a new class of restricted quantum membranesystems2nd World Congress “Genetics, Geriatrics and Neurodegenerative Diseases Re-search” (GeNeDis 2016)Sparta, Greece

Konstantinos Giannakis, Alexandros Singh, Kalliopi Kastampolidou, ChristosPapalitsas, and Theodore AndronikosOctober 20, 2016

Department of Informatics, Ionian University

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preview of our study

∙ Computing in an unconventional environment

∙ Membrane systems

∙ Quantum computational aspects

∙ Quantum evolution rules

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introduction

motivations

∙ Moore’s Law is reaching its physical limits.

∙ New computing paradigms?

∙ Redesign and revisit well-studied models and structures fromclassical computation.

∙ Unitarity in membrane systems.

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membrane computing

∙ Known as P systems with several proposed variants.∙ Evolution depicted through rewriting rules on multisets of theform u→v

∙ imitating natural chemical reactions.∙ u, v are multisets of objects.

∙ The hierarchical status of membranes evolves by constantlycreating and destroying membranes, by membrane division etc.

∙ Different types of communication rules:∙ symport rules (one-way passing through a membrane)∙ antiport rules (two-way passing through a membrane)

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examples

Membranes create hierarchical structures.

(a) Hierarchical nestedmembranes

(b) With simple objects and rules

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p systems evolution and computation

∙ Via purely non deterministic, parallel rules.

∙ Characteristics of membrane systems: the membrane structure,multisets of objects, and rules.

∙ They can be represented by a string of labelled matchingparentheses.

∙ Use of rules =⇒ transitions among configurations.∙ A sequence of transitions is interpreted as computation.∙ Accepted computations are those which halt and a successfulcomputation is associated with a result.

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rules used in membrane computing

...

...

b)

a)

c) exo

(a,in)aa(b,in)ba

(a,out)cab

c→abbbba

(a,out)caa

c→bbcca

=⇒

=⇒

=⇒

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definition

DefinitionA generic P system (of degree m, m ≥ 1) with the characteristics described above can be defined as a construct

Π=(V, T, C, H, µ, w1 , ..., wm, (R1 , ..., Rm), (H1 , ..., Hm) i0) ,

where

1. V is an alphabet and its elements are called objects.

2. T ⊆ V is the output alphabet.

3. C ⊆ V, C ∩ T = ⊘ are catalysts.

4. H is the set {pino, exo, mate, drip} of membrane handling rules.

5. µ is a membrane structure consisting of m membranes, with the membranes and the regions labeled in a one-to-one way withelements of a given set H.

6. wi , 1 ≤ i ≤ m, are strings representing multisets over V associated with the regions 1,2, ... ,m of µ.

7. Ri , 1 ≤ i ≤ m, are finite sets of evolution rules over the alphabet set V associated with the regions 1,2, ... , m of µ. These objectevolution rules have the form u → v.

8. Hi , 1 ≤ i ≤ m, are finite sets of membrane handling rules rules over the set H associated with the regions 1,2, ... , m of µ.

9. i0 is a number between 1 and m and defines the initial configuration of each region of the P system.

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advantages

X Inherent compartmentalization, easy extensibility and directintuitive appearance for biologists.

X Expression models and phenomena related toneurodegenerative diseases and malfunctions.

X Probability theory and stochasticity (many biological functionsare of stochastic nature).

X P systems: formal tools, with enhanced power and efficiency=⇒ could shed light to the problem of modeling complexbiological processes.

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computing in a quantum environment

∙ Quantum computing ⇒ Buzzword

∙ Moore’s Law is reaching its physical limits.

∙ New computing paradigms?

∙ Redesign and revisit well-studied models and structures fromclassical computation.

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consequences of moore’s law

∙ Continuously decreasing size of the computing circuits.

∙ Technological and physical limitations (limits of lithography inchip design).

∙ New technologies to overcome these barriers, with QuantumComputation being a possible candidate.

∙ Ability of these systems to operate at a microscopic level.

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basics of quantum computing

∙ QC considers the notion of computing as a natural, physicalprocess.

∙ It must obey to the postulates of quantum mechanics.

∙ Bit ⇒ Qubit.

∙ It was initially discussed in the works of Richard Feynman in theearly ’80s.

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dirac symbolism bra-ket notation

∙ State 0 is represented as ket |0⟩ and state 1 as ket |1⟩.

∙ Every ket corresponds to a vector in a Hilbert space.

∙ A qubit is in state |ψ⟩ described by:

|ψ⟩ = c0 |0⟩+ c1 |1⟩ (1)

∙ They are complex numbers for which |c0|2 + |c1|2 = 1.

