Interacting bosons or fermions give rise to some of the most fascinating phases of matter, including high-temperature superconductivity, the fractional quantum Hall effect, quantum spin liquids and Mott insulators. Although these systems are promising for technological applications, they also present conceptual challenges, as they require approaches beyond mean-field and perturbation theory. Here we develop a general framework for identifying the free theory that is closest to a given interacting model in terms of their ground-state correlations. Moreover, we quantify the distance between them using the entanglement spectrum. When this interaction distance is small, the optimal free theory provides an effective description of the low-energy physics of the interacting model. Our construction of the optimal free model is non-perturbative in nature; thus, it offers a theoretical framework for investigating strongly correlated systems.

We study how quantum states are scrambled via braiding in systems of non-Abelian anyons through the lens of entanglement spectrum statistics. In particular, we focus on the degree of scrambling, defined as the randomness produced by braiding, at the same amount of entanglement entropy. To quantify the degree of randomness, we define a distance between the entanglement spectrum level spacing distribution of a state evolved under random braids and that of a Haar-random state, using the Kullback-Leibler divergence DKL. We study DKL numerically for random braids of Majorana fermions (supplemented with random local four-body interactions) and Fibonacci anyons. For comparison, we also obtain DKL for the Sachdev-Ye-Kitaev model of Majorana fermions with all-to-all interactions, random unitary circuits built out of (a) Hadamard (H), π/8 (T), and CNOT gates, and (b) random unitary circuits built out of two-qubit Haar-random unitaries. To compare the degree of randomness that different systems produce beyond entanglement entropy, we look at DKL as a function of the Page limit-normalized entanglement entropy S/Smax. Our results reveal a hierarchy of scrambling among various models -even for the same amount of entanglement entropy -at intermediate times, whereas all models exhibit the same late-time behavior. In particular, we find that braiding of Fibonacci anyons randomizes initial product states more efficiently than the universal H+T+CNOT set.

We provide conceptual and mathematical foundations for near-term quantum natural language processing (QNLP), and do so in quantum computer scientist friendly terms. We opted for an expository presentation style, and provide references for supporting empirical evidence and formal statements concerning mathematical generality.We recall how the quantum model for natural language that we employ [42] canonically combines linguistic meanings with rich linguistic structure, most notably grammar. In particular, the fact that it takes a quantum-like model to combine meaning and structure, establishes QNLP as quantum-native, on par with simulation of quantum systems. Moreover, the now leading Noisy Intermediate-Scale Quantum (NISQ) paradigm for encoding classical data on quantum hardware, variational quantum circuits, makes NISQ exceptionally QNLP-friendly: linguistic structure can be encoded as a free lunch, in contrast to the apparently exponentially expensive classical encoding of grammar.Quantum speed-up for QNLP tasks has already been established in previous work [116]. Here we provide a broader range of tasks which all enjoy the same advantage.Diagrammatic reasoning is at the heart of QNLP. Firstly, the quantum model interprets language as quantum processes via the diagrammatic formalism of categorical quantum mechanics [38]. Secondly, these diagrams are via ZX-calculus translated into quantum circuits. Parameterisations of meanings then become the circuit variables to be learned:

Quantum Natural Language Processing (QNLP) deals with the design and implementation of NLP models intended to be run on quantum hardware. In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum (NISQ) computers for datasets of size ≥ 100 sentences. Exploiting the formal similarity of the compositional model of meaning by Coecke et al. ( 2010) with quantum theory, we create representations for sentences that have a natural mapping to quantum circuits. We use these representations to implement and successfully train two NLP models that solve simple sentence classification tasks on quantum hardware. We describe in detail the main principles, the process and challenges of these experiments, in a way accessible to NLP researchers, thus paving the way for practical Quantum Natural Language Processing.

Topological phases of matter remain a focus of interest due to their unique properties -fractionalisation, ground state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here we quantify the role of interactions in general classes of topological states of matter in all spatial dimensions, including parafermion chains and string-net models. Using the interaction distance [Nat. Commun. 8, 14926 (2017)], we measure the distinguishability of states of these models from those of free fermions. We find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible interaction distance. Our work opens the door to understanding the complexity of topological models and to applying new types of fermionisation procedures to describe their low-energy physics.

In this work, we propose a new and simple model that supports Chern semimetals. These new gapless topological phases share several properties with the Chern insulators like a well-defined Chern number associated to each band, topologically protected edge states and topological phase transitions that occur when the bands touch each, with linear dispersion around the contact points. The tightbinding model, defined on the Lieb lattice with intra-unit-cell and suitable nearest-neighbor hopping terms between three different species of spinless fermions, supports a single Dirac-like point. The dispersion relation around this point is fully relativistic and the 3 × 3 matrices in the corresponding effective Hamiltonian satisfy the Duffin-Kemmer-Petiau algebra. We show the robustness of the topologically protected edge states by employing the entanglement spectrum. Moreover, we prove that the Chern number of the lowest band is robust with respect to weak disorder. For its simplicity, our model can be naturally implemented in real physical systems like cold atoms in optical lattices. [3] are characterized by the topological Chern number that fixes the number of topologically protected edge modes. The former is an example of topological insulator in the class A, called also Chern insulator, where both time-reversal and particle-hole symmetries are broken while the latter is a topological superconductor living in the class D, where particle-hole symmetry is instead preserved. In the last years, part of the research of new topological phases has focused on gapless bulk systems like threedimensional Weyl [4] and Dirac semimetals [5] that represent three-dimensional versions of graphene. In these systems, the bands touch each other in a discrete set of points that can be seen as point-like defects in momentum space. These Weyl and Dirac points are topologically protected by well-defined Berry phases and the boundaries support suitable gapless modes. Clearly, when the time-reversal symmetry is broken in two dimensions, the Chern number can characterize the bands of the semimetals only when those bands do not touch each other, i.e. when there are no Dirac or Weyl cones. Some extended Haldane models with these characteristics have been recently analyzed [6,7].

We present lambeq, the first high-level Python library for Quantum Natural Language Processing (QNLP). The open-source toolkit offers a detailed hierarchy of modules and classes implementing all stages of a pipeline for converting sentences to string diagrams, tensor networks, and quantum circuits ready to be used on a quantum computer. lambeq supports syntactic parsing, rewriting and simplification of string diagrams, ansatz creation and manipulation, as well as a number of compositional models for preparing quantum-friendly representations of sentences, employing various degrees of syntax sensitivity. We present the generic architecture and describe the most important modules in detail, demonstrating the usage with illustrative examples. Further, we test the toolkit in practice by using it to perform a number of experiments on simple NLP tasks, implementing both classical and quantum pipelines.1 Stylised 'λambeq', pronounced "lambek". The name is a tribute to mathematician Joachim Lambek , whose seminal work lay at the intersection of mathematics, logic, and linguistics.

Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this Letter, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically, and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement, we show that highly entangled states supported across a system bipartition are largely disentangled from the rest of the system, thus, usually appearing as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems.