Magic and communication complexity

2025-10-08

AI Summary

This work investigates the intrinsic connection between “magic” (non-stabilizer resource) in quantum circuits and communication complexity, aiming to characterize the communication cost of low-magic functions. Method: We establish the first upper bound linking the number of magic gates in a quantum circuit to its deterministic simultaneous-message communication cost (R∥*); design a privacy-preserving quantum-to-classical protocol transformation framework; and integrate T-depth constraints, mid-circuit adaptive measurement modeling, and PSM* protocol construction. Contributions: We achieve the first exponential separation between R∥* and classical randomized communication (R) for Boolean functions. For n-bit generalized Toffoli gates and quantum multiplexers, we prove an Ω(n) magic lower bound, while constructing R∥* protocols with polylog(n) complexity—surpassing the performance limits of classical interactive randomized protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Agent Communication
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We establish novel connections between magic in quantum circuits and communication complexity. In particular, we show that functions computable with low magic have low communication cost. Our first result shows that the $mathsf{D}|$ (deterministic simultaneous message passing) cost of a Boolean function $f$ is at most the number of single-qubit magic gates in a quantum circuit computing $f$ with any quantum advice state. If we allow mid-circuit measurements and adaptive circuits, we obtain an upper bound on the two-way communication complexity of $f$ in terms of the magic + measurement cost of the circuit for $f$. As an application, we obtain magic-count lower bounds of $Omega(n)$ for the $n$-qubit generalized Toffoli gate as well as the $n$-qubit quantum multiplexer. Our second result gives a general method to transform $mathsf{Q}|^*$ protocols (simultaneous quantum messages with shared entanglement) into $mathsf{R}|^*$ protocols (simultaneous classical messages with shared entanglement) which incurs only a polynomial blowup in the communication and entanglement complexity, provided the referee's action in the $mathsf{Q}|^*$ protocol is implementable in constant $T$-depth. The resulting $mathsf{R}|^*$ protocols satisfy strong privacy constraints and are $mathsf{PSM}^*$ protocols (private simultaneous message passing with shared entanglement), where the referee learns almost nothing about the inputs other than the function value. As an application, we demonstrate $n$-bit partial Boolean functions whose $mathsf{R}|^*$ complexity is $mathrm{polylog}(n)$ and whose $mathsf{R}$ (interactive randomized) complexity is $n^{Omega(1)}$, establishing the first exponential separations between $mathsf{R}|^*$ and $mathsf{R}$ for Boolean functions.

Problem

• Linking quantum circuit magic to communication complexity
• Transforming quantum protocols into classical with polynomial cost
• Establishing exponential separations between communication complexity classes

Innovation

• Relates quantum circuit magic to communication complexity bounds
• Transforms quantum protocols into classical with polynomial overhead
• Establishes exponential separations between communication complexity classes