Emulation of quantum Turing machines Paulo Mateus SQIG - Instituto - PowerPoint PPT Presentation

Emulation of quantum Turing machines Paulo Mateus SQIG - Instituto de Telecomunicações DM -IST - U. Lisboa Joint work with A. Sernadas and A. Souto Friday 24 January 14

Context • Quantum automata • Open problems concerning QA (and other automata) and their importance • Category of bilinear automata • How Category Theory and (computational) Algebraic Theory of the ROF helped solving the OP • Quantum Turing machines as morphisms • Towards quantum Kolmogorov theory Friday 24 January 14

Quantum automata A quantum automaton is a tuple Q = h Σ , H, s i , U, O, ρ i where • Σ is a finite set of inputs, • H is a finite Hilbert space of states, • s i is a unitary vector in H denoting the initial state, • U is a Σ -indexed family { U σ } σ 2 Σ of unitary transformations in H , • O is a Hilbert space of outputs and P O : H ! O is a projection (there is a subspace H 0 of H isomorphic to O ). Friday 24 January 14

Quantum automata • A stochastic language over Σ is a map β : Σ ⇤ ! [0 , 1]. • The quantum behaviour of a quantum automaton Q is the map β Q : Σ ⇤ ! O where β Q ( ω ) = P O U ω s i with U ω = U σ k . . . U σ 1 and ω = σ 1 . . . σ k . • The stochastic behaviour of a quantum automaton Q is the stochastic language β Q : Σ ⇤ ! [0 , 1] where β Q ( ω ) = | P O U ω s i | 2 . Friday 24 January 14

Motivation • In practice quantum automata are the implementable quantum gadgets; • They are currently used to implement quantum protocols and quantum machines – A large spectrum of such gadgets is used to implement perfectly secure communications – There is already a large quantum computer • Engineering bottleneck: High dimensional quantum automata are hard to implement Friday 24 January 14

Open problems • How to obtain the minimal dimensional QA that behaves the same as a given one? [Moore and Crutchfield TCS 2000] • (How to find the minimal cover of a stochastic Mealy machines: Paz 1971) • Is it even decidable? • If so, what is the complexity. Friday 24 January 14

Categorical context Recall that C - Lin is a weak symmetric monoidal category furnished with N C as the monoidal operator and C as unit. A bilinear automaton over a finite alphabet Σ is a tuple A = h Q, δ, Γ , γ, I, λ i where: • Q 2 C - Lin (state object); • Γ 2 C - Lin (output object); • I 2 C - Lin (initialization object); N Q ) ! Q 2 C - Lin (next-state morphism); • δ : ( h Σ i C • γ : Q ! Γ 2 C - Lin (output morphism); • λ : I ! Q 2 C - Lin (initialization morphism). where h Σ i C denotes the C - linear space generated by Σ. Friday 24 January 14

Categorical context Remark 4.1 Since we have a natural bijection Q, Q ) ⇠ O hom C ( h Σ i C = hom C ( h Σ i C , hom C ( Q, Q )) , C N Q ) ! Q is the same as giving a morphism giving δ : ( h Σ i C δ ] : h Σ i C ! hom C ( Q, Q ) , that is uniquely defined by a finite family of morphisms { δ � : Q ! Q } � 2 Σ . Friday 24 January 14

� ? ? ✏ � ✏ Categorical context A morphism between two bilinear automata A = h Q, δ , Γ , γ , I, λ i and A 0 = h Q 0 , δ 0 , Γ , γ 0 , I, λ 0 i is a C - Lin morphism f : Q ! Q 0 such that the following diagram commutes � / Q h Σ i C N C Q � � C Γ id h Σ i C N C f f � 0 � 0 / Q 0 C Q 0 h Σ i C N � 0 Friday 24 January 14

? ✏ ? � � ✏ Categorical context N Or equivalently, such that the Σ -indexed family of commutative diagrams � σ / Q Q � � C Γ f f � 0 � 0 / Q 0 Q 0 � 0 σ We shall denote the resulting category of bilinear automata by BAut Γ C . N C Friday 24 January 14

o Categorical context The free ( h Σ i C N )-algebra generated by C is C ' ⌘ / h Σ i ⌦ C h Σ i ⌦ h Σ i C N C C C C h Σ i C ) L ... . where h Σ i ⌦ C = C L h Σ i C L ( h Σ i C N C ⇠ Observe that h Σ i ⌦ = h Σ ⇤ i C . Friday 24 January 14

~ ✏ / o ✏ Categorical context Given a bilinear automata A , the run map is the unique morphism ρ such that the following diagram commutes. ϕ η C h Σ i ⌦ h Σ i ⌦ h Σ i C N C C C N ρ id h Σ i C C ρ λ / Q h Σ i C N C Q δ If ρ is an epi, we say that A is reachable . We call β = γ � ρ : h Σ ⇤ i C ! Γ the behaviour of A . We denote the category of bilinear behaviours by Beh Γ C , which has only triv- ial morphisms, since automata connected by a morphism must have the same behaviour. Friday 24 January 14

