Quantum (UREM) P Systems: Background, Definition and Computational Power Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi

Quantum (UREM) P Systems: Background, Definition and

Computational Power

Dipartimento di Informatica, Sistemistica e Comunicazione

Università degli Studi di Milano – Bicocca

WMC 8Thessaloniki, June 25 – 28, 2007

Alberto Leporati

[email protected]

[email protected]

WMC 6 - Quantum Sequential P Systems 2

(deterministic) register machinessome notions of quantum mechanics

creation and annihilation operatorsphysical interpretation

classical P systems with unit rules and energy assigned to membranesquantum version of the above P systems

computational completeness simulation of register machinessolving 3-SAT

quantum register machines

Talk outline


at the beginning of 2004, the membrane community asks for a quantum membrane system

we had papers about quantum description of many-valued gates (using creation and annihilation operators)

we had energy-based P systems

other models of P systems with energy have been introduced since 2001

first try [Palma de Mallorca, 2004]: quantum version of energy-based P systems

Rudi defined (classical) UREM P systems

second try: quantum UREM P systems

Motivation


a deterministic n-register machine is a construct

M = (n, P, l0, lh )where:

n is the number of registers

P is a finite set of instructions with labels from lab(M)

l0 is the label of the first instruction

lh is the label of the final instruction

each register contains a non-negative integerwithout loss of generality, we assume:

lab(M) = {1, 2, …, m} l0 = 1 lh = m

Register machines


j : (INC(r), k)j, k lab(M) and 1 r nincrements the value of r, and jump to instruction k

j : (DEC(r), k, l)j, k, l lab(M) and 1 r nif the value of r is positive then decrement it and jump to instruction k else jump to instruction l

m : haltstop the machine

Register machines: instructions


any partial recursive function f:

can be computed by a (max{, }+2)-register machine M

input: (n1, …, n ) is initially stored in

registers 1 to if f (n1, …, n ) = (r1, …, r ), then:

the computation haltsthe first registers contain the resultthe other registers are empty

if f (n1, …, n ) is undefined, then the computation does not halt

Register machines: instructions


memory cells (registers) realized by qubits(two-level atoms, spin-½ particles, …)base states of qubits:

(computational basis of 2)

a qubit can also exist in superpositions:

with c0 ,c1 such that |c0|2 + |c1|2 = 1

0

10

1

01

10 10 cc

Quantum computers


if we measure , we obtain:

with probability |c0|2

with probability |c1|2

post-measurement state of : the observed base state measuring alters its state !

quantum register: collection of n qubits

base states of registers:

(orthonormal basis of n

2)

,nnn xxxx ,,11

0

1

1,0ixwith

Quantum computers


computations:the register is initialized with the boolean tuple (x1, x2 , …, xn )

a linear operator G: n

2 n

2 is applied to the contents of the register

the contents of the register are modified by the application of G

we interpret the state as the result of the computation

the action of G on superpositions is obtained by linearity

nn yyxxG ,,,, 11

Quantum computers


all these notions can be extended to qudits(d-valued versions of qubits)

base states (computational basis of d):

they represent the logical values:

1

0

0

0

1,

0

1

0

0

1

2,,

0

0

1

0

1

1,

0

0

0

1

0 d

d

d

1,

1

2,,

1

2,

1

1,0

d

d

ddLd

Quantum computers


a qudit can also exist in superpositions:

with such that

quantum register: collection of qudits

base states of registers:

(computational basis of )

11

10 1

1

10 cd

ccd

1

1

10 ,,, cccd

didn

n Lxxx ,,,1 withdn

dLi

ic 1|| 2

Quantum computers


computations are performed through linear

operators G: n

d n

d

let be a set of

equispaced real values:

to each v Ld we associate v d

thev can be viewed as the energy levels of a

(truncated) quantum harmonic oscillator, whose

Hamiltonian is (see next slides):

