We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithm
s as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to fs block sensitivity.
An obfuscator is an algorithm that translates circuits into functionally-equivalent similarly-sized circuits that are hard to understand. Efficient obfuscators would have many applications in cryptography. Until recently, theoretical progress has mai
nly been limited to no-go results. Recent works have proposed the first efficient obfuscation algorithms for classical logic circuits, based on a notion of indistinguishability against polynomial-time adversaries. In this work, we propose a new notion of obfuscation, which we call partial-indistinguishability. This notion is based on computationally universal groups with efficiently computable normal forms, and appears to be incomparable with existing definitions. We describe universal gate sets for both classical and quantum computation, in which our definition of obfuscation can be met by polynomial-time algorithms. We also discuss some potential applications to testing quantum computers. We stress that the cryptographic security of these obfuscators, especially when composed with translation from other gate sets, remains an open question.
We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data st
ructures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the $O(n^{35/27})$ and $O(n^{9/7})$ learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.
