D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing, pages 176--188, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is at least . 8) 5.2 Specific examples Let be a power of two n = 2 . Let C 1 be a [2 1] 2 m Extended Reed Solomon code and C 2 be a [2 1, 2 ] 2 m Extended Reed Solomon code. The CSS code obtained from C 1 , C 2 can correct 2 1 = n 2 1 erasure errors: [2 , 1, 2 ]] 2 m = n, 1, n 2] n . The related code obtained from our construction would need 2m 20s 2 log m #(m s) bits [9, 3] of quantum authentication key per component, to obtain # 2 # error probability. Each component also contains a secret share of n such keys, thus needs w (m ....

D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error rate. quantph /9906129. Preliminary version in STOC '97. Submitted to SIAM J. Comp., June 1999.

....codes for classical data, on the way to building multi party computing protocols. Let be a QECC that can correct d 1 n 2 arbitrary erasure errors: n, k, d] Such a code can be constructed over sufficiently large dimension Q; for instance, use a polynomial quantum code [1]. The coding space of is defined as Q ## # 3# . # Hn . We assume dim(H 1 ) dim(H 2 ) dim(Hn ) We construct a new code # over larger Hilbert spaces that can correct d except with small probability. Register i of the n component code # contains the following: k ....

....is at least . 8) 5.2 Specific examples Let be a power of two n = 2 . Let C 1 be a [2 1] 2 m Extended Reed Solomon code and C 2 be a [2 1, 2 ] 2 m Extended Reed Solomon code. The CSS code obtained from C 1 , C 2 can correct 2 1 = n 2 1 erasure errors: [2 , 1, 2 ]] 2 m = n, 1, n 2] n . The related code obtained from our construction would need 2m 20s 2 log m #(m s) bits [9, 3] of quantum authentication key per component, to obtain # 2 # error probability. Each component also contains a secret share of n such keys, thus needs w (m ....

D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proc. of 29th STOC, pages 176--188, El Paso, Texas, 4--6 May 1997. This is a preliminary version of [2].

....on the fact that at certain points in the computation, at most one error is likely to occur. Such algorithms will fail when errors are placed adversarially. Techniques from ftqc are nonetheless useful for multi party computing. We will draw most heavily on techniques due to Aharonov and Ben Or [2]. 1.2 Definitionsand Model In this paper, we use a simple simulation based framework for proving the security of quantum framework, similar to early classical definitions. We specify a task by giving a protocol for implementing it in an ideal model where players have access to a trusted third ....

....a classical code W , then we can write the CSS code as # FW . Thus, is the set of states of n qubits which yield a codeword of V when measured in the computational basis and a codeword of W when measured in the Fourier basis. Specifically, we will use quantum Reed Solomon codes from [2]. We specify a quantum RS code by a single parameter # n 2. The classical Reed Solomon code is the set of all vectors q = q(1) q(2) q(n) where q is any univariate polynomial of degree at most #. The related code V 0 is the subset of V corresponding to polynomials which ....

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D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error rate. quant-ph/9906129. Preliminary version in STOC '97. Submitted to SIAM J. Comp., June 1999.

....on the fact that at certain points in the computation, at most one error is likely to occur. Such algorithms will fail when errors are placed adversarially. Techniques from ftqc are nonetheless useful for multi party computing. We will draw most heavily on techniques due to Aharonov and Ben Or [3]. 1.2 Definitions and Model In this paper, we use a simple simulation based framework for proving the security of quantum framework, similar to early classical definitions [7] We specify a task by giving a protocol for implementing it in an ideal model where players have access to a trusted ....

....of that data, encoded with the dual code defined via the codes W,V . For quantum RS codes, rescaling each component of the dual code of produces the code , enabling us to perform the map ## ## E C is the encoding map for a code C. A nice result of fault tolerant computing [3, 20] is that when # n 3 (i.e. t n 6) one can in fact perform any operation on data encoded by a quantum RS code using only local operations and classical information transmitted between the components. So long as the classical communication is reliable, then these procedures can tolerate ....

