Improved Soundness for QMA with Multiple Provers

We present three contributions to the understanding of QMA with multiple provers: 1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM ‘09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a PCP). 2) We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV ‘10], thereby improving their result from a monolithic protocol where Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers k and a soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.) 3) We make progress towards an open question of [Aaronson et al., ToC ‘09] about what kinds of NP-complete problems are amenable to sublinear multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature - namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.

💡 Research Summary

The paper makes three main contributions to the study of QMA with multiple unentangled provers (QMA(k)). First, it revisits the Blier‑Tapp protocol for 2‑CSP (graph coloring) and provides a tighter soundness analysis. By refining the second‑moment bound and applying a one‑sided Chebyshev inequality, the authors improve the soundness gap from the previously known 1 − Ω(N⁻⁶) (or 1 − Ω(N⁻³⁻ε) in later work) to 1 − Ω(N⁻²) when the alphabet size K is constant. This result does not rely on a PCP with a constant gap, distinguishing it from earlier improvements that used PCP constructions.

Second, the paper gives a comprehensive analysis of the Chen‑Drucker protocol, which originally required Θ(√N) provers to obtain any non‑trivial soundness gap while using only LOCC (Bell) measurements. The authors show that for any number of provers k ≥ Ω(log N) the protocol achieves a soundness gap of Ω(k²/N). When k = Θ(√N) the original parameters are recovered; for smaller k the trade‑off is smooth and provably optimal (up to constant factors). The improvement stems from a more careful second‑moment calculation and the use of a one‑sided Chebyshev bound, which yields a tighter concentration result than the original analysis. The paper also explains why this technique cannot give a constant‑gap soundness when k is O(1), indicating that fundamentally new protocols would be needed for sublinear‑prover LOCC QMA with inverse‑polynomial soundness.

Third, the authors address an open question raised by Aaronson et al. concerning which NP‑complete problems admit sublinear‑size multi‑prover QMA protocols. By observing that many results from the PCP literature (notably the works of Ben‑Sasson & Sudan and Dinur) already imply quasi‑linear reductions from any language recognizable in nondeterministic quasi‑linear time on a random‑access machine (RAM) to 2‑CSP instances with constant soundness gap, they show that any language in NTIME_RAM(t) can be verified with perfect completeness and constant soundness using Θ(√t) unentangled quantum proofs, each of size Θ(log t) qubits. This generalizes the earlier result that only 3‑SAT enjoys such a sublinear multi‑prover QMA protocol.

The paper also discusses the implications of these results for the broader goal of constructing two‑prover LOCC QMA protocols for 3‑SAT with logarithmic proof length and a non‑trivial soundness gap. While the improved analysis of the Blier‑Tapp protocol yields a soundness gap of Ω(N⁻²) for two‑prover protocols, it does not achieve the required Ω(1/√N) gap, nor is it LOCC. The authors suggest that new techniques—perhaps avoiding the swap test or product test, which are non‑LOCC—will be necessary to bridge this gap.

Overall, the paper tightens soundness bounds for two key multi‑prover QMA protocols, establishes a smooth trade‑off between prover count and soundness for LOCC protocols, and broadens the class of languages known to admit sublinear‑size quantum proofs by leveraging existing PCP reductions. These advances clarify the power and limitations of QMA(k) and point toward future research directions in quantum proof systems.