Complexity lower bounds for computing the approximately-commuting operator value of non-local games to high precision
Bibliographic record
Abstract
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is $\mathrm{NP}$-complete to decide whether the classical value of a non-local game is 1 or $1- ε$. Furthermore, as long as $ε$ is small enough, this result does not depend on the gap $ε$. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap $ε$ decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of $\mathrm{MIP}^{co}(2,1,1,s)$ (proofs with two provers, one round, completeness probability $1$, soundness probability $s$, and commuting-operator strategies) can increase without bound as the gap $1-s$ gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class $\mathrm{PZK}$-$\mathrm{MIP}^{co}_δ(2,1,1,s)$ of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap $1-s$ gets arbitrarily small. While we do not know any computable time upper bound on the class $\mathrm{MIP}^{co}$, a result of the first author and Vidick shows that for $s = 1-1/\text{poly}(f(n))$ and $δ= 1/\text{poly}(f(n))$, the class $\mathrm{MIP}^{co}_δ(2,1,1,s)$, with constant communication from the provers, is contained in $\mathrm{TIME}(\exp(\text{poly}(f(n))))$. We give a lower bound of $\mathrm{coNTIME}(f(n))$ (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.045 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.006 | 0.015 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.002 | 0.009 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".