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).

We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.

We consider the decomposition of arbitrary isometries into a sequence of single-qubit and Controlled-not (C-not) gates. In many experimental architectures, the C-not gate is relatively 'expensive' and hence we aim to keep the number of these as low as possible. We derive a theoretical lower bound on the number of C-not gates required to decompose an arbitrary isometry from m to n qubits, and give three explicit gate decompositions that achieve this bound up to a factor of about two in the leading order. We also perform some bespoke optimizations for certain cases where m and n are small. In addition, we show how to apply our result for isometries to give a decomposition scheme for an arbitrary quantum operation via Stinespring's theorem, and derive a lower bound on the number of C-nots in this case too. These results will have an impact on experimental efforts to build a quantum computer, enabling them to go further with the same resources.

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-not (C-not) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-not gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-not gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-not count. We give explicit examples for small numbers of qubits that provide the lowest known C-not counts.

We introduce an open source software package UniversalQCompiler written in Mathematica that allows the decomposition of arbitrary quantum operations into a sequence of single-qubit rotations (with arbitrary rotation angles) and controlled-NOT (C-not) gates. Together with the existing package QI , this allows quantum information protocols to be analysed and then compiled to quantum circuits. Our decompositions are based on Phys. Rev. A 93, 032318 (2016), and hence, for generic operations, they are near optimal in terms of the number of gates required. UniversalQ-Compiler allows the compilation of any isometry (in particular, it can be used for unitaries and state preparation), quantum channel or positive-operator valued measure (POVM), although the run time becomes prohibitive for large numbers of qubits. The resulting circuits can be displayed graphically within Mathematica or exported to L A T E X. We also provide functionality to translate the circuits to OpenQASM, the quantum assembly language used, for instance, by the IBM Q Experience.

No abstract

No abstract

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.