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23
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 46 (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
An introduction to measurement based quantum computation, ArXiv: quantph/0508124
, 2005
"... In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement o ..."
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Cited by 18 (1 self)
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In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the socalled cluster model or oneway quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms. 1
Simulating quantum computation by contracting tensor networks
 SIAM Journal on Computing
, 2005
"... The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Am ..."
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Cited by 14 (1 self)
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The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Among many implications, we show efficient simulations for quantum formulas, defined and studied by Yao (Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 352–361, 1993), and for logdepth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that oneway quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188– 5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a smalltreewidth graph.
Level reduction and the quantum threshold theorem
 PH.D. THESIS, CALTECH, 2007, EPRINT ARXIV:QUANTPH/0703230
, 2007
"... ..."
Types for Quantum Computation
, 2007
"... This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract qua ..."
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Cited by 11 (5 self)
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This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in lowlevel formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly
Measurementbased quantum turing machines and their universality. Quantph/0404146
, 2004
"... Abstract. Quantum measurement is universal for quantum computation (Nielsen [8], Leung [5,6], Raussendorf [13,14]). This universality allows alternative schemes to the traditional threestep organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order ..."
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Cited by 9 (1 self)
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Abstract. Quantum measurement is universal for quantum computation (Nielsen [8], Leung [5,6], Raussendorf [13,14]). This universality allows alternative schemes to the traditional threestep organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurementbased computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurementbased Quantum Turing Machines. 1
How quantum computers can fail
"... We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system w ..."
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Cited by 4 (2 self)
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We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system with many highly correlated elements there will be a strong effect of error synchronization. These postulates appear to be damaging for quantum computers.
Codeword Stabilized Quantum Codes
, 708
"... Abstract — We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In pa ..."
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Cited by 4 (2 self)
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Abstract — We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, we find ((10,18, 3)) and ((10,20, 3)) codes. We also show how to construct encoding circuits for all codes within our framework. Index Terms — quantum error correction, nonadditive codes, stabilizer codes I.
Detrimental Decoherence
, 2007
"... We propose and discuss two conjectures on the nature of information leaks (decoherence) for quantum computers. These conjectures, if (or when) they hold, are damaging for quantum errorcorrection as required by faulttolerant quantum computation. The first conjecture asserts that information leaks ..."
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Cited by 2 (1 self)
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We propose and discuss two conjectures on the nature of information leaks (decoherence) for quantum computers. These conjectures, if (or when) they hold, are damaging for quantum errorcorrection as required by faulttolerant quantum computation. The first conjecture asserts that information leaks for a pair of substantially entangled qubits are themselves substantially positively correlated. The second conjecture asserts that in a noisy quantum computer with highly entangled qubits there will be a strong effect of error synchronization. We present a more general conjecture for arbitrary noisy quantum systems: prescribing (or describing) noisy quantum systems at a state ρ is subject to error E which “tends to commute ” with every unitary operator that stabilizes ρ.
Quantum Computers: Noise Propagation and Adversarial Noise Models
, 2009
"... In this paper we consider adversarial noise models that will fail quantum error correction and faulttolerant quantum computation. We describe known results regarding highrate noise, sequential computation, and reversible noisy computation. We continue by discussing highly correlated noise and the ..."
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Cited by 2 (1 self)
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In this paper we consider adversarial noise models that will fail quantum error correction and faulttolerant quantum computation. We describe known results regarding highrate noise, sequential computation, and reversible noisy computation. We continue by discussing highly correlated noise and the “boundary, ” in terms of correlation of errors, of the “threshold theorem. ” Next, we draw a picture of adversarial forms of noise called (collectively) “detrimental noise.” Detrimental noise is modeled after familiar properties of noise propagation. However, it can have various causes. We start by pointing out the difference between detrimental noise and standard noise models for two qubits and proceed to a discussion of highly entangled states, the rate of noise, and general noisy quantum systems. Research supported in part by an NSF grant, an ISF grant, and a BSF grant.