44 research outputs found
Non-deterministic Boolean Proof Nets
16 pagesInternational audienceWe introduce Non-deterministic Boolean proof nets to study the correspondence with Boolean circuits, a parallel model of computation. We extend the cut elimination of Non-deterministic Multiplicative Linear logic to a parallel procedure in proof nets. With the restriction of proof nets to Boolean types, we prove that the cut-elimination procedure corresponds to Non-deterministic Boolean circuit evaluation and reciprocally. We obtain implicit characterization of the complexity classes NP and NC (the efficiently parallelizable functions)
Distribution-Free Proofs of Proximity
Motivated by the fact that input distributions are often unknown in advance,
distribution-free property testing considers a setting in which the algorithmic
task is to accept functions having a certain property
and reject functions that are -far from , where the
distance is measured according to an arbitrary and unknown input distribution
. As usual in property testing, the tester is required to do so
while making only a sublinear number of input queries, but as the distribution
is unknown, we also allow a sublinear number of samples from the distribution
.
In this work we initiate the study of distribution-free interactive proofs of
proximity (df-IPP) in which the distribution-free testing algorithm is assisted
by an all powerful but untrusted prover. Our main result is a df-IPP for any
problem , with communication, sample, query,
and verification complexities, for any proximity parameter
. For such proximity parameters, this result matches the
parameters of the best-known general purpose IPPs in the standard uniform
setting, and is optimal under reasonable cryptographic assumptions.
For general values of the proximity parameter , our
distribution-free IPP has optimal query complexity but the
communication complexity is , which
is worse than what is known for uniform IPPs when . With
the aim of improving on this gap, we further show that for IPPs over
specialised, but large distribution families, such as sufficiently smooth
distributions and product distributions, the communication complexity can be
reduced to (keeping the query
complexity roughly the same as before) to match the communication complexity of
the uniform case
Modular quantum signal processing in many variables
Despite significant advances in quantum algorithms, quantum programs in
practice are often expressed at the circuit level, forgoing helpful structural
abstractions common to their classical counterparts. Consequently, as many
quantum algorithms have been unified with the advent of quantum signal
processing (QSP) and quantum singular value transformation (QSVT), an
opportunity has appeared to cast these algorithms as modules that can be
combined to constitute complex programs. Complicating this, however, is that
while QSP/QSVT are often described by the polynomial transforms they apply to
the singular values of large linear operators, and the algebraic manipulation
of polynomials is simple, the QSP/QSVT protocols realizing analogous
manipulations of their embedded polynomials are non-obvious. Here we provide a
theory of modular multi-input-output QSP-based superoperators, the basic unit
of which we call a gadget, and show they can be snapped together with LEGO-like
ease at the level of the functions they apply. To demonstrate this ease, we
also provide a Python package for assembling gadgets and compiling them to
circuits. Viewed alternately, gadgets both enable the efficient block encoding
of large families of useful multivariable functions, and substantiate a
functional-programming approach to quantum algorithm design in recasting QSP
and QSVT as monadic types.Comment: 15 pages + 9 figures + 4 tables + 45 pages supplement. For codebase,
see https://github.com/ichuang/pyqsp/tree/bet
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Separating NC along the δ axis
AbstractThe “δ axis” consists of certain classes Nk such that uniform NC=⋃kNk. A diagonalization is used to prove that Nk≠Nk+1 for all k⩾0. This separation of NC along the δ axis does not imply NCk≠NCk+1, because the relationship between NCk and Nk is not known. However, it is argued that the δ axis provides a natural subdivision of NC. To support this, a careful analogy with the classes DTIME(nk) is made, illuminating a surprisingly close relationship between PTIME and NC. The definition of Nk uses ramified constructs related to a combination of parallel time and space
A generic imperative language for polynomial time
The ramification method in Implicit Computational Complexity has been
associated with functional programming, but adapting it to generic imperative
programming is highly desirable, given the wider algorithmic applicability of
imperative programming. We introduce a new approach to ramification which,
among other benefits, adapts readily to fully general imperative programming.
The novelty is in ramifying finite second-order objects, namely finite
structures, rather than ramifying elements of free algebras. In so doing we
bridge between Implicit Complexity's type theoretic characterizations of
feasibility, and the data-flow approach of Static Analysis.Comment: 18 pages, submitted to a conferenc
Normalizer Circuits and Quantum Computation
(Abridged abstract.) In this thesis we introduce new models of quantum
computation to study the emergence of quantum speed-up in quantum computer
algorithms.
Our first contribution is a formalism of restricted quantum operations, named
normalizer circuit formalism, based on algebraic extensions of the qubit
Clifford gates (CNOT, Hadamard and -phase gates): a normalizer circuit
consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic
phase gates associated to a set , which is either an abelian group or
abelian hypergroup. Though Clifford circuits are efficiently classically
simulable, we show that normalizer circuit models encompass Shor's celebrated
factoring algorithm and the quantum algorithms for abelian Hidden Subgroup
Problems. We develop classical-simulation techniques to characterize under
which scenarios normalizer circuits provide quantum speed-ups. Finally, we
devise new quantum algorithms for finding hidden hyperstructures. The results
offer new insights into the source of quantum speed-ups for several algebraic
problems.
Our second contribution is an algebraic (group- and hypergroup-theoretic)
framework for describing quantum many-body states and classically simulating
quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism
(PSF), wherein quantum states are written as joint eigenspaces of stabilizer
groups of commuting Pauli operators: while the PSF is valid for qubit/qudit
systems, our formalism can be applied to discrete- and continuous-variable
systems, hybrid settings, and anyonic systems. These results enlarge the known
families of quantum processes that can be efficiently classically simulated.
This thesis also establishes a precise connection between Shor's quantum
algorithm and the stabilizer formalism, revealing a common mathematical
structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite
original papers if possible. Appendix E contains unpublished work on Gaussian
unitaries. If you spot typos/omissions please email me at JLastNames at
posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk:
https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism
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