D. Gottesman. Stabilizer codes and quantum error correction. PhD thesis, Cal. Inst. Tech, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of multiparticle entanglement. Highly entangled multiparticle states also occur in the relatively new context of quantum error correcting codes. In fact, the code subspace for an additive code can be described as the space of ground states of the Hamiltonian formed by the sum of the stabilizers [22]. For example, the code subspace for a 5 qubit code [5,1,3] encoding 1 qubit against single bit errors [23,24] is the space of ground states of the translation invariant Hamiltonian on a one dimensional lattice of five qubits: H [5,1,3] X j Z j 1 Z j 2 X j 3 , where the subscripts are to ....

D. Gottesman, Stabilizer Codes and Quantum Error Correction, Caltech Ph.D. thesis, physics (1997), quant-ph/9705052.

....produces only small errors is nonzero and close to that of the ideal (errorless) channel. This is indeed not even evident from most existing presentations of the theory of quantum error correcting codes. Papers which address this problem at least for special cases like depolarizing channels are [4, 6] and [15, Sec 7.16.2] while the general case is treated more recently in [7, 12] The purpose of this paper is less the presentation of new results but to show in an elementary and self contained way that small quantum errors can be corrected with an asymptotically small e ort. To this end the ....

D. Gottesman. Stabilizer codes and quantum error correction. Ph.D. thesis, California Institute of Technology (1997). quant-ph/9705052.

....the encoded state if both measurement results are the same. Lemma 2 (Random hashing) There exist purity testing protocols such that m = n s, 2 and which use ns s 1 bits of (classical) communication. This lemma actually follows from the observation that the set of all stabilizer codes [9] of dimension 2 is a purity testing code family with error 2 in the sense of [2] However, we give a direct proof with an explicit protocol description in Section 6.2 below. Barnum et al. provide a construction which achieves better communication complexity at the cost of increasing the ....

D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology, 1997.

.... Gamma 1 and correct any b d Gamma1 2 c errors. As a result it is desirable to keep d as large as possible. A strict definition of the minimum distance is given in the next section after introducing error bases. Remark For introductions to the theory of quantum error correcting codes see e.g. [13, 10, 14]. For a reader with a background in classical coding theory the papers [1, 2] have brief introductions to the field. 3 Error Bases A general quantum error of a p m ary quantum system, is a linear operator, say e, acting on the space C p m . If v is a state (a unit vector in the space) of the ....

D. Gottesman, "Stabilizer Codes and Quantum Error Correction," Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1997.

....plane we have demonstrated the existence of single qubit topological quantum codes. While two of the ones we find are new, the third is Shor s original 9 qubit code [9] this connects Kitaev s novel perspective [1,3] with the bulk of the work on quantum error correcting codes (see, for example, [10] and the references therein) One might ask whether the 5 qubit [11] and 7 qubit [12] single qubit codes are also equivalent to some projective plane quantum code. They are not there are no cellulations of RP 2 with 5 or 7 edges and lengths of all essential cycles and dual cycles at least 3. ....

D. Gottesman, Stabilizer Codes and Quantum Error Correction, Caltech Ph.D. thesis, physics (1997), quant-ph/9705052.

....after their discoverers [11, 31] and which contains codes that are much more efficient than this first one. For fault tolerance, which will be discussed next, we only need to use CSS codes. However, a more general framework that includes these codes was discovered simultaneously by two groups [19, 20, 10]. Substantial work on quantum error correcting codes has occurred since their discovery, much of it referenced in [10] In classical computers, error correcting codes have been found to be very useful for storing and transmitting information, but not for providing fault tolerant computing. It is ....

D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. Thesis, California Institute of Technology (1997). Also LANL e-print quantph /9705052, available online at http://xxx.lanl.gov/.

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D. Gottesman. Stabilizer codes and quantum error correction. PhD thesis, Cal. Inst. Tech, 1997.

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D. Gottesman, Stabilizer Codes and Quantum Error Correction, (Online preprint quant-ph/9705052), Caltech Ph.D. Thesis (1997).

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D. Gottesman, Stabilizer Codes and Quantum Error Correction, (Online preprint quantph /9705052), Caltech Ph.D. Thesis (1997).

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