2 These authors contributed equally to this work.While neural networks have been remarkably successful for a variety of practical problems, they are often applied as a black box, which limits their utility for scientific discoveries. Here, we present a neural network architecture that can be used to discover physical concepts from experimental data without being provided with additional prior knowledge. For a variety of simple systems in classical and quantum mechanics, our network learns to compress experimental data to a simple representation and uses the representation to answer questions about the physical system. Physical concepts can be extracted from the learned representation, namely: (1) The representation stores the physically relevant parameters, like the frequency of a pendulum.(2) The network finds and exploits conservation laws: it stores the total angular momentum to predict the motion of two colliding particles. (3) Given measurement data of a simple quantum mechanical system, the network correctly recognizes the number of degrees of freedom describing the underlying quantum state. (4) Given a time series of the positions of the Sun and Mars as observed from Earth, the network discovers the heliocentric model of the solar systemthat is, it encodes the data into the angles of the two planets as seen from the Sun. Our work provides a first step towards answering the question whether the traditional ways by which physicists model nature naturally arise from the experimental data without any mathematical and physical pre-knowledge, or if there are alternative elegant formalisms, which may solve some of the fundamental conceptual problems in modern physics, such as the measurement problem in quantum mechanics.Problem: Predict the position of a one-dimensional damped pendulum at different times. Physical model: Equation of motionSolution:Observation: Time series of positions: o = x(t i ) i∈{1,...,50} ∈ R 50 , with equally spaced t i . Mass m = 1kg, amplitude A 0 = 1m and phase δ 0 = 0 are fixed; spring constant κ ∈ [5, 10] kg/s 2 and damping factor b ∈ [0.5, 1] kg/s are varied between training samples. Question: Prediction times: q = t pred ∈ R.Correct answer: Position at time t pred : a cor = x(t pred ) ∈ R .Implementation: Network depicted in Figure 1b with 3 latent neurons. Key findings:• SciNet predicts the positions x(t pred ) with a root mean square error below 2% (with respect to the amplitude A 0 = 1m) (Figure 2a).• SciNet stores κ and b in two of the latent neurons, and does not store any information in the third latent neuron (Figure 2b).

Self-testing is a method to characterise an arbitrary quantum system based only on its classical input-output correlations, and plays an important role in device-independent quantum information processing as well as quantum complexity theory. Prior works on self-testing require the assumption that the system's state is shared among multiple parties that only perform local measurements and cannot communicate. Here, we replace the setting of multiple non-communicating parties, which is difficult to enforce in practice, by a single computationally bounded party. Specifically, we construct a protocol that allows a classical verifier to robustly certify that a single computationally bounded quantum device must have prepared a Bell pair and performed single-qubit measurements on it, up to a change of basis applied to both the device's state and measurements. This means that under computational assumptions, the verifier is able to certify the presence of entanglement, a property usually closely associated with two separated subsystems, inside a single quantum device. To achieve this, we build on techniques first introduced by Brakerski et al. (2018) and Mahadev (2018) which allow a classical verifier to constrain the actions of a quantum device assuming the device does not break post-quantum cryptography.

In device-independent quantum key distribution (DIQKD), an adversary prepares a device consisting of two components, distributed to Alice and Bob, who use the device to generate a secure key. The security of existing DIQKD schemes holds under the assumption that the two components of the device cannot communicate with one another during the protocol execution. This is called the no-communication assumption in DIQKD. Here, we show how to replace this assumption, which can be hard to enforce in practice, by a standard computational assumption from post-quantum cryptography: we give a protocol that produces secure keys even when the components of an adversarial device can exchange arbitrary quantum communication, assuming the device is computationally bounded. Importantly, the computational assumption only needs to hold during the protocol execution—the keys generated at the end of the protocol are information-theoretically secure as in standard DIQKD protocols.

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Quantum computations are typically performed as a sequence of basic operations, called quantum gates. Different gate sequences, called quantum circuits, can implement the same overall quantum computation. Since every additional quantum gate takes time and introduces noise into the system, it is important to find the smallest possible quantum circuit that implements a given computation, especially for near-term quantum devices that can execute only a limited number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching: given a large and small quantum circuit, we would like to find all maximal matches of the small circuit, called a pattern , in the large circuit, considering pairwise commutation of quantum gates. In this work, we present the first classical algorithm for pattern matching that provably finds all maximal matches and is efficient enough to be practical for circuit sizes typical for near-term devices. We demonstrate numerically 1 that combining our algorithm with known pattern-matching-based circuit optimization techniques reduces the gate count of a random quantum circuit by ∼ 30% and can further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.