P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A 52, 2493 (1995).  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

....codes. They play a role similar to the linear codes in classical coding theory. Quantum stabilizer codes have simple encoding algorithms, can be analyzed using classical coding theory, and yield methods for fault tolerant quantum computation. The first examples of quantum codes found by Shor [18], and Steane [21, 22] were quantum stabilizer codes. General quantum stabilizer codes were introduced by Gottesman [9] and Calderbank et al. 6] Later Calderbank et al. 7] gave the now standard connection between quantum stabilizer codes and classical selforthogonal codes, which was used to ....

P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, 52, p. 2493, 1995.

....of undetected error and weight enumerators to derive bounds on this probability for quantum codes. Index Terms Quantum code, measurement, undetected error, Shor Laflamme enumerators. 1 Introduction The possibility of correcting decoherence errors in entangled states was discovered by Shor [20] and Steane [21] Since then the theory of quantum codes has been a topic of intense study. Error processes in the depolarizing channel are characterized in [12] where it was shown that one can restrict attention to error operators given by Kronecker products of Pauli matrices. Present address: ....

P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, 52, no. 4, pp. 2493--2496, 1995.

....The protection of the memory against these errors is a crucial part in the construction of a resilient quantum computer. We describe in this chapter a generalization of stabilizer codes that allows to protect the encoded states. The rst quantum error correcting codes have been introduced by Shor [21] and Steane [24] about six years ago. The existence of such codes is a remarkable fact, since it shows that an in nite variety of errors a ecting a single quantum bit can be corrected by a nite number of operations. Moreover, the subsequent development of fault tolerant architectures [22] made it ....

....the base states jki into nine qudits by jki 7 1 d 3=2 d 1 X j=0 kj jjjji d 1 X j=0 kj jjjji d 1 X j=0 kj jjjji ; 1. 2) where k 2 f0; d 1g and = exp(2 i=d) This quantum error control code is a straightforward generalization of Shor s code [21] to the nonbinary case. The code (1.2) is able to correct an arbitrary error in one of the nine qudits. To see this, we note that the code is given by a concatenation of two codes. The inner code is a repetition code encoding a base state into three replicas jki 7 jki jki jki with k 2 f0; ....

P. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 2:2493-2496, 1995.

....probability will go to zero with the dephasing probability in the same way as with the bit flip probability for the previous codes. But just as the bit flip codes increased the probability of errors due to dephasing, these codes increase the probability of errors due to bit flips. Peter Shor [44] and, independently, Andrew Steane [45] discovered codes that can correct for both bit flip errors and dephasing, and indeed for arbitrary interactions of a single qubit with its environment. Shor s original code is particularly easy to understand given the background I have just developed: it is ....

P. W. Shor, "Scheme for reducing decoherence in quantum memory," Physical Review A, vol. 52, pp. 2493--2496, 1995.

....k = n Gamma d) into n qubit codewords. 02.70.Rw, 03.65.Bz, 89.80. h Typeset using REVT E X cleve cpsc.ucalgary.ca y gottesma theory.caltech.edu 1 I. INTRODUCTION Recently, significant progress has been made in the development of error correction schemes for quantum information systems [1 10]. This includes methods for converting classical error correcting codes to into quantum error correcting codes [2,3] formalizations of necessary and sufficient conditions for sets of states to form quantum codes [11,7,12] and a mathematical framework for a large class of quantum codes, known as ....

P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A 52, 2493 (1995).

....like 1 p 2 j00 : 0i 1 p 2 j11 : 1i 2 C 2 n ; 9 are available. However, if a single qubit is observed in this superposition, then it collapeses either to j00 : 0i or to j11 : 1i. A very simple code that encodes one qubit into nine qubits was proposed by Peter Shor [5]. This code is not particularly efficient, but it illustrates all essential features of a quantum error control code. I just give a rough outline, more details can be found in [3, 4, 5] The two states j0i and j1i are encoded as follows: j0i 7 j0i = 1 2 p 2 (j000i j111i) Omega (j000i ....

....j00 : 0i or to j11 : 1i. A very simple code that encodes one qubit into nine qubits was proposed by Peter Shor [5] This code is not particularly efficient, but it illustrates all essential features of a quantum error control code. I just give a rough outline, more details can be found in [3, 4, 5]. The two states j0i and j1i are encoded as follows: j0i 7 j0i = 1 2 p 2 (j000i j111i) Omega (j000i j111i) Omega (j000i j111i) j1i 7 j1i = 1 2 p 2 (j000i Gamma j111i) Omega (j000i Gamma j111i) Omega (j000i Gamma j111i) Suppose that a single qubit is affected by ....

P. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 2:2493--2496, 1995. 11

....of undetected error and weight enumerators to derive bounds on this probability for quantum codes. Index Terms Measurement, quantum code, Shor Laflamme enumerators, undetected error. I. INTRODUCTION T HE possibility of correcting decoherence errors in entangled states was discovered by Shor [20] and Steane [21] Since then the theory of quantum codes has been a topic of intense study. Error processes in the depolarizing channel are characterized in [12] where it was shown that one can restrict attention to error operators given by Kronecker products of Pauli matrices. This opened the ....

