Exponentially more precise quantum simulation of fermions in the configuration interaction representation

Abstract

We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As in previous work [Babbush et al., New Journal of Physics 18, 033032 (2016)], we employ a recently developed technique for simulating Hamiltonian evolution, using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The algorithm of this paper involves simulation under an oracle for the sparse, first-quantized representation of the molecular Hamiltonian known as the configuration interaction (CI) matrix. We construct and query the CI matrix oracle to allow for on-the-fly computation of molecular integrals in a way that is exponentially more efficient than classical numerical methods. Whereas second-quantized representations of the wavefunction require O(N ) qubits, where N is the number of single-particle spin-orbitals, the CI matrix representation requires O(η) qubits where η N is the number of electrons in the molecule of interest. We show that the gate count of our algorithm scales at most as O(η 2 N 3 1 We use the typical computer science convention that f ∈ Θ(g), for any functions f and g, if f is asymptotically upper and lower bounded by multiples of g, O indicates an asymptotic upper bound, O indicates an asymptotic upper bound suppressing any polylogarithmic factors in the problem parameters, Ω indicates the asymptotic lower bound and f ∈ o(g) implies f /g → 0 in the asymptotic limit. arXiv:1506.01029v3 [quant-ph] 25 May 2017 1. Represent the molecular Hamiltonian in Eq.(1) in first quantization using the CI matrix formalism. This requires selection of a spin-orbital basis set, chosen such that the conditions in Theorem 1 are satisfied.2. Decompose the Hamiltonian into sums of self-inverse matrices approximating the required molecular integrals via the method of Section IV.3. Query the CI matrix oracle to evaluate the above self-inverse matrices, which we describe in Section V.4. Simulate the evolution of the system over time t using the method of [27], which is summarized in Section VI.2 The basis of atomic orbitals is not necessarily orthogonal. However, this can be fixed using the efficient Lowdin symmetric orthogonalization procedure which seeks the closest orthogonal basis [16,36].

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