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Quantum Boolean Functions
, 2009
"... In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich ..."
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Cited by 10 (4 self)
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In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the GoldreichLevin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of
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.
Contents
, 2010
"... In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich ..."
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In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the GoldreichLevin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality
Formulas Resilient to ShortCircuit Errors
"... We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to shortcircuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a shortcircuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E compu ..."
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We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to shortcircuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a shortcircuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E computes the same function as F, as long as at most (1/10 − ε) of the gates on each path from the output to an input have been corrupted in E. The corruptions may be chosen adversarially, and may depend on the formula E and even on the input. We obtain our result by extending the KarchmerWigderson connection between formulas and communication protocols to the setting of adversarial error. This enables us to obtain errorresilient formulas from errorresilient communication protocols.
SURVEY ON THE BOUNDS OF THE QUANTUM FAULTTOLERANCE THRESHOLD
"... I rst brie y summarize the threshold theorem and describe the motivations for tightening the bounds on the threshold quantum decoherence rate. I then go on to summarize and organize recent results regarding both the lower and the upper ..."
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I rst brie y summarize the threshold theorem and describe the motivations for tightening the bounds on the threshold quantum decoherence rate. I then go on to summarize and organize recent results regarding both the lower and the upper
The Role of Correlated Noise in Quantum Computing
, 2011
"... This paper aims to give an overview of the current state of faulttolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent ..."
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This paper aims to give an overview of the current state of faulttolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent noise, following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also present a result by Terhal and Burkard [TB05] and later improved upon by Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be obtained for local nonMarkovian noise, where we allow the noise to be weakly correlated in space and time. We then turn to negative results, presenting work by BenAroya and TaShma [BT11] who showed conditional errors cannot be perfectly corrected. We end our survey by briefly mentioning some more speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11]. 1