2026-09-18T19:51:45Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1585162025-04-07T08:30:02Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Natarajan, Anand
Zhang, Tina
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2025-03-12T16:57:07Z
2025-03-12T16:57:07Z
2024-09
2025-03-04T18:48:36.290Z
https://hdl.handle.net/1721.1/158516
The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz [AIM14]. In the quantum world, there are two complexity classes that can be considered quantum analogues of classical free games: (1) AM⇤, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes.
1. We show a BellQMA(2) protocol for 3SAT on n variables, where the total amount of communication is Õ(√n). This answers an open question of Chen and Drucker [CD10] and also shows, conditional on ETH, that the algorithm of Brandao, Christandl and Yard [BCY10] for optimizing ˜ over separable states is tight up to logarithmic factors.
2. We show that AM*[ⁿprovers = 2, q = O(1), a = poly log(n)] = RE, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. (In contrast, [AIM14] shows that classical free games are much weaker than general classical games.) We show this using a question “hyper-compression” theorem that iteratively applies the introspection technique of Ji et al. [JNV⁺20]. Our result is a significant improvement over the headline result of Ji et al., whose MIP⇤ protocol for the halting problem has poly(n)-sized questions and answers.
3. By the same techniques, we obtain a zero-gap AM* protocol for a P2 complete language with constant-size questions and almost logarithmically (O(log n · log* n)) large answers, improving on the headline result of Mousavi, Nezhadi and Yuen [MNY21].
4. Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires W(log n) bits of communication. It follows that our results in item 3 are optimal up to an O(log* n) factor, and that the gapless compression theorems of [MNY21] are asymptotically optimal. We conjecture that these bounds can be saturated in the gapped case as well.
S.M.
S.M.
Massachusetts Institute of Technology
In Copyright - Educational Use Permitted
Copyright retained by author(s)
https://rightsstatements.org/page/InC-EDU/1.0/
Quantum free games
Thesis
application/pdf
Master
Master of Science
Master of Science in Electrical Engineering and Computer Science
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="e6beb155-9533-4755-82e4-d0f1c5c226f4">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Title>Quantum free games</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2024-09</PublicationDate>
<Authors>
<Author>
<DisplayName>Zhang, Tina</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>https://rightsstatements.org/page/InC-EDU/1.0/</License>
<Abstract>The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz [AIM14]. In the quantum world, there are two complexity classes that can be considered quantum analogues of classical free games: (1) AM⇤, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes.
1. We show a BellQMA(2) protocol for 3SAT on n variables, where the total amount of communication is Õ(√n). This answers an open question of Chen and Drucker [CD10] and also shows, conditional on ETH, that the algorithm of Brandao, Christandl and Yard [BCY10] for optimizing ˜ over separable states is tight up to logarithmic factors.
2. We show that AM*[ⁿprovers = 2, q = O(1), a = poly log(n)] = RE, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. (In contrast, [AIM14] shows that classical free games are much weaker than general classical games.) We show this using a question “hyper-compression” theorem that iteratively applies the introspection technique of Ji et al. [JNV⁺20]. Our result is a significant improvement over the headline result of Ji et al., whose MIP⇤ protocol for the halting problem has poly(n)-sized questions and answers.
3. By the same techniques, we obtain a zero-gap AM* protocol for a P2 complete language with constant-size questions and almost logarithmically (O(log n · log* n)) large answers, improving on the headline result of Mousavi, Nezhadi and Yuen [MNY21].
4. Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires W(log n) bits of communication. It follows that our results in item 3 are optimal up to an O(log* n) factor, and that the gapless compression theorems of [MNY21] are asymptotically optimal. We conjecture that these bounds can be saturated in the gapped case as well.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>