quantum automata for infinite periodicwords6th International Conference on Information, Intelligence,Systems and Applications (IISA 2015)

Konstantinos Giannakis, Christos Papalitsas, and Theodore AndronikosJuly 6, 2015

Department of Informatics, Ionian University

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preview of our study

∙ Quantum Computation

∙ Infinite inputs in Quantum Systems

∙ Periodic Measurements

∙ Transition matrices

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introduction

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.

∙ A novel definition for quantum computation with infinite horizon.

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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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explanation forget about schrödinger’s cat!

(a) Bank’srestrictions

(b)Observingpocket

(c) Observingcredit cards

∙ We are in a superposition of having money and not havingmoney!

∙ We have to “measure” the person to know the exact state!∙ Different “ways” to measure (we call them observables).∙ System collapses to the measured “state”!

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automata and computation (standard and quantum)

∙ Finite automata ⇒ simple models of computation.

∙ Finite quantum automata∙ A quantum system where each symbol represents the applicationof a unitary transformation.

∙ Proposed after the middle of the 1990s.∙ They can be seen as a generalization of probabilistic finiteautomata.

∙ Transitions are weighted with a probability amplitude ⇒ vectors ina Hilbert space.

∙ Probability semantics under which automata accept or reject.

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quantum automata variations

∙ Measure-many approach

∙ Measure-once approach

∙ There are regular languages not recognized by a quantumautomaton.

∙ We have to blame the reversibility of the quantum system!

∙ But they are space-efficient.

∙ 2-way variants are more powerful.

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ω-computability receiving infinite inputs

∙ ω-automata.∙ Infinite input∙ Acceptance conditions

∙ E.g. Büchi automata.∙ Büchi acceptance condition.∙ They accept the runs ρ for which In(ρ) ∩ F = ∅ (F ⊆ Q).

∙ Extension of the NFA with infinite inputs∙ The acceptance condition Acc declares how the infinite runs areaccepted by the automaton.

∙ The class of languages recognized from (almost) all the abovemachines are the ω-regular languages.

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main part

terminology needed for clarification

∙ Σ ⇒ the alphabet∙ Σ∗ ⇒ the set of all finite strings over Σ∙ Σω ⇒ the set of all infinite strings∙ If U is a n× n square matrix , U is its conjugate, and U† itstranspose and conjugate.

∙ Cn×n defines the set of all n× n complex matrices.∙ A unitary operator (or matrix) U is an orthogonal matrix withcomplex entries that preserves the norms of vectors.

∙ Equivalently, a matrix U is unitary if it has an inverse and if ∥Uψ∥= ∥ψ∥ for every vector ψ.

∙ Hn is an n-dimensional Hilbert space.∙ Bras ⟨ψ| and Kets |ψ⟩ in Dirac formalism for each state ψ.

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quantum computation states and formalism

∙ Two types of quantum states: pure and mixed states.∙ A pure state is a state represented by a single ket vector |ψ⟩ in aHilbert space over complex numbers.

∙ A mixed state is a statistical distribution of pure states (usuallydescribed with density matrices).

∙ 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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towards quantum automata using dirac formalism

∙ Each state of the machine is a superposition of the basis states|ψi⟩.

∙ They have the form |ψ⟩=c1 |ψ2⟩+ c2 |ψ2⟩+ · · ·+ cn |ψn⟩,

∙ The probability of observing the state|ψ′⟩=c1 |ψ′

2⟩+ c2 |ψ′2⟩+ · · ·+ cn |ψ′

n⟩ is p, with p =∑ψ′∈F

|ci|2 (F is the

set of accepting states).∙ In a MO-automaton the projection matrix P is applied strictlyonce.

∙ In MM-automata, there are three disjoint sets of states: the Qa

(accepting states), the Qr (rejecting states) and the Qn of neutralstates.

∙ Measurement after reading each symbol.

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towards quantum ω-automata definition

◦ A simple periodic, one-way quantum ω-automaton is a tuple (Q,Σ, Uδ , q0, π0, F, P, Acc) where:

1. Q is a finite set of states,2. Σ is the input alphabet,3. Ua is the n× n unitary matrix that describes the transitions among

the states for each symbol a ∈ Σ,4. q0 ∈ Q is the initial (pure) state,5. π0 is the initial vector,6. F ∈ Q is the set of final states,7. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and8. Acc is an acceptance condition.

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their functionality explanation

∙ It starts at its initial pure state q0, i.e. the state vector of thesystem is the π0.

