This document presents a new class of restricted quantum membrane systems. The key ideas are to define membrane systems that operate under strictly unitary quantum evolution rules, avoiding problems associated with transferring objects between membranes. A cascading membrane P system is defined, where membranes are arranged hierarchically with input and output spaces coupled in a pipeline. Computation proceeds by applying unitary operators to manipulate qubit registers representing object degrees in each membrane. This model is shown to be capable of simulating classical automata. The approach aims to combine variants of P systems with quantum computing techniques in a way that is consistent with underlying quantum physics.

1. a new class of restricted quantum membrane
systems
2nd
World Congress “Genetics, Geriatrics and Neurodegenerative Diseases Re-
search” (GeNeDis 2016)
Sparta, Greece
Konstantinos Giannakis, Alexandros Singh, Kalliopi Kastampolidou, Christos
Papalitsas, and Theodore Andronikos
October 20, 2016
Department of Informatics, Ionian University
0

2. preview of our study
∙ Computing in an unconventional environment
∙ Membrane systems
∙ Quantum computational aspects
∙ Quantum evolution rules
1

3. introduction

4. motivations
∙ Moore’s Law is reaching its physical limits.
∙ New computing paradigms?
∙ Redesign and revisit well-studied models and structures from
classical computation.
∙ Unitarity in membrane systems.
3

5. membrane computing
∙ Known as P systems with several proposed variants.
∙ Evolution depicted through rewriting rules on multisets of the
form u→v
∙ imitating natural chemical reactions.
∙ u, v are multisets of objects.
∙ The hierarchical status of membranes evolves by constantly
creating 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)
4

6. examples
Membranes create hierarchical structures.
(a) Hierarchical nested
membranes
(b) With simple objects and rules
5

7. 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 matching
parentheses.
∙ Use of rules =⇒ transitions among configurations.
∙ A sequence of transitions is interpreted as computation.
∙ Accepted computations are those which halt and a successful
computation is associated with a result.
6

8. rules used in membrane computing
...
...
b)
a)
c) exo
(a,in)aa(b,in)ba
(a,out)
cab
c→a
bbbba
(a,out)
caa
c→bb
cca
=⇒
=⇒
=⇒
7

9. definition
Definition
A 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 with
elements 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 object
evolution 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.
8

10. advantages
Inherent compartmentalization, easy extensibility and direct
intuitive appearance for biologists.
Expression models and phenomena related to
neurodegenerative diseases and malfunctions.
Probability theory and stochasticity (many biological functions
are of stochastic nature).
P systems: formal tools, with enhanced power and efficiency
=⇒ could shed light to the problem of modeling complex
biological processes.
9

11. 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 from
classical computation.
10

12. consequences of moore’s law
∙ Continuously decreasing size of the computing circuits.
∙ Technological and physical limitations (limits of lithography in
chip design).
∙ New technologies to overcome these barriers, with Quantum
Computation being a possible candidate.
∙ Ability of these systems to operate at a microscopic level.
11

13. basics of quantum computing
∙ QC considers the notion of computing as a natural, physical
process.
∙ It must obey to the postulates of quantum mechanics.
∙ Bit ⇒ Qubit.
∙ It was initially discussed in the works of Richard Feynman in the
early ’80s.
12

14. 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.
13

15. 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†
its
transpose and conjugate.
∙ Cn×n
defines the set of all n × n complex matrices.
∙ Hn is an n-dimensional Hilbert space.
14

16. quantum computation states and formalism
∙ The evolution of a quantum system is described by unitary
transformations.
∙ The states of an n-level quantum system are self-adjoint
positive mappings of Hn with unit trace.
∙ An observable of a quantum system is a self-adjoint mapping
Hn → Hn.
∙ Each state qi ∈ Q with |Q| = n can be represented by a vector
ei = (0, . . . , 1, . . . , 0).
15

17. quantum computation applying matrices, observables, and projection
∙ Each of the states is a superposition of the form
n∑
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 each
observable O an Hermitian matrix O.
∙ The possible outcomes of a measurement are the eigenvalues
of the observable.
∙ Transition from one state to another is achieved through the
application of a unitary operator Uσi
.
∙ The probability of obtaining a result p is ∥πPi∥, where π is the
current state (or a superposition) and Pi is the projection matrix
of the measured basis state.
∙ The state after the measurement collapses to the πPi
/
∥πPi∥.
16