Categorical context A quantum automaton is a bilinear automaton with initialization object C such that: • δ σ : Q ! Q is unitary for all σ 2 Σ with complete hermitean inner product for Q ; • γ is an orthogonal projection onto a subspace Γ 0 ✓ Q followed by an isomorphism to Γ (that is, Γ is a subobject of Q ); • λ is injective (or more generally any linear map, if we wish to include automata with trivially null behaviour) Friday 24 January 14

Categorical context We denote by QAut Γ C the full subcategory of BAut Γ C constituted by quantum automata. Similarly, we denote by QBeh Γ C the full subcategory of Beh Γ C with quantum behaviours. Friday 24 January 14

9 d o Categorical context Theorem For any behaviour β : h Σ i ⊗ C ! Γ there is a minimal realization for β and with initialization object C . C C - Lin Out Out’ B / Beh C BAut C ⊥ Min the functor that maps each behaviour to its min C ! Γ be a behaviour in QBeh Γ Theorem Let β : h Σ i ⊗ C . Then there exists a minimal realization in QAut Γ C for β . Friday 24 January 14

Computational algebra Theorem [Tarski, Renegar] Let P ( x ) be a predicate which is a Boolean function of atomic predicates either of the form f i ( x ) ≥ 0 or f j ( x ) > 0, with f 0 s being real polynomials. There is an algorithm to decide whether the set S = { x ∈ R n : P ( x ) } is nonempty in PSPACE in n, m, d , where n is the number of variables, m is the number of atomic predicates, and d is the highest degree among all atomic predicates of P ( x ). Moreover, there is an algorithm of time complexity ( md ) O ( n ) for this problem. To find a sample of S requires τ d O ( n ) space if all coe ffi cients of the atomic predicates use at most τ space. Friday 24 January 14

Computational algebra Theorem: Quantum automata (and SMM, QMM, etc...) can be minimized in EXPSPACE 13] P. Mateus, D. Qiu, and L. Li. On the complexity of minimizing probabilistic and quantum automata. Information and Computation , 218:36–53, 2012. 1. Firstly, for a given automaton A of some type (say probabilistic, quantum, etc.) with n states, we define the set = {A 0 : A 0 has n 0 states , is of the same type of A , and is equivalent to A} . S ( n 0 ) A 2. Next, we show that S ( n 0 ) can be described as the solution of a system of polynomial A equations and/or inequations if the automata can be bilinearized . Then there exists an algorithm to decide whether S ( n 0 ) is nonempty or not, and furthermore, if it A is nonempty, we can find a sample of it. Friday 24 January 14

Computational algebra Input: an automaton A with n states 0 , of the same type of A , and equivalent to A Output: a minimal automaton A Step 1: For i = 1 to n − 1 A is not empty) Return A 0 = sample S ( i ) If ( S ( i ) A Step 2: Return A 0 = A Friday 24 January 14

Applications 17] N. Paunkovic, J. Bouda, and P. Mateus. Fair and optimistic quantum contract signing. Physical Review A , 84(6):062331, 2011. 10] F. Assis, A. Stojanovic, P. Mateus, and Y. Omar. Improving classical authentication over a quantum channel. Entropy , 14(12):2531–2549, 2012. 7] L. Li, D. Qiu, and P. Mateus. Quantum secret sharing with classical Bobs. Journal of Physics A: Mathematical and Theoretical , 46(4):045304, 2013. Friday 24 January 14

Quantum Turing Machine • By a quantum Turing machine we mean a binary Turing machine with two tapes, one classical and the other with quantum contents, which are infinite in both directions. • Depending only on the state of the classical finite control automaton and the symbol being read by the classical head, the quantum head acts upon the quantum tape, a symbol can be written by the classical head, both heads can be moved independently of each other and the state of the control automaton can be changed. • A computation ends if and when the control automaton reaches the halting state ( q h ). Friday 24 January 14

Quantum Turing Machine Initially: • the QTM is in the starting state ( q s ); • the classical tape is filled with blanks (that is, with ⇤ ’s) outside the finite input sequence x of bits, • the classical head is positioned over the rightmost blank before the input bits, • the quantum tape contains three independent sequences of qubits – an infinite sequence of | 0 i ’s followed by the finite input sequence | ψ i of possibly entangled qubits followed by an infinite sequence of | 0 i ’s, • the quantum head is positioned over the rightmost | 0 i before the input qubits. Friday 24 January 14

Quantum Turing Machine The QTM is a partial map � : Q ⇥ A * U ⇥ D ⇥ A ⇥ D ⇥ Q where: • Q is the finite set of control states containing at least the two states q s and q h mentioned above; • A is the alphabet composed of 0, 1 and ⇤ ; • U is the set { Id , H , S , ⇡ / 8 , Sw , c-Not } of primitive unitary operators that can be applied to the quantum tape; and • D is the set { L , N , R } of possible head displacements – one position to the left, none, and one position to the right. Friday 24 January 14