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].

Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure.

No abstract

We provide a general method for efficiently simulating time-dependent Hamiltonian dynamics on a circuit-model based quantum computer. Our approach is based on approximating the truncated Dyson series of the evolution operator, extending the earlier proposal by Berry et al. [Phys. Rev. Lett. 114, 090502 (2015)] to evolution generated by explicitly time-dependent Hamiltonians. Two alternative strategies are proposed to implement time ordering while exploiting the superposition principle for sampling the Hamiltonian at different times. The resource cost of our simulation algorithm retains the optimal logarithmic dependence on the inverse of the desired precision. I. INTRODUCTIONSimulation of physical systems is envisioned to be one of the main applications for quantum computers [1]. Effective modeling of the dynamics and its generated time evolution is crucial to a deeper understanding of many-body systems, spin models, and quantum chemistry [2], and may thus have significant implications for many areas of chemistry and materials sciences. Simulation of the intrinsic Hamiltonian evolution of quantum systems was the first potential use of quantum computers suggested by Feynman [3] in 1982. Quantum simulation algorithms can model the evolution of a physical system with a complexity logarithmic in the dimension of the Hilbert space [4] (i.e. polynomial in the number of particles), unlike classical algorithms whose complexity is typically polynomial in the dimension, making simulations for practically interesting systems intractable for classical computers.The first quantum simulation algorithm was proposed by Lloyd [5] in 1996. There have been numerous advances since then providing improved performance [6][7][8][9][10][11][12][13][14][15]. One advance was to provide complexity that scales logarithmically in the error, and is nearly optimal in all other parameters [13]. Further improvements were provided by quantum signal processing methodology [14,16] as well as qubitization [15], which achieve optimal query complexity.An important case is that of time-dependent Hamiltonians. Efficient simulation of time-dependent Hamiltonians would allow us to devise better quantum control schemes [17] and describe transition states of chemical reactions [18]. Furthermore, simulation of dynamics generated by time-dependent Hamiltonians is a key component for implementing adiabatic algorithms [19] and the quantum approximate optimization algorithm [20] in a gate-based quantum circuit architecture.The most recent advances in quantum simulation algorithms are for time-independent Hamiltonians. Techniques for simulating time-dependent Hamiltonians based on the Lie-Trotter-Suzuki decomposition were developed in [7,9], but the complexity scales polynomially with error. More recent advances providing complexity logarithmic in the error [11,13] mention that their techniques can be generalized to time-dependent scenarios, but do not analyze this case. The most recent algorithms [14,15] are not directly applicable to the time-dependent case. H...

No abstract

The normalized state |ψ(t) = c 1 (t)|1 + c 2 (t)|2 of a single two-level system performs oscillations under the influence of a resonant driving field. It is assumed that only one realization of this process is available. We show that it is possible to approximately visualize in real time the evolution of the system as far as it is given by |c 2 (t)| 2 . For this purpose we use a sequence of particular unsharp measurements separated in time. They are specified within the theory of generalized measurements in which observables are represented by positive operator valued measures (POVM). A realization of the unsharp measurements may be obtained by coupling the two-level system to a meter and performing the usual projection measurements on the meter only. *

We develop a model for practical, entanglement-based long-distance quantum key distribution employing entanglement swapping as a key building block. Relying only on existing off-the-shelf technology, we show how to optimize resources so as to maximize secret key distribution rates. The tools comprise lossy transmission links, such as telecom optical fibers or free space, parametric down-conversion sources of entangled photon pairs, and threshold detectors that are inefficient and have dark counts. Our analysis provides the optimal trade-off between detector efficiency and dark counts, which are usually competing, as well as the optimal source brightness that maximizes the secret key rate for specified distances (i.e. loss) between sender and receiver.

We discuss a non-linear stochastic master equation that governs the time-evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. For the first time, we can prove that the calculated estimate almost always converges to the true state, also at low-efficiency measurements. We show that our single-state theory can be adapted to weak continuous ensemble measurements as well.