∙ Transitions among the states are expressed with complexamplitude.

∙ Acc defines the acceptance condition.

Periodic quantum acceptance condition

It defines that infinitely often the measurement of the quantum systemfinds with some probability the automaton in one of the final states.

Almost-sure periodic quantum acceptance condition

It defines that infinitely often the measurement of the quantum systemfinds the automaton in one of the final states with probability 1.

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periodic quantum automaton

∙ A simple m-periodic, 1-way quantum ω-automaton with periodicmeasurements is a tuple (Q, Σ, Uδ , q0, m, π0, F, P, Acc) where:

1. Q is a finite set of states,2. Σ is the input alphabet,3. Uα : Q × Σ −→ C[0,1] is the n× n unitary matrix that describes the

transitions among the states for each symbol a ∈ Σ,4. q0 ∈ Q is the (pure) initial state,5. m ∈ N defines the measurement period,6. π0 is the vector of the initial pure state q0,7. F ∈ Q is the set of final states,8. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and9. Acc is the almost-sure periodic quantum acceptance condition.

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transitions on quantum periodic automata

∙ The transition matrix for every symbol has the form of: Uϕ = i(cos(ϕ) sin(ϕ)− sin(ϕ) cos(ϕ)

)∙ ϕ defines the period (if m is the period of the transition, thenϕ = π/m).

∙ Counter-clockwise rotation.∙ We can reverse the rotation by transposing the Uϕ.

∙ Then we have UTϕ = i

(cos(ϕ) − sin(ϕ)sin(ϕ) cos(ϕ)

).

∙ Both return the system to its initial state after the same period.

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quantum periodic automata periodicity

∙ After m applications of the transition matrix U, the state of thesystem is Um |ψ⟩, where |ψ⟩ is the state of the system before them transitions.

∙ But Um=im(−1 00 −1

)since Um=im

(cos(mϕ) sin(mϕ)− sin(mϕ) cos(mϕ)

)and

ϕ = π/m.

∙ In 2m timesteps we obtain the U2m=(1 00 1

)∙ It is the same!∙ Their difference is a phase of π, since ϕ = 2mπ/m = 2π.

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examples

q0 q1

i sin(ϕ)

i cos(ϕ)

i cos(ϕ)

−i sin(ϕ)

Figure: A general form for a 2-state quantum automaton.

q0 q1

a, i√2

a, i√2 b,−i

a, i√2 b,−i

a,−i√2

Figure: For period m = 5, it accepts the ω-language (a4b)ω . Note thati cos(π/(m− 1)) = i sin(π/(m− 1)) = i

√2 for the symbol a, and

i cos(π/1) = −i for the symbol b.

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transitions of the state vector

0

π/4

π/2

3π/4

π

5π/4 7π/4

3π/2

r = 1

ϕϕ

ϕ ϕ

Figure: The vector is in the initial state and for every phase transition withangle ϕ = π/4 it is rotated counter-clockwise. After m− 1 (=4) transitionsthe system is in the state that is symmetric to the initial.

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simulation results for the first 30 transitions

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

transition #

prob

ability

State 1State 2

Figure: This simulation corresponds to the periodic quantum automatonwith period m = 4.

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concluding

future work and discussion

∙ Quantum automata with infinite computation are stillunexplored.

∙ Different variants of machines, distinguished either bymovement orientation or by the measurement mode.

∙ Need for models and verification processes for infinite QC.∙ Useful in the verification of quantum systems and the design ofquantum circuits.

∙ Space efficient for periodic ω-languages of the form (amb)ω .∙ Consistency with the underlying quantum physics.

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key references

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

Hirvensalo, M.Quantum automata theory–a review.In Algebraic Foundations in Computer Science. Springer, 2011, pp. 146–167.

Kondacs, A., and Watrous, J.On the power of quantum finite state automata.In Foundations of Computer Science, 1997. Proceedings., 38th Annual Symposium on (1997),IEEE, pp. 66–75.

Mereghetti, C., and Palano, B.Quantum Automata and Periodic Events.ch. 11, pp. 563–584.

Moore, C., and Crutchfield, J. P.Quantum automata and quantum grammars.Theoretical Computer Science 237, 1 (2000), 275–306.

Thomas, W.Automata on infinite objects.In Handbook of Theoretical Computer Science, vol. B: Formal Models and Semantics. ElsevierScience, 1990, pp. 133–192.

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Any Questions?

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