= a|0 + b |1 | + b |0 + a |1 | (1.86) = |0 | + |1 | , | = b | + a | . in .NET Add qr-codes in .NET = a|0 + b |1 | + b |0 + a |1 | (1.86) = |0 | + |1 | , | = b | + a | .

= a. generate, create none none on none projects Code 11 0 + b 1 . + b 0 + a 1 . (1.86) = . 0 . + . 1 . , . = b . + a .. where = a. b . , (1.87). We would like to choose the complex numbers a and b to make and orthogonal. The inner product is = a 2 . b 2 . + ab . (1.88).

If = 0, then setting to 0 gives a q uadratic equation for a/b , which has two complex solutions. If a in (1.85) is any nonzero complex number then either solution determines b , which, with a, gives a 1-Qbit unitary u for which.

u 1 . = . 0 . + . 1 . (1.89). where and are orthogonal. If = 0 then (1.84) is already of this form with u = 1. CBITS AND QBITS We can pick positive real numbers and so that = . / and = . / are unit vectors, making and an orthonormal pair. They are therefore related to 0 and 1 by a unitary t ransformation v: . = v. 0 , . = v. 1 . (1.90).

Equation (1.89) then gives12 = u v 0 . 0 + . 1 . 1 . (1.91).

We can write this as = u v C10 0 + . 1 . 0 . (1.92).

Since is a unit vector and unitary transformations preserve unit vectors, it follows from (1.91) that 0 + . 1 is a unit vector. It can therefore be obtained from 0 by a unitary transformation w. So = u v C10 w 1 0 . 0 = u1 v0 C10 w1 00 .. (1.93). We have thus established that a general 1-Qbit state can be construc none none ted out of three 1-Qbit unitaries and a single cNOT gate, acting on the standard state . 00 . This is an early example of the usefulness of cNOT gates. 1.12 Summary: Qbits versus Cbits Table 1.1 gives a concise comparison of the elementary properties of Cbits and Qbits. The table uses the term Bit, with an upper-case B, to mean Qbit or Cbit, which should be distinguished from bit, with a lower-case b, which means 0 or 1.

Alice (in the fth line of the table) is anybody who knows the relevant history of the Qbits their initial state preparation and the unitary gates that have subsequently acted on them.. 12 This form for none none a general vector in a space of 2 2 dimensions is a special case of a more general result for d d dimensions known as the polar (or Schmidt) decomposition theorem.. 1.12 SUMMARY: QBITS VERSUS CBITS Table 1.1. A summary of the features of Qbits, contrasted to the analogous features of Cbits Cbits States of n Bits Subsets of n Bits Reversible operations on states Can state be learned from Bits To learn state of Bits To get information from Bits Information acquired State after information acquired . x n , 0 x < 2n Always have states Permutations Yes Examine them Just look at them x Same: still x. Qbits x . x n , . x . 2 = 1 . Generally have no states Unitary transformations No Go ask Alice Measure them x with probability x . 2 Different: now x. 2 . General features and some simple examples 2.1 The general computational process A suitably progr none for none ammed quantum computer should act on a number x to produce another number f (x) for some speci ed function f . Appropriately interpreted, with an accuracy that increases with increasing k, we can treat such numbers as non-negative integers less than 2k . Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits.

If we specify the numbers x as n-bit integers and the numbers f (x) as m -bit integers, 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 quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall nd that this dual-register architecture can also be usefully 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 now, 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 take up the fundamental question of why the additional Qbits can be ignored in Section 2.3, only noting for now that it is the reversibility of the computation that makes this possible.

We de ne the transformation U f by specifying it as a reversible transformation taking computational-basis states into computationalbasis states. As noted in Section 1.6, the linear extension of such a classically meaningful transformation to arbitrary complex superpositions of computational-basis states is necessarily unitary.

The standard quantum-computational protocol, which we shall use repeatedly,.
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