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135
Reliable quantum computers
 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 ..."
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Cited by 127 (3 self)
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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 106, would be a formidable factoring engine. Even a smaller lessaccurate 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
A functional quantum programming language
 In: Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
, 2005
"... This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are inte ..."
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Cited by 47 (12 self)
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This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms
Unknown quantum states: the quantum de Finetti representation
 J. Math. Phys
"... We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanc ..."
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Cited by 45 (7 self)
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We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable densityoperator assignments and provides an operational definition of the concept of an “unknown quantum state ” in quantumstate tomography. This result is especially important for informationbased interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point. I.
Quantum analog of the MacWilliams identities in classical coding theory
, 1997
"... We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight ..."
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Cited by 36 (2 self)
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We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provided a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power. 89.70.+c,89.80.th,02.70.c,03.65.w Typeset using REVT E X The discovery of error correcting codes [1,2] for quantum computers has revolutionized the field of quantum information. Although quantum computing holds great promise, it is plagued by the fragility of quantum information [36]. Quantum error correction is a technique which enables one to encode quantum information in a robust way and therefore overcome this fragility. It is important...
Faulttolerant quantum computation with local gates
 Jour. of Modern Optics
"... I discuss how to perform faulttolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when nexttonearestneighbor gates are available, and present explicit constructions. In ..."
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Cited by 32 (0 self)
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I discuss how to perform faulttolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when nexttonearestneighbor gates are available, and present explicit constructions. In two or three dimensions, I also show how nearestneighbor gates can give a threshold result. In all cases, I simply demonstrate that a threshold exists, and do not attempt to optimize the error correction circuit or determine the exact value of the threshold. The additional overhead due to the faulttolerance in both space and time is polylogarithmic in the error rate per logical gate. 1
M.: Simple quantum errorcorrecting codes
 Phys. Rev. A
, 1996
"... Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C ⊥ 2 ⊆ C1, where C1 and C2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More ..."
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Cited by 32 (1 self)
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Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C ⊥ 2 ⊆ C1, where C1 and C2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to singleerror correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively. PACS numbers 03.65.Bz,89.70.+c Two recent papers have shown that efficient quantum errorcorrecting codes exist [1, 2]. A quantum errorcorrecting code is a method of storing or transmitting K bits of quantum information using n> K qubits[3], in such a way that if an arbitrary subset of the n qubits undergoes arbitrary errors, the transmitted quantum information can nevertheless be recovered exactly. The only condition is that the values of n and K, and
Graphs, quadratic forms and quantum codes
 In Proc. IEEE Int. Symp. Inf. Th. 2002. IEEE Inf. Th. Soc
, 2002
"... Abstract — We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic for ..."
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Cited by 26 (4 self)
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Abstract — We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms. We provide some simple examples to illustrate our results.
Information and Computation: Classical and Quantum Aspects
 REVIEWS OF MODERN PHYSICS
, 2001
"... Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely ..."
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Cited by 23 (2 self)
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Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental
Building quantum wires: the long and the short of it
 In Proc. International Symposium on Computer Architecture (ISCA 2003
, 2003
"... As quantum computing moves closer to reality the need for basic architectural studies becomes more pressing. Quantum wires, which transport quantum data, will be a fundamental component in all anticipated silicon quantum architectures. In this paper, we introduce a quantum wire architecture based up ..."
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Cited by 22 (8 self)
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As quantum computing moves closer to reality the need for basic architectural studies becomes more pressing. Quantum wires, which transport quantum data, will be a fundamental component in all anticipated silicon quantum architectures. In this paper, we introduce a quantum wire architecture based upon quantum teleportation. We compare this teleportation channel with the traditional approach to transporting quantum data, which we refer to as the swapping channel. We characterize the latency and bandwidth of these two alternatives in a deviceindependent way and describe how the advanced architecture of the teleportation channel overcomes a basic limit to the maximum communication distance of the swapping channel. In addition, we discover a fundamental tension between the scale of quantum effects and the scale of the classical logic needed to control them. This “pitchmatching ” problem imposes constraints on minimum wire lengths and wire intersections, which in turn imply a sparsely connected architecture of coarsegrained quantum computational elements. This is in direct contrast to the “sea of gates ” architectures presently assumed by most quantum computing studies. 1