QUnfold: Optimization for Detector Unfolding

• QUnfold is an open-source Python package that reframes detector deconvolution as a regularized quadratic optimization problem to reconstruct true distributions.
• It formulates the unfolding task by minimizing a penalized likelihood that integrates detector response with Tikhonov-style smoothing to overcome ill-conditioning.
• The package supports classical mixed-integer solvers and quantum annealing via QUBO mapping, enabling robust, multi-backend performance validation.

QUnfold is an open-source Python package for statistical unfolding in high-energy physics that recasts detector deconvolution as an optimization problem rather than treating it purely as a matrix-inversion or iterative deconvolution task. Its central objective is to reconstruct the underlying truth-level distribution of a physical observable from a detector-distorted measured spectrum by minimizing a regularized quadratic objective, in a form compatible with classical mixed-integer optimization as well as quantum-compatible formulations such as Quadratic Unconstrained Binary Optimization (QUBO). In this sense, QUnfold provides a unified optimization perspective on unfolding, with solver backends spanning classical mixed-integer methods and D-Wave’s hybrid quantum-classical workflow (Gasperini et al., 26 Feb 2026).

1. Detector response and the inverse problem

In the setting addressed by QUnfold, the true distribution is denoted $f(z)$, the measured distribution $g(\mu)$, and the detector response is encoded in a response function

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$

where $m(\mu|z)$ is the migration/resolution term and $\epsilon(z)$ is the efficiency. The detector-level distribution is

$g(\mu)=\int r(\mu|z) f(z)\,dz + \beta(\mu),$

with $\beta(\mu)$ representing background. After discretization into $M$ bins, the folding relation becomes

$\mu_i=\sum_{j=1}^{M} R_{ij} z_j + \beta_i,$

where $R\in\mathbb{R}^{M\times M}$ is the response matrix (Gasperini et al., 26 Feb 2026).

This formulation places unfolding squarely in the class of ill-posed inverse problems. A naive inversion of the response matrix is unstable because response matrices are typically ill-conditioned, and statistical fluctuations can be strongly amplified. QUnfold is designed around the premise that this instability should be handled directly at the objective-function level, with regularization incorporated into the reconstruction rather than appended afterward as a separate correction step.

2. Likelihood, regularization, and the quadratic objective

QUnfold formulates the reconstruction as the maximization of a penalized likelihood,

$g(\mu)$0

Assuming Poisson statistics in each measured bin,

$g(\mu)$1

with $g(\mu)$2 determined by $g(\mu)$3. The regularization term $g(\mu)$4 is used to suppress unphysical oscillations and stabilize the solution. QUnfold focuses on a smoothness prior based on curvature suppression. In the continuous limit,

$g(\mu)$5

typically with $g(\mu)$6, while in discrete form the penalty becomes

$g(\mu)$7

This is explicitly aligned with Tikhonov-style smoothing: the regularization strength is controlled by $g(\mu)$8, which mediates the variance-bias trade-off (Gasperini et al., 26 Feb 2026).

For optimization purposes, QUnfold replaces the Poisson log-likelihood by a Gaussian/$g(\mu)$9 approximation and obtains

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$0

where $r(\mu|z)=m(\mu|z)\,\epsilon(z),$1 is the discrete Laplacian operator implementing second-derivative smoothing. The reconstructed bin contents are constrained to be nonnegative integers, and each $r(\mu|z)=m(\mu|z)\,\epsilon(z),$2 carries an upper bound $r(\mu|z)=m(\mu|z)\,\epsilon(z),$3 estimated from efficiency corrections. Expanding the quadratic terms yields the integer quadratic form

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$4

This compact representation makes the role of detector response and smoothing fully explicit within a single objective.

3. QUBO representation and quantum compatibility

The QUBO reformulation is the mechanism by which QUnfold becomes quantum-compatible. QUBO is written in the standard form

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$5

with solution

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$6

Because QUBO is equivalent to an Ising energy minimization problem, it can be mapped to quantum annealing hardware or hybrid quantum-classical solvers (Gasperini et al., 26 Feb 2026).

To obtain this form, each integer bin content $r(\mu|z)=m(\mu|z)\,\epsilon(z),$7 is encoded as a bit string $r(\mu|z)=m(\mu|z)\,\epsilon(z),$8 with precision vector

$r(\mu|z)=m(\mu|z)\,\epsilon(z),$9

using

$m(\mu|z)$0

bits for bin $m(\mu|z)$1. Concatenating all bit strings yields the global binary vector $m(\mu|z)$2. Under this encoding, the integer quadratic objective becomes a binary quadratic objective,

$m(\mu|z)$3

which is equivalent to QUBO because $m(\mu|z)$4 for binary variables. The paper notes that the number of bits scales linearly with the number of histogram bins and logarithmically with the occupancy per bin, a practical property for discretized HEP spectra.

The significance of this mapping is not only hardware portability. It also shows that the same unfolding objective can be expressed in forms suitable for exact-ish classical solvers, hybrid orchestration, and quantum annealing-compatible execution, without changing the underlying statistical model.

