Error-Tolerant Version Space Algebra

ABSTRACT Application customization has been extensively researched in the field of Programming by Demonstration (PBD), and Version Space Algebra has proven itself to be a viable means of quickly learning precise action sequences from user demonstrations. However, this technique is not capable of handling user error in domains with actions that depend on parameters that accept myriad values. Activities such as image, audio and video editing require user actions that are difficult for users to precisely replicate in different circumstances. Demonstrations that are off by a single pixel or a split-second cause traditional composite Version Spaces to collapse. We present a method of incorporating error tolerance into Version Space algebra. This approach, termed Error-Tolerant Version Spaces, adapts Version Space Algebra to domains where the tactile capabilities of the user have a much greater chance of prematurely collapsing the hypothesis space that is being learned. The resulting framework is capable of quickly learning in domains where perfectly consistent user input can not be expected. We have successfully applied our technique in the domain of image redaction, allowing our users to quickly specify redactions that can be reliably applied to many images without the entry of explicit parameters

Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (*natural fault-tolerant computation*) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one or two-body interactions must leave the codespace during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus *asymptotically universal*.Comment: 40 pages (body: 30, appendices: 3, figures: 5, references: 2). Fixed problem with non-printing figures. New references added, minor typos correcte

Fault-tolerant logical gates in quantum error-correcting codes

Recently, Bravyi and K\"onig have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and applications of their characterization in various directions. First, we present a new no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. Second, we prove that the code distance of a D-dimensional local stabilizer code with non-trivial locality-preserving m-th level Clifford logical gate is upper bounded by $O(L^{D+1-m})$. For codes with non-Clifford gates (m>2), this improves the previous best bound by Bravyi and Terhal. Third we prove that a qubit loss threshold of codes with non-trivial transversal m-th level Clifford logical gate is upper bounded by 1/m. As such, no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes, and is not restricted to geometrically-local codes. Fourth we extend the result of Bravyi and K\"onig to subsystem codes. A technical difficulty is that, unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig is recovered.Comment: 13 pages, 4 figure

Correcting soft errors online in fast fourier transform

While many algorithm-based fault tolerance (ABFT) schemes have been proposed to detect soft errors offline in the fast Fourier transform (FFT) after computation finishes, none of the existing ABFT schemes detect soft errors online before the computation finishes. This paper presents an online ABFT scheme for FFT so that soft errors can be detected online and the corrupted computation can be terminated in a much more timely manner. We also extend our scheme to tolerate both arithmetic errors and memory errors, develop strategies to reduce its fault tolerance overhead and improve its numerical stability and fault coverage, and finally incorporate it into the widely used FFTW library - one of the today's fastest FFT software implementations. Experimental results demonstrate that: (1) the proposed online ABFT scheme introduces much lower overhead than the existing offline ABFT schemes; (2) it detects errors in a much more timely manner; and (3) it also has higher numerical stability and better fault coverage