Covariant Approximation Averaging in Lattice QCD

• Covariant Approximation Averaging (CAA) is a symmetry-based error reduction strategy in lattice QCD that produces unbiased estimators by combining multiple low-cost approximations with a single exact correction.
• CAA leverages lattice translation symmetry to average shifted approximations, significantly reducing computational cost and statistical variance in two- and three-point correlation functions.
• Its integration with techniques like the one-end trick and low-mode deflation makes CAA essential for precise all-to-all quark propagator calculations in hadron structure studies.

Covariant Approximation Averaging (CAA) is a symmetry-based error reduction strategy in lattice QCD that constructs improved stochastic estimators for correlation functions by combining a low-cost, symmetry-covariant approximation with a single unbiased correction from an exact computation. The main principle is to leverage exact lattice symmetries (such as translations) to generate multiple low-cost evaluations of an approximate observable, average them, and remove all bias via a single correction, yielding a statistically improved and unbiased estimator applicable to two-point, three-point, and more complex correlation functions. CAA is particularly effective in applications requiring all-to-all quark propagators and has been realized as all-mode averaging (AMA) in large-scale lattice computations (Shintani et al., 2014, 1212.5542, Akahoshi et al., 2021).

1. Formal Definition and Theoretical Framework

Let $S$ represent the exact quark propagator and $\mathcal{O}[S]$ any derived observable (e.g., a hadron $n$-point function). CAA constructs an improved estimator as follows. Define a group $G$ of lattice symmetries (typically site translations), and an approximation $S^{(\rm appx)}$ (commonly a low-mode truncated or a loose-tolerance CG solution) such that

$S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$

The observable splits accordingly. For each $g \in G$, $\mathcal{O}^{(\rm appx),g}$ is the observable constructed from $S^{(\rm appx)}$ at the $g$-shifted configuration or source. Defining

$\mathcal{O}[S]$0

the CAA/AMA estimator is

$\mathcal{O}[S]$1

By the covariance of the lattice action and approximation, CAA is strictly unbiased: $\mathcal{O}[S]$2 This construction reduces variance by exploiting the statistical independence among symmetry-related approximations while correcting any remaining bias from a single exact computation.

2. Methodology and Implementation

The canonical implementation uses translation symmetry on the lattice. The workflow proceeds as follows:

1. Eigenmode Computation: Compute $\mathcal{O}[S]$3 low eigenmodes $\mathcal{O}[S]$4 of the Hermitian Dirac kernel $\mathcal{O}[S]$5. Construct the low-mode part

$\mathcal{O}[S]$6

1. Approximate Propagator Construction: For each symmetry $\mathcal{O}[S]$7 (e.g., $\mathcal{O}[S]$8 or 64 spatial translations or shifts),
   • Generate $\mathcal{O}[S]$9 by injecting a shifted source and applying a low-cost approximation (e.g., low-mode truncation or relaxed CG, possibly combined with deflation), reusing precomputed modes.
   • Construct the approximate observable $n$0.
2. Exact Correction: At a reference point (typically a single source location), compute the full exact observable $n$1.
3. Estimator Assembly: The improved observable is the mean over all approximations plus the difference between the exact and its approximation at the reference:

$n$2

A typical pseudocode (as explicitly described in (Akahoshi et al., 2021)):

$S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$4 In large-scale applications such as the HAL QCD $n$3-resonance study, $n$4 and $n$5 were typical values (Akahoshi et al., 2021).

3. Performance and Error Reduction

The statistical error of the CAA estimator is determined by the correlation coefficient $n$6 between the exact and approximate observable: $n$7 and

$n$8

When $n$9 (highly correlated approximation), the error scales as $G$0, approaching the optimum for independent measurements at cost $G$1 of the naive approach (1212.5542, Shintani et al., 2014).

In the HAL QCD $G$2-resonance application, this yielded roughly a tenfold reduction in statistical error: uncertainties on the leading-order potential at $G$3 fm decreased from $G$4 to $G$5 per time-slice, sufficient to enable the first stable N$G$6LO extraction in a $G$7-wave channel (Akahoshi et al., 2021). Similarly, for nucleon two- and three-point functions, error-reduction factors close to $G$8 were observed, with overall cost savings of $G$9–$S^{(\rm appx)}$0 compared to standard methods (1212.5542, Shintani et al., 2014).

