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Fault-tolerant quantum computation with constant error, (1999)

by D Aharonov, M Ben-Or
Venue:STOC
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Quantum Error Correction Via Codes Over GF(4)

by A. R. Calderbank, E. M. Rains, P. W. Shor, N. J. A. Sloane , 1997
"... The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
Abstract - Cited by 304 (18 self) - Add to MetaCart
The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
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...computations on encoded qubits using imperfect gates have been extended to general additive codes by Gottesman [37]. However, the most efficient methods currently known for fault-tolerant computation =-=[2]-=-, [44], [48], [75] use only Calderbank–Shor–Steane codes (cf. Theorem 9). iv) It turns out that the proofs of the lower bounds on the capacity of quantum channels given in Bennett et al. [4], [5] and ...

Stabilizer Codes and Quantum Error Correction

by Daniel Gottesman , 1997
"... ..."
Abstract - Cited by 248 (3 self) - Add to MetaCart
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Fault-tolerant quantum computation by anyons

by A. Yu. Kitaev , 2003
"... A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
Abstract - Cited by 229 (3 self) - Add to MetaCart
A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.
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... occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation [2], with subsequent improvements =-=[3, 4, 5, 6]-=-. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortunately, the threshold value of δ is rather ...

Reliable quantum computers

by John Preskill - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 1998
"... The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mist ..."
Abstract - Cited by 165 (3 self) - Add to MetaCart
The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 10-6, would be a formidable factoring engine. Even a smaller less-accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence
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...to how long a computation can proceed until errors become likely. This limitation can be overcome by using a special kind of code, a concatenated code (Knill & Laflamme 1996; Knill et al. 1996, 1997; =-=Aharonov & Ben-Or 1996-=-; Kitaev 1996ab). To understand the concept of a concatenated code, imagine that we are using Steane’s quantum error-correcting code that encodes a single qubit as a block of 7 qubits. But if we look ...

A modular functor which is universal for quantum computation

by Michael H. Freedman, Michael Larsen, Zhenghan Wang - Comm. Math. Phys
"... Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based o ..."
Abstract - Cited by 117 (18 self) - Add to MetaCart
Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere. 1.
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...emanding threshold [P] required for software error correction. In this case modular functors and the usual theory of fault tolerance must be fitted together. This is possible using the perspective in =-=[AB]-=- and an argument for this sketched within the proof of Thm. 2.2. However, a comprehensive discussion of the interaction of the environment with topological degrees of freedom, and how computational st...

Efficient simulation of quantum systems by quantum computers

by Christof Zalka , 1998
"... We show that the time evolution of the wave function of a quantum-mechanical manyparticle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a per ..."
Abstract - Cited by 79 (0 self) - Add to MetaCart
We show that the time evolution of the wave function of a quantum-mechanical manyparticle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a performance which can’t be expected from any classical simulation of a quantum system. We then show how quantities of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers.
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... quantum computer is no longer in a pure quantum state. Another problem is the accumulation of errors due to inaccuracies in the induced unitary transformations. It has been shown (Knill et al. 1996; =-=Aharonov & Ben-Or 1996-=-; Zalka 1996a, b) that with fault tolerant quantum error correcting codes (Shor 1995) these problems can be suppressed very efficiently once the hardware reaches a certain quality (noise threshold). A...

Fast parallel circuits for the quantum Fourier transform

by Richard Cleve, John Watrous - PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00) , 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
Abstract - Cited by 72 (1 self) - Add to MetaCart
We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial-size, in combination with classical polynomial-time pre- and post-processing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with bounded-error probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the

Improved simulation of stabilizer circuits

by Scott Aaronson, Daniel Gottesman - Phys. Rev. Lett
"... The Gottesman-Knill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we ..."
Abstract - Cited by 66 (6 self) - Add to MetaCart
The Gottesman-Knill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely-available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any n-qubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements. 1
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...e should point out that fault-tolerance hardware is likely to consist mainly of CNOT, Hadamard, and phase gates, since the known fault-tolerant constructions (for example, that of Aharonov and Ben-Or =-=[2]-=-) are based on stabilizer codes. Although there has been some previous work on synthesizing CNOT circuits [16, 23] and general classical reversible circuits [24, 17], to our knowledge there has not be...

Quantum accuracy threshold for concatenated distance-3 codes

by Daniel Gottesman , 2005
"... ..."
Abstract - Cited by 65 (11 self) - Add to MetaCart
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...he case of concatenated distance-3 codes, and also, as extended in Theorem 4, to concatenation of codes with larger distance. Here we will present another proof, really an elaboration of the proof in =-=[2]-=-, which applies to concatenation of codes with distance 5 or more, but not to codes with distance 3. As in our previous proof, the key is to define notions of goodness and correctness so that “good im...

Nonbinary stabilizer codes over finite fields

by Avanti Ketkar, Santosh Kumar, Pradeep Kiran Sarvepalli - IEEE Trans. Inform. Theory , 2006
"... One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum comput ..."
Abstract - Cited by 50 (11 self) - Add to MetaCart
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper. 1
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...stabilizer codes is that there exist links to classical coding theory which ease the construction of good codes. More recently, some results were generalized to the case of nonbinary stabilizer codes =-=[1,3,4,16,23,24,34,35,41,47,49,56,64,70,79,83,85,86]-=-, but the theory is not nearly as complete as in the binary case. We recall the basic principles of nonbinary stabilizer codes over finite fields in the next section. In Section 3, we introduce a Galo...

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