QFrCoRT: Convex-Roof Optimization Framework

• QFrCoRT is a numerical framework that reformulates convex-roof optimization of quantum resource measures on the complex Stiefel manifold into an unconstrained Euclidean space problem.
• It unifies quantification of entanglement, coherence, and magic by converting the search over pure-state decompositions into a gradient-based optimization workflow using automatic differentiation.
• The framework tackles numerical stability issues and scales polynomially with system size, demonstrating high accuracy on benchmark states like Werner and Horodecki states.

QFrCoRT denotes a general numerical framework for evaluating a broad class of quantum resource measures defined by convex-roof extension. Introduced in the context of quantum resource theories, it reformulates the optimization over all pure-state ensemble decompositions of a mixed state as an optimization over a complex Stiefel manifold, and then removes the manifold constraint by a polar-projection trivialization so that standard gradient-based optimizers can be used in unconstrained Euclidean space (Zhu et al., 2024). The framework is presented as a unified workflow for entanglement, coherence, and magic-state quantification, and is also described as extendable to other convex-roof quantities in quantum information theory.

1. Resource-theoretic setting

In the paper’s formulation, a quantum resource theory is written as $\mathcal{R}=(\mathcal{F},\mathcal{O})$, where $\mathcal{F}(\mathcal{H})$ is the set of free states on Hilbert space $\mathcal{H}$, and $\mathcal{O}$ is the set of free operations (Zhu et al., 2024). A resource measure $R(\rho)$ is required to be nonnegative, to vanish exactly on free states, and to be monotone under free operations. QFrCoRT targets the class of resource measures obtained by convex-roof extension from a pure-state quantity.

The defining optimization problem is

$$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$

This is the formal problem solved by the framework. The difficulty is that the minimization ranges over all ensemble decompositions of a mixed state. The paper characterizes this as a difficult NP-hard decomposition problem and presents QFrCoRT as a route to converting that problem into a numerically tractable unconstrained optimization task (Zhu et al., 2024).

The significance of this starting point is structural rather than resource-specific. Entanglement of formation, geometric coherence, and stabilizer-based magic measures differ in their pure-state functionals, but they share the same convex-roof architecture. QFrCoRT is built precisely around that shared architecture.

2. Stiefel-manifold reformulation of the convex roof

The central reformulation begins from the spectral decomposition of a rank-$r$ state,

$$\rho=\sum_{j=1}^r \lambda_j|\lambda_j\rangle\langle \lambda_j|.$$

Any $n$-term pure-state decomposition $\rho=\sum_{i=1}^n p_i|\psi_i\rangle\langle\psi_i|$ is represented באמצעות unnormalized auxiliary states $\mathcal{F}(\mathcal{H})$0 satisfying

$\mathcal{F}(\mathcal{H})$1

and

$\mathcal{F}(\mathcal{H})$2

Here $\mathcal{F}(\mathcal{H})$3 is an $\mathcal{F}(\mathcal{H})$4 complex matrix with

$\mathcal{F}(\mathcal{H})$5

so $\mathcal{F}(\mathcal{H})$6, the complex Stiefel manifold (Zhu et al., 2024).

With this parametrization, the convex-roof optimization is equivalent to an optimization over the Stiefel manifold:

$\mathcal{F}(\mathcal{H})$7

This is the framework’s core geometric step. Instead of searching directly over arbitrary decompositions, one searches over matrices with orthonormal columns. The decomposition variables are therefore encoded in a constrained but standard manifold object.

This reformulation is the main unifying device in QFrCoRT. It does not depend on a particular resource theory beyond the requirement that the mixed-state quantity be a convex roof of a pure-state resource functional. That is why the same machinery can be reused across different resource measures.

3. Polar-projection trivialization and optimization workflow

QFrCoRT then eliminates the Stiefel-manifold constraint by a trivialization. The chosen trivialization is the polar projection

$\mathcal{F}(\mathcal{H})$8

which maps an unconstrained full-rank matrix $\mathcal{F}(\mathcal{H})$9 to its polar factor on the Stiefel manifold (Zhu et al., 2024). Since $\mathcal{H}$0 has $\mathcal{H}$1 real parameters, the resulting optimization is unconstrained in Euclidean space.

The paper states that the polar-projection trivialization is numerically better than matrix exponential or Euler–Hurwitz parameterizations. It also notes a practical caveat: when $\mathcal{H}$2, the polar map is ill-defined, but such cases are measure-zero and were not encountered in practice. Matrix square roots are computed by eigendecomposition, while gradients through the square root are obtained via a Sylvester equation.

The algorithmic workflow is deliberately uniform. One builds the objective function for the chosen resource, parameterizes the decomposition through $\mathcal{H}$3, computes gradients by automatic differentiation in PyTorch, and performs optimization using L-BFGS implemented in SciPy. The reported default choices are typically $\mathcal{H}$4 pure states in the decomposition, $\mathcal{H}$5 optimization repeats to reduce local-minimum issues, and a stopping tolerance of $\mathcal{H}$6 (Zhu et al., 2024).

