QuasiMoTTo: QMC Test-Time Scaling

• QuasiMoTTo is a sampling method that reparameterizes autoregressive generation as inverse-CDF sampling combined with quasi-Monte Carlo techniques to yield exact marginals.
• It replaces i.i.d. sampling with correlated QMC uniforms to cover the output space more evenly, reducing redundancy and nearly saturating the theoretical pass@k ceiling.
• Applied in both test-time scaling and policy-gradient RL, QuasiMoTTo achieves significant compute savings and faster convergence by enhancing sample diversity.

Searching arXiv for the specified paper and closely related references. Tool call: arxiv_search(query="(Li et al., 1 Jul 2026)", max_results=5) QuasiMoTTo, short for Quasi-Monte Carlo Test-Time Scaling, is a sampling method for autoregressive LLMs that replaces i.i.d. rollouts with correlated but exact samples entirely in parallel. The method is motivated by the observation that default test-time scaling and policy-gradient RL generate many parallel attempts per problem, yet independent sampling often wastes inference compute on redundant solutions because the model distribution is highly peaked. QuasiMoTTo addresses this inefficiency by reparameterizing autoregressive sampling as inverse-CDF sampling and drawing the underlying uniforms with quasi-Monte Carlo (QMC), so that the resulting samples cover the output space with far less redundancy while preserving the correct marginal distribution under the LLM (Li et al., 1 Jul 2026).

1. Problem setting and redundancy under i.i.d. sampling

In both test-time scaling and policy-gradient RL, the default procedure is to draw $k$ i.i.d. rollouts from the model $\pi_\theta$. The central problem identified for this regime is that independent samples can “clump” on the same high-probability modes. When $\pi_\theta$ places substantial mass on a few regions, many of the $k$ samples repeat closely related solution paths, so the batch covers the output space inefficiently.

The standard intuition is expressed by the i.i.d. pass@k relation

$\mathrm{pass@}k = 1-(1-p)^k,$

where $p=\mathrm{pass@}1$. For small $p$, this grows slowly. The same setting admits a general upper bound for any sampler that preserves the single-sample marginals:

$$P(k)\equiv \Pr[\ge 1\ \mathrm{success\ in}\ k\ \mathrm{rolls}] \le \min\{1,kp\}.$$

The paper describes this as a union-bound ceiling and argues that i.i.d. sampling generally falls strictly below that linear ceiling, leaving room for improvement by introducing dependence across samples (Li et al., 1 Jul 2026).

This framing is significant because it treats correlation not as a liability but as a degree of freedom. The usual reason to insist on independence is that it makes parallel sampling trivial to scale. QuasiMoTTo instead explores samplers that remain embarrassingly parallel while deliberately reducing within-batch redundancy.

2. Autoregressive sampling as inverse-CDF sampling

QuasiMoTTo relies on a reparameterization of autoregressive generation in which a full sequence $x=(x_1,\dots,x_T)$ corresponds to a sub-interval

$$I(x)\subset [0,1),\qquad |I(x)|=\pi_\theta(x)=\prod_{t=1}^T \pi_\theta(x_t\mid x_{<t}).$$

Under this view, sampling $\pi_\theta$0 is equivalent to drawing $\pi_\theta$1 and identifying the unique interval $\pi_\theta$2 that contains $\pi_\theta$3.

At token level, the decoding step is written as

$\pi_\theta$4

where

$\pi_\theta$5

After selecting the token, the uniform is rescaled according to

$\pi_\theta$6

At the end, $\pi_\theta$7 is discarded.

The key structural claim is that each $\pi_\theta$8 is a one-to-one, measure-preserving map, so if $\pi_\theta$9 then $\pi_\theta$0 exactly (Li et al., 1 Jul 2026). This is what allows QuasiMoTTo to modify the joint law of a batch of uniforms without changing the marginal law of each decoded sample.

