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Hamiltonian Monte Carlo from $q$ -deformed phase-space mechanics

This paper introduces a geometrically motivated $q$ -deformed Hamiltonian Monte Carlo framework that utilizes Jackson-type differences to construct a Metropolis-adjusted sampler, offering a robust alternative to classical HMC that avoids domain violations and scale mismatches while converging to standard dynamics as $q \to 1$ .

Original authors: Xiaomei Yang, Zhiliang Deng

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiaomei Yang, Zhiliang Deng

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the best spot to set up a campsite in a vast, foggy valley. You want to find the "sweet spot" (the most likely place based on your map), but the terrain is tricky. Sometimes the ground is flat, sometimes it's a steep cliff, and sometimes the ground is made of a strange, stretchy material that changes shape depending on how you look at it.

This is the problem scientists face when using a powerful tool called Hamiltonian Monte Carlo (HMC) to solve complex math puzzles. HMC is like a super-smart hiker who uses the laws of physics (specifically, how a ball rolls down a hill) to explore the valley efficiently. Usually, this hiker needs a perfect map of the slope (the "gradient") to know which way to roll. But what if you don't have a map? What if you only have a black box that tells you how high you are if you stand in one spot, but won't tell you how steep the hill is?

In this case, the hiker has to guess the slope by taking a tiny step forward and backward and seeing how the height changes. This is called a "finite difference."

The Problem: The "Additive" Step Trap

The traditional way to guess the slope is to take a fixed-size step. Imagine you always take a step of exactly 1 meter, no matter where you are.

If you are on a giant mountain (a large number), a 1-meter step is tiny and safe.
But if you are on a tiny pebble (a very small positive number), a 1-meter step is huge! You might step right off the edge of the world, or into a place where the math breaks down (like trying to take the square root of a negative number).

The paper shows that for problems involving positive numbers (like sizes, speeds, or concentrations that can't be zero or negative), this "fixed 1-meter step" strategy is a disaster. It causes the hiker to stumble, fall, and get stuck. The authors ran simulations showing that when the numbers get small, this method fails completely, with acceptance rates dropping to near zero and errors exploding.

The Solution: The "q-Deformed" Stretchy Ruler

The authors, Xiaomei Yang and Zhiliang Deng, propose a new way to take that step. Instead of a fixed ruler, they use a stretchy, multiplicative ruler based on something called q-calculus.

Think of it like this:

Old Way (Additive): "Move 1 meter to the right." (Dangerous on small scales!)
New Way (q-Jackson): "Move to a spot that is 5% larger or 5% smaller than where you are."

This new method, called q-HMC, doesn't care if you are on a giant mountain or a tiny pebble. If you are on a pebble, 5% is a tiny, safe step. If you are on a mountain, 5% is a big, useful step. It scales perfectly with the size of the number you are looking at.

How They Built It

The authors didn't just invent this rule out of thin air. They built it on a foundation of q-deformed phase-space mechanics.

The Theory: They started with a fancy mathematical framework involving "non-commutative geometry" (a world where the order of operations matters, like putting on socks before shoes vs. shoes before socks). They proved that in this weird, stretched-out mathematical world, the laws of physics still hold up perfectly. They showed that a special "symplectic form" (a way of measuring the shape of the space) stays invariant, just like in normal physics.
The Reality Check: Since we can't actually live in that weird mathematical world to run our simulations, they translated their fancy theory into a computer-friendly version using Jackson derivatives. This is just a fancy name for a specific type of "stretchy" difference calculation.
The Safety Net: Because this new "stretchy" step isn't a perfect physical law in our normal world, the authors added a Metropolis correction. Think of this as a bouncer at a club. After the hiker takes a step using the stretchy ruler, the bouncer checks: "Did you actually land in a good spot?" If the math says "No," the bouncer sends the hiker back to the start. This ensures the method is statistically valid, even if the step itself is a clever approximation.

