2019 Fiscal Year Annual Research Report
Theoretical Research on Quantum Supremacy
Project/Area Number 
19F19079

Research Institution  Nagoya University 
Host Researcher 
ルガル フランソワ 名古屋大学, 多元数理科学研究科, 准教授 (50584299)

Foreign Research Fellow 
ROSMANIS ANSIS 名古屋大学, 多元数理科学研究科, 外国人特別研究員

Project Period (FY) 
20190724 – 20210331

Keywords  Quantum computing / Complexity theory / Algorithms 
Outline of Annual Research Achievements 
In the past year, my main research achievements concern the precise characterization of the resources necessary for quantum computers to solve certain computational tasks. In particular, these tasks are the quantum approximate counting and the quantum coupon collection. The two papers that my collaborators and I wrote on these tasks were accepted and presented at the 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). In the approximate counting problem, there is a finite set with a subset of its elements marked, and one is asked to estimate the number of marked elements. In the quantum version of this problem, one can determine which elements are marked through various quantum computational resources. The study of how much of each resource is required to solve this task was started by Aaronson et al., but their characterization of resource requirements was incomplete. We managed to prove a complete characterization and, in doing so, to generalize one of the main methods for placing limitations on the power of quantum algorithms. In the quantum coupon collection, similarly, there is a set with some of its elements marked, but, for this problem, one is asked to find all the marked elements, only given specific quantum states that encode which of the elements are marked. We proved that, under certain conditions, this task can be solved faster by quantum computers than an analogue task by classical computers. In addition, we proved that our quantum algorithm is optimal, meaning that no further improvements of it are possible.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I believe my research is going well. In addition to research problems that we have already solved, there are others on which we have substantial progress. In the area of quantum query algorithms, one of the bestknown open problems is that of precisely characterizing the quantum query complexity of finding 3collisions in the input. We are currently working on adapting some of the recently introduced techniques for analyzing a closelyrelated task in the cryptographic setting, where random inputs are addressed. We have already obtained a partial adaptation for a slightly simpler task, that of finding 2collisions, and I believe that this approach can be further generalized. In the area of distributed quantum computing, we are investigating how many rounds of communication are required for solving locally checkable problems, that is, problems for which the correctness of the output can be quickly checked. In the case when the topology of the network corresponds to the cycle graph, we have already shown that quantum communication cannot significantly speedup computation of problems that are global in their nature. Also, for the cycle graph, no nontrivial limitations on the power of quantum networks were known for problems that are local in their nature. One such fundamental problem is the graph 3coloring. We have produced and executed computer tests that show that there are certain nontrivial limitations on the power of quantum distributed algorithms for 3coloring, and we are currently in the process of proving and strengthening these results analytically.

Strategy for Future Research Activity 
We will continue research on quantum query complexity, trying to establish tight query lower bounds for the 3collision problem. Some of the techniques that we plan to use have been recently used for analyzing quantum timespace tradeoffs. We plan to see if this might help to precisely characterize timespace tradeoffs for the 2collision problem. Also, recently researchers have started to study quantum query algorithms that can access the input in multiple diverse ways. Within the last year we have better understood how to analyze capabilities of such algorithms, but multiple questions remain open. One of them is to understand the power of these algorithms when their probability of success is not required to be large. In the area of distributed quantum computing, we will continue to investigate the round complexity of locally checkable problems, namely, how many rounds of communication are required to solve them. For classical networks, the round complexity classifies all locally checkable problems in few and very distinct classes. We would like to see if a similar classification can be established in the quantum setting. Quantum computation is performed by logical quantum circuits. There are significant similarities between studying quantum circuits and quantum networks, as in both scenarios one has to study how quickly and widely information can spread within the system. We plan to investigate what is the minimum depth of quantum circuits for constructing certain entangled quantum states, in particular, states that are codewords in quantum error correcting codes.

Research Products
(2 results)