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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [Grilo et al., 2014]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [Jeronimo and Wu, 2023]) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some other constant gap is equal to QMA. One interpretation is that Merlin’s ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆ NEXP.

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and Proofs Without Relative Phase. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bassirian_et_al:LIPIcs.ITCS.2024.9, author = {Bassirian, Roozbeh and Fefferman, Bill and Marwaha, Kunal}, title = {{Quantum Merlin-Arthur and Proofs Without Relative Phase}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.9}, URN = {urn:nbn:de:0030-drops-195370}, doi = {10.4230/LIPIcs.ITCS.2024.9}, annote = {Keywords: quantum complexity, QMA(2), PCPs} }

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**Published in:** LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function f, and present f in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in f has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness.

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. On the Power of Nonstandard Quantum Oracles. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 11:1-11:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bassirian_et_al:LIPIcs.TQC.2023.11, author = {Bassirian, Roozbeh and Fefferman, Bill and Marwaha, Kunal}, title = {{On the Power of Nonstandard Quantum Oracles}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {11:1--11:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-283-9}, ISSN = {1868-8969}, year = {2023}, volume = {266}, editor = {Fawzi, Omar and Walter, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.11}, URN = {urn:nbn:de:0030-drops-183215}, doi = {10.4230/LIPIcs.TQC.2023.11}, annote = {Keywords: quantum complexity, QCMA, expander graphs, representation theory} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study random constraint satisfaction problems (CSPs) at large clause density. We relate the structure of near-optimal solutions for any Boolean Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP’s predicate is, up to a constant, the mixture polynomial of the associated spin glass. We show two main consequences:
1) We prove that the maximum fraction of constraints that can be satisfied in a random Max-CSP at large clause density is determined by the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula [Parisi, 1980; Talagrand, 2006; Auffinger and Chen, 2017], we provide numerical values for some popular CSPs.
2) We prove that a Max-CSP at large clause density possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from [Huang and Sellke, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms [Chou et al., 2022].

Chris Jones, Kunal Marwaha, Juspreet Singh Sandhu, and Jonathan Shi. Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 77:1-77:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jones_et_al:LIPIcs.ITCS.2023.77, author = {Jones, Chris and Marwaha, Kunal and Sandhu, Juspreet Singh and Shi, Jonathan}, title = {{Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {77:1--77:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.77}, URN = {urn:nbn:de:0030-drops-175804}, doi = {10.4230/LIPIcs.ITCS.2023.77}, annote = {Keywords: spin glass, overlap gap property, constraint satisfaction problem, Guerra-Toninelli interpolation} }

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**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth p. We apply the QAOA to MaxCut on large-girth D-regular graphs. We give an iterative formula to evaluate performance for any D at any depth p. Looking at random D-regular graphs, at optimal parameters and as D goes to infinity, we find that the p = 11 QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these D-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-q-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as O(p² 4^p). This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to p = 20. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as p goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.

Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou. The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{basso_et_al:LIPIcs.TQC.2022.7, author = {Basso, Joao and Farhi, Edward and Marwaha, Kunal and Villalonga, Benjamin and Zhou, Leo}, title = {{The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.7}, URN = {urn:nbn:de:0030-drops-165144}, doi = {10.4230/LIPIcs.TQC.2022.7}, annote = {Keywords: Quantum algorithm, Max-Cut, spin glass, approximation algorithm} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of randomized classical algorithms.
1) We prove that every (quantum or classical) one-local algorithm (where the value of a vertex only depends on its and its neighbors' state) achieves on D-regular graphs of girth > 5 a maximum cut of at most 1/2 + C/√D for C = 1/√2 ≈ 0.7071. This is the first such result showing that one-local algorithms achieve a value that is bounded away from the true optimum for random graphs, which is 1/2 + P_*/√D + o(1/√D) for P_* ≈ 0.7632 [Dembo et al., 2017].
2) We show that there is a classical k-local algorithm that achieves a value of 1/2 + C/√D - O(1/√k) for D-regular graphs of girth > 2k+1, where C = 2/π ≈ 0.6366. This is an algorithmic version of the existential bound of [Lyons, 2017] and is related to the algorithm of [Aizenman et al., 1987] (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-local versions of QAOA on high-girth graphs [M. B. Hastings, 2019; Marwaha, 2021].
3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constant-locality QAOA for random D-regular graphs, as well as other natural instances, including graphs that do have short cycles.
While our theoretical bounds require the locality and girth assumptions, our experimental work suggests that it could be possible to extend them beyond these constraints. This points at the tantalizing possibility that O(1)-local quantum maximum-cut algorithms might be pointwise dominated by polynomial-time classical algorithms, in the sense that there is a classical algorithm outputting cuts of equal or better quality on every possible instance. This is in contrast to the evidence that polynomial-time algorithms cannot simulate the probability distributions induced by local quantum algorithms.

Boaz Barak and Kunal Marwaha. Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 14:1-14:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{barak_et_al:LIPIcs.ITCS.2022.14, author = {Barak, Boaz and Marwaha, Kunal}, title = {{Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {14:1--14:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.14}, URN = {urn:nbn:de:0030-drops-156105}, doi = {10.4230/LIPIcs.ITCS.2022.14}, annote = {Keywords: approximation algorithms, QAOA, maximum cut, local distributions} }

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