Author Correction: The effects of mutational processes and selection on driver mutations across cancer types.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Experimental learning of quantum states.

The number of parameters describing a quantum state is well known to grow exponentially with the number of particles. This scaling limits our ability to characterize and simulate the evolution of arbitrary states to systems, with no more than a few qubits. However, from a computational learning theory perspective, it can be shown that quantum states can be approximately learned using a number of measurements growing linearly with the number of qubits. Here, we experimentally demonstrate this linear scaling in optical systems with up to 6 qubits. Our results highlight the power of the computational learning theory to investigate quantum information, provide the first experimental demonstration that quantum states can be "probably approximately learned" with access to a number of copies of the state that scales linearly with the number of qubits, and pave the way to probing quantum states at new, larger scales.

Dynamic transcriptomic analysis reveals suppression of PGC1α/ERRα drives perturbed myogenesis in facioscapulohumeral muscular dystrophy.

Facioscapulohumeral muscular dystrophy (FSHD) is a prevalent, incurable myopathy, linked to epigenetic derepression of D4Z4 repeats on chromosome 4q, leading to ectopic DUX4 expression. FSHD patient myoblasts have defective myogenic differentiation, forming smaller myotubes with reduced myosin content. However, molecular mechanisms driving such disrupted myogenesis in FSHD are poorly understood. We performed high-throughput morphological analysis describing FSHD and control myogenesis, revealing altered myogenic differentiation results in hypotrophic myotubes. Employing polynomial models and an empirical Bayes approach, we established eight critical time points during which human healthy and FSHD myogenesis differ. RNA-sequencing at these eight nodal time points in triplicate, provided temporal depth for a multivariate regression analysis, allowing assessment of interaction between progression of differentiation and FSHD disease status. Importantly, the unique size and structure of our data permitted identification of many novel FSHD pathomechanisms undetectable by previous approaches. For further analysis here, we selected pathways that control mitochondria: of interest considering known alterations in mitochondrial structure and function in FSHD muscle, and sensitivity of FSHD cells to oxidative stress. Notably, we identified suppression of mitochondrial biogenesis, in particular via peroxisome proliferator-activated receptor gamma coactivator 1-α (PGC1α), the cofactor and activator of oestrogen-related receptor α (ERRα). PGC1α knock-down caused hypotrophic myotubes to form from control myoblasts. Known ERRα agonists and safe food supplements biochanin A, daidzein or genistein, each rescued the hypotrophic FSHD myotube phenotype. Together our work describes transcriptomic changes in high resolution that occur during myogenesis in FSHD ex vivo, identifying suppression of the PGC1α-ERRα axis leading to perturbed myogenic differentiation, which can effectively be rescued by readily available food supplements.

The effects of mutational processes and selection on driver mutations across cancer types.

Epidemiological evidence has long associated environmental mutagens with increased cancer risk. However, links between specific mutation-causing processes and the acquisition of individual driver mutations have remained obscure. Here we have used public cancer sequencing data from 11,336 cancers of various types to infer the independent effects of mutation and selection on the set of driver mutations in a cancer type. First, we detect associations between a range of mutational processes, including those linked to smoking, ageing, APOBEC and DNA mismatch repair (MMR) and the presence of key driver mutations across cancer types. Second, we quantify differential selection between well-known alternative driver mutations, including differences in selection between distinct mutant residues in the same gene. These results show that while mutational processes have a large role in determining which driver mutations are present in a cancer, the role of selection frequently dominates.

Quantum machine learning: a classical perspective.

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning (ML) techniques to impressive results in regression, classification, data generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets is motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed up classical ML algorithms. Here we review the literature in quantum ML and discuss perspectives for a mixed readership of classical ML and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in ML are identified as promising directions for the field. Practical questions, such as how to upload classical data into quantum form, will also be addressed.

PAX7 target genes are globally repressed in facioscapulohumeral muscular dystrophy skeletal muscle.

Facioscapulohumeral muscular dystrophy (FSHD) is a prevalent, incurable myopathy, linked to hypomethylation of D4Z4 repeats on chromosome 4q causing expression of the DUX4 transcription factor. However, DUX4 is difficult to detect in FSHD muscle biopsies and it is debatable how robust changes in DUX4 target gene expression are as an FSHD biomarker. PAX7 is a master regulator of myogenesis that rescues DUX4-mediated apoptosis. Here, we show that suppression of PAX7 target genes is a hallmark of FSHD, and that it is as major a signature of FSHD muscle as DUX4 target gene expression. This is shown using meta-analysis of over six FSHD muscle biopsy gene expression studies, and validated by RNA-sequencing on FSHD patient-derived myoblasts. DUX4 also inhibits PAX7 from activating its transcriptional target genes and vice versa. Furthermore, PAX7 target gene repression can explain oxidative stress sensitivity and epigenetic changes in FSHD. Thus, PAX7 target gene repression is a hallmark of FSHD that should be considered in the investigation of FSHD pathology and therapy.

Logic circuits from zero forcing.

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of "back forcing" as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits.

Increased signaling entropy in cancer requires the scale-free property of protein interaction networks.

One of the key characteristics of cancer cells is an increased phenotypic plasticity, driven by underlying genetic and epigenetic perturbations. However, at a systems-level it is unclear how these perturbations give rise to the observed increased plasticity. Elucidating such systems-level principles is key for an improved understanding of cancer. Recently, it has been shown that signaling entropy, an overall measure of signaling pathway promiscuity, and computable from integrating a sample's gene expression profile with a protein interaction network, correlates with phenotypic plasticity and is increased in cancer compared to normal tissue. Here we develop a computational framework for studying the effects of network perturbations on signaling entropy. We demonstrate that the increased signaling entropy of cancer is driven by two factors: (i) the scale-free (or near scale-free) topology of the interaction network, and (ii) a subtle positive correlation between differential gene expression and node connectivity. Indeed, we show that if protein interaction networks were random graphs, described by Poisson degree distributions, that cancer would generally not exhibit an increased signaling entropy. In summary, this work exposes a deep connection between cancer, signaling entropy and interaction network topology.

Intra-tumour signalling entropy determines clinical outcome in breast and lung cancer.

The cancer stem cell hypothesis, that a small population of tumour cells are responsible for tumorigenesis and cancer progression, is becoming widely accepted and recent evidence has suggested a prognostic and predictive role for such cells. Intra-tumour heterogeneity, the diversity of the cancer cell population within the tumour of an individual patient, is related to cancer stem cells and is also considered a potential prognostic indicator in oncology. The measurement of cancer stem cell abundance and intra-tumour heterogeneity in a clinically relevant manner however, currently presents a challenge. Here we propose signalling entropy, a measure of signalling pathway promiscuity derived from a sample's genome-wide gene expression profile, as an estimate of the stemness of a tumour sample. By considering over 500 mixtures of diverse cellular expression profiles, we reveal that signalling entropy also associates with intra-tumour heterogeneity. By analysing 3668 breast cancer and 1692 lung adenocarcinoma samples, we further demonstrate that signalling entropy correlates negatively with survival, outperforming leading clinical gene expression based prognostic tools. Signalling entropy is found to be a general prognostic measure, valid in different breast cancer clinical subgroups, as well as within stage I lung adenocarcinoma. We find that its prognostic power is driven by genes involved in cancer stem cells and treatment resistance. In summary, by approximating both stemness and intra-tumour heterogeneity, signalling entropy provides a powerful prognostic measure across different epithelial cancers.

ß-Catenin is central to DUX4-driven network rewiring in facioscapulohumeral muscular dystrophy.

Facioscapulohumeral muscular dystrophy (FSHD) is an incurable disease, characterized by skeletal muscle weakness and wasting. Genetically, FSHD is characterized by contraction or hypomethylation of repeat D4Z4 units on chromosome 4, which causes aberrant expression of the transcription factor DUX4 from the last repeat. Many genes have been implicated in FSHD pathophysiology, but an integrated molecular model is currently lacking. We developed a novel differential network methodology, Interactome Sparsification and Rewiring (InSpiRe), which detects network rewiring between phenotypes by integrating gene expression data with known protein interactions. Using InSpiRe, we performed a meta-analysis of multiple microarray datasets from FSHD muscle biopsies, then removed secondary rewiring using non-FSHD datasets, to construct a unified network of rewired interactions. Our analysis identified ß-catenin as the main coordinator of FSHD-associated protein interaction signalling, with pathways including canonical Wnt, HIF1-α and TNF-α clearly perturbed. To detect transcriptional changes directly elicited by DUX4, gene expression profiling was performed using microarrays on murine myoblasts. This revealed that DUX4 significantly modified expression of the genes in our FSHD network. Furthermore, we experimentally confirmed that Wnt/ß-catenin signalling is affected by DUX4 in murine myoblasts. Thus, we provide the first unified molecular map of FSHD signalling, capable of uncovering pathomechanisms and guiding therapeutic development.