3. QMC construction and the QuasiMoTTo sampler

Rather than draw $\pi_\theta$1 independent uniforms, QuasiMoTTo uses QMC to construct dependent $\pi_\theta$2 whose empirical distribution remains uniform but which “repel” one another and cover $\pi_\theta$3 more evenly. The paper highlights three examples.

Sampler	Construction	Property emphasized
Lattice	$\pi_\theta$4	Random shift of a regular grid
Stratified	One uniform per stratum $\pi_\theta$5 plus a random permutation	Uniform marginals with stratification
Sobol (token-level)	$\pi_\theta$6-dimensional points $\pi_\theta$7 with low star-discrepancy	Low-discrepancy coverage across coordinates

For Sobol, the discrepancy notion is

$\pi_\theta$8

The paper states that $\pi_\theta$9 can be the max sequence length, and coordinate $k$0 is decoded with the $k$1th inverse-CDF.

Once the QMC uniforms are obtained, each rollout decodes independently in $k$2 time and never communicates again. The procedure is therefore still parallel at the rollout level. The essential substitution is simple: QMC generates a low-discrepancy batch of uniforms, and the inverse-CDF autoregressive decoder pushes those uniforms through the LLM to produce a batch of exact $k$3-samples with higher coverage than i.i.d. sampling (Li et al., 1 Jul 2026).

A plausible implication is that QuasiMoTTo occupies a middle ground between two objectives that are often treated as incompatible: exactness of the marginal distribution and deliberate diversification of the batch.

4. Estimation, upper bounds, and coverage diagnostics

A technical complication of correlated sampling is that dependence breaks the standard i.i.d. pass@k estimator. For i.i.d. sampling, if $k$4 samples contain $k$5 successes, the unbiased estimator due to Chen et al. (2021) is

$k$6

The paper states directly that this formula breaks under dependent samples.

For lattice QMC with $k$7, the paper develops a dyadic bootstrap estimator. If

$k$8

then for any $k$9 and any $\mathrm{pass@}k = 1-(1-p)^k,$0, the stride-$\mathrm{pass@}k = 1-(1-p)^k,$1 subsequence

$\mathrm{pass@}k = 1-(1-p)^k,$2

is distributed as a fresh $\mathrm{pass@}k = 1-(1-p)^k,$3-point lattice. This allows pass@$\mathrm{pass@}k = 1-(1-p)^k,$4 to be estimated on the same batch by splitting into disjoint subsequences, computing pass@$\mathrm{pass@}k = 1-(1-p)^k,$5 on each, and averaging. The result is described as an unbiased estimator of pass@$\mathrm{pass@}k = 1-(1-p)^k,$6 under the lattice QMC joint law (Li et al., 1 Jul 2026).

The paper also formalizes the best possible success rate for any marginal-preserving sampler:

$\mathrm{pass@}k = 1-(1-p)^k,$7

No marginal-preserving sampler can exceed that linear-in-$\mathrm{pass@}k = 1-(1-p)^k,$8 ceiling.

Coverage and redundancy are assessed through several diagnostics rather than a single scalar metric:

• pass@k relative to the linear ceiling;
• count of “distinct solution chains”;
• in RL, the fraction of “zero-variance groups”, meaning groups where all $\mathrm{pass@}k = 1-(1-p)^k,$9 rollouts have identical reward.

The paper additionally notes that one can quantify average pairwise mutual information $p=\mathrm{pass@}1$0: i.i.d. gives $p=\mathrm{pass@}1$1, stratified gives $p=\mathrm{pass@}1$2, and lattice gives $p=\mathrm{pass@}1$3 because of full determinism given one point. In the exposition, higher $p=\mathrm{pass@}1$4 means stronger repulsion and higher coverage in $p=\mathrm{pass@}1$5. This suggests that dependence is being used as an explicit resource for coverage optimization rather than merely tolerated as an implementation artifact.

