6,013 research outputs found
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
Constant-Soundness Interactive Proofs for Local Hamiltonians
We give a quantum multiprover interactive proof
system for the local Hamiltonian problem in which there is a constant number of
provers, questions are classical of length polynomial in the number of qubits,
and answers are of constant length. The main novelty of our protocol is that
the gap between completeness and soundness is directly proportional to the
promise gap on the (normalized) ground state energy of the Hamiltonian. This
result can be interpreted as a concrete step towards a quantum PCP theorem
giving entangled-prover interactive proof systems for QMA-complete problems.
The key ingredient is a quantum version of the classical linearity test of
Blum, Luby, and Rubinfeld, where the function is
replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C),
the set of -dimensional Hermitian matrices that square to identity. The test
enforces that (i) each function is exactly linear,
\Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the
two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx
(-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page
Quantum interactive proofs and the complexity of separability testing
We identify a formal connection between physical problems related to the
detection of separable (unentangled) quantum states and complexity classes in
theoretical computer science. In particular, we show that to nearly every
quantum interactive proof complexity class (including BQP, QMA, QMA(2), and
QSZK), there corresponds a natural separability testing problem that is
complete for that class. Of particular interest is the fact that the problem of
determining whether an isometry can be made to produce a separable state is
either QMA-complete or QMA(2)-complete, depending upon whether the distance
between quantum states is measured by the one-way LOCC norm or the trace norm.
We obtain strong hardness results by proving that for each n-qubit maximally
entangled state there exists a fixed one-way LOCC measurement that
distinguishes it from any separable state with error probability that decays
exponentially in n.Comment: v2: 43 pages, 5 figures, completely rewritten and in Theory of
Computing (ToC) journal forma
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