The Ternary Pivot: Mathematical Modeling of NPU Systolic Array Utilization for BitNet b1.58 Quantization Kernels

By Rizowan Ahmed (@riz1raj)
Senior Technology Analyst | Covering Enterprise IT, Hardware & Emerging Trends

The tech industry’s focus on floating-point precision is shifting as hardware constraints meet the requirements of large-scale model deployment. For years, 16-bit and 8-bit precision were considered the standard for LLM inference. However, the BitNet b1.58 (ternary) architecture has demonstrated that lower precision is viable for consumer-grade hardware. The primary bottleneck has transitioned from raw compute to the physics of data movement across the silicon die. Optimizing for mathematical modeling of NPU systolic array utilization for BitNet b1.58 quantization kernels is now a critical requirement for efficient local inference.

The Shift to Ternary Compute

Neural Processing Units (NPUs) are often viewed as standard matrix-multiplication engines. While this model served architectures like Llama-2 or GPT-4, the rise of 1.58-bit (ternary) weights—where parameters are constrained to {-1, 0, 1}—renders standard Multiply-Accumulate (MAC) operations less efficient. In a ternary system, when a weight is zero, the multiplication is bypassed; when it is 1 or -1, the operation is simplified to addition or subtraction.

The challenge for modern hardware is that many systolic arrays are optimized for INT8 or FP16 density. Running BitNet b1.58 on these chips can lead to underutilization. Without hardware-aware kernels, a significant portion of systolic array cycles may remain idle while the memory controller manages data transfer. This represents a shift from compute-bound to logic-bound processing paradigms.

Mathematical Modeling of Systolic Array Utilization

To evaluate the efficiency of ternary kernels, we model the Arithmetic Intensity (AI) of the operation. In a standard INT8 General Matrix Multiply (GEMM), the AI is high. In a BitNet b1.58 kernel, the theoretical memory bandwidth requirement is reduced significantly, but the logic gates required for accumulation remain a factor in throughput.

The utilization $U$ of a systolic array of size $M \times N$ can be modeled as:

• $U = \frac{\sum_{i=1}^{k} Ops_{ternary}}{\text{Cycles} \times (M \times N)}$
• Where $Ops_{ternary}$ represents the non-zero weight interactions.
• In a ternary model, the distribution of zero weights impacts efficiency. Without zero-skipping hardware logic, utilization is limited by the hardware's ability to process sparse activations and weights.

SRAM Tiling and Memory Management

The gap between SRAM speed and DRAM bandwidth necessitates aggressive memory management. To achieve efficient sub-2-bit inference, developers employ Hardware-Aware KV-Cache Compression and SRAM Tiling. The objective is to maintain the active weight set and the current KV-cache tile within the NPU's local SRAM, which typically ranges from 4MB to 8MB in high-end consumer silicon.

SRAM Tiling for BitNet requires specific handling. Because weights are packed (using 2 bits to represent a ternary state), dequantization should ideally occur within the NPU's local buffers to avoid bandwidth bottlenecks. The kernel must fetch packed blocks and utilize Look-Up Table (LUT) based dequantization within the execution pipeline.

The KV-Cache Challenge

While weights can be reduced to 1.58 bits, the KV-Cache remains a significant memory consumer. For high-parameter models at large context windows, the KV-cache can exceed the available memory on consumer hardware if maintained at FP16. Solutions for modern kernels include:

• Grouped Query Attention (GQA): Reducing the number of Key/Value heads to save memory.
• Low-Bit Quantization: Scaling KV-cache precision dynamically based on attention requirements.
• Page-Based Memory Management: Handling fragmented KV-cache memory within SRAM.

The Roofline Model for Sub-2-Bit Kernels

Traditional Roofline models for NPUs measure performance in TOPS (Tera-Operations Per Second). For BitNet, an operation is defined as a 1.58-bit addition or subtraction.

The performance ceiling is often defined by the NPU's Dispatch Rate. In many architectures, the bottleneck is the instruction dispatcher's ability to feed the systolic array at a rate that matches the reduced data footprint of ternary weights. Asynchronous Dispatch and Kernel Fusion—combining dequantization, addition, and activation into a single pass—are essential for reaching theoretical peak performance.

Implementation Realities

Developing these kernels involves navigating various NPU APIs, such as CoreML or Qualcomm QNN. To maximize NPU systolic array utilization, developers often work with Intermediate Representation (IR) manipulation to overcome abstraction limitations.

Key technical considerations include:

• Bit-Level Packing: Efficiently storing 1.58-bit values within byte-aligned memory without excessive overhead.
• Tiling Alignment: Mapping ternary logic to systolic arrays that are typically power-of-two in dimension.
• Thermal Management: Factoring in localized heat generation during sustained high-throughput inference and adjusting frequency scaling within the tiling strategy.

The Outlook for Ternary Inference

The hardware market is seeing increased specialization for AI workloads. While data center accelerators continue to support a range of precisions for training, consumer NPUs are increasingly optimized for low-bit inference. The power-to-performance ratio of BitNet b1.58 offers significant advantages for local deployment. Mastery of SRAM tiling and systolic array utilization is becoming foundational for developing ubiquitous, high-performance local AI applications.

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