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terminology needed for clarification

∙ Σ ⇒ the alphabet

∙ Σ∗ ⇒ the set of all finite strings over Σ

∙ If U is an n× n square matrix , U is its conjugate, and U† itstranspose and conjugate.

∙ Cn×n defines the set of all n× n complex matrices.

∙ Hn is an n-dimensional Hilbert space.

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quantum computation states and formalism

∙ The evolution of a quantum system is described by unitarytransformations.

∙ The states of an n-level quantum system are self-adjointpositive mappings of Hn with unit trace.

∙ An observable of a quantum system is a self-adjoint mappingHn → Hn.

∙ Each state qi ∈ Q with |Q| = n can be represented by a vectorei = (0, . . . , 1, . . . , 0).

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quantum computation applying matrices, observables, and projection

∙ Each of the states is a superposition of the formn∑i=1

ciei.

∙ n is the number of states∙ ci ∈ C are the coefficients with |c1|2 + |c2|2 + · · ·+ |cn|2 = 1∙ ei denotes the (pure) basis state corresponding to i.

∙ Each symbol σi ∈ Σ a unitary matrix/operator Uσi and eachobservable O an Hermitian matrix O.

∙ The possible outcomes of a measurement are the eigenvaluesof the observable.

∙ Transition from one state to another is achieved through theapplication of a unitary operator Uσi .

∙ The probability of obtaining a result p is ∥πPi∥, where π is thecurrent state (or a superposition) and Pi is the projection matrixof the measured basis state.

∙ The state after the measurement collapses to the πPi/∥πPi∥.

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our main contribution

similar approaches

∙ Mainly by Leporati et al.∙ Inspired by classical energy-based P systems

∙ 2 models: based on strictly unitary rules and on non-unitaryoperations.

∙ Objects are represented by qudits, while multisets arecompositions of such individual systems.

∙ Energy units, associated with the objects, are incorporated in thesystem in the form of actual quanta of energy.

∙ Objects can change their state but can never cross membranes tomove to another region.

∙ Interactions happen through the modification of energy of theoscillators in each membrane.

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our key ideas

∙ No use of energy-based rules, oscillators, and non-unitary rules.

∙ We prefer more conventional quantum computing techniques.

∙ Our rules are strictly unitary.

∙ We avoid the problems associated with the notion of“transferring” systems/objects, which is inherent in similarworks.

∙ by providing registers with set “depths” that can easily bemanipulated with standard unitary operators.

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defining the cascading p systems

DefinitionA cascading P system is a tuple

Π = (Γ, µ,wm,Rm),

where

1. Γ is an alphabet, we call them objects.

2. µ is a membrane structure, in which membranes are nested inhierarchically arranged layers, in a way such that inputs and outputsform a pipeline through the layers. Each membrane consists of twoHilbert spaces, an input and an output one. The outermost membraneto contain the result of a computation.

3. Each wm describes the initial configuration of the m ∈ µ membrane’sstate. It is composed of |Γ| qubits.

4. Each element of Rm would be a unitary operator which acts in m ∈ µ.

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states and computation

∙ State:Each membrane in layer 0 has its own input space and a sharedoutput space. For each layer k > 0, the membranes of layer khave as inputs the output space of layer k− 1, and share anoutput space, which in turn is the input of k+ 1.

∙ Computation:For each membrane, we apply a set of rules. We, also, initialisethe i-th membrane’s input region with instances of objects asdefined by each wi. Computation starts from the innermostlayer (layer 0), applying the composition of rules Rm for all themembranes m ∈ layer 0 and continues with layer 1, layer 2 etc.The output space of the outermost layer contains the result ofthe computation.

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an example

M1

M2

M3

a

ba

∙ For each membrane, the input and output state kets|ab⟩ = |a⟩ ⊗ |b⟩ are composed of two qubits, whose valuesrepresent the “degrees of existence” for each letter. Forexample, M1’s initial state is |10⟩ = |1⟩ ⊗ |0⟩.

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the rules

∙ Membrane 1 rule: R1 = |10⟩M1in ⊗ |00⟩M1out ↔ |00⟩M1in ⊗ |10⟩M1out

∙ Membrane 2 rule: R2 = |11⟩M2in ⊗ |10⟩M2out ↔ |00⟩M2in ⊗ |11⟩M2out

∙ Membrane 3 rule: R3 = |11⟩M3in ⊗ |00⟩M3out ↔ |00⟩M3in ⊗ |11⟩M3out

The actual rules would work on the whole spaceM1in⊗M2in⊗M1out/M2out/M3in⊗M3out.

