We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilson's Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.

Tensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given.

We present a novel combination of generative and predictive machine learning models for discovering unique protein inhibitors. The new method is assessed on its ability to generate unique inhibitors for the cancer associated protein kinase, CDK9. We validate our method by performing biochemical assays, attaining a hit rate of more than 10%, demonstrating the method to be a notable improvement upon a more standard, and somewhat naive approach. Moreover, we imposed the additional challenge of finding inhibitors that are readily synthesized. Importantly, two new inhibitors are found, with one being distinct from reported CDK9 inhibitors. We discuss the results in the context of modern machine learning principles and the desire expressed by the rational drug design community to secure molecules that are structurally different, yet with high binding affinities, to structurally determined protein targets.

Architectures for sparse hierarchical representation learning have recently been proposed for graph-structured data, but so far assume the absence of edge features in the graph. We close this gap and propose a method to pool graphs with edge features, inspired by the hierarchical nature of chemistry. In particular, we introduce two types of pooling layers compatible with an edge-feature graph-convolutional architecture and investigate their performance for molecules relevant to drug discovery on a set of two classification and two regression benchmark datasets of MoleculeNet. We find that our models significantly outperform previous benchmarks on three of the datasets and reach state-of-the-art results on the fourth benchmark, with pooling improving performance for three out of four tasks, keeping performance stable on the fourth task, and generally speeding up the training process.

This paper demonstrates a method for tensorizing neural networks based upon an efficient way of approximating scale invariant quantum states, the Multi-scale Entanglement Renormalization Ansatz (MERA). We employ MERA as a replacement for the fully connected layers in a convolutional neural network and test this implementation on the CIFAR-10 and CIFAR-100 datasets. The proposed method outperforms factorization using tensor trains, providing greater compression for the same level of accuracy and greater accuracy for the same level of compression. We demonstrate MERA layers with 14000 times fewer parameters and a reduction in accuracy of less than 1% compared to the equivalent fully connected layers, scaling like O(N).

Quantum circuits with hierarchical structure have been used to perform binary classification of classical data encoded in a quantum state. We demonstrate that more expressive circuits in the same family achieve better accuracy and can be used to classify highly entangled quantum states, for which there is no known efficient classical method. We compare performance for several different parameterizations on two classical machine learning datasets, Iris and MNIST, and on a synthetic dataset of quantum states. Finally, we demonstrate that performance is robust to noise and deploy an Iris dataset classifier on the ibmqx4 quantum computer.

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.

We study a restricted class of correlator product states (CPS) for a spin-half chain in which each spin is contained in just two overlapping plaquettes. This class is also a restriction upon matrix product states (MPS) with local dimension $2^n$ ($n$ being the size of the overlapping regions of plaquettes) equal to the bond dimension. We investigate the trade-off between gains in efficiency due to this restriction against losses in fidelity. The time-dependent variational principle formulated for these states is numerically very stable. Moreover, it shows significant gains in efficiency compared to the naively related matrix product states - the evolution or optimisation scales as $2^{3n}$ for the correlator product states versus $2^{4n}$ for the unrestricted matrix product state. However, much of this advantage is offset by a significant reduction in fidelity. Correlator product states break the local Hilbert space symmetry by the explicit selection of a local basis. We investigate this dependence in detail and formulate the broad principles under which correlator product states may be a useful tool. In particular, we find that scaling with overlap/bond order may be more stable with correlator product states allowing a more efficient extraction of critical exponents - we present an example in which the use of correlator product states is several orders of magnitude quicker than matrix product states.

In this paper we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to $(1+1)$ dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant field theories describing the critical models under investigation. The algorithm is applied to three exemplary models; the cMPS version to the non-relativistic Lieb-Liniger model and the relativistic massless boson, and MPS version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter. This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results.

We introduce a variational method for calculating dispersion relations of translation invariant (1 + 1)-dimensional quantum field theories. The method is based on continuous matrix product states and can be implemented efficiently. We study the critical Lieb-Liniger model as a benchmark and excellent agreement with the exact solution is found. Additionally, we observe solitonic signatures of Lieb's type II excitation. In addition, a nonintegrable model is introduced where a U(1)-symmetry breaking term is added to the Lieb-Liniger Hamiltonian. For this model we find evidence of a nontrivial bound-state excitation in the dispersion relation.