5. Empirical performance on reasoning benchmarks

The empirical study evaluates QuasiMoTTo on four reasoning benchmarks:

Benchmark	Task description
Countdown	Arithmetic target from numbers
Maze	2D grid, output a path U/D/L/R
Sudoku	9×9 fill-in
1D-ARC	Abstract sequence transformations

Across all four tasks, lattice-based QuasiMoTTo is reported to lie above i.i.d. pass@k and to nearly saturate the linear ceiling. The reported averages over problems include two explicit operating points:

• At $p=\mathrm{pass@}1$6: i.i.d. pass@16 $p=\mathrm{pass@}1$7, QuasiMoTTo $p=\mathrm{pass@}1$8, ceiling $p=\mathrm{pass@}1$9.
• At $p$0: i.i.d. $p$1, QuasiMoTTo $p$2, ceiling $p$3.

The central sample-efficiency claim is that, to achieve the same pass@k as i.i.d. at a given target level such as $p$4, QuasiMoTTo requires 25–47% fewer samples on these tasks (Li et al., 1 Jul 2026).

These results are presented as evidence that the method does not merely diversify outputs in a superficial sense. The gain is tied specifically to higher coverage with lower redundancy, and the comparison to the union-bound ceiling gives a concrete sense of how much of the theoretically available improvement is being realized.

6. Integration into policy-gradient RL and compute savings

The same correlated-sampling idea is applied to policy-gradient RL (GRPO). In that setting, for each prompt one collects $p$5 rollouts $p$6, computes rewards $p$7, and uses the group-baseline policy gradient

$p$8

Because QuasiMoTTo preserves the marginal $p$9, the paper states that this estimator remains unbiased in expectation, since any linear-in-marginals estimator is fine. The within-group rollouts are correlated, but that correlation is reported to produce higher variance in $$P(k)\equiv \Pr[\ge 1\ \mathrm{success\ in}\ k\ \mathrm{rolls}] \le \min\{1,kp\}.$$0 and thus stronger learning signal (Li et al., 1 Jul 2026).

The empirical RL evaluation is conducted on Maze and Sudoku, filtered to “hard but solvable” problems. The headline result is that QuasiMoTTo trains to the same pass@1 in roughly 50% fewer gradient steps than i.i.d. sampling. The paper gives two associated observations:

• Zero-variance groups drop from approximately 30% to 10%.
• Learning curves, plotted as pass@1 versus steps, show the QuasiMoTTo curve rising twice as fast as i.i.d.

Within the paper’s framing, the RL benefit and the inference-time benefit share the same mechanism: reducing redundancy within a batch yields either a better chance of encountering a correct solution or a stronger reward contrast for policy-gradient updates.

7. Scope, exactness, and terminological boundaries

A defining property of QuasiMoTTo is that it uses correlated samples as a drop-in replacement for i.i.d. samples while preserving the correct marginal law of each sample. This is why the same batch can be used for both pass@k-style evaluation and policy-gradient training. The method does not alter $$P(k)\equiv \Pr[\ge 1\ \mathrm{success\ in}\ k\ \mathrm{rolls}] \le \min\{1,kp\}.$$1 itself; it changes how a finite batch of uniforms is coupled before inverse-CDF decoding. The paper’s key takeaway is explicit: by replacing independent sampling with low-discrepancy QMC sampling plus inverse-CDF decoding, QuasiMoTTo reduces redundancy, boosts coverage, and translates those improvements into both inference compute savings and RL compute savings (Li et al., 1 Jul 2026).

A possible source of confusion is terminological. The arXiv paper “Quasi-morphisms on cotangent bundles and symplectic homogenization” (Monzner et al., 2011) concerns homogeneous quasi-morphisms on Hamiltonian diffeomorphism groups and symplectic homogenization, not quasi-Monte Carlo test-time scaling. That work belongs to symplectic topology rather than language-model inference. Accordingly, QuasiMoTTo in current ML usage refers to the 2026 sampling method, whereas the cotangent-bundle paper is an unrelated mathematical work (Monzner et al., 2011).