On Bounded Advice Classes (2405.18155v1)
Abstract: Advice classes in computational complexity have frequently been used to model real-world scenarios encountered in cryptography, quantum computing and machine learning, where some computational task may be broken down into a preprocessing and deployment phase, each associated with a different complexity. However, in these scenarios, the advice given by the preprocessing phase must still be generated by some (albeit more powerful) bounded machine, which is not the case in conventional advice classes. To better model these cases we develop `bounded advice classes', where a more powerful Turing machine generates advice for another, less powerful, Turing machine. We then focus on the question of when various classes generate useful advice, to answer this we connect bounded advice to unary languages. This connection allows us to state various conditional and unconditional results on the utility of advice generated by $\mathsf{EXP}$, $\mathsf{NP}$, $\mathsf{BQP}$, $\mathsf{PSPACE}$, and more. We study the relations between bounded advice classes, quantum bounded advice classes, and randomised bounded advice. We also examine how each of these concepts interact with recently introduced classes, like $\mathsf{BPP/samp}$. Our results also improve the state of the art in existing research on the complexity of advice functions.
- Scott Aaronson. Nsa in p/poly: The power of precomputation, 2015. Published on May 22nd, 2015.
- A qubit, a coin, and an advice string walk into a relational problem. arXiv preprint arXiv:2302.10332, 2023.
- A full characterization of quantum advice. SIAM Journal on Computing, 43(3):1131–1183, 2014.
- Complexity zoo, 2023. Accessed: date.
- Imperfect forward secrecy: How diffie-hellman fails in practice. In Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, pages 5–17, 2015.
- Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences, 52(3):421–433, 1996.
- Does provable absence of barren plateaus imply classical simulability? or, why we need to rethink variational quantum computing. arXiv preprint arXiv:2312.09121, 2023.
- R. Gavalda. The complexity of learning with queries. In Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory, pages 324–337, 1994.
- R. Gavalda. Bounding the complexity of advice functions. Journal of Computer and System Sciences, 50(3):468–475, 1995.
- Juris Hartmanis. On sparse sets in np–p. Information Processing Letters, 16(2):55–60, 1983.
- Power of data in quantum machine learning. Nature communications, 12(1):2631, 2021.
- Shadows of quantum machine learning. arXiv preprint arXiv:2306.00061, 2023.
- On circuit-size complexity and the low hierarchy in np. SIAM Journal on Computing, 14(1):41–51, 1985.
- Complexity-restricted advice functions. SIAM Journal on Computing, 23(2):261–275, 1994.
- New collapse consequences of np having small circuits. SIAM Journal on Computing, 28(1):311–324, 1998.
- Classically approximating variational quantum machine learning with random fourier features. arXiv preprint arXiv:2210.13200, 2022.
- Alexander Meduna. Turing Transducers, pages 833–887. Springer London, London, 2000.
- N. Rajgopal and R. Santhanam. On the structure of learnability beyond P/Poly. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), volume 207 of Leibniz International Proceedings in Informatics (LIPIcs), pages 46:1–46:23, Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. Sections 2.4, 3.
- Classical surrogates for quantum learning models. Physical Review Letters, 131(10):100803, 2023.
- Ilya Volkovich. On learning, lower bounds and (un) keeping promises. In International Colloquium on Automata, Languages, and Programming, pages 1027–1038. Springer, 2014.
- John Watrous. Quantum computational complexity. arXiv preprint arXiv:0804.3401, 2008.