Spectral Hammer: Dual Frameworks in Physics

• Spectral Hammer is defined as two distinct methods: one for amplitude-level reweighting in heavy-flavor decays and another for fidelity enhancement in noisy quantum circuits.
• In semileptonic decay analyses, HAMMER computes reweighting factors by factorizing amplitude tensors, enabling efficient extraction of new physics parameters without SM-template bias.
• For quantum circuits, the HAMMER technique leverages Hamming-space clustering to improve output fidelity, boosting metrics like PST and IST by factors up to 1.38× in experimental settings.

The term "Spectral Hammer" refers, in contemporary scientific literature, to multiple unrelated frameworks and software tools that combine spectral, amplitude, or Hamming-space methods for boosting fidelity and interpretability in disparate scientific domains. Notably, the designation encompasses two major, distinct methodologies: (1) the HAMMER software for efficient, model-consistent reweighting in heavy-flavor semileptonic decays, and (2) HAMMER as a post-processing technique for fidelity improvement in noisy quantum circuits (sometimes referred to as "Hamming Reconstruction"). Both use the "HAMMER" moniker as an acronym related to their respective amplitude or Hamming-centric algorithmic approaches, but are unrelated in subject matter. This article provides a comprehensive overview of both instantiations, emphasizing methodological innovation, software architecture, practical applications, and their impact on contemporary research workflows.

1. HAMMER for New Physics Interpretations in Semileptonic Decays

HAMMER (Helicity Amplitude Module for Matrix Element Reweighting) is a software toolkit designed for the amplitude-level reweighting of Monte Carlo (MC) samples in analyses of semileptonic decays, particularly those involving $b \to c\tau\nu$ transitions. Its purpose is to enable the reinterpretation of previously generated (and computationally expensive) MC events—originally simulated with Standard Model (SM) or simplified phase-space assumptions—for arbitrary new physics (NP) scenarios or for updated hadronic matrix element parametrizations (Bernlochner et al., 2020).

Key capabilities include:

• Amplitude-level Reweighting: For each event, HAMMER computes a weight ratio,

$$r_{(I)} = \frac{d\Gamma^{\text{new}}/d\text{PS}}{d\Gamma^{\text{old}}/d\text{PS}},$$

where $d\Gamma/d\text{PS}$ denotes the differential decay rate in the new or old physics/model hypothesis, evaluated as a function of the event's four-momenta.

• Tensorization: The code factorizes the computationally expensive amplitude and tensor-level calculations. Amplitude tensors, generalized over NP Wilson coefficients and form factor parameters, are stored and reused, enabling rapid recalculation under new hypotheses.
• Plug-and-Play in Experimental Frameworks: HAMMER operates on event-level truth four-momentum information, facilitating integration with analyses at Belle II, LHCb, and other heavy-flavor experiments.

This approach contrasts with previous methods in which analysts fitted new physics parameters to observables extracted from SM-based templates, potentially introducing substantial bias due to altered acceptance and decay kinematics under NP.

2. Methodology and Mathematical Framework

The foundation of HAMMER's renormalization lies in a decomposition of the decay matrix element,

$$\mathcal{M}(\{s\}, \{q\}) = \sum_{(\alpha, i)} c_\alpha F_i(\{q\}) \mathcal{A}_{\alpha i}(\{q\}),$$

where $c_\alpha$ denotes the NP or SM Wilson coefficients, $F_i(\{q\})$ are hadronic form factors, and $\mathcal{A}_{\alpha i}$ are constructed amplitudes.

The corresponding (polarized) differential decay rate is: $$\frac{d\Gamma(\{s\})}{d\text{PS}} = \sum_{(\alpha,i),(\beta,j)} c_\alpha c_\beta^* F_i(\{q\}) F_j^*(\{q\}) \mathcal{W}_{\alpha i \beta j},$$ where $\mathcal{W}_{\alpha i \beta j}$ is a weight tensor. This structure decouples the costly phase-space integration from the evaluation of new Wilson coefficient or form factor variations, ensuring computational efficiency.

The API provides specification of decay chains ("include" and "forbid") and form factor schemes, while processing routines manage statistical uncertainties, histogramming (with ROOT support), and multi-core parallelization.

3. Bias Avoidance and Self-consistent Forward Folding

Conventional analyses may utilize SM-based MC templates for observable extraction and, subsequently, reinterpret these measurements as NP constraints—a process shown to induce significant statistical bias when NP alters event kinematics or detector response. Using toy Asimov datasets generated under benchmark NP models (e.g., two-Higgs-doublet, tensor, or leptoquark scenarios), the recovered values of $R(D)$ and $R(D^*)$ can deviate from their NP "truth" by up to $3$–$4\sigma$.