Observable	Standard Error	CAA/AMA Error Ratio	Normalized Cost Ratio
$S^{(\rm appx)}$1	156 MeV	0.17	0.04
$S^{(\rm appx)}$2	12 MeV	0.36	0.19
$S^{(\rm appx)}$3	176 MeV	0.20	0.06

4. Applications in Lattice QCD

CAA is widely applicable in the context of all-to-all propagators, essential for multi-hadron and resonance studies with spatially extended operators and non-localities. Major applications include:

• HAL QCD Method: Used for high-precision, non-local potential extraction, especially in channels requiring the Nambu–Bethe–Salpeter (NBS) wave function with explicit spatial dependence. In the $S^{(\rm appx)}$4-resonance calculation, CAA was combined with the one-end trick and sequential propagator techniques, enabling efficient construction of many spatially displaced correlators with minimal additional cost while rigorously correcting for all bias at a single point (Akahoshi et al., 2021).
• Hadron Structure: CAA/AMA yields an order-of-magnitude reduction in cost-to-error for nucleon and meson two-point/three-point functions, form factors, and disconnected diagrams (1212.5542, Shintani et al., 2014).
• Broader Observables: The method is compatible with a wide array of fermion discretizations (domain-wall, Wilson, twisted-mass, overlap) and can be extended to multi-current matrix elements and flavor singlet channels (Shintani et al., 2014).

5. Advantages, Limitations, and Extensions

Advantages

• Unbiasedness: CAA preserves the expectation value of the target observable exactly by construction under the relevant group symmetry (Shintani et al., 2014).
• Dramatic Variance Reduction: Effective statistics are amplified by $S^{(\rm appx)}$5 with only one exact measurement per configuration required.
• Compatibility with Other Tricks: Integrates seamlessly with the one-end trick and sequential propagator techniques, eliminating stochastic noise at the sink (Akahoshi et al., 2021).
• Reduced Solver Count: Substantial decrease in the number of Dirac-operator inversions compared to fully stochastic all-to-all approaches.

Limitations

• Quality of Approximation: The approximation $S^{(\rm appx)}$6 must be highly correlated with $S^{(\rm appx)}$7; otherwise, residual fluctuations can dominate the estimator's variance. The balance between low-mode computation ($S^{(\rm appx)}$8) and number of symmetry shifts ($S^{(\rm appx)}$9) is critical to optimize performance (Akahoshi et al., 2021, Shintani et al., 2014).
• Computation of Low Modes: Calculating and storing many low eigenvectors incurs significant initial cost, albeit amortized over many measurements.
• Symmetry Scope: In channels with additional symmetry mixing (rotations, parity), $S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$0 may need to be expanded beyond simple translations.

Extensions

• Inclusion of Rotational/Reflection Symmetries: Amplifies effective statistics beyond translations (Akahoshi et al., 2021).
• Combination with Hierarchical Probing, High-Mode Deflation: Targets further reduction of residual variance.
• Generalization to Disconnected Diagrams, Multi-current Matrix Elements, High-momentum Baryon Observables: Demonstrated in nucleon and meson matrix element calculations (1212.5542, Shintani et al., 2014).

6. Practical Considerations and Outlook

Implementation of CAA requires careful attention to covariance under lattice group actions; in practice, fixed iteration counts or randomized shifts can be used to eliminate rounding-induced biases (Shintani et al., 2014). The cost of low-mode computation becomes negligible as the scope of observable averaging grows.

In all tested scenarios—including $S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$1-resonance potentials, nucleon form factors, and meson correlators—CAA and its variants (notably AMA) enabled factors of $S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$2–$S = S^{(\rm appx)} + (S - S^{(\rm appx)}).$3 statistical error reduction for fixed computational resources, establishing CAA as an essential tool in modern lattice QCD (Shintani et al., 2014, 1212.5542, Akahoshi et al., 2021). A plausible implication is that the broad applicability and systematic unbiasedness of CAA make it broadly relevant to forthcoming high-precision studies in hadronic physics, spectrum determination, and nuclear interactions.