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An Introduction to Quantum Algorithms 2.5 Quantum logic gates As mentioned earlier, each possible bit configuration in the quantum superposition is denoted by the tensor product of its counterpart qubits. Consider | 101 i , the bit string that represents the integer value 5: | 101 i = | 1 i ⊗ | 0 i ⊗ | 1 i = 0 1 1 0 0 1 = 0 0 0 0 0 1 0 0 T As with single qubits, the squared absolute value of the amplitude associated with a given bit string is the probability of observing that bit string upon collapsing the register to a classical state, and the the sqares of the absolute values of the amplitudes of all 2 n possible bit configuations of an n -bit register sum to unity: 2 n - 1 X i =0 | a i | 2 = 1 Quantum registers are a relatively straightforward extension of quantum bits. 2.5 Quantum logic gates Understanding how quantum registers work mathematically and how they differ from classical registers, it is now possible to think about how the state of a quantum register can evolve over time, changing to reach some ultimate goal. In classical computing, one way of thinking about algorithm design and computation is via universal Turing machines. Quantum universal Turing machines were first described by David Deutsch in 1985 [16], but designing algorithms for quantum Turing machines is even more difficult and tedious than doing so for classical Turing machines; both a quantum Turing machine’s tape and its read-write head exist in superpositions of an exponential number states! As in classical computer science, instead of using the Turing machine as a computational model, operations on a quantum computer are most often described using quantum circuits made up of qubits and quantum logic gates, a concept also introduced by Deutsch a few years after his specification of the quantum analog to a Turing machine [ 17 ]. In classical computer science but arguably even more so in quantum computing, although circuits are computationally equivalent to Turing machines, they are usually much simpler to depict, manipulate and understand. In classical computing, binary values, as stored in a register, pass through logic gates that, given a certain binary input, produce a certain binary output. Mathematically, classical logic gates are described using boolean algebra. Quantum logic gates act in a similar way, in that quantum logic gates applied to quantum registers map the quantum superposition to another, together allowing the evolution of the system to some desired Page 12 of 35
An Introduction to Quantum Algorithms 2.5 Quantum logic gates final state, a correct answer.
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• Fall '13
• Xue
• Hilbert space, Quantum algorithms

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