7 control passes out from the upper branch of the

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Unformatted text preview: box uses CNOTs to copy the external input into M ¶5. M operates, generating the result in an internal register. M contains garbage. ¶6. The f = 1 flag directs control into the upper branch (resetting f = 0), which uses CNOTs to copy the result into the external output register Out. ¶7. Control passes out from the upper branch of the switch down and back into the lower branch, which negates f , setting f = 1. ¶8. Control passes back into the machine through the lower switch branch (resetting f = 0), and backwards through M , clearing out all the garbage, restoring all the registers to 0s. ¶9. It passes backwards through the Copy box, copying the input back from M to the external input register In. This restores the internal register to 0s. ¶10. Control passes out through the lower branch of the left switch (setting f = 1), but it negates f again, so f = 0. It arrives at the terminal program atom t. ¶11. At the end of the process, everything is reset to the initial conditions, except that we have the result in the Out register. ¶12. Subroutines etc.: F discusses how to do subroutines and other programming constructs. F.2 Benio↵ ’s quantum Turing machine ¶1. In 1980 Paul Benio↵ published the first design for a universal quantum computer, which was based on the Turing machine. ¶2. Tape: The tape is represented by a finite lattice of quantum spin systems with eigenstates corresponding to the tape symbols. (Therefore, he cannot implement an open-ended TM tape, but neither can an ordinary digital computer.) ¶3. Head: The head is a spinless system that moves along the lattice. F. UNIVERSAL QUANTUM COMPUTERS 185 ¶4. State: The state of the TM was represented by another spin system. ¶5. He defined unitary operators for doing the various operations (e.g., changing the tape). ¶6. In 1982 he extended his model to erase the tape, as in Bennett’s model. ¶7. Computation step: Each step was performed by measuring the tape state under the head and the internal state (thus collapsing them) and using this to control the unitary operator applied to the tape and state. ¶8. As a consequence, the computer does not make much use of superposition. F.3 Deutsch’s universal quantum computer This section is based on Deutsch, D., “Quantum theory, the Church-Turing principle, and the universal quantum computer. Proc. Royal Soc. London A, 400 (1985), pp. 97–119. ¶1. Benio↵ ’s computer is e↵ectively classical; it can be simulated by a classical TM. ¶2. Feynman’s construction is not a true universal computer, since you need to construct it for each computation, and it’s not obvious how to get the required dynamical behavior. ¶3. Deutsch seeks a broader definition of quantum computation, and a universal quantum computer Q. ¶4. Processor: “The processor consists of M 2-state observables, {ni }” ˇ (i 2 M), where M = {0, . . . , M 1}. Collectively they are called n. ˇ ¶5. Memory: “The memory consists of an infinite sequence {mi...
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