19,840 research outputs found
Universality in Learning from Linear Measurements
We study the problem of recovering a structured signal from independently and identically drawn linear measurements. A convex penalty function f(â‹…) is considered which penalizes deviations from the desired structure, and signal recovery is performed by minimizing f(â‹…) subject to the linear measurement constraints. The main question of interest is to determine the minimum number of measurements that is necessary and sufficient for the perfect recovery of the unknown signal with high probability. Our main result states that, under some mild conditions on f(â‹…) and on the distribution from which the linear measurements are drawn, the minimum number of measurements required for perfect recovery depends only on the first and second order statistics of the measurement vectors. As a result, the required of number of measurements can be determining by studying measurement vectors that are Gaussian (and have the same mean vector and covariance matrix) for which a rich literature and comprehensive theory exists. As an application, we show that the minimum number of random quadratic measurements (also known as rank-one projections) required to recover a low rank positive semi-definite matrix is 3nr, where n is the dimension of the matrix and r is its rank. As a consequence, we settle the long standing open question of determining the minimum number of measurements required for perfect signal recovery in phase retrieval using the celebrated PhaseLift algorithm, and show it to be 3n
Continuous-variable quantum neural networks
We introduce a general method for building neural networks on quantum
computers. The quantum neural network is a variational quantum circuit built in
the continuous-variable (CV) architecture, which encodes quantum information in
continuous degrees of freedom such as the amplitudes of the electromagnetic
field. This circuit contains a layered structure of continuously parameterized
gates which is universal for CV quantum computation. Affine transformations and
nonlinear activation functions, two key elements in neural networks, are
enacted in the quantum network using Gaussian and non-Gaussian gates,
respectively. The non-Gaussian gates provide both the nonlinearity and the
universality of the model. Due to the structure of the CV model, the CV quantum
neural network can encode highly nonlinear transformations while remaining
completely unitary. We show how a classical network can be embedded into the
quantum formalism and propose quantum versions of various specialized model
such as convolutional, recurrent, and residual networks. Finally, we present
numerous modeling experiments built with the Strawberry Fields software
library. These experiments, including a classifier for fraud detection, a
network which generates Tetris images, and a hybrid classical-quantum
autoencoder, demonstrate the capability and adaptability of CV quantum neural
networks
Universal blind quantum computation
We present a protocol which allows a client to have a server carry out a
quantum computation for her such that the client's inputs, outputs and
computation remain perfectly private, and where she does not require any
quantum computational power or memory. The client only needs to be able to
prepare single qubits randomly chosen from a finite set and send them to the
server, who has the balance of the required quantum computational resources.
Our protocol is interactive: after the initial preparation of quantum states,
the client and server use two-way classical communication which enables the
client to drive the computation, giving single-qubit measurement instructions
to the server, depending on previous measurement outcomes. Our protocol works
for inputs and outputs that are either classical or quantum. We give an
authentication protocol that allows the client to detect an interfering server;
our scheme can also be made fault-tolerant.
We also generalize our result to the setting of a purely classical client who
communicates classically with two non-communicating entangled servers, in order
to perform a blind quantum computation. By incorporating the authentication
protocol, we show that any problem in BQP has an entangled two-prover
interactive proof with a purely classical verifier.
Our protocol is the first universal scheme which detects a cheating server,
as well as the first protocol which does not require any quantum computation
whatsoever on the client's side. The novelty of our approach is in using the
unique features of measurement-based quantum computing which allows us to
clearly distinguish between the quantum and classical aspects of a quantum
computation.Comment: 20 pages, 7 figures. This version contains detailed proofs of
authentication and fault tolerance. It also contains protocols for quantum
inputs and outputs and appendices not available in the published versio
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