Suppressing decoherence of quantum algorithms by jump codes

The stabilizing properties of one-error correcting jump codes are explored under realistic non-ideal conditions. For this purpose the quantum algorithm of the tent-map is decomposed into a universal set of Hamiltonian quantum gates which ensure perfect correction of spontaneous decay processes under ideal circumstances even if they occur during a gate operation. An entanglement gate is presented which is capable of entangling any two logical qubits of different one-error correcting code spaces. With the help of this gate simultaneous spontaneous decay processes affecting physical qubits of different code spaces can be corrected and decoherence can be suppressed significantly

Attacking the Knudsen-Preneel Compression Functions

Abstract. Knudsen and Preneel (Asiacrypt’96 and Crypto’97) introduced a hash function design in which a linear error-correcting code is used to build a wide-pipe compression function from underlying blockciphers operating in Davies-Meyer mode. Their main design goal was to deliver compression functions with collision resistance up to, and even beyond, the block size of the underlying blockciphers. In this paper, we (re)analyse the preimage resistance of the Knudsen-Preneel compression functions in the setting of public random func-tions. We give a new preimage attack that is based on two observations. First, by using the right kind of queries it is possible to mount a non-adaptive preimage attack that is optimal in terms of query complexity. Second, by exploiting the dual code the subsequent problem of reconstructing a preimage from the queries can be rephrased as a problem related to the generalized birthday problem. As a consequence, the time complexity of our attack is intimately tied to the minimum distance of the dual code. Our new attack consistently beats the one given by Knudsen and Preneel (in one case our preimage attack even beats their collision attack) and demonstrates that the gap between their claimed collision resistance and the actual preimage resistance is surprisingly small. Moreover, our new attack falsifies their (conjectured) preimage resistance security bound and shows that intuitive bounds based on the number of ‘active ’ components can be treacherous. Complementing our attack is a formal analysis of the query complexity (both lower and upper bounds) of preimage-finding attacks. This analysis shows that for many concrete codes the time complexity of our attack is optimal.

Qudit Colour Codes and Gauge Colour Codes in All Spatial Dimensions

Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate, and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and K\"onig's bound [Phys. Rev. Lett. 110, 170503 (2013)]. Furthermore, by using gauge fixing techniques, an effective set of Clifford gates can be achieved, removing the need for state distillation. We show that the qudit color code can support the qudit analogues of these gates, and show that in higher spatial dimensions a color code can support a phase gate from higher levels of the Clifford hierarchy which can be proven to saturate Bravyi and K\"onig's bound in all but a finite number of special cases. The methodology used is a generalisation of Bravyi and Haah's method of triorthogonal matrices [Phys. Rev. A 86 052329 (2012)], which may be of independent interest. For completeness, we show explicitly that the qudit color codes generalize to gauge color codes, and share the many of the favorable properties of their qubit counterparts.Comment: Authors' final cop

A Behavioral and Neural Evaluation of Prospective Decision-Making under Risk

Making the best choice when faced with a chain of decisions requires a person to judge both anticipated outcomes and future actions. Although economic decision-making models account for both risk and reward in single-choice contexts, there is a dearth of similar knowledge about sequential choice. Classical utility-based models assume that decision-makers select and follow an optimal predetermined strategy, regardless of the particular order in which options are presented. An alternative model involves continuously reevaluating decision utilities, without prescribing a specific future set of choices. Here, using behavioral and functional magnetic resonance imaging (fMRI) data, we studied human subjects in a sequential choice task and use these data to compare alternative decision models of valuation and strategy selection. We provide evidence that subjects adopt a model of reevaluating decision utilities, in which available strategies are continuously updated and combined in assessing action values. We validate this model by using simultaneously acquired fMRI data to show that sequential choice evokes a pattern of neural response consistent with a tracking of anticipated distribution of future reward, as expected in such a model. Thus, brain activity evoked at each decision point reflects the expected mean, variance, and skewness of possible payoffs, consistent with the idea that sequential choice evokes a prospective evaluation of both available strategies and possible outcomes