P.W. Shor and J.A. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome", quant-ph/9604006.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....transmission, maximized over input density operators, does not in general constitute an upper bound; rather, the large block limit must be taken (and the block size n divided out) in the expression for capacity. Cases are known where rates greater than the one block expression may be achieved [69], 60] This superadditivity of the quantum expresssion is curious, and renders the expression much less useful for computation of channel capacity, both because large block sizes make computation cumbersome and because at present not enough is known about the rate of convergence of this ....

.... channel results are all upper bounds or equivalences, is an important open question; the best existing results along these lines are those of Bennett, DiVincenzo, Smolin and Wootters for channels acting on qubits, using entanglement purification as discussed in Chapter 7, and the improvements in [69], 60] as well as the related 202 results for higher dimensional channels mentioned in that Chapter. Further results along these lines may well require an understanding of random quantum codes which are not of the usual stabilizer variety, perhaps codes based on non unitary error bases. I ....

P. Shor and J. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome," LANL e-print quant-ph/9706061, 1996.

.... authors have worked on the problem of quantum information transmission through quantum channels; some of this work calculates or or bounds the capacity we study here, for particular channels or classes of channels: an incomplete list that could serve as an entry to the literature includes [17] [23], 20] 18] 19] 7] 9] 24] Some of the extensive literature on the more algebraic approach to quantum coding also yields information about the quantum capacity. II. Quantum sources and channel capacity A. Mathematical preliminaries and notation The e ect of encoding procedures, ....

P. Shor and J. Smolin, \Quantum error-correcting codes need not completely reveal the error syndrome," LANL e-print quantph /9706061, 1996.

....that QC can be done with bounded decoherence error, assuming the error correction mechanism is without error itself. Bennett et al. [BDS 96] Laflamme [LMP 96] gave the first optimal 5 qubit codes, leading to asymptotically optimal (for large code blocks) quantum error correction codes. Shor [Sho96] and Kitaev [KY96,Kit97] extended these techniques to do fault tolerant quantum computation on quantum networks, in the presence of bounded decoherence error, even if the error correction mechanism also suffers from error decoherence errors. A final innovation (Gottesman et al. [GEK 96] Aharonov, ....

....D.R. Franceschetti, and S.E. Stevens, Jr. A DNA Based Implementation of an Evolutionary Search for Good Encodings for DNA Computation, ICEC 97 Special Session on DNA Based Computation, Indiana, April 1997. DHK96] Delcher, A. L. L. Hood, R.M. Karp, Report on the DNA Biomolecular Computing Workshop, June 1996. DHS97] Deputat, M. G. Hajduczok, E. Schmitt, On Error Correcting Structures Derived from DNA, 3rd DIMACS Meeting on DNA Based Computers, Univ. of Penns. June, 1997) DDSPL 93] Drmanac, R, S. Drmanac, Z. Strezoska, T. Paunesku, I. Labat, M. Zeremski, J. Snoddy, W. K. Funkhouser, B. Koop, ....

Shor, P. and Smolin, J., Quantum error-correcting codes need not completely reveal the error syndrome. (Online preprint quant-ph/9604006.), submitted to Phys. Rev. Lett. (1996).

.... that the ffi 1=6 threshold bound does extend to nondegenerate codes, and this will be appear in a forthcoming paper [22] It is interesting to note that there exist some degenerate stabilizer codes that outperform all known nondegenerate codes on the depolarizing channel, for some values of ffi [5]. The best lower bound for this channel that we are aware of is 1 Gamma H(2ffi) Gamma 2ffi log 2 3 [8] For the depolarizing channel with error probability ffi, our results imply that, for stabilizer codes, the capacity is upper bounded by 1 Gamma H(ffi) the bound for the classical binary ....

