We introduce novel algorithms for the quantum simulation of fermionic systems which are dramatically more efficient than those based on the Lie-Trotter-Suzuki decomposition. We present the first application of a general technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The key difficulty in applying algorithms for general sparse Hamiltonian simulation to fermionic simulation is that a query, corresponding to computation of an entry of the Hamiltonian, is costly to compute. This means that the gate complexity would be much higher than quantified by the query complexity. We solve this problem with a novel quantum algorithm for on-the-fly computation of integrals that is exponentially faster than classical sampling. While the approaches presented here are readily applicable to a wide class of fermionic models, we focus on quantum chemistry simulation in second quantization, perhaps the most studied application of Hamiltonian simulation. Our central result is an algorithm for simulating an N spin-orbital system that requires N tgates. This approach is exponentially faster in the inverse precision and at least cubically faster in N than all previous approaches to chemistry simulation in the literature.

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].

Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule.We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing "qsamples" instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution.In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.

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