This derivation is exemplary of the convenience of

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This derivation is exemplary of the convenience of Dirac notation; the same equivalence is made clear in much fewer steps and through much less tedious math in Dirac notation: 1 = h ψ | ψ i = ( a 0 h 0 | + a 1 h 1 | ) · ( a 0 | 0 i + a 1 | 1 i ) = | a 0 | 2 h 0 | 0 i + | a 1 | 2 h 1 | 1 i + a 1 a 0 h 1 | 0 i + a 0 a 1 h 0 | 1 i = | a 0 | 2 + | a 1 | 2 Thus the probability of observing a single possible state from the superposition is obtained by squaring the absolute value of its amplitude; the probability of the qubit being in the state | 0 i is | a 0 | 2 , and the probability that the qubit will be measured as | 1 i is | a 1 | 2 , or 1 - | a 0 | 2 . Remember that the contents of Dirac bras and kets are labels that describe the underlying vectors. | 0 i and | 1 i may be transformed into any two vectors that form an orthonormal basis in C 2 . The most common basis used in quantum computing is called the computational basis : | 0 i = 1 0 , | 1 i = 0 1 But any other orthonormal basis could be used. For example, the basis vectors: | + i = | 0 i + | 1 i 2 = 1 2 1 1 , |-i = | 0 i - | 1 i 2 = 1 2 1 - 1 Page 10 of 35
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An Introduction to Quantum Algorithms 2.4 Quantum registers Provide a slightly different but equivalent way of expressing of a qubit: | ψ i = a 0 | 0 i + a 1 | 1 i = a 0 | + i + |-i 2 + a 1 | + i - |-i 2 = a 0 + a 1 2 | + i + a 0 + a 1 2 |-i Here, instead of measuring the states | 0 i and | 1 i each with respective probabilities | a 0 | 2 and | a 1 | 2 , the states | + i and |-i would be measured with probabilities | a 0 + a 1 | 2 / 2 and | a 0 - a 1 | 2 / 2. Because the computational basis tends to be the most straightforward basis for computing and understanding quantum algorithms, the rest of this tutorial will assume the computational basis is being used unless otherwise stated. 2.4 Quantum registers It is hard to do any interesting computation with only a single qubit. Like classical computers, quantum computers use quantum registers made up of multiple qubits. When collapsed, quantum registers are bit strings whose length determines the amount of information they can store. In superposition, each qubit in the register is in a superposition of | 1 i and | 0 i , and consequently a register of n qubits is in a superposition of all 2 n possible bit strings that could be represented using n bits. The state space of a size- n quantum register is a linear combination of n basis vectors, each of length 2 n : | ψ n i = 2 n - 1 X i =0 a i | i i Here i is the base-10 integer representation of a length- n number in base-2. A three-qubit register would thus have the following expansion: | ψ 2 i = a 0 | 000 i + a 1 | 001 i + a 2 | 010 i + a 3 | 011 i + a 4 | 100 i + a 5 | 101 i + a 6 | 110 i + a 7 | 111 i Or in vector form, using the computational basis: | ψ 2 i = a 0 1 0 0 0 0 0 0 0 + a 1 0 1 0 0 0 0 0 0 + a 2 0 0 1 0 0 0 0 0 + a 3 0 0 0 1 0 0 0 0 + a 4 0 0 0 0 1 0 0 0 + a 5
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  • Fall '13
  • Xue
  • Hilbert space, Quantum algorithms

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