1

1

2

1

10 ,,,, d

d

d

d

1

1

2

1

10

d

d

d

Quantum computers


Quantum harmonic oscillator


is the

eigenvector of the state of energy k 0 + k

spectral decomposition:

where represents the event: a measure of

system energy yields the value 0 + k

)1(00

00

00

0

0

0

d

H

1,,1,0,1

dkd

kH k

Quantum harmonic oscillator

k

d

kPkH

1

0 0 )(

1

d

kPPk


to modify a qudit, we can use creation and annihilation operators on

d:

0000

1000

0200

0010

0100

0020

0001

0000

d

a

d

a

action of a+ on the base states of qudits:

0

1

2,,1,0,1

11

1

a

dkd

kk

d

ka for

Creation and annihilation


action of a on the base states of qudits:

0

0

1,,2,1,1

1

1

a

dkd

kk

d

ka for

we can also define the operator:

N is self-adjointeigenvalues: 0, 1, …, d1

eigenvectors:

1000

0200

0010

0000

d

aaN


1

d

kkN


N describes the number of particles of physical systems consisting of a maximum number of d–1 particlesto add a particle to the system (if not already full) apply creation:

to remove a particle from the system (if not already empty) apply annihilation:

First physical interpretation of N, a+ and a

01

2...,,1,011

dNa

dkforkNkkNa

00

1...,,2,11

Na

dkforkNkkNa


Hamiltonian H can be expressed as:

H = N = a+a

Thus,a+ realizes the transition from the eigenstate of energy k to the “next” eigenstate of energy k+1 for 0 k d–1, while it collapses the eigentstate of energy d–1 to the null vector

a transforms the eigenstate of energy k to the

“previous” eigenstate of energy k–1, while it collapses

the eigenstate of energy 0 to the null vector

Second physical interpretation of N, a+ and a


the collection of all linear operators G: d

d is a d2-dimensional linear space, whose canonical basis is:

dyx LyxxyE ,:,

xLzzE

yxxyxE

dyx

yx

all for 0,

,

behavior of Ex,y :

every Ei/(d-1),j/(d-1) is represented by an order d

square matrix with the element at position (j+1,i+1) equal to 1 and all other elements equal to 0



if we define:

where u,v {a, a+} and p,q,r non negative integers, then:

Realization of Ex,y with a+ and a

uvvvvvvApqr

rqpvu **,,

,


The angular momentum interpretation

fixed d 2, and setting j = (d–1)/2, the angular momentum on

d is the triple of self-adjoint operators J = (Jx, Jy, Jz)

j(j+1) is an eigenvalue of J 2 = Jx

2 + Jy2 + Jz

2

matrix representation of Jz w.r.t. the orthonormal basis of its eigenvectors:

2

1000

02

300

002

30

0002

1

d

d

d

d

J z



thus, Jz can assume d possible eigenvalues:

with corresponding eigenvectors:

let us consider the operators:

(non-Hermitian and adjoint of each other)

1,,1,02

)12(

dkfor

kdm

12

)12(

d

kkdJ z

yxyx JiJJJiJJ


in matrix form:

00000

10000

0)2(2000

00)2(200

00010

d

d

d

d

J

Spin-rising and spin-lowering

01000

00)2(200

000)2(20

00001

00000

d

d

d

d

J

spin-rising

spin-lowering



behavior of J+, for m = –j, …, +j:

action on the vectors of the basis (qudits):

to switch Jz to the “next” value (if possible), apply J+ on the system

this corresponds to switching to the previous truth value

1)1()1( mJmmjjmJJ zz

00

1...,,2,11

)(1

J

dkford

kkdk

d

kJ



behavior of J-, for m = –j, …, +j:

action on the vectors of the basis (qudits):

to switch Jz to the “previous” value (if possible), apply J- on the system

this corresponds to switching to the next truth value

1)1()1( mJmmjjmJJ zz

01

2...,,1,01

1))1()(1(

1

J

dkford

kkdk

d

kJ


Example: d = 3

spin-rising and spin-lowering operators:

effect of J+:

