An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory

TL;DR

The paper proves an in-principle, super-polynomial quantum advantage for approximating certain combinatorial optimization problems by combining cryptographic hardness with learning-theoretic reductions. It constructs an explicit chain of reductions from DFA learning to formula colouring and then to integer linear programming, under RSA-inversion hardness, and shows a polynomial-time quantum algorithm (via Shor's factoring) can efficiently approximate these hard instances. The authors map these problems to quantum Hamiltonians, clarifying connections to variational quantum algorithms while highlighting an explicit, end-to-end construction of advantage-bearing instances. The work illuminates fundamental limits and capabilities of fault-tolerant quantum devices for optimization tasks and provides concrete guidance for building quantum-versus-classical benchmarks in this domain.

Abstract

Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It is still unclear, however, to what extent quantum algorithms can actually outperform classical algorithms for this type of problems. In this work, by resorting to computational learning theory and cryptographic notions, we prove that quantum computers feature an in-principle super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant and introducing a new reduction, we identify special types of problems that are hard for classical computers to approximate up to polynomial factors. At the same time, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The core of the quantum advantage discovered in this work is ultimately borrowed from Shor's quantum algorithm for factoring. Concretely, we prove a super-polynomial advantage for approximating special instances of the so-called integer programming problem. In doing so, we provide an explicit end-to-end construction for advantage bearing instances. This result shows that quantum devices have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms. Our results also give clear guidance on how to construct such advantage-bearing problem instances.

Figures (6)

• Figure 1: Overview of the setting of our work.A A diagrammatic sketch of the travelling salesperson problem aimed at finding the shortest possible route that visits each city (represented as vertices) exactly once and returns to the origin city. B Venn diagram that depicts the sense in which a quantum advantage---symbolized in C---is proven in our work for integer programming problems. The grey set contains all instances of integer linear programming, and the subsets contain the hard or respectively easy to solve instances. By hard to approximate we mean that there is no polynomial time algorithm that approximates the size of the optimal solution up to a factor of $opt^{\alpha} \cdot |I|^{\beta}$, where $|I|$ is the instance size, $\alpha, \beta$ are constants such that $\alpha \geq 0$, $0 \leq \beta < 1$ and $opt$ denotes the size of the optimal solution. Whether the dotted line holds true, i.e., whether there exists a problem that can be solved exactly by a polynomial-time quantum algorithm, but are hard to approximate classically, is left for further research.
• Figure 2: Example of a deterministic finite automaton. The DFA is represented as a quintuple $(Q, \Sigma, \lambda, q_0, \omega)$, where $Q = \{q_0, q_1, q_2\}$, $\Sigma = \{a, b\}$, $\lambda$ is defined by the transitions (e.g., $\lambda(q_0, a) = q_1$, $\lambda(q_0, b) = q_2$, etc.), $q_0$ is the initial state, and $\omega = \{q_2\}$ is the set of accept states.
• Figure 3: The interplay between representations and concept. The domain $X$ can formally be seen as a set of finite bit strings. Concepts are subsets of the domain, which can be described by representations $c \in C$. Together with the map $\sigma$, mapping representations to concepts, the tuple $(\sigma, C)$ is called a representation class.
• Figure 4: The reduction chain from the consistency problem to combinatorial optimization problems. In Section \ref{['subsec:classhard_bc']}, we introduce Boolean circuits, whose sizes are hard to approximate by $|h|$, where $h$ is a hypothesis that is consistent with a sample labeled by the circuits. This directly implies the approximation hardness of $Con(\operatorname{DFA},\operatorname{DFA})$. In Section \ref{['subseq:classhard_fc']}, we present an approximation-preserving reduction from $Con(\operatorname{DFA}, \operatorname{DFA})$ to formula colouring kearns_boolfunc_1993. We then extend the results of Ref. kearns_boolfunc_1993 by showing in Section \ref{['subsec:classhard_ilp']} an approximation-preserving reduction from formula colouring to integer linear programming, yielding the approximation hardness for ILP.
• Figure 5: A Boolean circuit in the class C-RSA. The input to the circuit in $\operatorname{C-RSA}$ is the power sequence of the RSA ciphertext of $\operatorname{RSA}(x,N,e)=y$. The circuit computes the LSB of $x$ by simply performing modular multiplication on the $2^i$'th powers of the power sequence where the secret key bit $d_i=1$, for the secret key $d$. Thereby the secret key $d$ is hard-wired into the circuit and the decryption $x=y^d \text{ mod } N$ is explicitly performed. This can be done in an $O(\log(n))$ deep circuit beame_iteratedproducts_1986.

Theorems & Definitions (24)

• Definition 2.1: Formula colouring problem $\operatorname{FC}$ kearns_boolfunc_1993
• Definition 2.2: Integer linear programming problem ($\operatorname{ILP}$)
• Theorem 2.3: Classical hardness of approximation for integer linear programming
• Theorem 2.4: Quantum efficiency for $\operatorname{ILP-RSA}$
• Definition 4.1: Evaluation problem $Eval(C)$
• Definition 4.2: Consistency problem $Con(C,H)$ kearns_boolfunc_1993
• Theorem 4.3: Occam's razor blumer_occam_1987kearns_boolfunc_1993
• Definition 4.4: Formula colouring problem $\operatorname{FC}$ kearns_boolfunc_1993
• Definition 4.5: Boolean circuit for the LSB of RSA kearns_boolfunc_1993
• Theorem 4.6: Classical approximation hardness of $\operatorname{C-RSA}$ kearns_boolfunc_1993