Q UANTUM C OMPUTATION AND I NFORMATION S EMINAR S ECURITY AND Q UANTUM I NFORMATION G ROUP T ECHNICAL U NIVERSITY OF L ISBON 9 October 2009 A mathematical view of quantum computation J UANJO R UÉ AND S EBASTIAN X AMBÓ F ACULTAT DE M ATEMÀTIQUES I E STADÍSTICA U NIVERSITAT P OLITÈCNICA DE C ATALUNYA 08028 B ARCELONA (S PAIN )
2 J OINT WORK WITH J UANJO R UÉ Quantum Computation: Foundations and State of the Art (Master Thesis UPC)
3 M AIN P OINTS Introduction Quantum computation in mathematical terms q ‐ computation and q ‐ algorithms q ‐ computer and q ‐ programs Comments about some q ‐ algorithms Quantum computation in physical terms Main characteristics of quantum phenomena States and observables. Example: q ‐ bits (or qubits) Quantum computer. Ending remarks
4 A BSTRACT A mathematical model of a quantum computer, or q ‐ computer, will be presented, together with related concepts such as q ‐ gates, q ‐ computations and q ‐ algorithms/programs. Emphasis will be given to examples, such as the q ‐ Fourier transform and q ‐ algorithm of Shor to factor integers in poly ‐ nomial time. The possible physical realizations of the model will be analyzed using an axiomatic version of quantum me ‐ chanics. At the end, a few lines for future work will be men ‐ tioned.
5 I NTRODUCTION C OMPUTACIÓN Level C LASSICAL Q UANTUM Mathematical Mathematical logic Linear algebra Vectors Turing machine Boole algebra (Shannon) Matrices von Neumann machines Algorithmic theory Parallel computing Physical theory Mechanics Quantum Mechanics Electromagnetism (basic axioms) Technology Circuits, transistors, … Ionic traps, … Economics Ubiquity of processors Future computers mobile phones digital cameras, …
6 Q UANTUM COMPUTATION IN MATHEMATICAL TERMS Notations � positive integer (number of bits or q ‐ bits ) � positive integer in the range 0 ∙∙ 2 � � 1 If � � � � �� � � , � ��� � ��� � � � � � binary expression of � we write � � � � � �� ( � � � � � � � 2 � � � � ��� 2 ��� ) ( conjugate of � ) � ��� space of � ‐ vectors of order � : � � ∑ � � � � � ∑ � � |�� , � � � � � � These are complex vectors of 2 � components: � � 1 0 0 � � 0 1 0 � , |2 � � 1� � � � ; |0� � � � , |1� � � � � � � � � � � � � � �� 0 0 1 If � � ∑ � |�� is another � ‐ vector, and � � � , � � � � � � ∑ �� � � � |�� , �� � ∑ �� � |�� , ��|�� � ∑ � � � � � � . � � � � �� (we say that |0� , |1� , …, |2 � � 1� is an orthonormal basis ) ��|�� � �
7 Example ( � � 1 ) � � � � |0� � � � |1� � �� � � � � � � � �1 0� � � � �0 1� . Example ( � � 2 ) � � � � |0� � � � |1� � � � |2� � � � |3� � � �� |00� � � �� |01� � � �� |10� � � �� |11� � � � �� 1 0 0 0 � � � �� 0 1 0 0 � � � � � � � � �� � � � � �� � � � � �� � � � � �� � � � � � �� 0 0 1 0 � � � �� 0 0 0 1 � � � � Proposition . |��� � |�� � |�� � |��|�� , where �� � � � � � �� � � � � � � � � � � � . � � � � � � � � In general, |� ��� � � � � � � � |� ��� � � � � |� � � � |� � � � ��� � � ��� � � � � ��� � |� ��� � � |� � �|� � �
8 Proof 1 |0� � |0� � �1 0� � �1 0 � � |00� 0� � � 0 0 0 |0� � |1� � �1 0� � �0 1 � � |01� 1� � � 0 0 0 |1� � |0� � �0 1� � �1 0 � � |10� 0� � � 1 0 0 |1� � |1� � �0 1� � �0 0 � � |11� 1� � � 0 1
9 q ‐ Computation If � � �� �� � is a matrix, its transpose is � � � �� �� � and its adjoint � � � �� � � � � . � �� � � � � A � ‐ computation of order n is a matrix � � �� �� � ���,��� � , � �� � � , such that �� � � � � � �� (that is, � is a unitary matrix of order 2 � � 1 : � � ��2 � � � � ��� ). If �, � � � ��� , �� � � ��� and � �� � � � . In other words, Composition. The composition of two q ‐ computations of order � is a q ‐ computation of order � ; and Reversibility. The inverse of a q ‐ computation of order � is a q ‐ computation of order � .