P. W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, vol. 52, no. 4, pp. 2493--2496, 1995.

....dealt with decoherence errors by extending classical error correction techniques to quantum analogs. Generally, there is assumed a decoherence error model where the errors introduced are assumed to be uniform random with bounded magnitude, independently for each qubit. 17 ffl Quantum Codes. Shor[Sho95] and Steane[Ste96a] gave the first techniques for reducing quantum decoherence, by the addition of extra qubits which are then projected via observation operations to eliminate errors in the superposition. Calderbank, Shor [CS95] and Steane [Ste96b] then proved that QC can be done with bounded ....

P. Shor, Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493, (1995).

....of successful recovery for any k bits of data is at least 1 Gamma an ( ffi) error correcting code. The subject of error correcting codes for quantum information is much younger, developing within the past couple of years, though it has received considerable attention during this time [1 14]. Much of the above terminology extends naturally to quantum information by considering qubits instead of bits. Call a quantum code that maps k qubit data to n qubit codewords an ( n; k) code. We need to specify the behavior of noisy quantum channels. A natural quantum analogue of the first ....

P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A Vol. 52, No. 4, pp. 2493--2496 (1995).

....to resist and reverse the effects of decoherence. This discovery has important consequences for quantum computation, 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 ....

Shor, P. 1995 Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493.

....had to take place immediately after the required preparation. Thus, although working prototypes of quantum teleportation have recently been demonstrated [8,9] quantum teleportation across significant time and space will have to await a technology that allows for the efficient long term storage [28,17,30] and purification [6,7] of quantum information. Nevertheless, it may be that shortdistance quantum teleportation will play a role in transporting quantum information inside quantum computers. Thus we see that the fates of quantum computing and quantum teleportation are inseparably entangled 2 ....

Shor, Peter W., "Scheme for reducing decoherence in quantum memory", Physical Review A, Vol. 52, no. 4, October 1995, pp. 2493--2496.

....dealt with decoherence errors by extending classical error correction techniques to quantum analogs. Generally, there is assumed a decoherence error model where the errors introduced are assumed to be uniform random with bounded magnitude, independently for each qubit. 4.3 Quantum Codes. Shor[Sho95] and Steane[Ste96a] gave the first techniques for reducing quantum decoherence, by the addition of extra qubits which are then projected via observation operations to eliminate errors in the superposition. Calderbank, Shor [CS95] and Steane [Ste96b] then proved that QC can be done with bounded ....

P. Shor, Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493, (1995).

....appear to protect classical information by duplicating it, so because of the theorem that a quantum bit cannot be cloned, it was widely believed that these techniques could not be applied to quantum information. That quantum error correcting codes could indeed exist was recently shown by one of us [32]. At around the same time, Bennett et al. 2] discovered that two experimenters each holding one component of many noisy Einstein Podolsky Rosen (EPR) pairs could purify them, yielding fewer nearly perfect EPR pairs. The resulting pairs can then be used to teleport quantum information from one ....

P. W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, 52, p. 2493 (1995).

....computer [7] It has widely been assumed that the quantum no cloning theorem [8] makes error correction impossible in quantum communication and computation because redundancy cannot be obtained by duplicating quantum bits. This argument was shown to be in error for quantum communication in Ref. [9], where a code was given that mapped one qubit (two state quantum system) into nine qubits so that the original qubit could be recovered perfectly even after arbitrary decoherence of any one of these nine qubits. This gives a quantum code on 9 qubits with rate 1 9 that protects against one ....

....jA e i X e 00 E jA 0 e 00 i X e 0 E ( Gamma1) e 0 Deltae 00 ja e 0 ;e i; which is just jc w i tensored with a state of the ancillae and the environment that does not depend on w. We have thus unitarily restored the original state and corrected t decohered bits. 2 As in Ref. [9], we correct the error by measuring the decoherence without disturbing the encoded information. Intuitively, what we do is to measure the decoherence without observing the encoded state; this then lets us correct the decoherence while leaving the encoded state unchanged. In our decoding procedure, ....

P. W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A 52, 2493 (1995).

....appear to protect classical information by duplicating it, so because of the theorem that a quantum bit cannot be cloned, it was widely believed that these techniques could not be applied to quantum information. That quantum error correcting codes could indeed exist was recently shown by one of us [65]. Two of us [17] then showed that a class of good quantum codes could be obtained by using a construction that starts with a binary linear code C containing its dual C . Independently, Steane also discovered the existence of quantum codes [73] and the same construction [72] At around the same ....

P. W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, 52, p. 2493 (1995).

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P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A 52, 2493 (1995).

No context found.

P.W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A Vol. 52, No. 4, pp. 2493--2496 (1995).

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P. W. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 52:2493, 1995.

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Shor, P., "Scheme for reducing decoherence in quantum memory." PR A. 52:2493.

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