....is stronger than the previously established upper bound of 1 Gamma 4ffi [6] though the latter bound applies to nonstabilizer codes as well. The best lower bound that we are aware of for this channel is 1 Gamma H(ffi) Gamma ffi log 2 3 [6] and a slightly larger value for some values of ffi [5]. Should any improvements to the upper bounds in [17] for classical coding occur, they will automatically apply to quantum stabilizer codes. Our results demonstrate interesting 4 connections between quantum stabilizer codes and classical linear codes, and, for some instances of channels, yield ....

P.W. Shor and J.A. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome", e-print quant-ph/9604006.

....but it will also have broader ramifications. Here are some the milestones that have been reached this year: That quantum error correcting codes exist was first pointed out by Peter Shor (1995) and Andrew Steane (1996a) in the fall of 95. By early 96, Steane (1996b) and Calderbank and Shor (1996) had shown that good codes exist, that is, codes that are capable of correcting many errors. We learned from the work on random codes by Lloyd (1997) and by Bennett, DiVincenzo, Smolin, and Wootters (1996) that if we want to store quantum information for a while, then we can find a code that will ....

....carry out these operations. Thus, we need to find methods for recovering from errors that are sufficiently robust that we can still recover the quantum information with high accuracy even when we make some errors during the recovery step. This is the problem of fault tolerant recovery, and Peter Shor (1996) showed in a pioneering paper written last May that fault tolerant recovery is possible if the error rate is not too high. 2 Of course, we want more than just to store quantum information; we want to be able to process the information and build up an interesting quantum computation. So we must ....

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Shor, P. & and Smolin, J. 1996 Quantum error-correcting codes need not completely reveal the error syndrome. (Online preprint quant-ph/9604006.) Steane, A. M. 1996a Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793.

....that QC can be done with bounded decoherence error, assuming the error correction mechanism is without error itself. Bennett et al. [BDS 96] Laflamme [LMP 96] gave the first optimal 5 qubit codes, leading to asymptotically optimal (for large code blocks) quantum error correction codes. Shor [Sho96] and Kitaev [KY96,Kit97] extended these techniques to do fault tolerant quantum computation on quantum networks, in the presence of bounded decoherence error, even if the error correction mechanism also suffers from error decoherence errors. A final innovation (Gottesman et al. [GEK 96] Aharonov, ....

Shor, P. and Smolin, J., Quantum error-correcting codes need not completely reveal the error syndrome. (Online preprint quant-ph/9604006.), submitted to Phys. Rev. Lett. (1996).

....which is not permitted in the quantum case: two different errors may be indistinguishable by their error syndromes, but may nevertheless be both correctable (see Sec. VI) Spurred by intuitive ideas of how this degeneracy might improve the capacity of the quantum channel, Shor and Smolin [14] explored some non random coding strategies, and found a range of depolarizing channels (very noisy ones) for which the obvious analog of the Shannon bound is violated; a higher capacity is attained than for random codes. The main point of this paper is to present the Shor Smolin discovery using ....

....a compact formula for the attainable capacity for code states, and establishing the identity of the coherent information with the ShorSmolin quantum channel capacity. III. SHOR SMOLIN CONCATENATION PROCEDURE In order to formulate the main result, we first review the Shor Smolin procedure [14] for sending reliable qubit states, with a finite capacity, over a depolarizing channel. Just as in conventional channel coding, it involves an additive code specified above by S. In conventional channel coding shown in Fig. 2, the additive code is used as follows: the state ji to be ....

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P. W. Shor and J. A. Smolin, "Quantum Error-Correcting Codes Need Not Completely Reveal the Error Syndrome," Report No. quant-ph/9604006 . 22

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P.W. Shor and J.A. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome", quant-ph/9604006.

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P.W. Shor and J.A. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome", e-print quant-ph/9604006.

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Shor, P., and Smolin, J. Quantum error-correcting codes need not completely reveal the error syndrome. http://xxx.lanl.gov/abs/quant-ph/ 9604006, 1996.

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P.W. Shor and J.A. Smolin, "Quantum error-correcting codes need not completely reveal the error syndrome", quant-ph/9604006.

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