The framework is explicitly contrasted with three established classes of methods. SDP methods relax the convex-roof problem into semidefinite constraints and provide lower bounds but scale poorly. Manually derived gradient methods require case-by-case analytical gradients and are not unified. Seesaw strategies alternate optimization blocks but may struggle for nearly pure multipartite states. QFrCoRT’s stated advantage is that the same optimization machinery applies across different measures, with gradients handled automatically.

4. Numerical stabilization and differentiable objective design

A substantial part of QFrCoRT is devoted to numerical stability. For entropy-based objectives, direct evaluation can produce $\mathcal{H}$7 or division-by-zero issues. The paper therefore rewrites objectives in terms of the unnormalized auxiliary states and truncates small values below a threshold $\mathcal{H}$8 (Zhu et al., 2024).

For entanglement of formation, the objective is rewritten as

$\mathcal{H}$9

where

$\mathcal{O}$0

The same stabilization logic is reported for linear-entropy and stabilizer-purity objectives.

For geometric coherence, the nondifferentiable $\mathcal{O}$1 operation is replaced by a log-sum-exp smoothing:

$\mathcal{O}$2

This preserves compatibility with automatic differentiation while keeping the computation within the same unconstrained optimization pipeline (Zhu et al., 2024).

These implementation choices are not ancillary. They are part of the framework’s claim to being unified in practice, not only in formal geometry. A plausible implication is that the success of the method depends as much on stable differentiable objective construction as on the manifold reformulation itself.

5. Resource measures treated and benchmark problems

The paper demonstrates QFrCoRT on entanglement, coherence, and magic-state quantification (Zhu et al., 2024).

Resource	Pure-state quantity	Benchmark state or family
Entanglement of formation	$\mathcal{O}$3	$\mathcal{O}$4 Werner states
Linear entropy of entanglement	$\mathcal{O}$5	$\mathcal{O}$6 Horodecki bound-entangled states
Geometric measure of coherence	$\mathcal{O}$7	Noisy coherent states
Stabilizer entropy / linearized version	$\mathcal{O}$8	Qubit states, $\mathcal{O}$9

For entanglement, the mixed-state entanglement of formation is defined by the convex roof of the von Neumann entropy of the reduced state,

$R(\rho)$0

with $R(\rho)$1. The linear entropy of entanglement is treated analogously, with

$R(\rho)$2

The reported test families are the Werner states

$R(\rho)$3

where $R(\rho)$4 is the swap operator, and the $R(\rho)$5 Horodecki bound-entangled states $R(\rho)$6.

For coherence, the paper uses the geometric measure of coherence, defined on pure states by

$R(\rho)$7

and benchmarks it on the noisy coherent state

$R(\rho)$8

for which the quoted analytic value is

$R(\rho)$9

For magic, the framework is applied to stabilizer entropy. For a pure $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$0-qubit state,

$$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$1

with stabilizer purity

$$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$2

The mixed-state extension uses a convex roof for $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$3, and the paper also discusses the linearized version $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$4. The reported demonstrations use qubit states with $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$5, where stabilizer states form an octahedron inside the Bloch sphere.

6. Empirical behavior, scope, and nomenclature

The reported numerical results emphasize both accuracy and efficiency (Zhu et al., 2024). For $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$6 Werner states, the computed entanglement of formation agrees closely with analytical results and reproduces the separable/entangled boundary. For the Horodecki $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$7 bound-entangled states, the gradient-based method yields accurate upper bounds on the linear entropy of entanglement, complementing SDP lower bounds. For coherence, the method matches analytic values for noisy coherent states and is reported to be substantially faster than SDP; for $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$8 to $$R(\rho)=\min_{\{p_i,|\psi_i\rangle\}:\,\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|}\sum_i p_i\,R(|\psi_i\rangle).$$9, a fit gives time scaling $r$0 with $r$1 for gradient-based optimization and $r$2 for SDP, while numerical errors stay below $r$3 even at relatively large dimensions. For linear entropy of entanglement, the paper states that polar projection is much faster than PPT/SDP relaxation, which becomes unavailable in larger dimensions such as $r$4. For magic, the heatmaps of linear stabilizer entropy clearly separate stabilizer and magic regions, consistent with the Bloch-sphere octahedron picture.

The paper also reports polynomial scaling with system size in the studied examples. For entanglement of formation on Werner states, the parameter count satisfies $r$5. This does not alter the NP-hardness of the original decomposition problem, but it does indicate that the practical optimization variables scale in a controlled way for the cases examined.

The stated overall scope is broader than the three showcased resources. The methodology is described as readily extendable to other convex-roof quantities beyond resource theories, including constrained Holevo capacity and quantum Fisher information. This suggests that the unifying principle is the convex-roof structure itself, not any specific operational interpretation of entanglement, coherence, or magic.

In unrelated natural-language-processing literature, the acronym QFrCoRT is also used for the Quebec-French Corpus of Regional Terms, a zero-shot multiple-choice benchmark for dialect understanding composed of 171 single-word Quebec regional terms, each paired with 10 answer options (Beauchemin et al., 6 Oct 2025). That usage is distinct from the quantum-information framework discussed above and reflects an acronym collision rather than a conceptual connection.