QuArC

In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations for non-abelian quasiparticles described by the Fibonacci anyon model. We develop a probabilistically polynomial algorithm that outputs a braid pattern to approximate a given single-qubit unitary to a desired precision. We also classify the single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern and present an efficient algorithm to produce their braid patterns. Our techniques produce braid patterns that meet the uniform asymptotic lower bound on the compiled circuit depth and thus are depth-optimal asymptotically. Our compiled circuits are significantly shorter than those output by prior state-of-the-art methods, resulting in improvements in depth by factors ranging from 20 to 1000 for precisions ranging between 10 −10 and 10 −30 .

User behavior on the Web changes over time. For example, the queries that people issue to search engines, and the underlying informational goals behind the queries vary over time. In this paper, we examine how to model and predict user behavior over time. We develop a temporal modeling framework adapted from physics and signal processing that can be used to predict time-varying user behavior using smoothing and trends. We also explore other dynamics of Web behaviors, such as the detection of periodicities and surprises. We develop a learning procedure that can be used to construct models of users' activities based on features of current and historical behaviors. The results of experiments indicate that by using our framework to predict user behavior, we can achieve significant improvements in prediction compared to baseline models that weight historical evidence the same for all queries. We also develop a novel learning algorithm that explicitly learns when to apply a given prediction model among a set of such models. Our improved temporal modeling of user behavior can be used to enhance query suggestions, crawling policies, and result ranking.

Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a 2.5 times reduction in expected depth over ancilla-free techniques for single-qubit unitary decomposition. However, the previously best-known algorithm to synthesize RUS circuits requires exponential classical runtime. In this work we present an algorithm to synthesize an RUS circuit to approximate any given singlequbit unitary within precision ε in probabilistically polynomial classical runtime. Our synthesis approach uses the Clifford+T basis, plus one ancilla qubit and measurement. We provide numerical evidence that our RUS circuits have an expected T -count on average 2.5 times lower than the theoretical lower bound of 3 log 2 (1/ε) for ancilla-free single-qubit circuit decomposition.

Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of T gates of about 1/3 of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+T gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford+T gate set. The protocol, which we call Probabilistic Quantum Circuits with Fallback (PQF), implements a walk on a discrete Markov chain in which the target unitary is an absorbing state and in which transitions are induced by multi-qubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols terminate after a finite number of steps. Specifically, we apply our method to the Clifford+T , Clifford+V , and Clifford+π/12 gate sets to achieve decompositions with expected gate counts of log b (1/ε)+O(log(log(1/ε))), where b is a quantity related to the expansion property of the underlying universal gate set. arXiv:1409.3552v1 [quant-ph]