000

200

020

020

002

000

JJ

00

10

01

00,2 JaJad and for


let

(u,v {a, a+} and p,q,r non negative integers), then:

Realization of Ex,y with J+ and J–

uvvvvvvApqr

rqpvu **,,

,


defined as in the classical case, as a four-tuple M = (n, P, l0, lh )

each register is an infinite-dimensional quantum harmonic oscillator with base states |0, |1, …the program counter is realized with a quantum system with base states { |x : x Lm }

configuration: program counter + the contents of registers

described as a (base) vector in the Hilbert space

m

(n H )

program counter

register

Quantum register machines


for simplicity, instructions are denoted by j : (INC(r), k) and j : (DEC(r), k, l)

instruction j : (INC(r), k) is defined as the operator:

instruction j : (DEC(r), k, l) is defined as the operator:

IdId1,,

rnrjk

INCkrj appO

IdId

IdId

1

001

,,,

rnrjk

rnrjl

DEClkrj

app

ppO



program P is formally defined as the operator:

OP describes a computation step of M

P halts when the program counter becomes |pm = |1

OP would act as the null operator

we can apply any operator (ex: a projection) when the program counter becomes |pm = |1

krj lkrj

DEClkrj

INCkrjP OOO

,, ,,,,,,,,



a classical P system with Unit Rules and Energy assigned to Membranes (UREM) is a construct:

= (A, , e0, …, ed, w0, …, wd, R0, …, Rd)

where:A is the alphabet of objects

is the membrane structure, with membranes labelled by 0, 1, …, d in a one-to-one manner

e0, …, ed are the initial energy values assigned to membranes (non-negative integers)

w0, …, wd are the initial multisets of objects

R0, …, Rd are the sets of rules

Classical UREM P systems


each rule of Ri has the form ( : a, e, b), where {in, out}, a, b A, e instead of ( : a, e, b) Ri we can write (i : a, e, b) and define:

R = {(i : a, e, b) : ( : a, e, b) Ri , 0 i d}

initial configuration: e0, …, ed and w0, …, wd

input: e0, …, ed computation step:

choose non-deterministically one rule from some Ri execute the rule

sequential model

Classical UREM P systems


(ini: a, e, b)

object a enters into membrane iit is changed to b

the energy ei of membrane i is changed to ei+erule is active if:

o a is in the region surrounding membrane i

o ei+e 0

i

ei

a b

ei + e

Classical UREM P systems: rules


b a

(outi: a, e, b)

object a exits from membrane iit is changed to b

the energy ei of membrane i is changed to ei+erule is active if:

o a is in the region delimited by membrane i

o ei+e 0

i

eiei + e



in classical UREM P systems, we use some sort of local priorities:

if two or more rules are active in membrane i, then one of those with max |e| is applied

computation steps induce transitions between configurationscomputation: a sequence of transitionsthe computation halts when no rule can be applied

input: expressed in e0, …, ed

output: distribution of energy among membranes



this model computationally complete? [Freund et al., 2004]: every partial recursive function f:

can be computed, using at most max{, }+3 membranessketch of the proof:

simulation of register machines

let M = (n, P, 1, m ) be the register machine to be simulated

let (x1, x2, …, x ) be the input vector

the system contains a single object, which stores the label of the current instruction

Classical UREM P systems:completeness


membrane structure:

instruction j : (INC(i), k) is simulated in two steps:

)~,1,:( jji ppin

),0,~:( kji ppout

i

eipj jp~

ei+1jp~

pk

1

x1

x

+1

0

n

0

p1

1

x1

x

+1

0

n

0

p1



instruction j : (DEC(i), k, l) is also simulated in two steps:

when object pm appears, the computation halts

pj

)~,0,:( jji ppin

),1,~:( kji ppout

),0,~:( lji ppout

pj

ei > 0

ei = 0i

i

ei

0

jp~pk

ei 1

jp~

jp~ jp~pl



if we use local priorities, thensimulation is performed correctlyUREM P systems are computationally complete

if we don’t use priorities, thenUREM P systems are not complete

they characterize the family PsMAT of

Parikh sets generated by context-free matrix grammars (with -rules)



a quantum version of classical UREM P systems

objects of A: classical states of a quantum system

if |A| = d 2, we put:

the system can also be in a superpositionmultiset: collection of quantum systems, each in its own stateevery membrane is associated with an infinite quantum harmonic oscillator