Hearing the shape of the Ising model with a programmable superconducting-flux annealer.

Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform experiments on a programmable annealer with superconducting flux technology. Departing from the paradigm of adiabatic computation, we take advantage of a noisy evolution of the device to generate statistics of low energy states. The graphs considered in the experiments have the same classical partition functions, but different quantum spectra. The data obtained from the annealer distinguish non-isomorphic graphs via information contained in the classical refinements of the functions but not via the differences in the quantum spectra.

Graph-theoretic approach to quantum correlations.

Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.

Cellular network entropy as the energy potential in Waddington's differentiation landscape.

Differentiation is a key cellular process in normal tissue development that is significantly altered in cancer. Although molecular signatures characterising pluripotency and multipotency exist, there is, as yet, no single quantitative mark of a cellular sample's position in the global differentiation hierarchy. Here we adopt a systems view and consider the sample's network entropy, a measure of signaling pathway promiscuity, computable from a sample's genome-wide expression profile. We demonstrate that network entropy provides a quantitative, in-silico, readout of the average undifferentiated state of the profiled cells, recapitulating the known hierarchy of pluripotent, multipotent and differentiated cell types. Network entropy further exhibits dynamic changes in time course differentiation data, and in line with a sample's differentiation stage. In disease, network entropy predicts a higher level of cellular plasticity in cancer stem cell populations compared to ordinary cancer cells. Importantly, network entropy also allows identification of key differentiation pathways. Our results are consistent with the view that pluripotency is a statistical property defined at the cellular population level, correlating with intra-sample heterogeneity, and driven by the degree of signaling promiscuity in cells. In summary, network entropy provides a quantitative measure of a cell's undifferentiated state, defining its elevation in Waddington's landscape.

Network transfer entropy and metric space for causality inference.

A measure is derived to quantify directed information transfer between pairs of vertices in a weighted network, over paths of a specified maximal length. Our approach employs a general, probabilistic model of network traffic, from which the informational distance between dynamics on two weighted networks can be naturally expressed as a Jensen Shannon divergence. Our network transfer entropy measure is shown to be able to distinguish and quantify causal relationships between network elements, in applications to simple synthetic networks and a biological signaling network. We conclude with a theoretical extension of our framework, in which the square root of the Jensen Shannon Divergence induces a metric on the space of dynamics on weighted networks. We prove a convergence criterion, demonstrating that a form of convergence in the structure of weighted networks in a family of matrix metric spaces implies convergence of their dynamics with respect to the square root Jensen Shannon divergence metric.

Estimating entanglement monotones with a generalization of the Wootters formula.

Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.

Differential network entropy reveals cancer system hallmarks.

The cellular phenotype is described by a complex network of molecular interactions. Elucidating network properties that distinguish disease from the healthy cellular state is therefore of critical importance for gaining systems-level insights into disease mechanisms and ultimately for developing improved therapies. By integrating gene expression data with a protein interaction network we here demonstrate that cancer cells are characterised by an increase in network entropy. In addition, we formally demonstrate that gene expression differences between normal and cancer tissue are anticorrelated with local network entropy changes, thus providing a systemic link between gene expression changes at the nodes and their local correlation patterns. In particular, we find that genes which drive cell-proliferation in cancer cells and which often encode oncogenes are associated with reductions in network entropy. These findings may have potential implications for identifying novel drug targets.

Number-theoretic nature of communication in quantum spin systems.

The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety of control methods and network topologies have been proposed, on the basis that transfer with perfect fidelity-i.e., deterministic and without information loss-is impossible through unmodulated spin chains with more than a few particles. Solving the original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine the exact number of qubits in unmodulated chains (with an XY Hamiltonian) that permit transfer with a fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer. We prove that this happens if and only if the number of nodes is n = p - 1, 2p - 1, where p is a prime, or n = 2(m) - 1. The result highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers and, in this case, primality.

Approximate entropy of network parameters.

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

Shannon and von Neumann entropy of random networks with heterogeneous expected degree.

Entropic measures of complexity are able to quantify the information encoded in complex network structures. Several entropic measures have been proposed in this respect. Here we study the relation between the Shannon entropy and the von Neumann entropy of networks with given expected degree sequence. We find in different examples of network topologies that when the degree distribution contains some heterogeneity, an intriguing correlation emerges between the two entropic quantities. This results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks, which respectively corresponds to the von Neumann and the Shannon entropy.