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lect notes on quantum computation(lect2)

lect notes on quantum computation(lect2) - Last revised...

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Last revised 4/5/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2006 c 2006, N. David Mermin II. Quantum Computation: General features and some simple examples A. The general computational process We would like a suitably programmed quantum computer to act on a number x to produce another number f ( x ) for some specified function f . Appropriately interpreted, with an accuracy that increases with increasing k , we can treat all such numbers as non- negative integers less than 2 k . Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits. If we specify x as an n -bit integer and f ( x ) as an m -bit integer, then we shall need at least n + m Qbits: a set of n -Qbits, called the input register , to represent x , and another set of m -Qbits, called the output register , to represent f ( x ). Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f ( x ) assigns the same value to different values of x , as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f ( x ). Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since (as remarked in Chapter 1 and expanded on in Section C of this chapter) quantum computers must operate reversibly (except for measurement gates) to perform their magic, quantum computers are generally designed to operate with both input and output registers. We shall find that this dual-register procedure can be exploited by a quantum computer in some strikingly nonclassical ways. The computational process will generally require many Qbits besides the n + m in the input and output registers, but we shall ignore these additional Qbits for the moment, viewing a computation of f as doing nothing more than applying a unitary transformation, U f to the n + m Qbits of the input and output registers. We shall return to the fundamental question of why the additional Qbits can be ignored in Section C, only noting for now that it will be the reversibility of the computation that makes it possible for them to be ignored. To define unitary transformations U it is enough to define their action on any basis, since any other state | Ψ i must be a linear superposition of basis states, and therefore, since unitary transformations are linear, their action on | Ψ i is entirely determined by their action on the basis. In most cases of interest one takes the basis in which U is defined to be the computational (classical) basis. In many cases (such as the definition below of U f ) the action of U on each computational basis state is simply to produce another computational basis state. Since U must have an inverse, it therefore acts on the entire 1
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computational basis as a permutation, just as a classical transformation does. Since linear
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