Measurement-based quantum computation (MBQC) is a model of quantum computation, in which computation proceeds via adaptive single qubit measurements on a multiqubit quantum state. It is computationally equivalent to the circuit model. Unlike the circuit model, however, its classical analog is little studied. Here we present a classical analog of MBQC whose computational complexity presents a rich structure. To do so, we identify uniform families of quantum computations [refining the circuits introduced by Bremner Proc. R. Soc. A 467, 459 (2010)] whose output is likely hard to exactly simulate (sample) classically. We demonstrate that these circuit families can be efficiently implemented in the MBQC model without adaptive measurement and, thus, can be achieved in a classical analog of MBQC whose resource state is a probability distribution which has been created quantum mechanically. Such states (by definition) violate no Bell inequality, but, if widely held beliefs about computational complexity are true, they, nevertheless, exhibit nonclassicality when used as a computational resource—an imprint of their quantum origin.

A central result in the study of Quantum Hamiltonian Complexity is that the k-local hamiltonian problem is QMA-complete [1]. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models based on measurement instead of unitary evolution, furthermore we can use post-selected measurement as an additional computational tool. In this work, we generalise Kitaev's construction to allow for non-unitary evolution including post-selection. Furthermore, we consider a type of post-selection under which the construction is consistent, which we call tame post-selection. We consider the computational complexity consequences of this construction and then consider how the probability of an event upon which we are post-selecting affects the gap between the ground state energy and the energy of the first excited state of its corresponding Hamiltonian. We provide numerical evidence that the two are not immediately related, by giving a family of circuits where the probability of an event upon which we post-select is exponentially small, but the gap in the energy levels of the Hamiltonian decreases as a polynomial.

Computer vision has a wide range of applications from medical image analysis to robotics. Over the past few years, the field has been transformed by machine learning and stands to benefit from potential advances in quantum computing. The main challenge for processing images on current and near-term quantum devices is the size of the data such devices can process. Images can be large, multidimensional and have multiple color channels. Current machine learning approaches to computer vision that exploit quantum resources require a significant amount of manual pre-processing of the images in order to be able to fit them onto the device. This paper proposes a framework to address the problem of processing large scale data on small quantum devices. This framework does not require any dataset-specific processing or information and works on large, grayscale and RGB images. Furthermore, it is capable of scaling to larger quantum hardware architectures as they become available. In the proposed approach, a classical autoencoder is trained to compress the image data to a size that can be loaded onto a quantum device. Then, a Restricted Boltzmann Machine (RBM) is trained on the D-Wave device using the compressed data, and the weights from the RBM are then used to initialize a neural network for image classification. Results are demonstrated on two MNIST datasets and two medical imaging datasets.

Measurement-Based Quantum Computing (MBQC) is an alternative to the quantum circuit model, whereby the computation proceeds via measurements on an entangled resource state. Noise processes are a major experimental challenge to the construction of a quantum computer. Here, we investigate how noise processes affecting physical states affect the performed computation by considering MBQC on a one-dimensional cluster state. This allows us to break down the computation in a sequence of building blocks and map physical errors to logical errors. Next, we extend the Matrix Product State construction to mixed states (which is known as Matrix Product Operators) and once again map the effect of physical noise to logical noise acting within the correlation space. This approach allows us to consider more general errors than the conventional Pauli errors, and could be used in order to simulate noisy quantum computation.

Measurement-Based Quantum Computing (MBQC) is an alternative to the quantum circuit model, whereby the computation proceeds via measurements on an entangled resource state. Noise processes are a major experimental challenge to the construction of a quantum computer. Here, we investigate how noise processes affecting physical states affect the performed computation by considering MBQC on a one-dimensional cluster state. This allows us to break down the computation in a sequence of building blocks and map physical errors to logical errors. Next, we extend the Matrix Product State construction to mixed states (which is known as Matrix Product Operators) and once again map the effect of physical noise to logical noise acting within the correlation space. This approach allows us to consider more general errors than the conventional Pauli errors, and could be used in order to simulate noisy quantum computation.