In this paper we study the relation between the target space geometry of (1, 1) sigma models and the factorization of the associated superconformal theory, at the classical level. It turns out that the superconformal currents T± factorize as , with and conserved separately, provided that the target space admits projectors P(±) and Q(±) (that is, a pair of almost-product structures), compatible with the Riemannian structure of the target space, and covariantly constant with respect to the two torsionful connections ∇(±) that arise naturally from the sigma model. It is a surprising result that the integrability of the projectors is not an obstruction for the associated symmetries and to form copies of the superconformal algebra. While one expects to be able to define a superconformal theory associated with a particular Riemannian submanifold defined by an integrable projector, a consequence of the above result is that there are no obstructions to defining a superconformal theory associated with non-integrable projectors. We show that this notion of non-geometry encompasses the locally non-geometric examples that arise in the T-duality inspired doubled formulations. In addition, we derive the general conditions for (2, 2) supersymmetry to be realized in the projective sense. This extends the relation between (2, 2) sigma models and bi-Hermitian geometry to the non-geometric setting. For the bosonic subsector we propose a BRST-type approach to defining the physical degrees of freedom in the non-geometric scenario.

The symmetries of two-dimensional supersymmetric sigma models on target spaces with covariantly constant forms associated to special holonomy groups are analysed. It is shown that each pair of such forms gives rise to a new one, called a Nijenhuis form, and that there may be further reductions of the structure group. In many cases of interest there are also covariantly constant one-forms which also give rise to symmetries. These geometries are of interest in the context of heterotic supergravity solutions and the associated reductions are studied from a spacetime point of view via the Killing spinor equations.

We show that the superconformal symmetries of the (1, 1) sigma model decompose into a set of more refined symmetries when the target space admits projectors P (±) , and the orthogonal complements Q (±) , covariantly constant with respect to the two natural torsionful connections ∇ (±) that arise in the sigma model. Surprisingly the new symmetries still close to form the superconformal algebra, even when the projectors are not integrable, so one is able to define a superconformal theory not associated with a particular geometry, but rather with non-integrable projectors living on a larger manifold. In the bosonic subsector we discuss the appropriate generalization of the coupling of the worldsheet metric for the non-geometric cases, and the complications that occur when the (+) holomorphic and (−) anti-holomorphic projectors don't commute.

We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. There are two natural deformations of the A-model in the AKSZ language: 1) the Zucchini model, which can be defined on a generalized complex manifold and reduces to the A-model when the generalized complex structure comes from a symplectic structure, and 2) a topological membrane model, which naturally accommodates fluxes, and reduces to the Zucchini model on the boundary of the membrane when the fluxes are turned off. We show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from H, in the sense that satisfying the AKSZ master equation implies precisely the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. In addition, the master equation imposes conditions on the fluxes that generalize dH = 0. The membrane model can be defined on a large class of U(m)- and U(m) × U(m)-structure manifolds relevant for string theory, including geometries inspired by (1, 1) supersymmetric σ-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. In addition we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. When only the closed 3-form flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau. We argue that deformations from the standard A-model are due to the choice of gauge fixing fermion, rather than a flux deformation of the AKSZ action. The particularly interesting cases arise when the fermion depends on auxiliary fields, the simplest possibility being due to the (2, 0)+(0, 2) component of a non-trivial b-field. The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. Deformations due to H-flux can be more radical, completely preventing auxiliary fields from being integrated out.

We study a generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) formulation of the A- and B-models which involves a doubling of coordinates, and can be understood as a complexification of the Poisson $σ$-model underlying these. In the flat space limit the construction contains models obtained by twisting an N=2 supersymmetric $σ$-model on Hull's doubled geometry. The curved space generalization involves a product of two diffeomorphic Calabi-Yau manifolds, and the $O(d,d)$ metric can be understood as a complexification of the CY metric. In addition, we consider solutions that can not be obtained by twisting the above $σ$-model. For these it is possible to interpolate between a model evaluated on holomorphic maps and one evaluated on constant maps by different choices of gauge fixing fermion. Finally, we discuss some intriguing similarities between aspects of the doubled formulation and topological M-theory, and a possible relation with results from the theory of Lie and Courant algebroids, where a doubled formulation plays a role in relating two- and three-dimensional topological theories.

Two-dimensional sigma-models describing superstrings propagating on manifolds of special holonomy are characterized by symmetries related to covariantly constant forms that these manifolds hold, which are generally non-linear and close in a field dependent sense. The thesis explores various aspects of the special holonomy symmetries.

In addition to superconformal symmetry, (1, 1) supersymmetric two-dimensional sigma models on special holonomy manifolds have extra symmetries that are in one-to-one correspondence with the covariantly constant forms on these manifolds. The superconformal algebras extended by these symmetries close as W-algebras, i. e. they have field-dependent structure functions. It is shown that it is not possible to write down cohomological equations for potential quantum anomalies when the structure functions are field-dependent. In order to do this it is necessary to linearise the algebras by treating composite currents as generators of additional symmetries. It is shown that all cases can be linearised in a finite number of steps, except for G(2) and SU(3). Additional problems in the quantisation procedure are briefly discussed.

A study of (1, 1) supersymmetric two-dimensional non-linear sigma models with boundary on special holonomy target spaces is presented. In particular, the consistency of the boundary conditions under the various symmetries is studied. Models both with and without torsion are discussed.