Extractors for Circuit Sources

We give an algorithm that learns any monotone Boolean function $\fisafunc$ to any constant accuracy, under the uniform distribution, in time polynomial in n and in the decision tree size of $f.$ This is the first algorithm that can learn arbitrary monotone Boolean functions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of $f.$ A key ingredient of the result is a new bound showing that the average sensitivity of any monotone function computed by a decision tree of size s must be at most $\sqrt{\log s}$. This bound has proved to be of independent utility in the study of decision tree complexity [O. Schramm, R. O'Donnell, M. Saks, and R. Servedio, Every decision tree has an influential variable, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2005, pp. 31–39]. We generalize the basic inequality and learning result described above in various ways—specifically, to partition size (a stronger complexity measure than decision tree size), p-biased measures over the Boolean cube (rather than just the uniform distribution), and real-valued (rather than just Boolean-valued) functions.

We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are: (1) We extract k(k/nd) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> bits with exponentially small error from n-bit sources of min-entropy k that are generated by functions f : {0,1} → {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> where each output bit depends on ≤ d input bits. In particular, we extract from NC sources, corresponding to d = O(1). (2) We extract k(k/n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+γ</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> bits with super-polynomially small error from ri-bit sources of min-entropy k that are generated by poly(n)-size AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sup> circuits, for any γ >; 0. As our starting point, we revisit the connection by Trevisan and Vadhan (FOCS 2000) between circuit lower bounds and extractors for sources generated by circuits. We note that such extractors (with very weak parameters) are equivalent to lower bounds for generating distributions (FOCS 2010; with Lovett, CCC 2011). Building on those bounds, we prove that the sources in (1) and (2) are (close to) a convex combination of high-entropy "bit-block" sources. Introduced here, such sources are a special case of affine ones. As extractors for (1) and (2) one can use the extractor for low-weight affine sources by Rao (CCC 2009). Along the way, we exhibit an explicit boolean function b : {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> → {0,1} such that poly(n)-size AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sup> circuits cannot generate the distribution (Y, b(Y)), solving a problem about the complexity of distributions. Independently, De and Watson (RANDOM 2011) obtain a result similar to (1) in the special case d = o(lg n).

The 2-way quantum finite automaton introduced by Kondacs and Watrous [Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite automata, was introduced by Moore and Crutchfield [ Theoret. Comput. Sci., 237 (2000), pp. 275--306], and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous [Proceedings of the38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75]. We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measure-many automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable sets can be accepted with bounded error by a measure-many quantum finite automaton, introducing new construction techniques for quantum automata in the process.

We show that Nechiporuk's method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an $\Omega(n^2/\log^2 n)$ lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.

The strict avalanche criterion was introduced by Webster and Tavares [] in order to combine the ideas of completeness and the avalanche effect. A cryptographic transformation is complete if each output bit depends on all the input bits, and it exhibits the avalanche effect if an average of one half of the output bits change whenever a single input bit is complemented. To fulfil the strict avalanche criterion, each output bit should change with probability one half whenever a single input bit is complemented. This means, in particular, that there is no good lower order (fewer bits) approximation to the function. This is clearly a desirable cryptographic property since such an approximation would enable a corresponding reduction in the amount of work needed for an exhaustive search.

We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an f-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al. (In: SODA’14, 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor (Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS) 2002, pp. 481--489. IEEE Computer Society, 2002). Our main technical tool for establishing the approximation guarantee is a separation lemma that certifies the existence of a strong partition for solutions that are basic feasible in an extended version of the natural LP. We believe that our rounding technique is of independent interest when hard constraints are considered.

This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when $\eta$, the probability for an error in a qubit or a gate, is smaller than a constant threshold $\eta_c$. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate $\eta$ decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with $p>2$ states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be $\eta_c\simeq 10^{-6}$, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.

Quantum technology application is a rapidly developing discipline in future-generation research. It has grown its appeal due to several unique aspects which cannot be encountered in classical domains. Quantum properties can be used to create powerful and effective technology. The processing capability of a quantum computer is triggered by managing quantum particles in a quantum computer, which improves transformation in many domains. Quantum logic operations have the significant benefit of being reversible. Unlike classical irreversible processing, quantum logic gates are reversible and unitary, allowing input bits to be recovered via a one-to-one connection between input and output bits. In QCADesigner, as well as in the Quantum environment, different reversible logic gates may be constructed. Some other promising features include Superposition (Quantum Parallelism), Entanglement, etc. This study uses IBM quantum to create the QCA-based Majority voter gate.

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

This paper considers the basic $\mathcal{PULL}$ model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of $n$ agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast. We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property that uses $\ell$-bits messages to one that uses only $\log \ell$-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer $T$, within $\tilde O(\log n\log T)$ rounds w.h.p., and using messages that contain $3$ bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in $\tilde O(\log n)$ time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.

This work presents the coder and decoder of block mBnB of the type 1B2B or Manchester. In the coder 1B2B each block/word of 1 input bit is coded in the block / word of 2 output bits. In the decoder 1B2B happen the inverse, each block of 2 input bits is newly converted in the original block of 1 output bit. The coder injects in the transmission line a number of 1’s exactly equal to the number of 0’s, what guarantees a DC constant component and maximizes the transitions number. The objective is to implement the pair coder and decoder 1B2B so that it improves the transmission quality and increases the information security. Keywords: Block codes, Digital systems, Transmission lines

The authors theoretically demonstrate the implementation of a low-power 4-state universal logic gate (NOR) using a linear array of three dipole-coupled magnetostrictive-piezoelectric multiferroic nanomagnets (e.g. Ni/PZT) with biaxial magnetocrystalline anisotropy. The two peripheral nanomagnets in the array encode the 4-state input bits in their magnetization orientations and the central nanomagnet's magnetization orientation represents the output bit. Numerical simulations are performed to confirm that the 4-state output bit is the Boolean NOR function of the two 4-state inputs bits when the array reaches its ground state. A voltage pulse alternating between −0.2 and +0.2 V, applied to the piezoelectric layer of the central nanomagnet, generates alternating tensile and compressive stress in its magnetostrictive layer. This drives the array to the correct ground state where dipole interaction between the magnets ensures that the output is the NOR function of the input. For the system considered, the gate operation is executed while dissipating only ∼33 000 kT (0.138 fJ) of energy.

With the growing availability of experimental loophole-free Bell tests1–5, it has become possible to implement a new class of device-independent random number generators whose output can be certified6,7 to be uniformly random without requiring a detailed model of the quantum devices used8–10. However, all these experiments require many input bits to certify a small number of output bits, and it is an outstanding challenge to develop a system that generates more randomness than is consumed. Here we devise a device-independent spot-checking protocol that consumes only uniform bits without requiring any additional bits with a specific bias. Implemented with a photonic loophole-free Bell test, we can produce 24% more certified output bits (1,181,264,237) than consumed input bits (953,301,640). The experiment ran for 91.0 h, creating randomness at an average rate of 3,606 bits s–1 with a soundness error bounded by 5.7 × 10−7 in the presence of classical side information. Our system allows for greater trust in public sources of randomness, such as randomness beacons11, and may one day enable high-quality private sources of randomness as the device footprint shrinks. Device-independent randomness expansion is demonstrated in an experiment that is secure in the presence of a classical eavesdropper who does not share any entanglement with the setup.

Multiplicative complexity of vector valued Boolean functions

48th Annual IEEE Symposium on Foundations of Computer Science - Title