1,,

1

2,

1

1,0:

ddLaaA d

Quantum UREM P systems


each oscillator (membrane) lives in a different Hilbert spacethe global state is described in the tensor product of such Hilbert spacesto modify the state of the oscillator we use (infinite dimensional version of) a+ and a

a+: switch to next energy levela : switch to previous energy level

rules are defined as (n, d)-functions f: An An

such functions are not necessarily bijections the corresponding operators are not necessarily unitary



to write the operators, we use an extension of the Conditional Quantum Control technique [BDEJ, 1995]

the operators are sums of local operatorseach local operator is a tensor product of a suitable composition of a+ and a

we realize a controlled behavior by operators

, with X Ld

example: (2, 2)-functionfirst and second qubit: control and target, respectively

if control = |1 then apply O1 to the target

if control = |0 then apply O0 to the target

XXE XX ,

Controlled quantum behavior


the control qubit is left unchangedrules realized by the following operator:

we can generalize to (n, d)-functions, with (the first) k control qudits and nk target qudits

if control qudits are in base states then apply OX : nk d nk d to target qudits

the global operator which describes the function (that is, the rule) has the form:

1011,100,0 1100 OOOEOE

kxxX ,,1

110

1

0

111100

k

k

d

kkX

d

X

OddOOOXX

Controlled quantum behavior


as in the classical case, rules are associated to membranes

each rule of Ri is an operator of the form:

|yx| O, with x, y Ld

where O can be expressed by an appropriate composition of a+ and athe rule is active iff a qudit in state |x occurs outside membrane i

the state of the system is changed to |yoperator O is applied to the quantum harmonic oscillator

guard



if the guard is not satisfied, the null vector is obtained

also O can produce the null operator (if ei +e < 0)

membrane structure: defined as in the classical case

if membrane i has two or more rules, they are summed only one rule for each membrane, with many guards no need to use local priorities

the result of a computation step on an object is a superposition of the results of each active guard



if |x is in a region with two or more membranes, then all their rules are applied in parallelthe object which activates a rule never crosses the membrane objects never move we don’t need “magic” transportation systemsthe application of rules determines transitions between configurationscomputation: sequence of transitionsa computation halts when it reaches a halting configuration (no rule can be applied)

result: distribution of energies in the halting configuration



a non-halting computation produces no resultinput: distribution of energies in the initial configuration we can compute partial functions

theorem: every partial recursive function f:

can be computed, using at most max{, }+3 membranessketch of the proof:

simulation of register machines

let M = (n, P, 1, m ) be the register machine to be simulatedlet (x1, x2, …, x ) be the input vector

Quantum UREM P systems:completeness



input values are in the first registers at the beginning of the computation

output values are in registers 1 to at the end of computation (if it halts)

the system contains a single object (in region 0), which stores the label of the current instructionthe quantum UREM P system has the form:


objects:

input: | x1, | x2, … , | x | x+1 = … = | xn = 0 (the null vector)

rule of membrane i:

where the Oij’s are local operators which simulate instructions j : (INC(i), k) and j : (DEC(i), k, l) that affect register i rules depend upon the set P of instructions

m

jii j

OR1

mLjjA :



instructions j : (INC(i), k) are simulated by guarded rules:

if |pj occurs in region 0, then

it is transformed to |pkthe energy level of the harmonic oscillator contained in membrane i is incremented

instructions j : (DEC(i), k, l) are simulated by guarded rules:

ijk Rapp

ijkjl Rapppp 00


Note: 1

1

m

kpk


if the oscillator is in state |0:

program counter = |pl new state of oscillator: |0

00

0

000

0

lkl

jjk

jjl

j

ppp

papp

ppp

pO

0



if the oscillator is in state | x, with x > 0:

program counter = |pk new state of oscillator: | x-1

simulation of m: Halt:do nothing (null operator) when |pm appears

1

00

xkxkl

xjjk

xjjl

xj

papp

papp

ppp

pO

0



it is clear that we can simulate classical register machinesquantum UREM P systems can also simulate quantum register machines, hence:

main difference between quantum UREM P systems and quantum register machines:

local vs. global operators

deterministic register machines

quantum register machines

quantum UREM P systems

Computational power


The 3-SAT problem

variables X = {x1, x2, …, xn}

literal Xi: directed (xi) or negated (xi) variable

3-clause: disjunction of exactly three literals

assignment: function a : X {false, true}

The 3-SAT problem:instance: set C = {C1,C2,…,Cm} of 3-clauses built on X

(equiv., = C1 C2 … Cm)

question: is there an assignment which satisfies all the clauses? (equiv., which satisfies )

Note: m (2n) 3 = 8n3


Solving 3-SAT

semi-uniform solutionwith quantum register machineswith quantum UREM P systems

brute force approachthe constructions uses non-unitary operatorscrucial assumption: the external observer is able to:

apply a projector E1,1 = |11|

discriminate between a null and a non-null vector


Solving 3-SAT with QRMs

let n be an instance of 3-SAT (built on n variables)

we first use a classical register machine with n+1 registers (one for each variable + one for the result)structure of the program:



if we obtain:

3,2,1, iiii XXXC

If the variable occurs

negated, then thecomparison is madewith 0



each instruction:

is translated as:

each instruction:

is translated as:



the translation from n to program P is “mechanical”

program P evaluates n for a given assignment

we can initialize a quantum register machine with:

then we apply operator OP for 23m+2 = 6m+2 times

operator OP now contains also the term:

0102

1

0011

11

2

1001

n

nn H



if n is not satisfiable, then the contents of the

output register is |0, which is transformed to the null vector

if n is satisfiable, then the output register contains 0|0 + 1|1, with 1 0; by applying OP we obtain:



Conclusion:

We are able to solve 3-SAT if it is possible to:

apply the operator to the output registerdiscriminate between a null and a non-null vector

aaNE 1,111


Solving 3-SAT with quantum

UREM P systems

structure of the system: like the one for Turing completeness, but with n+1 subsystems

(Note: Hilbert spaces are not “structured”)

input values: energies |x1, …, |xn , |0 assigned

to harmonic oscillators

alphabet A: possible values of the program counter of the QRM

only one object in the system, initially |p1 = |0

we could build P and simulate the register machine, but we can simplify



UREM P systems

given the instruction:

instead of simulating it with:

we can simplify it to:

= i, j

)1(,, kjiO



UREM P systems

similarly, instruction:

can be simulated with:

value |pk+1 is no longer used, hence we can

compact A and obtain:



UREM P systems

the “goto end” instructions can be transformed to ifs: for example, given:

we obtain:

that is, added to membrane 2,3

the last instruction ( = 1) is translated to

added to membrane n+1

)0(8,3,2

)1(7,3,2 OO and



UREM P systems

each Ri is obtained by summing the operators that

affect xi

we initialize the first n harmonic oscillators with a

superposition of all classical assignments

when the object in the system reaches the state |pend>, we apply the operator:

to extract the result (just like with QRMs)

same assumptions made with QRMs


precisely assess the computational power of quantum UREM P systems and quantum register machines

solving EXP-complete problems ?

study the effect of superpositions

study the effect of entanglement

should measurements in P systems have local or global effects ?

simulation of a quantum Turing machine

implementation of quantum algorithms …

Directions for future research

Documents

Quantum (UREM) P Systems: Background, Definition and Computational Power Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi