chap6 - Chapter 6 Quantum Computation 6.1 Classical...

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Unformatted text preview: Chapter 6 Quantum Computation 6.1 Classical Circuits The concept of a quantum computer was introduced in Chapter 1. Here we will specify our model of quantum computation more precisely, and we will point out some basic properties of the model. But before we explain what a quantum computer does, perhaps we should say what a classical computer does. 6.1.1 Universal gates A classical (deterministic) computer evaluates a function: given n-bits of input it produces m-bits of output that are uniquely determined by the input; that is, it finds the value of f : { , 1 } n → { , 1 } m , (6.1) for a particular specified n-bit argument. A function with an m-bit value is equivalent to m functions, each with a one-bit value, so we may just as well say that the basic task performed by a computer is the evaluation of f : { , 1 } n → { , 1 } . (6.2) We can easily count the number of such functions. There are 2 n possible inputs, and for each input there are two possible outputs. So there are altogether 2 2 n functions taking n bits to one bit. 1 2 CHAPTER 6. QUANTUM COMPUTATION The evaluation of any such function can be reduced to a sequence of elementary logical operations. Let us divide the possible values of the input x = x 1 x 2 x 3 ...x n , (6.3) into one set of values for which f ( x ) = 1, and a complementary set for which f ( x ) = 0. For each x ( a ) such that f ( x ( a ) ) = 1, consider the function f ( a ) such that f ( a ) ( x ) = braceleftBigg 1 x = x ( a ) 0 otherwise (6.4) Then f ( x ) = f (1) ( x ) ∨ f (2) ( x ) ∨ f (3) ( x ) ∨ ... . (6.5) f is the logical OR ( ∨ ) of all the f ( a ) ’s. In binary arithmetic the ∨ operation of two bits may be represented x ∨ y = x + y- x · y ; (6.6) it has the value 0 if x and y are both zero, and the value 1 otherwise. Now consider the evaluation of f ( a ) . In the case where x ( a ) = 111 ... 1, we may write f ( a ) ( x ) = x 1 ∧ x 2 ∧ x 3 ... ∧ x n ; (6.7) it is the logical AND ( ∧ ) of all n bits. In binary arithmetic, the AND is the product x ∧ y = x · y. (6.8) For any other x ( a ) ,f ( a ) is again obtained as the AND of n bits, but where the NOT ( ¬ ) operation is first applied to each x i such that x ( a ) i = 0; for example f ( a ) ( x ) = ( ¬ x 1 ) ∧ x 2 ∧ x 3 ∧ ( ¬ x 4 ) ∧ ... (6.9) if x ( a ) = 0110 ... . (6.10) 6.1. CLASSICAL CIRCUITS 3 The NOT operation is represented in binary arithmetic as ¬ x = 1- x. (6.11) We have now constructed the function f ( x ) from three elementary logi- cal connectives: NOT, AND, OR. The expression we obtained is called the “disjunctive normal form” of f ( x ). We have also implicitly used another operation, COPY, that takes one bit to two bits: COPY : x → xx. (6.12) We need the COPY operation because each f ( a ) in the disjunctive normal form expansion of f requires its own copy of x to act on....
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chap6 - Chapter 6 Quantum Computation 6.1 Classical...

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