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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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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 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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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.
Quantum circuits of Tdepth one
 Physical Review A
"... We give a Clifford+T representation of the Toffoli gate of Tdepth 1, using four ancillas. More generally, we describe a class of circuits whose Tdepth can be reduced to 1 by using sufficiently many ancillas. We show that the cost of adding an additional control to any controlled gate is at most 8 ..."
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We give a Clifford+T representation of the Toffoli gate of Tdepth 1, using four ancillas. More generally, we describe a class of circuits whose Tdepth can be reduced to 1 by using sufficiently many ancillas. We show that the cost of adding an additional control to any controlled gate is at most 8 additional Tgates, and Tdepth 2. We also show that the circuit THT does not possess a Tdepth 1 representation with an arbitrary number of ancillas initialized to 0〉. 1
‘Computational Complexity of Quantum Hamiltonian Systems ’ in Leiden.
, 2008
"... A survey of classical simulation methods ..."
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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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
Dequantizing Readonce Quantum Formulas
"... Quantum formulas, defined by Yao [FOCS’93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any readonce quantum formula over a gate set that contains all singlequbit gates is equivalent to a readonce classical formula of th ..."
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Quantum formulas, defined by Yao [FOCS’93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any readonce quantum formula over a gate set that contains all singlequbit gates is equivalent to a readonce classical formula of the same size and depth over an analogous classical gate set. For example, any readonce quantum formula over Toffoli and singlequbit gates is equivalent to a readonce classical formula over Toffoli and not gates. We then show that the equivalence does not hold if the readonce restriction is removed. To show the power of quantum formulas without the readonce restriction, we define a new model of computation called the onequbit model and show that it can compute all boolean functions. This model may also be of independent interest.