Binary First Quantized Mapping

• Binary First Quantized Mapping is a technique that converts high-dimensional real data into a binary {+1, –1} format with optimal scaling to minimize quantization error.
• It underpins efficient algorithms in deep learning, digital signal processing, quantum simulation, and metric-preserving embeddings by enabling aggressive compression and fast bitwise operations.
• The method achieves quantization error reduction and hardware acceleration by leveraging proximal operations, scaling optimizations, and structured regularization in both algorithmic and physical implementations.

A binary first quantized mapping (BQFM) is a transformation that encodes real-valued or high-dimensional data into a domain of discrete, binary variables—typically {+1, –1}—at the earliest viable point in the processing pipeline. This quantization paradigm is foundational in resource-constrained deep learning, digital signal processing, quantum simulation, and rapid metric-preserving embeddings. BQFM provides aggressive compression, enables highly efficient hardware execution via bitwise logic, and can offer essential guarantees—statistical, numerical, or physical—about downstream behavior. Modern works clarify both the mathematical optimality and practical algorithmics of binary quantization across neural networks, generative models, statistical learning, embedded systems, and quantum platforms.

1. Mathematical Foundations of Binary Quantization

A canonical BQFM transforms a real vector $w \in \mathbb{R}^N$ into a binary representation $q \in \{\pm 1\}^N$ with an optimal scale $\alpha$ to minimize Euclidean error. The least-squares solution (Pouransari et al., 2020) is

$$Q(w) = \alpha^*\,\mathrm{sign}(w),\quad \alpha^* = \frac{1}{N}\sum_{i=1}^N |w_i|,$$

where $\mathrm{sign}(\cdot)$ operates elementwise. This mapping is provably optimal in the mean square sense and generalizes to multi-bit “foldable” binary decompositions: $$Q_k(x) = \sum_{i=1}^k v_i\,s_i(x),\,\, s_i(x)\in\{\pm 1\},\,\, v_1\ge v_2\ge \dots\ge v_k\ge 0,$$ with explicit nonlinear equations for $v_i$. In hardware, inner products between such binary vectors are computed efficiently via XNOR and population counts (popcount), e.g. $$\langle u,v \rangle = 2\,\mathrm{popcount}(\mathrm{XNOR}(u,v)) - N$$.

In optimization contexts, BQFM arises as a proximal operator with a quantization-inducing regularizer (Bai et al., 2018): $$\min_\theta L(\theta) + \lambda R(\theta),\qquad R(\theta) = \|\theta - \mathrm{sign}(\theta)\|_1,$$ with the prox step

$$\mathrm{prox}_{\lambda R}(w) = \mathrm{sign}(w) + \mathrm{sign}(w-\mathrm{sign}(w))\odot [|w - \mathrm{sign}(w)| - \lambda]_+,$$

softly thresholding weights toward $\pm 1$.

2. Algorithmic Realizations in Deep Learning and Signal Processing

Distinct BQFM algorithms exploit the above quantization in training and inference pipelines:

• ProxQuant-Binary (Bai et al., 2018): Alternates stochastic gradient steps on full-precision weights with a binary-prox step, yielding stability and $O(1/T)$ convergence to stationarity. Empirically, it reduces sign-flip volatility and error compared to straight-through methods (BinaryConnect).
• Blended Coarse Gradient Descent (BCGD) (Yin et al., 2018): Introduces a blend factor $\rho$ for sufficient descent, updating auxiliary full-precision and binary weights, and employs proxy gradients for non-differentiable quantized activations. This enables efficient training of fully quantized networks.
• Dynamic Grouped Block Quantization for LLMs (Zheng et al., 3 Sep 2025): Formulates an objective minimizing variance within adaptive sub-matrix blocks with binary values, regularized to avoid pathological fragmentation. The method partitions weights, applies optimal scaling per block, and achieves 1.007 average bits per parameter while preserving model quality and requiring minimal computation.

3. BQFM in Quantum Simulation

Binary first quantized mapping plays a crucial role in quantum simulations for encoding physical states and operator actions efficiently (Mikkelsen et al., 13 Nov 2025, Baker et al., 8 Oct 2025):

• Bosonic Systems: For $N$ bosons in $M=2^n$ modes, each particle register is $n$ qubits encoding a mode index in binary. A state $|l^{(1)},...,l^{(N)}\rangle$ is the tensor product over these registers, and operators (such as $a_\ell^\dagger a_m$) become sums over Pauli strings via projector expansions, e.g., $|l_i\rangle\langle m_i|$ expands into $(X\pm iY)/2$ or $(I\pm Z)/2$. BQFM achieves exponentially lower qubit cost than unary encodings and competitive gate counts for Trotter steps—$O(Mn)$ $R_z$ and $O(Mn^2)$ CNOT gates.
• Initial State Preparation: Universal mapping from second-quantized Fock states to first-quantized wavefunctions uses the Jordan–Schwinger homomorphism and Schur–Weyl duality. Block-encoded superpositions are prepared and the inverse quantum Schur transform reconstructs the required many-particle state with polynomial resource scaling.

