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Level reduction and the quantum threshold theorem
 PH.D. THESIS, CALTECH, 2007, EPRINT ARXIV:QUANTPH/0703230
, 2007
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ACCURACY THRESHOLD FOR POSTSELECTED QUANTUM COMPUTATION
, 2008
"... We prove an accuracy threshold theorem for faulttolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated errordetecting code ..."
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Cited by 14 (2 self)
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We prove an accuracy threshold theorem for faulttolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated errordetecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated errorcorrecting codes) not to stronglycorrelated adversarial noise. Our rigorously established lower bound on the accuracy threshold, 1.04 × 10 −3, is well below Knill’s numerical estimates.
Upper Bounds on the Noise Threshold for Faulttolerant Quantum Computing
, 2008
"... We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kqubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary onequbit gates that are essentially noisefree. We assume that the ..."
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Cited by 8 (0 self)
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We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kqubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary onequbit gates that are essentially noisefree. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p> 1 − Θ(1 / √ k), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k = 2, our bound is p> 35.7%. Moreover, if the only allowed gate on more than one qubit is the twoqubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds. 1
Quantum universality by state distillation
, 2009
"... Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This “magic states distillation ” question is closely related to quantum fault tolerance. Lower bounds on the noi ..."
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Cited by 5 (1 self)
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Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This “magic states distillation ” question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for faulttolerant computation. Upper bounds show the limits of threshold upperbound arguments based on the GottesmanKnill theorem. We extend the range of singlequbit mixed states that are known to give universality, by using a simple paritychecking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure. No distillation upper bounds are known beyond those given by the GottesmanKnill theorem. One might ask whether distillation upper bounds reduce to upper bounds for singlequbit ancilla states. For multiqubit pure states and previously considered twoqubit ancilla states, the answer is yes. However, we exhibit twoqubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multiqubit state distillation methods. 1
Accuracy threshold for postselected quantum computation
 Quantum Information and Computation
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On the Structure of Protocols for Magic State Distillation
, 908
"... Abstract. We present a theorem that shows that all useful protocols for magic state distillation output states with a fidelity that is upperbounded by those generated by a much smaller class of protocols. This reduced class consists of the protocols where multiple copies of a state are projected ont ..."
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Abstract. We present a theorem that shows that all useful protocols for magic state distillation output states with a fidelity that is upperbounded by those generated by a much smaller class of protocols. This reduced class consists of the protocols where multiple copies of a state are projected onto a stabilizer codespace and the logical qubit is then decoded. 1
Accepted by.........................................................
, 2008
"... This thesis explores the use of entangled states in quantum computation and quantum information science. Entanglement, a quantum phenomenon with no classical counterpart, has been identified as an important and quantifiable resource in many areas of theoretical quantum information science, including ..."
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This thesis explores the use of entangled states in quantum computation and quantum information science. Entanglement, a quantum phenomenon with no classical counterpart, has been identified as an important and quantifiable resource in many areas of theoretical quantum information science, including quantum error correction, quantum cryptography, and quantum algorithms. We first investigate the equivalence classes of a particular class of entangled states (known as graph states due to their association with mathematical graphs) under local operations. We prove that for graph states corresponding to graphs with neither cycles of length 3 nor 4, the equivalence classes can be characterized in a very simple way. We also present software for analyzing and manipulating graph states. We then study quantum errorcorrecting codes whose codewords are highly entangled states. An important area of investigation concerning QECCs is to determine which resources are necessary in order to carry out any computation on the code to an arbitrary degree of accuracy, while simultaneously maintaining a high degree