HAMMER enables a self-consistent, forward-folding methodology: NP Wilson coefficients are directly fit using reweighted MC, so the distribution of observables incorporates the full NP-dependent kinematic and acceptance variation. Likelihood contours for real and imaginary parts of Wilson coefficients recover the true NP benchmarks and eliminate template-driven bias (Bernlochner et al., 2020).

4. Software Architecture, Performance, and Integration

HAMMER is implemented in C++ with a Python interface. Major classes include:

• Hammer: Orchestrates event processing and handles user configuration.
• Process: Encodes the decay process tree.
• Particle: Manages PDG codes, four-momenta, and decay vertices/braches.
• IOBuffer: Facilitates fast input/output.

The system reads generic event-level data (truth four-vectors), can process multiple decay processes in parallel, and outputs high-dimensional histograms with statistical uncertainty tracking.

Memory requirements and runtime scale efficiently with the number of outcome types and phase space points, permitting practical deployment on real-world experimental datasets.

5. Applications and Extensions

Primary applications are in high-precision flavor physics, including the direct extraction of NP Wilson coefficients in $b \to c\ell\nu$ decays at Belle II and LHCb. The amplitude–tensor separation allows for rapid adaptation to new form factor models or NP scenarios without re-simulating detector-level events.

Additional intended extensions include:

• Generalization to rare or baryonic decays.
• Providing input to global NP fits.
• Future integration with advanced detector simulations for anomaly characterization.

A plausible implication is that this approach enables more robust reinterpretation of anomalies and findings in flavor physics; however, current implementation focuses primarily on $b \to c\ell\nu$ transitions.

6. HAMMER for Fidelity Boosting in Noisy Quantum Circuits

A distinct, unrelated instantiation of "HAMMER" appears in quantum information science as a post-processing technique for reconstructing output distributions of noisy quantum circuits, exploiting Hamming-space structure in erroneous outcomes (Tannu et al., 2022).

Here, the "HAMMER" (Hamming Reconstruction) approach leverages the empirical observation that on near-term (NISQ) devices, erroneous outcome bitstrings cluster in Hamming space near the correct result. Rather than being distributed uniformly at random (which would yield an expected Hamming distance of $n/2$ for $n$-qubit systems), the noisy output has most probability mass in neighborhoods close to the true answer.

The method proceeds as follows:

1. Hamming Spectrum Analysis: Outcomes are bucketed by their Hamming distance from the (unknown) correct answer, yielding the empirical cumulative Hamming strength (CHS) vector.
2. Neighborhood Scoring: For each outcome $x$, a score $S(x)$ is computed based on weighted CHS contributions from nearby bitstrings, restricting to those of lower or similar original probability.
3. Likelihood Update: The new likelihood for $x$ is given by

$L(x) = P(x) \times S(x),$

where $P(x)$ is the measured probability.

1. Normalization: The updated distribution is renormalized and used for downstream inference.

Empirical results over 500+ quantum circuits on IBM and Google devices show average improvement in solution quality by a factor of $1.37\times$, with the Probability of Successful Trial (PST) and Inference Strength (IST) metrics increasing by up to $1.38\times$ and (in some cases) $5\times$, respectively.

7. Comparative Analysis, Limitations, and Impact

Both HAMMER frameworks stand out in their respective fields by offering computationally efficient, post hoc modification of experimentally generated distributions to recover fidelity (in quantum outputs) or eliminate bias (in flavor-observable measurements).

• In particle physics, HAMMER addresses a core methodological gap by enabling NP-consistent extraction of Wilson coefficients without SM-template bias.
• In quantum information, the Hamming Reconstruction-based HAMMER exploits intrinsic noise structure without hardware modification or explicit error channel modeling, positioning it as complementary to standard hardware-level or calibration-based error mitigation approaches.

Both methodologies scale efficiently to large problem sizes: in quantum circuits, the algorithm has linear memory scaling in qubit number and quadratic scaling in the number of unique measured bitstrings; in Monte Carlo reweighting, tensor and amplitude caching decouples computational burdens from the number of NP/FF hypotheses.

A stated limitation is that, for the quantum HAMMER method, efficacy requires that erroneous outcomes are Hamming-clustered—a property confirmed on current NISQ devices but subject to change with evolving hardware. In the context of new physics fits, applicability is predicated on availability of unbiased truth-level MC samples.

In summary, "Spectral Hammer" encompasses distinct scientific tools embodying the principle of information recovery or adjustment via spectral, amplitude, or Hamming-space methodologies. These tools are notable for their modularity, computational efficiency, and the measurable impact on fidelity or bias in quantum and high-energy physics analyses.