QMA Lower Bounds for Batch Verification via Approximate Degree

2026-07-09

AI Summary

This work investigates the trade-off between witness length and query complexity in the QMA model when batch-verifying multiple instances of Boolean functions. By developing a general lower-bound framework based on approximate degree, it establishes—for the first time—that saving even a constant factor in witness length for certain DNF formulas necessarily incurs a significant increase in query complexity. The approach integrates techniques from quantum query complexity, approximation theory via polynomials, and lifting methods from communication complexity. This unified framework yields new lower bounds for read-once CNF formulas, surjectivity, and k-element distinctness, and further extends these results to the communication complexity setting, thereby revealing fundamental scaling laws governing resource consumption as the number of instances grows in batch verification protocols.

Abstract

We study batch verification in QMA query and communication complexity, where the goal is to understand how the resources needed to verify $m$ copies of a Boolean function $f$ depend on $m$. We give a general technique for proving lower bounds on the witness-query tradeoff needed to batch verify a function $f$ in terms of its approximate degree. Applying this technique to an explicit family of DNF formulas $f$, we show that attempting to save even a constant factor on the witness length of the baseline approach to batch verifying $f$ necessitates a large polynomial increase in the query cost. We also obtain new lower bounds on the QMA query complexity of read-once CNF formulas and on the surjectivity and $k$-element distinctness functions. Our lower bounds also lift to give communication analogs of these results.

Problem

• batch verification
• QMA
• approximate degree
• query complexity
• communication complexity

Innovation

• batch verification
• QMA complexity
• approximate degree
• query lower bounds
• communication complexity