What the Experiments Showed

The authors tested this new hiker against the old one in several scenarios:

Tiny Positive Numbers: When the target numbers were very small (like $10^{-3}$ ), the old "fixed step" hiker failed miserably, with errors 100 times larger than the new method. The new q-HMC hiker glided right through.
Big Positive Numbers: When the numbers were large, both methods worked fine. The new method didn't break anything; it just matched the old one.
Mixed Landscapes: They tested a tricky mix of tiny positive numbers and regular signed numbers (which can be negative). The old method struggled to find a step size that worked for both. The new q-HMC method handled the mix naturally, keeping the positive numbers positive and the signed numbers moving smoothly.
Real-World Physics: They even tested it on a complex problem involving heat diffusion (a PDE inverse problem). Here, the new method acted like a "black box" version of a very sophisticated physics tool (the adjoint method), giving almost the same results without needing the complex physics equations.

What This Paper Does NOT Say

It is important to know what this paper doesn't claim:

It is not a magic bullet for everything. The authors explicitly state that if you already have a perfect map (an exact gradient), you should still use the standard HMC. The new method is for when you don't have the map and have to guess the slope.
It is not a "breakthrough" that solves all convergence problems. The paper proves the method is valid (it won't give you the wrong answer) and that it can explore the space (it won't get stuck), but it doesn't claim to be faster than every other method in every situation.
It doesn't replace the "Adjoint Method" in physics. In the diffusion experiment, they showed the new method mimics the adjoint method well, but they clarify that if you have a real adjoint solver, you should still use that. The q-HMC is a great backup when you only have a black box.

The Bottom Line

The paper suggests that by borrowing ideas from a strange, stretched-out version of physics (q-calculus), we can build a better hiker for exploring complex, positive-number landscapes. Instead of taking rigid, fixed steps that might knock you off a cliff, the q-HMC method takes flexible, relative steps that grow and shrink with the terrain.

In their simulations, this approach avoided the "domain violations" (stepping off the map) and "scale mismatches" (taking a step too big for a tiny pebble) that plagued the old methods. It's a clever, geometrically inspired way to make sure your computer doesn't trip over its own feet when trying to solve tricky math problems.

Technical Summary: Hamiltonian Monte Carlo from $q$ -deformed phase-space mechanics

Problem Statement
Hamiltonian Monte Carlo (HMC) is a standard method for sampling from probability distributions $\pi(x) \propto \exp[-U(x)]$ by simulating Hamiltonian dynamics in an extended phase space. However, in practical "black-box" scenarios where analytical gradients of the potential $U(x)$ are unavailable, HMC relies on finite-difference approximations. The paper identifies a critical limitation in standard additive finite-difference methods when applied to positive-scale parameters (e.g., variances, diffusion coefficients). Additive perturbations ( $x \to x \pm h$ ) can violate domain constraints (e.g., producing negative values for strictly positive variables) or suffer from severe scale mismatch, where a fixed step size is either too large for small-scale regions or too small for large-scale regions. This leads to large force errors, poor energy conservation, and low Metropolis acceptance rates.

Methodology
The authors propose a $q$ -deformed HMC (q-HMC) framework motivated by $q$ -deformed phase-space mechanics and non-commutative geometry. The methodology proceeds in three stages:

