lect notes on quantum computation

lect notes on quantum computation - Last revised LECTURE...

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Last revised 1/31/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2006 c 2006, 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 exploits certain very special transformations of its internal state. The laws of quantum mechanics allow these peculiar transformations under very carefully controlled conditions. For a computer to be a quantum computer the physical systems that encode the individual 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 shall call “classical” when we wish to contrast it to a quantum computer — introduce potentially catastrophic disruptions into the operation of a quantum computer. Such disastrous interactions can include interactions with the external 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 that represent bits with other degrees of freedom of those same systems, associated with computationally irrelevant features of their internal structure. All such interactions between what is computationally relevant and what is not are said to result in “decoherence”, which is fatal to a quantum computation. To avoid decoherence individual bits cannot be encoded in physical systems of macro- scopic 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 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 unavailably large amounts of energy to excite. Such atomic-scale systems must also be decoupled from their surroundings except for the com- pletely 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 by 1
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extraneous interactions can actually be corrected, provided they occur at a sufficently low rate. While error correction is routine for bits represented by classical systems, quantum
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