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Unformatted text preview: Last revised 2/14/05 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 c 2005, N. David Mermin I. Fundamental Properties of Cbits and Qbits It is tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behavior of all physical phenomena, this temptation must be resisted. Your laptop operates under the laws of quantum mechanics, but it is not a quantum computer. A quantum computer is one whose operation takes advantage of certain kinds of transformations of its internal state. The laws of quantum mechanics allow these peculiar transformations under very special circumstances. For a computer to be a quantum computer the physical systems that encode the in- dividual bits must have no physical interactions whatever that are not under the complete control of the program. All other interactions, however irrelevant they might be in an ordinary computer which we call classical when we wish to contrast it to a quan- tum computer introduce potentially catastrophic disruptions into the operation of a quantum computer. Such disastrous interactions can include interactions with the exter- nal environment air molecules bouncing off the physical systems that represent bits, or those systems absorbing a minute amount of ambient radiant thermal energy. There can even be disruptive interactions between the computationally relevant degrees of freedom of the physical systems representing bits with other irrelevant degrees of freedom of those same systems associated with their internal structure. All such interactions between what is computationally relevant and what is not are said to result in decoherence, which is death to a quantum computation. What this means is that individual bits cannot be encoded in physical systems of macroscopic size, because such systems cannot be isolated from their own irrelevant internal degrees of freedom. The bits must be encoded in a very small number of quantum states of a system of atomic size, where extra internal degrees of freedom do not come into play because they do not exist, or because they require unavailable amounts of energy to excite. Such atomic-scale systems must be decoupled from all of their surroundings except for the completely controlled interactions that are associated with the computational process itself. Two things keep the situation from being hopeless. First, because the separation between the discrete energy levels of a system on the atomic scale can be enormously larger than the separation between the levels of a large system, the dynamical isolation of an atomic system is easier to achieve. It can take a substantial kick to knock an atom out of its ground state. The second reason for hope is the discovery that errors induced 1 by extraneous interactions can actually be corrected, provided they occur at a sufficently low rate. While error correction is routine for classical bits, quantum error correction islow rate....
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- Spring '05