On Bounded Advice Classes

TL;DR

The paper introduces bounded-advice classes, where a more powerful machine generates advice for a weaker one, to better model preprocessing scenarios in cryptography, quantum computing, and ML. It leverages a tight link between bounded advice and unary languages, establishing that bounded advice equals unary advice via $P/poly^B = P^{Un(P^B)}$, and then derives conditions under which such advice is useful for classical and quantum classes. Key contributions include conditional and unconditional results connecting bounded advice to $EXP$, $NP$, $BQP$, and $PSPACE$, a characterization of usefulness through unary-language results, and explicit bounds for quantum preparation such as $BPP/samp^{BQP}$ and $BQP/qpoly^A$ in relation to $QMA$-type containment. The findings sharpen our understanding of when preprocessing-based advice expands computable languages and illuminate the interaction of bounded advice with quantum and randomized models, offering new insights and improvements to the state of the art in advice-function complexity. All mathematical statements are presented with proper notation, including relationships like $P/poly^B = P^{Un(P^B)}$ and the various bounded-advice classes, to enable precise interpretation and application in theoretical and practical contexts.

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.

Figures (1)

• Figure 1: Step of the procedure for finding all the elements of a sparse language using $L'$. We begin with a possibly incomplete list, $A$, consisting of elements of $L$ below to some given size, $n$. In step a. we represent $A$ with a prefix tree, each layer representing a possible prefix of the string. If a string is a prefix for an element in $\mathsf{A}$ its branch continues to a green node. b. We add the possible unexplored children to the tree. We then use algorithm \ref{['Alg:binary search variant']} starting from each unexplored child prefixes to find possible new elements of $A$. Either one is found and added to $A$ (d.) or none are found, implying there are none left to find and $A$ is complete (c.).

Theorems & Definitions (33)

• Definition 1: Oracular classes
• Definition 2: Double oracles
• Definition 3: Advice classes
• Definition 4: Polynomial-sized bounded advice classes
• Definition 5: $\mathsf{BQP/qpoly^C}$
• Theorem 6: Bshouty1996oraclesKobler1998new
• Theorem 7
• Lemma 8
• proof
• Lemma 9