'Should conditionals be emergent ...': asyndetic conditionals in English and German as a Challenge to Grammaticalization Research

The present article examines asyndetic or conjunctionless conditionals in German and English. According to Jespersen’s Model (1940), this construction arose diachronically from a paratactic discourse sequence with a polar interrogative, but more recently Harris and Campbell (1995) have claimed that this model lacks any theoretical and empirical foundation. To demonstrate how asyndetic conditionals may emerge from discourse, this study reframes Jespersen’s Model in grammaticalization terms and adduces several constructional features in order to show that a grammaticalization process has actually taken place. In particular, this is achieved by applying traditional grammaticalization parameters such as bondedness, paradigmatic variability and specialization to synchronic and diachronic variation patterns with regard to clause integration, the finite verb of the protasis and the possible-world categories realis, potentialis, irrealis. The article also explores the relevance of speech-situation evocation to the formation of interrogative-based conditionals

Efficient classical simulations of quantum Fourier transforms and normalizer circuits over Abelian groups

The quantum Fourier transform (QFT) is sometimes said to be the source of various exponential quantum speed-ups. In this paper we introduce a class of quantum circuits which cannot outperform classical computers even though the QFT constitutes an essential component. More precisely, we consider normalizer circuits. A normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. We prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. We subsequently discuss several aspects of normalizer circuits. First we show that our result generalizes the Gottesman-Knill theorem. Furthermore we highlight connections to Shor's factoring algorithm and to the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.Comment: 23 pages + appendice

Universal quantum computation with little entanglement

We show that universal quantum computation can be achieved in the standard pure-state circuit model while, at any time, the entanglement entropy of all bipartitions is small---even tending to zero with growing system size. The result is obtained by showing that a quantum computer operating within a small region around the set of unentangled states still has universal computational power, and by using continuity of entanglement entropy. In fact an analogous conclusion applies to every entanglement measure which is continuous in a certain natural sense, which amounts to a large class. Other examples include the geometric measure, localizable entanglement, smooth epsilon-measures, multipartite concurrence, squashed entanglement, and several others. We discuss implications of these results for the believed role of entanglement as a key necessary resource for quantum speed-ups

Classical simulation complexity of extended Clifford circuits

Clifford gates are a winsome class of quantum operations combining mathematical elegance with physical significance. The Gottesman-Knill theorem asserts that Clifford computations can be classically efficiently simulated but this is true only in a suitably restricted setting. Here we consider Clifford computations with a variety of additional ingredients: (a) strong vs. weak simulation, (b) inputs being computational basis states vs. general product states, (c) adaptive vs. non-adaptive choices of gates for circuits involving intermediate measurements, (d) single line outputs vs. multi-line outputs. We consider the classical simulation complexity of all combinations of these ingredients and show that many are not classically efficiently simulatable (subject to common complexity assumptions such as P not equal to NP). Our results reveal a surprising proximity of classical to quantum computing power viz. a class of classically simulatable quantum circuits which yields universal quantum computation if extended by a purely classical additional ingredient that does not extend the class of quantum processes occurring.Comment: 17 pages, 1 figur

A monomial matrix formalism to describe quantum many-body states

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M-spaces encompass various important state families, such as all Pauli stabilizer states and codes, the AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M-spaces can transparently be understood by manipulating their monomial stabilizer groups. In particular we derive a unified procedure to construct an eigenbasis of any M-space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M-spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally we prove that a large subclass of M-spaces---containing in particular most of the aforementioned examples---can be simulated efficiently classically with a unified method.Comment: 11 pages + appendice

Quantum simulation of classical thermal states

We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) system of qubits arranged on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengthes of two-body interactions, which allow one to choose the type and dimension of the classical model as well as the interaction strength and temperature.Comment: 4 pages, 1 figur

Asynchronous grammaticalization: V1-conditionals in present-day English and German

The present paper contrasts verb-first (= V1-)conditionals in written usage in present-day English and German. Based on the hypothesis that V1-protases originated in independent interrogatives and then grammaticalized as conditional subordinate clauses in an asynchronous fashion in both languages, we use data from the British National Corpus (BNC) and the Deutsches Referenzkorpus (DeReKo) to investigate the lexical overlap of V1-protases with interrogatives and their functional overlap with ‘if-/wenn’-conditionals. The results show, inter alia, that English V1-conditionals are highly divergent from polar interrogatives and occupy a functional niche with respect to ‘if-’conditionals, with their German counterparts showing more transitional characteristics in both respects; they also suggest a special role for V1-protases with ‘should/sollte’ in expressing a subtype of neutral, rather than tentative, conditionality. Finally, prospects are discussed for future research regarding possible synchronic (i.e. discourse-functional) and diachronic (i.e. systemic) motivations for the differences and similarites observed between V1-conditionals in the two present-day languages

Classical simulations of Abelian-group normalizer circuits with intermediate measurements

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian group $G$ is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [arXiv:1201.4867] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To appear in Quantum Information and Computation, Vol.14 No.3&4, 201