18. our main contribution

19. similar approaches
∙ Mainly by Leporati et al.
∙ Inspired by classical energy-based P systems
∙ 2 models: based on strictly unitary rules and on non-unitary
operations.
∙ Objects are represented by qudits, while multisets are
compositions of such individual systems.
∙ Energy units, associated with the objects, are incorporated in the
system in the form of actual quanta of energy.
∙ Objects can change their state but can never cross membranes to
move to another region.
∙ Interactions happen through the modification of energy of the
oscillators in each membrane.
18

20. 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 similar
works.
∙ by providing registers with set “depths” that can easily be
manipulated with standard unitary operators.
19

21. defining the cascading p systems
Definition
A 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 in
hierarchically arranged layers, in a way such that inputs and outputs
form a pipeline through the layers. Each membrane consists of two
Hilbert spaces, an input and an output one. The outermost membrane
to contain the result of a computation.
3. Each wm describes the initial configuration of the m ∈ µ membrane’s
state. It is composed of |Γ| qubits.
4. Each element of Rm would be a unitary operator which acts in m ∈ µ.
20

22. states and computation
∙ State:
Each membrane in layer 0 has its own input space and a shared
output space. For each layer k > 0, the membranes of layer k
have as inputs the output space of layer k − 1, and share an
output space, which in turn is the input of k + 1.
∙ Computation:
For each membrane, we apply a set of rules. We, also, initialise
the i-th membrane’s input region with instances of objects as
defined by each wi. Computation starts from the innermost
layer (layer 0), applying the composition of rules Rm for all the
membranes m ∈ layer 0 and continues with layer 1, layer 2 etc.
The output space of the outermost layer contains the result of
the computation.
21

23. 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 values
represent the “degrees of existence” for each letter. For
example, M1’s initial state is |10⟩ = |1⟩ ⊗ |0⟩.
22

24. 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 space
M1in ⊗ 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
23

25. simulating classical automata i
∙ Given a depth k ∈ N, we are able to build a P system that
simulates an automaton running on words of length l = k.
∙ Construction:
We build a cascading P system whose alphabet consists of the
alphabet of the automaton we are simulating, plus all its states
(represented as tokens/letters).
Consider k nested membranes, with input/output spaces
coupled as before. Each space consists of two components: a
letter qudit and a state qudit so that it looks something like this:
|letter⟩ ⊗ |state⟩
Starting from the inner membrane, we initialise the letter kets to
the value of the corresponding letter of the input word such
that the k-th membrane contains the k-th letter.
24

26. simulating classical automata ii
∙ All state kets are initialised to |q0⟩.
∙ Then to each membrane is assigned the sum of n = |Σ| rules of
the form:
|letter⟩ ⟨letter| ⊗ U,
where |Σ| is the length of the automaton’s alphabet and U
changes the output state’s ket to |newState⟩ based on the
automaton’s transition function:
δ(letter, currentState) = newState
25

27. 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
26

28. 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 flip
is the operator that “flips” a qubit’s value.
27

29. concluding

30. conclusions
∙ An effort to mix of variants of P systems with quantum evolution
rules.
∙ Membrane systems that operate under unitary transformations.
∙ A novel methodology regarding the construction of the quantum
rules.
∙ Unlike related works, our approach involves the use of strictly
unitary rules.
∙ Consistency with the underlying quantum physics.
∙ Potential application of our proposed variants in other
disciplines.
∙ The description of actual algorithms based on these computation
machines.
∙ Connection with game-theoretic aspects of computing.
∙ Relation to quantum game theory.
∙ Implementation of similar approaches, in order to model and
describe actual complex biological models.
29

31. 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 Experimental
Medicine and Biology. Springer International Publishing, 2015, pp. 163–172.
Leporati, A.
(UREM) P systems with a quantum-like behavior: background, definition, and computational
power.
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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32. Any Questions?
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