14 research outputs found
Logarithmic SAT Solution with Membrane Computing
P systems have been known to provide efficient polynomial (often linear) deterministic solutions to hard problems. In particular, cP systems have been shown to provide very crisp and efficient solutions to such problems, which are typically linear with small coefficients. Building on a recent result by Henderson et al., which solves SAT in square-root-sublinear time, this paper proposes an orders-of-magnitude-faster solution, running in logarithmic time, and using a small fixed-sized alphabet and ruleset (25 rules). To the best of our knowledge, this is the fastest deterministic solution across all extant P system variants. Like all other cP solutions, it is a complete solution that is not a member of a uniform family (and thus does not require any preprocessing). Consequently, according to another reduction result by Henderson et al., cP systems can also solve k-colouring and several other NP-complete problems in logarithmic time
P systems with evolutional symport and membrane creation rules solving QSAT
P systems are computing devices based on sets of rules that dictate how they work.
While some of these rules can change the objects within the system, other rules can even
change the own structure, like creation rules. They have been used in cell-like membrane
systems with active membranes to efficiently solve NP-complete problems. In this work,
we improve a previous result where a uniform family of P systems with evolutional
communication rules whose left-hand side (respectively, right-hand side) have most 2
objects (resp., 2 objects) and membrane creation solved SAT efficiently, and we obtain
an efficient solution to solve QBF-SAT or QSAT (a PSPACE-complete problem) having at
most 1 object (respectively, 1 object) in their left-hand side (resp., right-hand side) and not
making use of the environmentMinisterio de Ciencia e Innovación TIN2017-89842-
Evaluating space measures in P systems
P systems with active membranes are a variant of P systems where membranes can be created by division of existing membranes, thus creating an exponential amount of resources in a polynomial number of steps. Time and space complexity classes for active membrane systems have been introduced, to characterize classes of problems that can be solved by different membrane systems making use of different resources. In particular, space complexity classes introduced initially considered a hypothetical real implementation by means of biochemical materials, assuming that every single object or membrane requires some constant physical space (corresponding to unary notation). A different approach considered implementation of P systems in silico, allowing to store the multiplicity of each object in each membrane using binary numbers. In both cases, the elements contributing to the definition of the space required by a system (namely, the total number of membranes, the total number of objects, the types of different membranes, and the types of different objects) was considered as a whole. In this paper, we consider a different definition for space complexity classes in the framework of P systems, where each of the previous elements is considered independently. We review the principal results related to the solution of different computationally hard problems presented in the literature, highlighting the requirement of every single resource in each solution. A discussion concerning possible alternative solutions requiring different resources is presented
Quantum Hamiltonian Complexity
Constraint satisfaction problems are a central pillar of modern computational
complexity theory. This survey provides an introduction to the rapidly growing
field of Quantum Hamiltonian Complexity, which includes the study of quantum
constraint satisfaction problems. Over the past decade and a half, this field
has witnessed fundamental breakthroughs, ranging from the establishment of a
"Quantum Cook-Levin Theorem" to deep insights into the structure of 1D
low-temperature quantum systems via so-called area laws. Our aim here is to
provide a computer science-oriented introduction to the subject in order to
help bridge the language barrier between computer scientists and physicists in
the field. As such, we include the following in this survey: (1) The
motivations and history of the field, (2) a glossary of condensed matter
physics terms explained in computer-science friendly language, (3) overviews of
central ideas from condensed matter physics, such as indistinguishable
particles, mean field theory, tensor networks, and area laws, and (4) brief
expositions of selected computer science-based results in the area. For
example, as part of the latter, we provide a novel information theoretic
presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include
brief introduction to quantum information, brief list of some recent
developments added, minor changes throughou
Sublinear P system solutions to NP-complete problems
Many membrane systems (e.g. P System), including cP systems (P Systems with compound terms), have been used to solve efficiently many NP-hard problems, often in linear time. However, these solutions have been independent of each other and have not utilised the theory of reductions. This work presents a sublinear solution to k-SAT and demonstrates that k-colouring can be reduced to k-SAT in constant time. This work demonstrates that traditional reductions are efficient in cP systems and that they can sometimes produce more efficient solutions than the previous problem-specific solutions
Guidable Local Hamiltonian Problems with Implications to Heuristic Ans\"atze State Preparation and the Quantum PCP Conjecture
We study 'Merlinized' versions of the recently defined Guided Local
Hamiltonian problem, which we call 'Guidable Local Hamiltonian' problems.