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D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error rate. Los Alamos eprint quant-ph/9906129. Journal version of [2] (submitted to SIAM J. Comp.), June 1999.

....Quantum Protocols Relatively little work exists on multi party cryptographic protocols for quantum computers. Secret sharing with a quantum secret was rst studied by Cleve, Gottesman and Lo [CGL99] They suggested a generalization of the Shamir scheme, which is also used by Aharonov and Ben Or [AB99] as an error correcting code. One of the contributions of [CGL99] was that to point out the strong connection between secret sharing and error correcting codes in the quantum setting (see Section 1.3.2) Our vqss protocol is based on the [CGL99] scheme, using a modi cation of the techniques of ....

....the computation, at most one error is likely to occur. Such algorithms will fail in a model of adversarially placed errors. Techniques from ftqc are nonetheless useful for multi party computing. Considerable research has been done on ftqc. We rely mainly on the techniques of Aharonov and Ben Or [AB99] which were based on those of Shor [Sho96] Using css quantum error correcting codes, Shor showed that fault tolerance was possible so long as the error rate in the computer decreased logarithmically with the size of the computation being performed. Aharonov and Ben Or showed that by using ....

[Article contains additional citation context not shown here]

Dorit Aharonov and Michael Ben-Or. Fault tolerant quantum computation with constant error rate. Los Alamos eprint quant-ph/9906129. Journal version of [AB97] (submitted to SIAM J. Comp.), June 1999.

....on the fact that at certain points in the computation, at most one error is likely to occur. Such algorithms will fail when errors are placed adversarially. Techniques from ftqc are nonetheless useful for multi party computing. We will draw most heavily on techniques due to Aharonov and Ben Or [2]. 1.2 Definitions and Model In this paper, we use a simple simulation based framework for proving the security of quantum framework, similar to early classical de nitions. We specify a task by giving a protocol for implementing it in an ideal model where players have access to a trusted third ....

....a classical code W , then we can write the CSS code as C = V FW . Thus, C is the set of states of n qubits which yield a codeword of V when measured in the computational basis and a codeword of W when measured in the Fourier basis. Speci cally, we will use quantum Reed Solomon codes from [2]. We specify a quantum RS code by a single parameter n=2. The classical Reed Solomon code is the set of all vectors q = q(1) q(2) q(n) where q is any univariate polynomial of degree at most . The related code V 0 is the subset of V corresponding to polynomials which ....

[Article contains additional citation context not shown here]

D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error rate. quant-ph/9906129. Preliminary version in STOC '97. Submitted to SIAM J. Comp., June 1999.

....since the 1.2. MOTIVATION 3 popular implementation models of quantum computing cavity QED, trapped cold ions, or bulk spin NMR all allow, at least in principle, to use more than just two level quantum systems. Moreover, the concatenation of codes used in fault tolerant architectures [1] is most naturally understood in terms of quantum codes with bigger alphabets. We allow arbitrary alphabet sizes for that reason. 1.2 Motivation A quantum computer stores its information in the state of quantum systems. The computational state space of a quantum system is a nite dimensional ....

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. of the 29th Annual ACM Symposium on Theory of Computation (STOC), pages 176-188, New York, 1997. ACM.

....physically realizable . Accuracy Threshold Theorem 1: Assume the requirements for scalable computing. If the error per gate (including no op ) is less than a threshold, then it is possible to efficiently quantum compute arbitrarily accurately. Shor 1996[1] Kitaev 1996[2] Aharonov Ben Or 1996[3], Knill al. 1996[4] 4 TOC Finite Quantum Systems I A finite quantum system Q is determined by: A Hilbert space Q. dimQ = N . Logical orthonormal basis: e 1 , e N . Unit vectors of Q ## pure states of Q. Examples (N = 2) #i e 1 .6 e 1 .8 e 2 # = i e 1 ) # ( 6 e 1 ....

