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PhD Proposal: Robust self-testing and the complexity of constant-sized quantum correlations
Honghao Fu
Virtual via Zoom (click here): https://umd.zoom.us/j/7856728753
Monday, March 23, 2020, 2:00-4:00 pm Calendar
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Abstract
Quantum computers can achieve unprecedented efficiency when solving certain computational tasks over classical computers. The recent technology advances have made the future of a universal quantum computer more promising. At the same time, the following question is inevitably raised: how can a classical verifier test a quantum device with only classical interactions? Previous robust self-testing results have established that certain measurement statistics imply that the underlying quantum system and the measurements applied can be certified. The only assumption behind such assertions is that the quantum device contains two spatially separated components, i.e., nonlocal components.

The goal of my thesis is to develop an understanding about the phenomenon of self-testing concerning the number of questions and answers required, and its implication on the set of quantum correlations. Specifically, we aim to prove that constant number of questions and answers are sufficient for robust self-testing of maximally entangled states with unbounded local dimension. Building on the self-testing result, we plan to prove that there exists a family of constant-sized correlations such that the problem of determining if a member of this family is quantum is undecidable.

We show that for any prime odd integer d with smallest primitive root r, there exists a correlation of size Θ(r) that can robustly self-test a maximally entangled state of dimension 4d-4. Since there are infinitely many prime numbers whose smallest multiplicative generators are at most 5 (The Mathematical Intelligencer, 10.4 (1988)), our result implies that constant-sized correlations are sufficient for robust self-testing of maximally entangled states with unbounded local dimension.

In the next step, we plan to modify the Kharlampovich-Myasnikov-Sapir group that can simulate any Minsky Machine (Bulletin of Mathematical Sciences, Vol. 7.2 (2017)), such that in the modified group there is a special group element which is nontrivial if and only if the Minsky Machine does not accept some input. The modified group can be embedded into the solution group of a linear system game by applying Slofstra's embedding theorem (Forum of Mathematics, Pi. Vol. 7, (2019)), and the resulting solution group is of fixed size. Then we combine the winning correlation with each member of the family of correlations from the self-testing result to form a new family of constant-sized correlations. Using techniques from the self-testing result and group theory, we would like to prove that any member of the family of the combined correlations is quantum if and only if the Minsky Machine does not halt on some input.

Examining Committee: 
 
                          Chair:               Dr. Carl Miller
                          Dept rep:         Dr.  William Gasarch
                          Members:        Dr. Andrew Childs
Bio

Honghao Fu is a Ph.D. student at the University of Maryland's Department of Computer Science, working under the supervision of Prof. Carl Miller. He is interested in questions about quantum correlations and, in particular, the complexity aspect of quantum correlations.

This talk is organized by Tom Hurst