Unlike their guided counterparts, these problems do not have a guiding state
provided as a part of the input, but merely come with the promise that one
exists. We consider in particular two classes of guiding states: those that can
be prepared efficiently by a quantum circuit; and those belonging to a class of
quantum states we call classically evaluatable, for which it is possible to
efficiently compute expectation values of local observables classically. We
show that guidable local Hamiltonian problems for both classes of guiding
states are -complete in the inverse-polynomial precision
setting, but lie within (or ) in the constant
precision regime when the guiding state is classically evaluatable.
Our completeness results show that, from a complexity-theoretic perspective,
classical Ans\"atze selected by classical heuristics are just as powerful as
quantum Ans\"atze prepared by quantum heuristics, as long as one has access to
quantum phase estimation. In relation to the quantum PCP conjecture, we (i)
define a complexity class capturing quantum-classical probabilistically
checkable proof systems and show that it is contained in
for constant proof queries; (ii) give a no-go
result on 'dequantizing' the known quantum reduction which maps a
-verification circuit to a local Hamiltonian with constant
promise gap; (iii) give several no-go results for the existence of quantum gap
amplification procedures that preserve certain ground state properties; and
(iv) propose two conjectures that can be viewed as stronger versions of the
NLTS theorem. Finally, we show that many of our results can be directly
modified to obtain similar results for the class .Comment: 61 pages, 6 figure
In Memoriam, Solomon Marcus
This book commemorates Solomon Marcus’s fifth death anniversary with a selection of articles in mathematics, theoretical computer science, and physics written by authors who work in Marcus’s research fields, some of whom have been influenced by his results and/or have collaborated with him
Uniformity is weaker than semi-uniformity for some membrane systems
We investigate computing models that are presented as families of finite
computing devices with a uniformity condition on the entire family. Examples of
such models include Boolean circuits, membrane systems, DNA computers, chemical
reaction networks and tile assembly systems, and there are many others.
However, in such models there are actually two distinct kinds of uniformity
condition. The first is the most common and well-understood, where each input
length is mapped to a single computing device (e.g. a Boolean circuit) that
computes on the finite set of inputs of that length. The second, called
semi-uniformity, is where each input is mapped to a computing device for that
input (e.g. a circuit with the input encoded as constants). The former notion
is well-known and used in Boolean circuit complexity, while the latter notion
is frequently found in literature on nature-inspired computation from the past
20 years or so.
Are these two notions distinct? For many models it has been found that these
notions are in fact the same, in the sense that the choice of uniformity or
semi-uniformity leads to characterisations of the same complexity classes. In
other related work, we showed that these notions are actually distinct for
certain classes of Boolean circuits. Here, we give analogous results for
membrane systems by showing that certain classes of uniform membrane systems
are strictly weaker than the analogous semi-uniform classes. This solves a
known open problem in the theory of membrane systems. We then go on to present
results towards characterising the power of these semi-uniform and uniform
membrane models in terms of NL and languages reducible to the unary languages
in NL, respectively.Comment: 28 pages, 1 figur
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Many-Body Quantum Dynamics and Non-Equilibrium Phases of Matter
Isolated, many-body quantum systems, evolving under their intrinsic dynamics, exhibit a multitude of exotic phenomena and raise foundational questions about statistical mechanics. A flurry of theoretical work has been devoted to understanding how these systems reach thermal equilibrium in the absence of coupling to an external bath and, when thermalization does not occur, investigating the emergent non-equilibrium phases of matter. With the advent of synthetic quantum systems, such as ultra-cold atoms in optical lattices or trapped ions, these questions are no longer academic and can be directly studied in the laboratory. This dissertation explores the non-equilibrium phenomena that stem from the interplay between interactions, disorder, symmetry, topology, and external driving. First, we study how strong disorder, leading to many-body localization, can arrest the heating of a Floquet system and stabilize symmetry-protected topological order that does not have a static analogue. We analyze its dynamical and entanglement properties, highlight its duality to a discrete time crystal, and propose an experimental implementation in a cold-atom setting.Quenched disorder and the many-body localized state are crucial ingredients in protecting macroscopic quantum coherence. We explore the stability of many-body localization in two and higher dimensions and analyze its robustness to rare regions of weak disorder.We then study a second example of non-thermal behavior, namely integrability. We show that a class of random spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram, which includes regions of integrability, classical chaos, and of a novel integrable structure whose conservation laws are reminiscent of the integrals of motion found in a many-body localized phase.The third group of disordered, non-ergodic systems we consider, spin glasses, have fascinating connections to complexity theory and the hardness of constraint satisfaction. We define a statistical ensemble that interpolates between the classical and quantum limits of such a problem and show that there exists a sharp boundary separating satisfiable and unsatisfiable phases