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computation (STOC), pages 176--188, New York, New York, 1996. ACM Press.

....shown that the set consisting of a Hadamard gate, a c NOT gate, and a phase rotation gate of angle =4 is universal. In order to form a practical basis for quantum computation, a universal set must also be able to operate in a noisy environment, and therefore has to be fault tolerant[Sho95, Sho96, AB97, Kit97, KLZ98] The above set of three gates has the additional advantage of also being fault tolerant. Experimental procedures for determining the properties of quantum black boxes were given by Chuang and Nielsen[CN97] and Poyatos, Cirac and Zoller[PCZ97] however these procedures implicitly ....

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. 29th STOC, pp. 46--55, 1997.

.... , X n are independent with X i having a # a i distribution then (1) X 1 X 1 X 2 , X 1 X 2 X 1 X 2 X 3 , X 1 X n 1 X 1 X n , X 1 X n are independent with X 1 X i X 1 X i 1 having density #(A, B; x) #(A) #(A)#(B) x A 1 (1 x) B 1 on [0, 1] for A = a 1 a i , B = a i 1 and S n = X 1 X n having density # a1 an . 2) S n is independent of the vector # X 1 S n , X n S n # . BOUNDS FOR KAC S MASTER EQUATION 7 Lemma 2.2. Let Z 1 , Z 2 , Z n be independent standard Gaussian random variables. Then ....

..... Z n be independent standard Gaussian random variables. Then (1) # Z 1 # Z 2 1 Z 2 n , Z n # Z 2 1 Z 2 n # is uniformly distributed on the n sphere with first coordinate having density #( n 2 ) #( 1 2 )#( n 1 2 ) 1 x 2 ) n 3 2 on [ 1, 1]. 2) Let W 1 , W n 1 be standard Gaussian variables, independent of each other and of Z 1 , Z n in (1) Let A = W 1 # W 2 1 W 2 n 1 , B = # W 2 2 W 2 n 1 W 2 1 W 2 n 1 = # 1 A 2 . Then A, B # Z 1 # Z 2 1 Z 2 n , ....

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Aharonov, D. and Ben-Or, M. (1997), Fault tolerant quantum computation with constant error, Proc. 29 th S.T.O.C., Assoc. Comp. Mach. New-York, 176-188.

....have recently shown that the set consisting of a Hadamard gate, a c NOT gate, and a phase rotation gate of angle # 4 is universal. In order to form a practical basis for quantum computation, a universal set must also be able to operate in a noisy environment, and therefore has to be fault tolerant[32, 2, 20, 22]. The above set of three gates has the additional advantage of also being fault tolerant. Experimental procedures for determining the properties of quantum black boxes were given by Chuang and Nielsen[12] and Poyatos, Cirac and Zoller[26] however these procedures implicitly require apparatus ....

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. 29th STOC, pp. 46--55, 1997.

....In particular, we introduce a new method for parallel multiprecision phase estimation. An immediate benefit of the QFT circuits from Theorem 1 regards fault tolerant implementations of the QFT. Using the most efficient techniques known for fault tolerant implementation of quantum circuits (see [1, 25, 30]) our circuits for the QFT can be implemented with a size increase of only a poly logarithmic factor, to O(n(log n) c ) In contrast, these techniques result in at least a linear increase in size for any linear depth approximate QFT for instance, for the approximate QFT circuits in [13] the ....

D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing, pages 176--188, 1997.

....for all the results in this paper. The technique is known as the hybrid argument among cryptographers. Using quantum error correction techniques, faulttolerant quantum circuits can be created that are resilient to constant error in the rotation gates, independent of the size of the circuit [AB, Go, KLZ]. How does one explain the power of quantum computation The dimension of the Hilbert space associated with an n qubit system is 2 n . Therefore, just describing the state of this system requires 2 n complex numbers. Moreover, Nature must update the 2 n complex numbers to evolve the system ....

