Quantum channels as a categorical completion

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201

Denotationally Correct, Purely Functional, Efficient Reverse-mode Automatic Differentiation

Reverse-mode differentiation is used for optimization, but it introduces references, which break the purity of the underlying programs, making them notoriously harder to optimize. We present a reverse-mode differentiation on a purely functional language with array operations. It is the first one to deliver a provably efficient, purely functional, and denotationally correct reverse-mode differentiation. We show that our transformation is semantically correct and verifies the cheap gradient principle. Inspired by PROPs and compilation to categories, we introduce a novel intermediate representation that we call 'unary form'. Our reverse-mode transformation is factored as a compilation scheme through this intermediate representation. We obtain provably efficient gradients by performing general partial evaluation optimizations after our reverse-mode transformation, as opposed to manually derived ones. For simple first-order programs, the obtained output programs resemble static-single-assignment (SSA) code. We emphasize the modularity of our approach and show how our language can easily be enriched with more optimized primitives, as required for some speed-ups in practice.Comment: 34 pages, 17 figure

Higher Order Automatic Differentiation of Higher Order Functions

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.Comment: 34 pages, 5 figures, submitted at LMCS 2020. arXiv admin note: substantial text overlap with arXiv:2001.0220

Crossing beyond scattering amplitudes

We find that different asymptotic measurements in quantum field theory can be related to one another through new versions of crossing symmetry. Assuming analyticity, we conjecture generalized crossing relations for multi-particle processes and the corresponding paths of analytic continuation. We prove them to all multiplicity at tree-level in quantum field theory and string theory. We illustrate how to practically perform analytic continuations on loop-level examples using different methods, including unitarity cuts and differential equations. We study the extent to which anomalous thresholds away from the usual physical region can cause an analytic obstruction to crossing when massless particles are involved. In an appendix, we review and streamline historical proofs of four-particle crossing symmetry in gapped theories.Comment: 108 page

ADEV: Sound Automatic Differentiation of Expected Values of Probabilistic Programs

Optimizing the expected values of probabilistic processes is a central problem in computer science and its applications, arising in fields ranging from artificial intelligence to operations research to statistical computing. Unfortunately, automatic differentiation techniques developed for deterministic programs do not in general compute the correct gradients needed for widely used solutions based on gradient-based optimization. In this paper, we present ADEV, an extension to forward-mode AD that correctly differentiates the expectations of probabilistic processes represented as programs that make random choices. Our algorithm is a source-to-source program transformation on an expressive, higher-order language for probabilistic computation, with both discrete and continuous probability distributions. The result of our transformation is a new probabilistic program, whose expected return value is the derivative of the original program's expectation. This output program can be run to generate unbiased Monte Carlo estimates of the desired gradient, which can then be used within the inner loop of stochastic gradient descent. We prove ADEV correct using logical relations over the denotations of the source and target probabilistic programs. Because it modularly extends forward-mode AD, our algorithm lends itself to a concise implementation strategy, which we exploit to develop a prototype in just a few dozen lines of Haskell (https://github.com/probcomp/adev).Comment: to appear at POPL 202

Cooperative binding of ApiAP2 transcription factors is crucial for the expression of virulence genes in Toxoplasma gondii

International audienceToxoplasma gondii virulence depends on the expression of factors packed into specific organelles such as rhoptry and microneme. Although virulence factor expression is tightly regulated, the molecular mechanisms controlling their regulation remain poorly understood. ApiAP2 are a family of conserved transcription factors (TFs) that play an important role in regulating gene expression in apicomplexan parasites. TgAP2XI-5 is able to bind to transcription-ally active promoters of genes expressed during the S/M phase of the cell cycle, such as virulence genes (rhoptries and micronemes genes). We identified proteins interacting with TgAP2XI-5 including a cell cycle-regulated ApiAP2 TF, TgAP2X-5. Using an inducible knock-down strategy and RNA-seq, we demonstrated that the level of expression of number of virulence factors transcripts is affected by the disruption of TgAP2X-5 expression. While TgAP2X-5 disruption has mild effect on parasite invasion, it leads to the strain avirulence in mice. To better understand the molecular mechanisms at stake, we investigated the binding of TgAP2XI-5 at promoters in the TgAP2X-5 mutant strain in a genome-wide assay. We show that disruption of TgAP2X-5 expression leads to defects in TgAP2XI-5 binding to multiple rhoptry gene promoters. Taken together, these data suggest a cooperative contribution of two ApiAP2 TF in the regulation of virulence genes in T. gondii

Crystal structure of Saccharomyces cerevisiae mitochondrial GatFAB reveals a novel subunit assembly in tRNA-dependent amidotransferases

Yeast mitochondrial Gln-mtRNAGln is synthesized by the transamidation of mischarged Glu-mtRNAGln by a non-canonical heterotrimeric tRNA-dependent amidotransferase (AdT). The GatA and GatB subunits of the yeast AdT (GatFAB) are well conserved among bacteria and eukaryota, but the GatF subunit is a fungi-specific ortholog of the GatC subunit found in all other known heterotrimeric AdTs (GatCAB). Here we report the crystal structure of yeast mitochondrial GatFAB at 2.0 Å resolution. The C-terminal region of GatF encircles the GatA-GatB interface in the same manner as GatC, but the N-terminal extension domain (NTD) of GatF forms several additional hydrophobic and hydrophilic interactions with GatA. NTD-deletion mutants displayed growth defects, but retained the ability to respire. Truncation of the NTD in purified mutants reduced glutaminase and transamidase activities when glutamine was used as the ammonia donor, but increased transamidase activity relative to the full-length enzyme when the donor was ammonium chloride. Our structure-based functional analyses suggest the NTD is a trans-acting scaffolding peptide for the GatA glutaminase active site. The positive surface charge and novel fold of the GatF-GatA interface, shown in this first crystal structure of an organellar AdT, stand in contrast with the more conventional, negatively charged bacterial AdTs described previousl

The oncometabolite 2-hydroxyglutarate activates the mTOR signalling pathway

The identification of cancer-associated mutations in the tricarboxylic acid (TCA) cycle enzymes isocitrate dehydrogenases 1 and 2 (IDH1/2) highlights the prevailing notion that aberrant metabolic function can contribute to carcinogenesis. IDH1/2 normally catalyse the oxidative decarboxylation of isocitrate into α-ketoglutarate (αKG). In gliomas and acute myeloid leukaemias, IDH1/2 mutations confer gain-of-function leading to production of the oncometabolite R-2-hydroxyglutarate (2HG) from αKG. Here we show that generation of 2HG by mutated IDH1/2 leads to the activation of mTOR by inhibiting KDM4A, an αKG-dependent enzyme of the Jumonji family of lysine demethylases. Furthermore, KDM4A associates with the DEP domain-containing mTOR-interacting protein (DEPTOR), a negative regulator of mTORC1/2. Depletion of KDM4A decreases DEPTOR protein stability. Our results provide an additional molecular mechanism for the oncogenic activity of mutant IDH1/2 by revealing an unprecedented link between TCA cycle defects and positive modulation of mTOR function downstream of the canonical PI3K/AKT/TSC1-2 pathway