4. Metric-Preserving Embeddings, Compressed Sensing, and Statistical Sketches

BQFM generalizes to metric spaces and statistical models via binary embeddings and quantized sketching (Huynh et al., 2018, Schellekens et al., 2021):

• Fast Binary Embeddings: Points $x\in\mathbb{R}^n$ are mapped to $q\in\{\pm 1\}^m$ by fast Johnson–Lindenstrauss transforms (e.g., Hadamard, partial circulant FFT) followed by structured noise-shaping quantization (ΣΔ, β-encoding). The induced pseudo-metric

$d_V(q,r) = \|\hat{V}(q - r)\|_2$

controls distortion, achieving polynomial (ΣΔ) or exponential ($\beta$-encoding) decay in error with $m$.

• Asymmetric Compressive Learning: Binary sketch maps φ₁($x$) = sign(cos(ωᵀ$x$ + ξ)) quantize random Fourier features, yielding sketches $s₁$ for large-scale statistical learning. The Limited Projected Distortion property ensures statistical risk bounds, matching those of full-precision sketches up to the quantization penalty.

5. Hardware Architectures and Implementation Principles

Cutting-edge hardware realizations embody BQFM mapping for neural and probabilistic systems (Kingra et al., 2022, Redfern et al., 2020):

• Hybrid CMOS–OxRAM: Binary/ternary neural network weights trained in software are mapped onto multi-level conductance states of single-device OxRAM synapses. Input vectors are decomposed into positive/negative masks and multiplexed read phases accumulate dot products, consumed by CMOS sigmoid neurons. Calibration ensures sub-5% accuracy loss on Fashion-MNIST, achieving 16×–32× memory savings.
• Fully Binary Convolutional Networks: All convolution and matrix multiplies use {+1,–1} operands. Bit-masks hold these binary values, and hardware exploits XNOR and popcount instructions for maximal inference speed; batch normalization and residual connections are efficiently interleaved.

6. Performance Benchmarks, Guarantees, and Trade-offs

Empirical and theoretical analysis reveals distinct trade-offs:

• Deep Learning: BQFM mappings stabilized by prox-gradient or least-squares scaling consistently outperform heuristic binarization, reducing quantization error (angular deviation $\sim$20°–25° vs 37°) and accuracy gaps (e.g. LS 1-bit at 58.9% top-1 ImageNet accuracy, LS 2-bit at 63.4%) (Pouransari et al., 2020).
• LLMs: Dynamic grouped methods sustain perplexity close to float (e.g. LLAMA 3B: 8.23 vs 7.81) and QA accuracy (0.627 vs 0.648) at true 1-bit storage (Zheng et al., 3 Sep 2025).
• Quantum Simulation: BQFM resource counts for gate-based simulation of bosonic Hamiltonians match the best unary approaches (provided $N\ll M=2^n$), but with an exponentially lighter qubit footprint (Mikkelsen et al., 13 Nov 2025).
• Embedded Hardware: Binary first quantized mapping to OxRAM achieves 83.1% accuracy (BNN) with $<0.2\%$ drop from ideal, and inference energy $<100$ pJ/pass (Kingra et al., 2022).

A plausible implication is that the combination of optimal scaling, adaptive regularization, and block-through hardware mapping is essential to avoid quantization collapse and maintain high performance.

7. Extensions, Implementation Considerations, and Limitations

The efficacy of binary first quantized mapping depends on domain-specific adaptation:

• In deep networks, BQFM mappings must integrate scaling and/or regularization to offset quantization error accumulation. Proximal and blended methods yield provable convergence and empirical stability.
• For quantum and statistical embedding, fast transforms and noise-shaped quantization realize efficient encoding and metric control.
• Hardware implementations relying on binary logic and bitwise operations require calibration protocols for device variation, with single-device architectures yielding maximal memory and area savings.
• Limitations include increased optimization complexity for multi-bit or block-adaptive mappings, sensitivity to device and quantizer variation, and constrained accuracy for highly nonlinear tasks in 1-bit regimes.

In summary, binary first quantized mapping constitutes a unifying framework harnessing optimal sign-based reduction, dynamic scaling, and structured adaptive regularization, enabling efficient and effective compression, training, simulation, and inference in modern computational paradigms across deep learning, quantum simulation, compressed sensing, and hardware design.