Aggregation Operations from First Example: . . . Quantum Computing - PowerPoint PPT Presentation

Fuzzy Logic and . . . Why Quantum . . . Quantum States: . . . A Natural Relation . . . Aggregation Operations from First Example: . . . Quantum Computing Second Example: . . . Third Example: Union . . . Fourth Example: . . . Lidiana Visintin 1 , Renata Hax Sander Reiser 1 Superposition Leads . . . Adriano Maron 1 , Vladik Kreinovich 2 Home Page 1 Universidade Federal de Pelotas, Brazil Title Page { lvisintin, reiser, akmaron } @inf.ufpel.edu.br ◭◭ ◮◮ 2 University of Texas at El Paso, USA ◭ ◮ vladik@utep.edu Page 1 of 17 Go Back Full Screen Close Quit

Fuzzy Logic and . . . Why Quantum . . . 1. Fuzzy Logic and Quantum Computing Quantum States: . . . • In the traditional Boolean (2-valued) logic, every state- A Natural Relation . . . ment is either true (1) or false (0). First Example: . . . Second Example: . . . • Both fuzzy logic and quantum computing extend the Third Example: Union . . . usual 2-valued logic, to handle uncertainty. Fourth Example: . . . • In both cases, we need to extend usual Boolean oper- Superposition Leads . . . ations (“and”, ”or”, “not”, etc.). Home Page • In fuzzy logic, many different extensions are possible; Title Page it is often not clear which extension to use. ◭◭ ◮◮ • Quantum logic provides a unique way of extending ◭ ◮ Boolean operations to more general case. Page 2 of 17 • In this talk, we describe a correspondence between fuzzy logic and quantum computing. Go Back Full Screen • We use this correspondence to describe corresponding fuzzy aggregation operations. Close Quit

Fuzzy Logic and . . . Why Quantum . . . 2. Why Quantum Computing Quantum States: . . . • The speed of all processes is limited by the speed of A Natural Relation . . . light c . First Example: . . . Second Example: . . . • To send a signal across a 30 cm laptop, we need at least Third Example: Union . . . 1 ns; this corresponds to only 1 Gflop. Fourth Example: . . . • If we want to make computers faster, we need to make Superposition Leads . . . processing elements smaller. Home Page • Already, each processing cell consists of a few dozen Title Page molecules. ◭◭ ◮◮ • If we decrease the size further, we get to the level of ◭ ◮ individual atoms and molecules. Page 3 of 17 • On this level, physics is different, it is quantum physics. Go Back • One of the properties of quantum physics is its proba- Full Screen bilistic nature (example: radioactive decay). Close Quit

Fuzzy Logic and . . . Why Quantum . . . 3. Why Quantum Computing (cont-d) Quantum States: . . . • At first glance, this interferes with our desire to make A Natural Relation . . . reproducible computations. First Example: . . . Second Example: . . . • However, scientists learned how to make lemonade out Third Example: Union . . . of this lemon. Fourth Example: . . . • First main discovery: Grover’s quantum search algo- Superposition Leads . . . rithm. Home Page • To search for an object in an unsorted array of n ele- Title Page ments, we need, in the worst case, at least n steps. ◭◭ ◮◮ • Reason: if we use fewer steps, we do not cover all the ◭ ◮ elements, and thus, we may miss the desired object. • In quantum physics, we can find an element in √ n Page 4 of 17 steps. Go Back • For a Terabyte database, we get a million times speedup. Full Screen • Main idea: we can use superposition of different searches. Close Quit

Fuzzy Logic and . . . Why Quantum . . . 4. Why Quantum Computing (cont-d) Quantum States: . . . • Another discovery: Shor’s cracking RSA coding. A Natural Relation . . . First Example: . . . • The RSA algorithm is behind most secure transactions. Second Example: . . . • A person selects two large prime numbers p 1 and p 2 , Third Example: Union . . . and advertises their product n = p 1 · p 2 . Fourth Example: . . . • By using this open code n , anyone can encode their Superposition Leads . . . message. Home Page • To decode this message, one needs to know the factors Title Page p 1 and p 2 . ◭◭ ◮◮ • Factoring a large integer is known to be a computa- ◭ ◮ tionally difficult problem. Page 5 of 17 • It turns out that with quantum computers, we can fac- Go Back tor fast and thus, read all encrypted messages. Full Screen • The situation is not so bad: there is also a quantum encryption which cannot be easily cracked. Close Quit

Fuzzy Logic and . . . Why Quantum . . . 5. Quantum States: Case of a Single Qubit Quantum States: . . . (= Qu antum Bit ) A Natural Relation . . . • A bit is a system which has two possible states 0 and 1. First Example: . . . Second Example: . . . • In quantum physics, in addition to � 0 | and � 1 | , we also Third Example: Union . . . have superpositions Fourth Example: . . . α 0 � 0 | + α 1 � 1 | . Superposition Leads . . . • For each state, as a result of measurement, we always Home Page get either 0 or 1. Title Page • The probability of observing 0 is equal to | α 0 | 2 , and ◭◭ ◮◮ the probability of observing 1 is equal to | α 1 | 2 . ◭ ◮ • The total probability should be equal to 1: Page 6 of 17 | α 0 | 2 + | α 1 | 2 = 1 . Go Back • In general, α i re complex numbers. Full Screen • (In quantum computing, only real values α i are used.) Close Quit

