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Unformatted text preview: Last revised 4/7/05 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 V. Quantum Error Correction. c 2004, N. David Mermin The discovery in 1995 of quantum error correction by Peter Shor and, independently, Andrew Steane, had an enormous impact on the prospects for the field of quantum com- putation. It changed the dream of building a working quantum computer from something that was clearly unattainable, to something that merely poses an immense technological challenge. In a classical computer individual bits are embodied in physical systems that are immense on the atomic scale. The two states representing 0 and 1 are so grossly different that the probability is infinitessimal for flipping from one state to the other as a result of thermal fluctuations, mechanical vibrations, or other irrelevant extraneous interactions. In a classical computer error correction is not an issue. It does become an issue, even classically, in the transmission of information over large distances, where the signal gets weaker and weaker. One can deal with this in a variety of straightforward or clever ways. The crudest would be to encode each logical bit in three actual bits, replacing | i and | 1 i with the codewords | i = | i| i| i and | 1 i = | 1 i| 1 i| 1 i . One could then monitor each codeword, checking for flips in any of the individual bits and restoring them by the principle of majority rule, whenever a flip was detected. The rate at which monitoring took place would have to be high enough to make negligible the probability of two flips taking place between inspections. The situation is quite different in a quantum computer for several reasons: (a) The qubits are carried by different quantum states of individual atomic-scale phys- ical systems such as atoms, photons, trapped ions, nuclear magnetic moments, etc. Any coupling to any other degrees of freedom, not under the explicit control of the computer and its program, can substantially alter the state of a physical qubit, thereby disrupting the computation. In the absence of error correction, the system embodying each qubit would have to be impossibly well isolated both from computationally irrelevant interactions with other parts of the computer and, more generally, from interactions with anything else in the environment of the physical qubit. (b) Bit flips are not the only errors. There are entirely nonclassical sources of trouble. Phase errors, for example the alteration of | i + | 1 i to | i - | 1 i , can be just as damaging as bit flip errors. (c) In contrast to the classical situation, diagnosing an error is problematic. In a quantum computer to monitor a qubit is to measure it, which alters its state (if it has 1 one of its own) and destroys its quantum correlations with other qubits (if it is entangled with them). Quantum error correction requires one to identify errors in a way that reveals no information about the uncorrupted state of the qubits, yet enables one to undo the...
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This note was uploaded on 09/28/2008 for the course PHYS 481 taught by Professor Anon during the Spring '05 term at Cornell University (Engineering School).
- Spring '05