752 research outputs found
Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used
tool in theoretical computer sciences to obtain structural results about
computational problems and complexity classes by indirect proofs. The Uniform
Diagonalization Theorem allows the construction of problems outside complexity
classes while still being reducible to a specific decision problem. This paper
provides a generalization of the Uniform Diagonalization Theorem by extending
it to promise problems and the complexity classes they form, e.g. randomized
and quantum complexity classes. The theorem requires from the underlying
computing model not only the decidability of its acceptance and rejection
behaviour but also of its promise-contradicting indifferent behaviour - a
property that we will introduce as "total decidability" of promise problems.
Implications of the Uniform Diagonalization Theorem are mainly of two kinds:
1. Existence of intermediate problems (e.g. between BQP and QMA) - also known
as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class
is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the
original Uniform Diagonalization Theorem the extension applies besides BQP and
QMA to a large variety of complexity class pairs, including combinations from
deterministic, randomized and quantum classes.Comment: 15 page
Can biological quantum networks solve NP-hard problems?
There is a widespread view that the human brain is so complex that it cannot
be efficiently simulated by universal Turing machines. During the last decades
the question has therefore been raised whether we need to consider quantum
effects to explain the imagined cognitive power of a conscious mind.
This paper presents a personal view of several fields of philosophy and
computational neurobiology in an attempt to suggest a realistic picture of how
the brain might work as a basis for perception, consciousness and cognition.
The purpose is to be able to identify and evaluate instances where quantum
effects might play a significant role in cognitive processes.
Not surprisingly, the conclusion is that quantum-enhanced cognition and
intelligence are very unlikely to be found in biological brains. Quantum
effects may certainly influence the functionality of various components and
signalling pathways at the molecular level in the brain network, like ion
ports, synapses, sensors, and enzymes. This might evidently influence the
functionality of some nodes and perhaps even the overall intelligence of the
brain network, but hardly give it any dramatically enhanced functionality. So,
the conclusion is that biological quantum networks can only approximately solve
small instances of NP-hard problems.
On the other hand, artificial intelligence and machine learning implemented
in complex dynamical systems based on genuine quantum networks can certainly be
expected to show enhanced performance and quantum advantage compared with
classical networks. Nevertheless, even quantum networks can only be expected to
efficiently solve NP-hard problems approximately. In the end it is a question
of precision - Nature is approximate.Comment: 38 page
Computer Program Simulation of a Quantum Turing Machine with Circuit Model
Molina and Watrous present a variation of the method to simulate a quantum Turing machine employed in Yaoâs 1995 publication âQuantum Circuit Complexityâ. We use a computer program to implement their method with linear algebra and an additional unitary operator defined to complete the details. Their method is verified to be correct on a quantum Turing machine
Pathways to cellular supremacy in biocomputing
Synthetic biology uses living cells as the substrate for performing human-defined computations. Many current implementations of cellular computing are based on the âgenetic circuitâ metaphor, an approximation of the operation of silicon-based computers. Although this conceptual mapping has been relatively successful, we argue that it fundamentally limits the types of computation that may be engineered inside the cell, and fails to exploit the rich and diverse functionality available in natural living systems. We propose the notion of âcellular supremacyâ to focus attention on domains in which biocomputing might offer superior performance over traditional computers. We consider potential pathways toward cellular supremacy, and suggest application areas in which it may be found.A.G.-M. was supported by the SynBio3D project of the UK Engineering and Physical Sciences Research Council (EP/R019002/1) and the European CSA on biological standardization BIOROBOOST (EU grant number 820699). T.E.G. was supported by a Royal Society University Research Fellowship (grant UF160357) and BrisSynBio, a BBSRC/ EPSRC Synthetic Biology Research Centre (grant BB/L01386X/1). P.Z. was supported by the EPSRC Portabolomics project (grant EP/N031962/1). P.C. was supported by SynBioChem, a BBSRC/EPSRC Centre for Synthetic Biology of Fine and Specialty Chemicals (grant BB/M017702/1) and the ShikiFactory100 project of the European Unionâs Horizon 2020 research and innovation programme under grant agreement 814408
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