OptRot+: Rotation Methods in Quantum Science

• OptRot+ is a class of rotation-type methods that optimize operator mappings using adjoint and dual maps for precise quantum state reconstruction.
• It is applied in quantum tomography, entanglement detection, orbital optimization, and post-training quantization to improve measurement and performance.
• Its constrained optimization framework, leveraging orthogonality and unitarity, enables reduced qubit requirements and robust numerical convergence.

OptRot$^{+}$ denotes a class of rotation-type methods and maps found in quantum information, quantum chemistry, and machine learning. It refers to both operator-symbol mappings in quantum tomography (Amosov et al., 2011), positive optimal maps for entanglement detection (Rutkowski et al., 2015), orthogonally-constrained orbital optimization in quantum algorithms (Bierman et al., 2023), and data-dependent rotations for post-training quantization in neural networks (Gadhikar et al., 30 Dec 2025). The defining characteristic is the use of rotation or unitary structure—often as an adjoint or dual map—to optimize or encode key physical or numerical properties.

1. Dual Symbol Maps in Quantum Tomography

OptRot$^{+}$ in quantum optics is the adjoint (dual) of the optical-tomogram quantization map. In this framework, states are represented by their tomograms $w(X,\theta)$, which are the probability densities for rotated quadratures $X_\theta = \hat{q}\cos\theta + \hat{p}\sin\theta$ at phase $\theta$. The dual OptRot$^{+}$ map assigns to any operator $\hat{A}$ its “dual tomographic symbol”,

$$w_A^+(X,\theta) = \operatorname{Tr}\left[\hat{A}\, \hat{D}(X,\theta)\right]$$

with the quantizer $\hat{D}(X,\theta)$ defined by

$$\hat{D}(X,\theta) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\eta\, |\eta|\, \exp\left[i\eta(X - \hat{q}\cos\theta - \hat{p}\sin\theta)\right].$$

Expectation values become direct integrals over measured tomograms: $$\operatorname{Tr}\left[\hat{\rho}\,\hat{A}\right] = \int d\theta\, dX\, w(X,\theta)\, w_A^+(X,\theta).$$ This algebraic structure defines a full star-product quantization on the $(X,\theta)$ manifold, with OptRot$^{+}$ providing an explicit adjoint to the tomographic (dequantization) mapping (Amosov et al., 2011). Reconstruction of operators and observables, as well as their direct physical interpretation in optical homodyne tomography, is central to this formalism.

2. Positive Optimal and Indecomposable Maps

In operator algebra and quantum information, OptRot$^{+}$ refers to the Miller–Olkiewicz “rotation-type” map and its generalization to $M_d(\mathbb{C})$, the $d$-dimensional complex matrix algebra (Rutkowski et al., 2015). The map $\Phi_{d}: M_{d}(\mathbb{C}) \to M_{d}(\mathbb{C})$ is defined by

$$\Phi_{d}(A) = \sum_{i=1}^{d-1} \left[\frac{1}{d-1}\sum_{k=1}^{d-1} a_{kk}\right] E_{ii} + a_{dd} E_{dd} - \frac{1}{d-1} \sum_{i=1}^{d-1} (a_{id} E_{id} + a_{di} E_{di})$$

where $E_{ij}$ are the standard matrix units.

Key properties include:

• Positivity: Proven by Schur-complement analysis on rank-one operators.
• Bistochasticity: $\Phi_{d}(I) = I$ and trace preservation.
• Optimality: Product vectors parametrized by determinant-zero conditions span $\mathbb{C}^d \otimes \mathbb{C}^d$ and guarantee optimality of the witness operator $$W_d = (\mathrm{id}\otimes\Phi_d)(|\Omega\rangle\langle\Omega|)$$ for entanglement detection.
• Indecomposability: $\Phi_d$ witnesses entanglement in PPT states that are undetectable by decomposable maps. Explicit PPT-entangled states are constructed via block structures and trace computations.

In $d=3$, this directly induces the original Miller–Olkiewicz OptRot$^{+}$ map. The general construction enables detection of PPT entanglement classes and serves as an analytic tool in quantum information theory (Rutkowski et al., 2015).

3. Orthogonally-Constrained Orbital Optimization in Quantum Algorithms

OptRot$^{+}$ also denotes a state-averaged, orthogonally-constrained gradient-projection method for orbital optimization in variational quantum excited-state solvers (Bierman et al., 2023). Here, electronic structure Hamiltonians are rotated into optimally chosen active spaces by a partial unitary $U \in \mathbb{R}^{M\times N}$ satisfying $U^\top U = I_N$, where $M$ is the number of spin-orbitals and $N < M$ the number of active orbitals.