If we apply the sequence R3 · R2 · R1 to the initial state:

|10⟩M1in ⊗ |11⟩M2in ⊗ |00⟩M1M2out/M3in ⊗ |00⟩M3out

we get the final state:

|00⟩M1in ⊗ |00⟩M2in ⊗ |00⟩M1M2out/M3in ⊗ |11⟩M3out

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simulating classical automata i

∙ Given a depth k ∈ N, we are able to build a P system thatsimulates an automaton running on words of length l = k.

∙ Construction:We build a cascading P system whose alphabet consists of thealphabet of the automaton we are simulating, plus all its states(represented as tokens/letters).Consider k nested membranes, with input/output spacescoupled as before. Each space consists of two components: aletter qudit and a state qudit so that it looks something like this:

|letter⟩ ⊗ |state⟩Starting from the inner membrane, we initialise the letter kets tothe value of the corresponding letter of the input word suchthat the k-th membrane contains the k-th letter.

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simulating classical automata ii

∙ All state kets are initialised to |q0⟩.

∙ Then to each membrane is assigned the sum of n = |Σ| rules ofthe form:

|letter⟩ ⟨letter| ⊗ U,where |Σ| is the length of the automaton’s alphabet and Uchanges the output state’s ket to |newState⟩ based on theautomaton’s transition function:

δ(letter, currentState) = newState

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simulation example i

Consider the following classical automaton:

∙ Σ = {a,b}∙ Q = {s0, s1}∙ δ(a, s0) = s1, δ(b, s0) = s0, δ(a, s1) = s1, δ(b, s1) = s0

Let us simulate a run at depth k = 2 for the word “ab”.

Our membrane system’s initial global state is:

|a⟩m1in ⊗ |q0⟩m1in ⊗ |b⟩m1out/m2in ⊗ |q0⟩m1out/m2in ⊗ |a⟩m2out ⊗ |s0⟩m2out

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simulation example ii

∙ The first membrane’s rule is:

|a⟩ ⟨a| ⊗ |s0⟩ ⟨s0| ⊗ I⊗ flip⊗ I +|b⟩ ⟨b| ⊗ |s0⟩ ⟨s0| ⊗ I⊗ I⊗ I +|a⟩ ⟨a| ⊗ |s1⟩ ⟨s1| ⊗ I⊗ I⊗ I +|b⟩ ⟨b| ⊗ |s1⟩ ⟨s1| ⊗ I⊗ I⊗ I

∙ While the second one’s is:

I⊗ I⊗ |a⟩ ⟨a| ⊗ |s0⟩ ⟨s0| ⊗ flip +I⊗ I⊗ |a⟩ ⟨a| ⊗ |s0⟩ ⟨s0| ⊗ I +I⊗ I⊗ |a⟩ ⟨a| ⊗ |s1⟩ ⟨s1| ⊗ flip +I⊗ I⊗ |a⟩ ⟨a| ⊗ |s1⟩ ⟨s1| ⊗ I

In the above expression, I denotes the identity operator and flipis the operator that “flips” a qubit’s value.

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concluding

conclusions

∙ An effort to mix of variants of P systems with quantum evolutionrules.

∙ Membrane systems that operate under unitary transformations.∙ A novel methodology regarding the construction of the quantumrules.

∙ Unlike related works, our approach involves the use of strictlyunitary rules.

∙ Consistency with the underlying quantum physics.∙ Potential application of our proposed variants in otherdisciplines.

∙ The description of actual algorithms based on these computationmachines.

∙ Connection with game-theoretic aspects of computing.∙ Relation to quantum game theory.∙ Implementation of similar approaches, in order to model anddescribe actual complex biological models.

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key references

Calude, C.Unconventional computing: A brief subjective history.Tech. rep., Department of Computer Science, The University of Auckland, New Zealand, 2015.

Feynman, R. P.Simulating physics with computers.International journal of theoretical physics 21, 6 (1982), 467–488.

Giannakis, K., and Andronikos, T.Mitochondrial fusion through membrane automata.In GeNeDis 2014, P. Vlamos and A. Alexiou, Eds., vol. 820 of Advances in ExperimentalMedicine and Biology. Springer International Publishing, 2015, pp. 163–172.

Leporati, A.(UREM) P systems with a quantum-like behavior: background, definition, and computationalpower.In International Workshop on Membrane Computing (2007), Springer, pp. 32–53.

Leporati, A., Mauri, G., and Zandron, C.Quantum sequential P systems with unit rules and energy assigned to membranes.In International Workshop on Membrane Computing (2005), Springer, pp. 310–325.

Păun, G.Computing with membranes: Attacking NP-complete problems.In Unconventional models of Computation, UMC’2K. Springer, 2001, pp. 94–115.

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Any Questions?

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