Lecture Notes 3.F

# Here the eigenvectors are labeled by their

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Unformatted text preview: } (i 2 Z) ˇ of 2-state observables.” Collectively the sequence is called m. ˇ ¶6. Tape position: An observable x, with spectrum Z, represents the ˇ tape position (address) of the head. 186 CHAPTER III. QUANTUM COMPUTATION ¶7. Computational basis states: The computational basis states have the form: def |x; n; mi = |x; n0 , n1 , . . . , nM 1 ; . . . , m 1 , m 0 , m1 , . . . i . Here the eigenvectors are labeled by their eigenvalues x, n, and m. ¶8. Dynamics: The dynamics of computation is described by a unitary operator U : | (nT )i = U n | (0)i. ¶9. Initial tape: Initially, only a ﬁnite number of these are prepared in a non-zero state. X X | (0)i = | m | 2 = 1, m |0; 0; mi, where m m “where only a ﬁnite number of the m are non-zero and whenever an inﬁnite number of the m are non-zero.” Note that this may be a superposition of initial tapes. m vanishes ¶10. The non-zero entries are the program and its input. ¶11. Unitary operator: The matrix elements of U (relating the new state to the current state) have the form: hx0 ; n0 ; m0 | U | x; n; mi =[ x+1 + 0 0 x0 U (n , mx |n, mx ) + x1 x0 U (n0 , m0x |n, mx )] Y my mx . y 6=x U + and U represent moves to the right and left, respectively. The ﬁrst two s ensure that the tape position cannot move by more than one position in a step. The ﬁnal product of deltas ensures that all the other tape positions are unchanged; it’s equivalent to: 8y 6= x : my = mx . ¶12. The U + and U functions deﬁne the actual transitions of the machine in terms of the processor state and the symbol under the tape head. Each choice deﬁnes a quantum computer Q[U + , U ]. F. UNIVERSAL QUANTUM COMPUTERS 187 ¶13. Halting: The machine cannot be observed before it has halted, since it will generally alter its state. Therefore one of the processor’s bits is chosen as a halt indicator. It can be observed from time to time without a↵ecting the computation. ¶14. Power: Q can simulate TMs, but also any other quantum computer to arbitrary precision. It can simulate any ﬁnitely realizable physical system to arbitrary precision. It can simulate some physical systems that go beyond the power of TMs (hypercomputation)....
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