4. Package architecture and execution modes

QUnfold is implemented as a Python package organized around the construction and transformation of the unfolding objective. Its package architecture comprises five steps: ingest measured histograms and response matrices; construct the quadratic objective $m(\mu|z)$5; optionally encode integer variables into QUBO binary variables; dispatch the problem to either a classical mixed-integer solver or D-Wave’s hybrid solver; and return the unfolded histogram together with the corresponding objective value or solution diagnostics (Gasperini et al., 26 Feb 2026).

The package integrates multiple backends and is explicitly described as an abstraction layer around unfolding as an optimization problem, not just a quantum tool.

Backend	Formulation	Role
Gurobi	Quadratic integer objective	Classical mixed-integer baseline and validation
D-Wave hybrid quantum-classical solver	Quantum-compatible optimization workflow	Hybrid counterpart for the same objective
QUBO mapping	Binary quadratic objective	Quantum annealing-compatible representation

The role of the classical mixed-integer solver is central. It provides a high-quality baseline and a practical way to solve the integer quadratic objective directly without binary expansion. The hybrid D-Wave solver, by contrast, demonstrates that the same reconstruction task can be embedded in a quantum-enhanced workflow. This division of labor is important: QUnfold does not replace classical unfolding practice with a hardware-specific alternative, but instead exposes a common optimization core across solver modalities.

5. Benchmark configuration and empirical findings

The benchmark study uses four synthetic truth distributions—normal, exponential, gamma, and Breit-Wigner—with each sample containing $m(\mu|z)$6 generated events distributed in $m(\mu|z)$7 equally spaced bins. Detector distortions are intentionally controlled and include Gaussian smearing, a fixed bias shift, and limited detection efficiency. The response matrix is built from Monte Carlo truth–reco pairs, and the measured histogram is generated by applying Poisson fluctuations to the expected detector-level spectrum. QUnfold is compared against Matrix Inversion (MI), Iterative Bayesian Unfolding (IBU), Singular Value Decomposition unfolding (SVD), quadratic integer optimization solved with Gurobi (GRB), and QUBO solved with the D-Wave hybrid solver (HYB). Performance is evaluated with the Pearson $m(\mu|z)$8 statistic, with lower values indicating better reconstruction (Gasperini et al., 26 Feb 2026).

The reported behavior is structurally consistent across the tested spectra. Direct matrix inversion exhibits the expected failure mode: it is highly oscillatory in low-statistics bins and very sensitive to statistical noise. IBU and SVD stabilize the reconstruction and substantially improve over inversion, but residual biases remain, especially in regions with steep gradients, narrow features, or low occupancy. The optimization-based methods, GRB and HYB, achieve the lowest $m(\mu|z)$9 values across the tested distributions and produce bin-by-bin ratios $\epsilon(z)$0 close to unity over most of the spectrum.

The paper emphasizes several qualitative features of the optimization-based solutions. They preserve localized structures, handle heavy tails and resonant peaks well, and avoid the strong oscillations seen in matrix inversion. At the same time, they do not over-smooth the distribution as aggressively regularized methods sometimes do. A further result is that the hybrid D-Wave solver reproduces the classical Gurobi solutions closely, which indicates that the QUBO mapping preserves the underlying quadratic optimization problem faithfully. The principal trade-off remains the standard unfolding tension between accuracy, regularization, and computational strategy: $\epsilon(z)$1 controls the statistical bias-variance compromise, whereas the solver choice affects computational efficiency and scalability rather than the definition of the objective itself.

6. Research context, scope, and open problems

Within the broader unfolding literature, QUnfold belongs to the class of detector-response deconvolution methods that operate on discretized spectra and make regularization explicit at the objective-function level. Its immediate comparison set is the established histogram-based toolbox represented in the benchmarks by Matrix Inversion, Iterative Bayesian Unfolding, and Singular Value Decomposition unfolding. A distinct direction is pursued by "Moment Unfolding," which directly unfolds distribution moments as a function of another observable without first discretizing the data and is reported to be more precise than bin-based approaches in its illustrative jet-substructure study (Desai et al., 2024).

QUnfold’s main contribution is therefore best understood as a change in formulation. It replaces the conventional separation between inverse reconstruction and regularization by a single regularized quadratic optimization problem, and it exposes that problem in forms compatible with both classical mixed-integer solvers and quantum-ready QUBO workflows. This also explains why the paper describes its contribution as more than a quantum annealing demonstration: the unifying element is the optimization viewpoint itself (Gasperini et al., 26 Feb 2026).

The present scope remains limited. The benchmarks are synthetic and low-dimensional; larger bin counts, higher-dimensional observables, and realistic experimental systematics are not yet tested. The authors identify the need for systematic pseudo-experiment studies to quantify bias-variance behavior more rigorously, and they treat scalability as a major issue for practical HEP use. They also suggest that more sophisticated regularizers and data-driven hyperparameter selection could improve performance. A plausible implication is that QUnfold is currently most informative as a framework for methodological exploration—especially where discrete event counts, explicit constraints, and solver interoperability matter—rather than as a final answer to all unfolding regimes.