10 A q ‐ input for a q ‐ computation � is a vector � � � � such that ��|�� � 1 (unitary vector). Example: � ��� � �|0� � |1� � � � |2 � � 1��/√2 � The q ‐ output of a q ‐ computation � is the (unitary) vector � � �� . Examples ( � � 1 ) . A q ‐ computation of order 1 is a matrix � � � ��� , i.e., a matrix of the form � � � �� � � � � � � � � , � � � , � � , � � � � , � � � � � � � � � � � � 1 . �� � � � � �� � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � Note. It is easy to check that � �� � � � � � � � � is unitary. The claim is that �� � � � any unitary matrix of order 2 has this form.
11 Special cases: a) Pauli matrices � � �1 0 0 1� , � � � � � �0 1 1 0� , � � � � � �0 �� 0� , � � � � � �1 0 �1� � 0 � �|0� � |1� , �|1� � |0� � � ��� � � Note. The Pauli matrices are self ‐ adjoint: � � � � � � � � � � . b) Hadamard matrix �1� �|0� � �|0� � |1��/√2 √� �1 1 � � � � 1 |1� � �|0� � |1��/√2 c) Fase matrices � � � �1 � �� � � � ��/� �� ���/� 0 0 � ��/� � 0 0 In particular, � �/� � �1 0 �� and � �/� � �1 0 � ��/� � 0 0
12 Examples � � 2 Let � � � ��� . The we define � �� ��� � � ��� as follows: � �� ���|0�� �|0�� , � �� ���|1�� �|1��|�� . � � �� �� � �� � �� � �� � If , then 1 0 0 0 0 1 0 0 � �� ��� � � � 0 0 � �� � �� 0 0 � �� � �� In particular we set � �� � � �� ��� : � �� |0�� �|0�� , � �� |1�� �|1�|1 � �� : 1 0 0 0 Leaves the second bit unchanged or negates 0 1 0 0 � �� � � � it according to whether the first bit is 0 or 1. 0 0 0 1 It is a conditional negation (C ONTROLED ‐ N OT ) 0 0 1 0
13 � �� ��� is defined in an analogous way. For example, 0 1 0 0 1 0 0 0 � �� � � �� ��� � � � 0 0 1 0 0 0 0 1 1 0 0 0 0 � �� 0 0 � ��,� � � �� �� � � � � � 0 0 1 0 0 0 0 1
14 q ‐ Computer A q ‐ computer of order � is a system that allows to perform the following operations: Initialization or Input � 1. ���� Selects the unitary vector � � � ��� . 2. � �,� , � � � ��� Action � on the j ‐ th bit � � � � � |� � � � � � �|� � � � | � � 3. � �,� Negationof the k ‐ th bit if the j ‐ th bit is |1� (see examples � �,� and � �,� above for � � 2 ) | � 1 � � 0 � � � � | � 1 � � 1 � � � | � 1 � � 1 � � � � | � 1 � � 0 � � � Observation of � 4. ���� , � � � ��� unitary Returns � � 0 �� �2 � � 1� with probability |� � | � and resets as ��|��� .
15 A q ‐ algorithm is a sequence � � , … , � � � � ��� such that each � � is either of � �,� or of type � �,� , and we say that it performs the q ‐ computation � � � � � � � ( � is called the complexity 1 0 0 0 0 0 1 0 of the algorithm). S WAP [0,1] � � � 0 1 0 0 Example 0 0 0 1 Swap (trasposition of 2 bits) S WAP ��, �� � �,� , � �,� , � �,� Indeed, � � � � � � � |� � � � � � � � � �,� : |� � � � � � � � � � � � |� � � � � � � � � �,� : |� � � � � � � �� � � � � � � � � � � � � � � |� � � � � � � � � � � � |� � � � � � � � � �,� : |� � � � �
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