ ) o ( 1 ) gate complexity, where N is the number of grid points, ϵ is target precision, and t is the duration of time evolution. This is roughly a speedup by O ( η ) over most prior algorithms. We provide lower complexity than all prior work for N < η 7 (the regime of practical interest), with only first-quantized interaction-picture simulations providing better performance for N > η 7 . As with the classical fast multipole method, large particle numbers η ≳ 10 3 would be needed to realize this advantage.
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O~(N3/2λ)\widetilde{\cal O}(N^{3/2} \lambda)O(N3/2λ) T complexity, where NNN is number of orbitals and λ\lambdaλ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O ~ ( N 3 / 2 λ ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.