The optimization alternates classical steps (updating $U$ via first-order projected-gradient descent on the Stiefel manifold) and quantum steps (optimizing ansatz parameters $\theta$ via a quantum eigensolver):

• Gradient Projection: Enforces $U^TU = I_N$ via tangent-space projection, $G_{\text{tangent}} = G - U\,\mathrm{Sym}(U^T G)$.
• Update and Re-Orthonormalization: $U_{\text{new}} = \text{orth}(V)$ is obtained by spectral decomposition of $V^T V$.
• Objective: Minimize average energy over $K$ weighted reference states using rotated Hamiltonian and corresponding reduced density matrices.

Compared to conventional CASSCF-style orbital optimization (second-order Newton steps via anti-Hermitian exponentiation), OptRot$^{+}$ operates directly in $M\times N$ space, uses only first-order gradients, avoids Hessian calculations, and empirically provides more robust convergence and reduced qubit requirements. Benchmark studies demonstrate near-FCI accuracy for small molecular systems with substantial (>75%) qubit reductions (Bierman et al., 2023).

4. Data-Dependent Rotations for Post-Training Quantization

In post-training quantization (PTQ) for LLMs, OptRot$^{+}$ generalizes data-free rotation methods through activation covariance-aware objectives (Gadhikar et al., 30 Dec 2025). The optimization seeks rotation matrices $(R_1,\{R_{2,\ell}\})$ minimizing

$$\sum_{\ell, s} UB_\ell \|\mathrm{vec}(R_\ell^\top W^{(\ell, s)} R'_\ell)\|_4$$

subject to $R_\ell^\top R_\ell = I$, where each $UB_\ell$ is a layerwise upper bound on quantization error incorporating

• Weight Incoherence: Quantified via the element-wise 4-norm—$\|\mathrm{vec}(W)\|_4$—to penalize large outliers.
• Activation Covariance: $$UB(H) = \operatorname{tr} H - \frac{1}{2\,\operatorname{tr} H} \sum_{i \ne j} H_{ij}^2$$ based on measured input activity.

The learning procedure precomputes per-layer $UB_\ell$ from calibration data and applies Cayley-SGD on the Stiefel manifold for efficient rotation updates. All learned rotations are fusible into the original weights for zero inference overhead.

Empirical studies demonstrate that OptRot$^{+}$ matches or slightly outperforms data-free and activation-aware alternatives (Hadamard, SpinQuant) for 4-bit weight and 8-bit activation schemes. For more aggressive quantization (W4A4), accuracy may degrade, reflecting a trade-off between dispersion reduction and feature alignment (Gadhikar et al., 30 Dec 2025).

5. Comparative Analysis and Domain-Specific Recommendations

OptRot$^{+}$-type maps and procedures are characterized by principled extremization/rotation steps on constrained manifolds (orthogonality, trace preservation, or unitary structure)—either in physical operator symbol mappings, entanglement detection, quantum orbital truncation, or machine learning weight alignment. The common mathematical construction involves adjoint or dual mappings that optimize over rotation-like degrees of freedom.

Practical guidelines for deployment across domains:

• Quantum tomography: OptRot$^{+}$ enables direct extraction of expectation values from measured tomograms, bypassing intermediate density operator or Wigner reconstructions (Amosov et al., 2011).
• Quantum information: The Miller–Olkiewicz OptRot$^{+}$ map and its generalization allow systematic construction of optimal indecomposable entanglement witnesses, especially for high-dimensional PPT states (Rutkowski et al., 2015).
• Quantum chemistry/NISQ algorithms: Orthogonally-constrained projection (OptRot$^{+}$) offers more scalable and robust orbital optimization than second-order approaches, significantly reducing qubit numbers while preserving accuracy (Bierman et al., 2023).
• LLM quantization: Data-dependent OptRot$^{+}$ provides an efficient layerwise post-processing tool for weight rotations, with negligible computational overhead and consistent accuracy gains for moderate quantization settings (Gadhikar et al., 30 Dec 2025).

6. Limitations and Trade-Offs

Although OptRot$^{+}$ methods have achieved notable practical and theoretical utility, specific limitations arise:

• In LLM quantization, OptRot$^{+}$ may degrade accuracy under aggressive quantization schemes (W4A4), as minimizing weight incoherence alone shifts activation distributions unfavorably.
• In quantum algorithmic contexts, while OptRot$^{+}$ avoids second-order expansions, it may require substantial measurement and classical computational resources for large $N$ or $K$ states.
• In operator algebraic settings, the generalization of OptRot$^{+}$ maps relies on explicit block structures and spanning-product vectors, which may not translate directly to more exotic or infinite-dimensional operator systems.

A plausible implication is that OptRot$^{+}$ approaches, given their manifold-constrained optimization structures, are likely extensible to further applications involving orthogonality or unitary group actions—especially wherever rank-preserving, rotation-based adjoint mappings are required. However, domain-specific calibration is essential for maximizing efficacy and avoiding unintended degradation of associated observables or metrics.