Geometric Formulation:
- Starting from a Lagrangian formulation on a $q$ -commutative quantum plane (where coordinates $\hat{x}$ and momenta $\hat{p}$ satisfy $\hat{p}\hat{x} = q\hat{x}\hat{p}$ ), the authors derive the $q$ -Hamiltonian equations.
- They prove the formal invariance of the $q$ -symplectic form under the $q$ -Hamiltonian flow. This establishes a geometric origin for the dynamics, distinguishing it from ad hoc finite-difference modifications.
- The formal operator-valued dynamics are realized computationally using Jackson derivatives. Specifically, the standard derivatives in the Hamiltonian flow are replaced by Jackson-type finite differences.
Computable Realization:
- The formal $q$ -differential operators are mapped to classical Euclidean variables via a realization involving dilation and Jackson derivative operators.
- For numerical implementation, the authors utilize a symmetric log-normalized Jackson derivative. For a variable $x$ , the force approximation is defined as:
  $D^{\text{sym}}_{\delta, x} f(x) = \frac{f(e^{\delta/2}x) - f(e^{-\delta/2}x)}{\delta x}$
  where $\delta = |\log q|$ . This corresponds to a centered finite difference in the logarithmic coordinate ( $\log x$ ) but is implemented via multiplicative perturbations in the original variable ( $x \to e^{\pm \delta/2}x$ ).
Algorithm Construction:
- A Metropolis-adjusted q-Hamiltonian sampler is constructed. The proposal is generated using a $q$ -leapfrog integrator (composed of explicit shear maps based on the Jackson force and velocity).
- Because the computable realization is not a canonical Hamiltonian flow in the ordinary Euclidean sense, a Metropolis correction is applied to ensure the transition kernel satisfies detailed balance with respect to the target distribution.
- The authors establish conditions for the irreducibility of the marginal position chain, ensuring the sampler can explore the support of the target distribution.

Key Contributions

Geometric Derivation: The paper provides a rigorous geometric derivation of HMC dynamics from $q$ -deformed mechanics, proving the invariance of the $q$ -symplectic form, rather than treating the method as a heuristic modification.
Scale-Consistent Perturbation: The $q$ -Jackson force acts as a relative (multiplicative) perturbation mechanism. For positive variables, this is mathematically equivalent to a centered finite difference in log-coordinates but avoids the need for explicit log-reparametrization of the target function.
Validity Proof: The authors prove that the resulting Markov transition satisfies detailed balance and provide sufficient conditions for irreducibility, validating the method as a rigorous MCMC sampler.
Black-Box Applicability: The method is designed specifically for scenarios where the potential is a black-box function of positive parameters, and gradients are unavailable.

Numerical Results
The paper evaluates q-HMC against exact-gradient HMC, raw additive finite-difference HMC, and log-coordinate finite-difference HMC across several benchmarks:

1D and High-Dimensional Positive Targets: In cases involving small-scale positive variables (e.g., lognormal distributions with small means), raw additive finite differences fail due to domain violations and large force errors. q-HMC maintains high acceptance rates and low Hamiltonian errors, performing comparably to log-coordinate finite differences.
Variance-Component Inference: In Bayesian inference for variance components, raw additive methods degrade significantly as dimension increases or scales become small. q-HMC remains stable, tracking the performance of exact-gradient benchmarks.
Mixed Log-Trap Target: In a target with both small positive and signed variables, where a global log-transformation is impossible, q-HMC applied directly to the mixed coordinates outperforms partial-log and scaled additive methods, avoiding "bad rejections" caused by domain violations.
PDE-Based Inverse Problem: For a diffusion coefficient inverse problem, the q-Jackson force (multiplicative perturbation of the coefficient) reproduces the accuracy of the adjoint sensitivity method (the exact gradient) much more closely than raw additive finite differences, which suffer from scale mismatch.

Significance and Claims
The paper claims that the primary significance of this work is connecting $q$ -deformed Hamiltonian mechanics with a valid Metropolis-corrected sampling scheme. The authors explicitly state that q-HMC is not intended to replace exact-gradient HMC or adjoint-based methods when analytical gradients are available.

Instead, the method's significance lies in providing a scale-consistent black-box force approximation for positive functional parameters. The $q$ -Jackson force naturally handles the geometry of scale parameters through multiplicative perturbations, avoiding the domain violations and scale mismatches inherent in additive finite differences. The paper positions this as a robust alternative for Bayesian inverse problems and statistical learning tasks involving constrained, multiscale, or mixed-coordinate parameters where gradients are inaccessible.