Learning DNFs under product distributions via {\mu}-biased quantum Fourier sampling

We show that DNF formulae can be quantum PAC-learned in polynomial time under product distributions using a quantum example oracle. The best classical algorithm (without access to membership queries) runs in superpolynomial time. Our result extends the work by Bshouty and Jackson (1998) that proved that DNF formulae are efficiently learnable under the uniform distribution using a quantum example oracle. Our proof is based on a new quantum algorithm that efficiently samples the coefficients of a {\mu}-biased Fourier transform.Comment: 17 pages; v3 based on journal version; minor corrections and clarification

Stabilisers as a design tool for new forms of Lechner-Hauke-Zoller Annealer

In a recent paper Lechner, Hauke and Zoller (LHZ) described a means to translate a Hamiltonian of $N$ spin-$\frac{1}{2}$ particles with 'all-to-all' interactions into a larger physical lattice with only on-site energies and local parity constraints. LHZ used this mapping to propose a novel form of quantum annealing. Here we provide a stabiliser-based formulation within which we can describe both this prior approach and a wide variety of variants. Examples include a triangular array supporting all-to-all connectivity, and moreover arrangements requiring only $2N$ or $N\log N$ spins but providing interesting bespoke connectivities. Further examples show that arbitrarily high order logical terms can be efficiently realised, even in a strictly 2D layout. Our stabilisers can correspond to either even-parity constraints, as in the LHZ proposal, or as odd-parity constraints. Considering the latter option applied to the original LHZ layout, we note it may simplify the physical realisation since the required ancillas are only spin-$\frac{1}{2}$ systems (i.e. qubits, rather than qutrits) and moreover the interactions are very simple. We make a preliminary assessment of the impact of this design choices by simulating small (few-qubit) systems; we find some indications that the new variant may maintain a larger minimum energy gap during the annealing process.Comment: A dramatically expanded revision: we now show how to use our stabiliser formulation to construct a wide variety of new physical layouts, including ones with fewer than Order N^2 spins but custom connectivities, and a means to achieve higher order coupling even in 2

Modelling Non-Markovian Quantum Processes with Recurrent Neural Networks

Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that recurrent neural networks are accurate over different times and regimes.Comment: 10 pages, 8 figure

Learning hard quantum distributions with variational autoencoders

Studying general quantum many-body systems is one of the major challenges in modern physics because it requires an amount of computational resources that scales exponentially with the size of the system.Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a new representation of states based on variational autoencoders. Variational autoencoders are a type of generative model in the form of a neural network. We probe the power of this representation by encoding probability distributions associated with states from different classes. Our simulations show that deep networks give a better representation for states that are hard to sample from, while providing no benefit for random states. This suggests that the probability distributions associated to hard quantum states might have a compositional structure that can be exploited by layered neural networks. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterising states of the size expected in first generation quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect to v1 (rewriting of section "hard and easy quantum states", extended discussion on comparison with tensor networks

Statistical Limits of Supervised Quantum Learning

Within the framework of statistical learning theory it is possible to bound the minimum number of samples required by a learner to reach a target accuracy. We show that if the bound on the accuracy is taken into account, quantum machine learning algorithms for supervised learning---for which statistical guarantees are available---cannot achieve polylogarithmic runtimes in the input dimension. We conclude that, when no further assumptions on the problem are made, quantum machine learning algorithms for supervised learning can have at most polynomial speedups over efficient classical algorithms, even in cases where quantum access to the data is naturally available.Comment: v3: 6 pages, journal version, title changed (previous title "The Statistical Limits of Supervised Quantum Learning"), other minor improvements; v2: 6 pages, title changed (previous title "Fast quantum learning with statistical guarantees"), format changed to two-columns, typos corrected, remarks that better clarify the limitations of our analysis adde

A chemical survey of exoplanets with ARIEL

Thousands of exoplanets have now been discovered with a huge range of masses, sizes and orbits: from rocky Earth-like planets to large gas giants grazing the surface of their host star. However, the essential nature of these exoplanets remains largely mysterious: there is no known, discernible pattern linking the presence, size, or orbital parameters of a planet to the nature of its parent star. We have little idea whether the chemistry of a planet is linked to its formation environment, or whether the type of host star drives the physics and chemistry of the planet’s birth, and evolution. ARIEL was conceived to observe a large number (~1000) of transiting planets for statistical understanding, including gas giants, Neptunes, super-Earths and Earth-size planets around a range of host star types using transit spectroscopy in the 1.25–7.8 μm spectral range and multiple narrow-band photometry in the optical. ARIEL will focus on warm and hot planets to take advantage of their well-mixed atmospheres which should show minimal condensation and sequestration of high-Z materials compared to their colder Solar System siblings. Said warm and hot atmospheres are expected to be more representative of the planetary bulk composition. Observations of these warm/hot exoplanets, and in particular of their elemental composition (especially C, O, N, S, Si), will allow the understanding of the early stages of planetary and atmospheric formation during the nebular phase and the following few million years. ARIEL will thus provide a representative picture of the chemical nature of the exoplanets and relate this directly to the type and chemical environment of the host star. ARIEL is designed as a dedicated survey mission for combined-light spectroscopy, capable of observing a large and well-defined planet sample within its 4-year mission lifetime. Transit, eclipse and phase-curve spectroscopy methods, whereby the signal from the star and planet are differentiated using knowledge of the planetary ephemerides, allow us to measure atmospheric signals from the planet at levels of 10–100 part per million (ppm) relative to the star and, given the bright nature of targets, also allows more sophisticated techniques, such as eclipse mapping, to give a deeper insight into the nature of the atmosphere. These types of observations require a stable payload and satellite platform with broad, instantaneous wavelength coverage to detect many molecular species, probe the thermal structure, identify clouds and monitor the stellar activity. The wavelength range proposed covers all the expected major atmospheric gases from e.g. H2O, CO2, CH4 NH3, HCN, H2S through to the more exotic metallic compounds, such as TiO, VO, and condensed species. Simulations of ARIEL performance in conducting exoplanet surveys have been performed – using conservative estimates of mission performance and a full model of all significant noise sources in the measurement – using a list of potential ARIEL targets that incorporates the latest available exoplanet statistics. The conclusion at the end of the Phase A study, is that ARIEL – in line with the stated mission objectives – will be able to observe about 1000 exoplanets depending on the details of the adopted survey strategy, thus confirming the feasibility of the main science objectives.Peer reviewedFinal Published versio