Aharanov, D., and Ben-Or, M., "Fault tolerant quantum computation with constant error", quant-ph/9611025.

....a (x)i. A quantum circuit for their algorithm (slightly improved [14] is shown in Fig. 2. The first set of gates is the same as The tensor factors need not be two dimensional, i.e. qubits. Higher dimensional factors have been considered in the context of error correction [20] and fault tolerance [21]. But in every case the dimension is bounded and scaling to larger problems is achieved using polynomially many tensor factors. Ahn, Weinacht Bucksbaum have demonstrated single factor operations [11] gate operations analogous to controlled NOT on two Rydberg atoms would be required for such an ....

D. Aharonov and M. Ben-Or, "Fault tolerant quantum computation with constant error", quant-ph/9611025; D. Gottesman, "Fault-tolerant quantum computation with higher-dimensional systems ", Chaos, Solitons and Fractals 10 (1999) 1749--1758.

....have recently shown that the set consisting of a Hadamard gate, a c NOT gate, and a phase rotation gate of angle =4 is universal. In order to form a practical basis for quantum computation, a universal set must also be able to operate in a noisy environment, and therefore has to be fault tolerant[32, 2, 20, 22]. The above set of three gates has the additional advantage of also being fault tolerant. Experimental procedures for determining the properties of quantum black boxes were given by Chuang and Nielsen[12] and Poyatos, Cirac and Zoller[26] however these procedures implicitly require apparatus ....

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. 29th STOC, pp. 46-55, 1997.

....Quantum Computation Steane (University of Oxford) and Shor discovered quantum error correcting codes [77, 72, 33] that opened the way to fault tolerant computation [73] and partially solved the problem of decoherence. This initial work was extended by Aharonov and Ben Or (participant HUJI) [1, 61]. Cleve (participant Univ.Calgary) also did initial work on quantum error correcting codes [38, 34] Ekert and Macchiavello gave the analogues of the Hamming and Gilbert Varshamov bounds in the quantum case [46] Quantum Computation and Testing [80] was initiated by participants (LRI, CWI, UW) ....

D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of 29th STOC, pages 176--188, 1997. quant-ph/9611025.

....seriously affects the effectiveness of the quantum error correction. This shows that with present technology large scale factorization, or other demanding applications, seem out of reach. In the second part we address the problem of so called accuracy thresholds in quantum computation [13,14]. In [13,14] it is noted that arbitrarily complicated (long) quantum computations can be performed, once the error rate of a quantum gate can be pushed below a certain threshold. We will discuss whether the required thresholds presented in [13,14] can be achieved or if spontaneous emission rules ....

....seriously affects the effectiveness of the quantum error correction. This shows that with present technology large scale factorization, or other demanding applications, seem out of reach. In the second part we address the problem of so called accuracy thresholds in quantum computation [13,14] In [13,14] it is noted that arbitrarily complicated (long) quantum computations can be performed, once the error rate of a quantum gate can be pushed below a certain threshold. We will discuss whether the required thresholds presented in [13,14] can be achieved or if spontaneous emission rules out this ....

[Article contains additional citation context not shown here]

D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, preprint (1996).

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D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing, pages 176--188, 1997.

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D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error, Proc. ACM STOC, pp. 176--188, 1997. quant-ph/9906129.

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D. Aharonov and M. Ben-Or, \FaultTolerant Quantum Computation with Constant Error," Proc. of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 46-55, 1997.

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D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error, in Proc. ACM STOC'1997.

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D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error rate. submitted to SIAM journal of computation. quant-ph/9906129.

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D. Aharonov and M. Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 176--188, El Paso, Texas, 4--6 May 1997. This is a preliminary version of [3].

No context found.

Dorit Aharonov and Michael Ben-Or. Fault tolerant quantum computation with constant error. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 176-188, El Paso, Texas, 4-6 May 1997. This is a preliminary version of [AB99].

No context found.

D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. 29th STOC, pp. 46--55, 1997.

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