Bevezetés a kvantum-informatikába és kommunikációba 2014/2015 tavasz

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1 Bevezetés a kvantum-informatikába és kommunikációba 2014/2015 tavasz Kvantumkapuk, áramkörök február 26.

3 A kvantummechanika posztulátumai (1) 1. Állapotleírás Zárt fizikai rendszer aktuális állapota egy olyan állapotvektorral írható le, amely komplex együtthatókkal rendelkezik, egységnyi hosszú a H Hilbert-térben (egy komplex lineáris vektortérben,amelyben értelmezve van a belső szorzat). 2. A rendszer időbeli fejlődése A zárt rendszer időbeli fejlődése unitér transzformációval írható le, amely csak a kezdő és végállapottól függ.

4 A kvantummechanika posztulátumai (2) 3. A mérés Legyen X a mérés lehetséges eredményeinek a halmaza. Egy mérés a mérési operátorok halmazával adható meg: Μ Μ, x X, Μ x Ha a megmérendő rendszer állapota, akkor annak a valószínűsége, hogy a mérés az x eredményt adja: P A mérés után a rendszer állapota az alábbi lesz x T X M x M x Μ x p x

5 A kvantummechanika posztulátumai (3) 4. Összetett rendszer Ha V és Y a két kvantumrendszerhez rendelt Hilbert-tér, akkor az ebből a két rendszerből álló összetett rendszerhez a V YHilbert-tér rendelhető. W

6 1 st Postulate (state space) The actual state of any closed physical system can be described by means of a so called state vector v having complex coefficients and unit length in a Hilbert space V, i.e. a complex linear vector space (state space) equipped with inner product.

7 2 nd Postulate (evolution) The evolution of any closed physical system in time can be characterized by means of unitary transforms depending only on the starting and finishing time of the evolution. 2nd Postulate can be interpreted as v (t 2 ) = U(t 1, t 2 )v(t 2 ) and v V. The above definition describes the evolution between discrete time instants, which is more suitable in context of quantum computing. Its original continuous-time form is known as Schrödinger equation Relationship between H and U

8 3 rd Postulate (measurement) Any quantum measurement can be described bymeans of a set of measurement operators {M m }, where m stands for the possible results of the measurement. The probability of measuring m if the system is in state v can be calculated as and the system after measuring m gets in state Because classical probability theory requires that Completeness relation:

9 4 th Postulate (composite systems) The state space of a composite physical system W can be determined using the tensor product of the individual systems W = V Y. Furthermore having defined v V and y Y then the joint state of the composite system is w = v y.

10 Slides for Quantum Computing and Communications An Engineering Approach Chapter 2 Quantum Computing Basics Sándor Imre Ferenc Balázs

11 Elementary Quantum Gates "Excellent!" I cried, "Elementary" said he. Watson and Holmes, in "The Crooked Man", The Memoirs of Sherlock Holmes, Sir Arthur Conan Doyle

12 Pauli gates Input: Pauli X (bit-flip) gate: Pauli Z (phase-flip) gate:

13 Pauli gates Pauli Y (???-flip) gate: Geometrical interpretation of Pauli X gate: rotation around axis x in the Bloch sphere

14 Phase gate:

15 Hadamard gate Hadamard gate is Hermitian i.e., furthermore: It is worth memorising:

16 Hadamard gate n-qbit Hadamard gate whit input n-qbit Hadamard gate with arbitrary classical input

17 Hadamard gate and the superposition principle

18 General Description of the Interferometer "An idea is always a generalization, and generalization is a property of thinking. To generalize means to think." Georg Hegel

25 Analysis 5

26 Analysis 6 : idealistic scenario : fully random operation

27 Entanglement "Wonder is from surprise, and surprise stops with experience." Bishop Robert South

28 A surprising quantum state Based on the 4 th Postulate, decompose the following two-qubit state a 00 b 11?? No such decomposition exists! Two types of quantum states product entangled

30 The CNOT gate Truth table Matrix Master equation

31 The CNOT gate as classical copy machine Provided the data input is initialized permanently with then the CNOT gate emits a copy of the control input on each output! Let s try to copy the following state! The input joint state is Using the superposition principle the output becomes which is nothing else then an entangled pair!

37 Bell states Bell states are orthogonal!!!

39 Remarks Only one of the entangled qbits is enough to entangle another qbit to the previous set of qbits. Entanglement cannot be produced using only classical communication!

40 Decoherence Entanglement with the environment The first two postulates are valid only for closed systems therefore entanglement with the environment can be very dangerous! In order to demonstrate it we use the well-known quantum interferometer and assume that the photon traveling trough the interferometer is entangled with a butterfly outside the laboratory (i.e. the environment). The environment is represented by The flying rule of butterfly

42 Decoherence and the interferometer Before butterfly s action After butterfly s action Now, the Hadamard gate comes

43 Decoherence and the interferometer Attention! and are not necessarily orthogonal Assuming is real

44 Decoherence and the interferometer = If then the entanglement disappears. However, if then the operation becomes fully random instead of being deterministic while the observer becomes miss leaded.

45 Schrödinger s cat "When I hear about Schrödinger s cat, I reach for my gun." Stephen Hawking Source:

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