Small Depth Quantum Circuits

Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058

A Generalized Quantum Branching Program

Classical branching programs are studied to understand the space complexity of computational problems. Prior to this work, Nakanishi and Ablayev had separately defined two different quantum versions of branching programs that we refer to as NQBP and AQBP. However, none of them, to our satisfaction, captures the intuitive idea of being able to query different variables in superposition in one step of a branching program traversal. Here we propose a quantum branching program model, referred to as GQBP, with that ability. To motivate our definition, we explicitly give examples of GQBP for n-bit Deutsch-Jozsa, n-bit Parity, and 3-bit Majority with optimal lengths. We the show several equivalences, namely, between GQBP and AQBP, GQBP and NQBP, and GQBP and query complexities (using either oracle gates and a QRAM to query input bits). In way this unifies the different results that we have for the two earlier branching programs, and also connects them to query complexity. We hope that GQBP can be used to prove space and space-time lower bounds for quantum solutions to combinatorial problems.Comment: 21 page

Few Quantum Algorithms on Amplitude Distribution

Amplitude filtering is concerned with identifying basis-states in a superposition whose amplitudes are greater than a specified threshold; probability filtering is defined analogously for probabilities. Given the scarcity of qubits, the focus of this work is to design log-space algorithms for them. Both algorithms follow a similar pattern of estimating the amplitude (or, probability for the latter problem) of each state, in superposition, then comparing each estimate against the threshold for setting up a flag qubit upon success, finally followed by amplitude amplification of states in which the flag is set. We show how to implement each step using very few qubits by designing three subroutines. Our first algorithm performs amplitude amplification even when the "good state'' operator has a small probability of being incorrect -- here we improve upon the space complexity of the previously known algorithms. Our second algorithm performs "true amplitude estimation'' in roughly the same complexity as that of "amplitude estimation'', which actually estimates a probability instead of an amplitude. Our third algorithm is for performing amplitude estimation in parallel (superposition) which is difficult when each estimation branch involves different oracles. As an immediate reward, we observed that the above algorithms for the filtering problems directly improved the upper bounds on the space-bounded query complexity of problems such as non-linearity estimation of Boolean functions and $k$-distinctness.Comment: 35 page