Fuzzy Logic and . . . Why Quantum . . . 6. A Natural Relation with Fuzzy Quantum States: . . . • Traditional probability theory describes objective prob- A Natural Relation . . . abilities – frequencies of different events. First Example: . . . Second Example: . . . • Fuzzy logic describes subjective opinions, what proba- Third Example: Union . . . bilists call subjective probabilities. Fourth Example: . . . • It is therefore reasonable to associate a fuzzy degree Superposition Leads . . . f ∈ [0 , 1] with subjective probability. Home Page • In a quantum state α 0 � 0 | + α 1 � 1 | , the probability of Title Page “true” is | α 1 | 2 . ◭◭ ◮◮ • If we identify this value with f , we get α 2 1 = f and ◭ ◮ α 2 0 = 1 − α 2 1 = 1 − f . • Thus, α 0 = √ 1 − f , α 1 = √ f , and the fuzzy degree f Page 7 of 17 Go Back is associated with a state Full Screen � � 1 − f � 0 | + f � 1 | . Close Quit

Fuzzy Logic and . . . Why Quantum . . . 7. Quantum States of Multi-Qubit Systems Quantum States: . . . • In classical physics, a 2-qubit system has 4 possible A Natural Relation . . . states: 00, 01, 10, and 11. First Example: . . . Second Example: . . . • In quantum physics, we can have a superposition: Third Example: Union . . . α 00 � 00 | + α 01 � 01 | + α 10 � 10 | + α 11 � 11 | . Fourth Example: . . . Superposition Leads . . . • Here, the probability of observing 00 is | α 00 | 2 , etc., so Home Page that | α 00 | 2 + | α 01 | 2 + | α 10 | 2 + | α 11 | 2 = 1 . Title Page ◭◭ ◮◮ • A system of two independent qubits ψ = α 0 � 0 | + α 1 � 1 | and ψ ′ = α ′ 0 � 0 | + α ′ ◭ ◮ 1 � 1 | is described by a tensor product ψ ⊗ ψ ′ = α 0 · α ′ Page 8 of 17 0 � 00 | + α 0 · α ′ 1 � 01 | + α 1 · α ′ 0 � 00 | + α 1 · α ′ 1 � 11 | . Go Back • A similar description holds for 3-, 4-, . . . , N -qubit sys- Full Screen tems. Close Quit

Fuzzy Logic and . . . Why Quantum . . . 8. Resulting Relation with Fuzzy Quantum States: . . . • We decided to associate a fuzzy degree f with a state A Natural Relation . . . First Example: . . . � � 1 − f � 0 | + f � 1 | . Second Example: . . . • A membership function f ( x ) is described by several Third Example: Union . . . fuzzy degrees f ( x 1 ), . . . , f ( x n ). Fourth Example: . . . Superposition Leads . . . • It is reasonable to assume that these degrees are, in Home Page some reasonable sense, independent. Title Page • Thus, we associate a membership function with a ten- sor product of the corresponding quantum states: ◭◭ ◮◮ �� � ◭ ◮ � ⊗ n 1 − f ( x i ) � 0 | + f ( x i ) � 1 | . i =1 Page 9 of 17 • For example, for n = 2, we get Go Back � � � � 1 − f ( x 1 ) · 1 − f ( x 2 ) � 00 | + 1 − f ( x 1 ) · f ( x 2 ) � 01 | + Full Screen � � � � f ( x 1 ) · 1 − f ( x 2 ) � 10 | + f ( x 1 ) · f ( x 2 ) � 11 | . Close Quit

Fuzzy Logic and . . . Why Quantum . . . 9. Quantum Operations and Transformations Quantum States: . . . • Quantum transformations should preserve superposi- A Natural Relation . . . tion, so they should be linear. First Example: . . . Second Example: . . . • Quantum transformations should preserve the require- Third Example: Union . . . ment that the total probability is 1. Fourth Example: . . . • Such transformations are called unitary . Superposition Leads . . . • The consequence is that all quantum transformations Home Page are reversible. Title Page • We cannot have a simple “and”-operation for which ◭◭ ◮◮ f (0 , 0) = f (0 , 1) = 0. ◭ ◮ • As a result, a quantum implementation of a function Page 10 of 17 y = f ( x 1 , . . . , x n ) requires an extra bit x 0 : def Go Back U f : � x 1 , . . . , x n , x 0 | → � x 1 , . . . , x n , y | , y = x 0 ⊕ f ( x 1 , . . . , x n ) . Full Screen • Here, ⊕ is “xor”: 0 ⊕ 1 = 1 ⊕ 0 = 1, 0 ⊕ 0 = 1 ⊕ 1 = 0. This U f is reversible: x 0 = y ⊕ f ( x 1 , . . . , x n ) . Close Quit