Bevezetés a kvantum-informatikába és kommunikációba 2015/2016 tavasz Kvantumkapuk, áramkörök 2016. március 3.
A kvantummechanika posztulátumai (1-2) 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.
A kvantummechanika posztulátumai (3) 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
A kvantummechanika posztulátumai (4) 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
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.
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
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:
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.
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
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
Generalised interferometer Copyright 2005 John Wiley & Sons Ltd.
Abstract quantum circuit of the generalised interferometer Copyright 2005 John Wiley & Sons Ltd.
Analysis 1 Copyright 2005 John Wiley & Sons Ltd. 0
Analysis 2 Copyright 2005 John Wiley & Sons Ltd.
Analysis 3 Copyright 2005 John Wiley & Sons Ltd.
Analysis 4 Copyright 2005 John Wiley & Sons Ltd.
Analysis 5
Analysis 6 : idealistic scenario : fully random operation
Entanglement "Wonder is from surprise, and surprise stops with experience." Bishop Robert South
Tensor product Applying the 4th postulate
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
The CNOT gate Copyright 2005 John Wiley & Sons Ltd. Upper wire: control Lower wire: data
The CNOT gate Truth table Matrix Master equation
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!
Copyright 2005 John Wiley & Sons Ltd. The SWAP gate 1
Copyright 2005 John Wiley & Sons Ltd. The SWAP gate 2
Copyright 2005 John Wiley & Sons Ltd. The SWAP gate 3
Copyright 2005 John Wiley & Sons Ltd. The SWAP gate 4
EPR paradox