Coherence Theory#

This document explains the mathematical foundations of PhaseLab’s coherence calculations.

Circular Statistics#

Phase data requires circular statistics because phases are periodic (0 = 2π).

Circular Mean

The circular mean handles wrap-around correctly:

\[\bar{\theta} = \text{atan2}\left(\frac{1}{N}\sum_{i=1}^N \sin(\theta_i), \frac{1}{N}\sum_{i=1}^N \cos(\theta_i)\right)\]

Standard (linear) mean fails for phases near 0/2π boundary.

Circular Variance

Circular variance measures phase spread:

\[V_\phi = 1 - R\]

where R is the mean resultant length:

\[R = \frac{1}{N}\sqrt{\left(\sum_i \cos(\theta_i)\right)^2 + \left(\sum_i \sin(\theta_i)\right)^2}\]

Properties

V_φ = 0: All phases identical (perfect alignment)
V_φ = 1: Phases uniformly distributed (complete disorder)

Phase Extraction#

Different data sources require different phase extraction methods.

From Quantum Expectations

Pauli expectation values E ∈ [-1, 1] map to phases:

\[\theta = \arccos(E)\]

This maps:

E = 1 → θ = 0
E = 0 → θ = π/2
E = -1 → θ = π

From Hamiltonian Coefficients

Hamiltonian terms have coefficients that encode structure:

\[\theta_i = \arctan2(\text{Im}(c_i), \text{Re}(c_i))\]

For real coefficients:

\[\theta_i = \pi \cdot \mathbb{1}[c_i < 0]\]

Coherence Computation#

From Phase Variance

The coherence R-bar is:

\[\bar{R} = e^{-V_\phi / 2}\]

From Expectation Values (Direct)

For N expectation values:

\[\bar{R} = \frac{1}{N}\sqrt{\left(\sum_i \cos(\arccos(E_i))\right)^2 + \left(\sum_i \sin(\arccos(E_i))\right)^2}\]

Simplifies to:

\[\bar{R} = \frac{1}{N}\sqrt{\left(\sum_i E_i\right)^2 + \left(\sum_i \sqrt{1-E_i^2}\right)^2}\]

Heuristic Mode

For fast screening, PhaseLab uses coefficient variance as a proxy:

\[V_{coeff} = \text{Var}(|c_1|, |c_2|, ..., |c_n|)\]

This correlates with but is not identical to true phase variance.

ATLAS-Q Acceleration#

ATLAS-Q provides accelerated coherence computation through:

IR Measurement Grouping: Groups commuting Pauli terms to reduce measurements by 5x
Batch Processing: Vectorized computation over multiple circuits
GPU Kernels: Custom Triton kernels for phase statistics

The acceleration applies to quantum mode, not heuristic mode.

Variance Reduction#

IR measurement grouping reduces variance:

Without Grouping

Each Pauli term measured independently:

\[\sigma^2_{total} = \sum_i \sigma^2_i / N_{shots}\]

With Grouping

Commuting terms measured together:

\[\sigma^2_{grouped} = \sum_g \sigma^2_g / N_{shots}\]

Since |groups| < |terms|, variance decreases.

Typical improvement: 5x variance reduction for molecular Hamiltonians.

Statistical Considerations#

Sample Size

More measurements improve coherence estimates:

\[\sigma_{\bar{R}} \propto \frac{1}{\sqrt{N}}\]

Minimum recommended: N > 100 measurements

Confidence Intervals

For large N, R-bar is approximately normal:

\[\bar{R} \pm 1.96 \cdot \frac{\sigma_R}{\sqrt{N}}\]

Bias Correction

Small samples overestimate R. Correction factor:

\[\bar{R}_{corrected} = \bar{R} - \frac{1-\bar{R}^2}{2N}\]

Implementation Details#

PhaseLab implements coherence in phaselab/quantum/coherence.py:

def compute_coherence_from_expectations(expectations, use_atlas_q=True):
    # Convert to phases
    phases = np.arccos(np.clip(expectations, -1, 1))

    # Circular statistics
    cos_sum = np.sum(np.cos(phases))
    sin_sum = np.sum(np.sin(phases))

    # Mean resultant length
    R = np.sqrt(cos_sum**2 + sin_sum**2) / len(phases)

    # Phase variance
    V_phi = 1 - R

    # Coherence
    R_bar = np.exp(-V_phi / 2)

    return CoherenceResult(R_bar=R_bar, V_phi=V_phi, ...)