Quantum Matrix Multiplication and Effects on Large Language Models

Abstract

Matrix multiplication is essential for modern deep learning architectures. It is used extensively in transformers and large language models. Major tech companies are greatly increasing model sizes and pushing towards increased computational requirements. Therefore, optimizing matrix multiplication becomes essential for both training and running inference, especially as GPUs and semiconductor manufacturing are becoming scarce. This paper discusses efficient matrix multiplication techniques and explores how quantum algorithms can lead to future computational improvements in AI and ML. We distinguish between verification algorithms and actual multiplication algorithms, and examine how quantum approaches, sparsity techniques, and algorithmic optimizations can lead to reduced computation in LLMs.

I. Introduction

Matrix multiplication is the backbone of almost all AI and neural network operations. Feedforward layers and attention mechanisms in transformers all rely on extensive matrix multiplications. In large language models, matrices represent embeddings, weights, and activations — repeatedly multiplied, leading to substantial computational costs.

Originally, classical matrix multiplication algorithms have provided improvements over the naive O(n³) approach. Strassen's algorithm gives O(n^2.81), and the Coppersmith-Winograd algorithm provides O(n^2.376). However, these classical improvements have practical limitations due to large constant factors and numerical stability issues.

II. Verification vs. Multiplication

It is crucial to distinguish between two different types of matrix algorithms: those that perform matrix multiplication and those that verify it has been done correctly.

Verification algorithms are useful for distributed computing scenarios and error detection, but they do not directly help compute matrix products faster. For LLMs and ML, we need algorithms that actually perform multiplication efficiently.

III. Quantum Algorithms for Matrix Multiplication

Recent developments in quantum computing offer promising approaches for computing matrix products using:

• Block encoding: Representing matrices as quantum states
• Quantum Singular Value Transformation (QSVT): Applying matrix functions
• Quantum amplitude amplification: Enhancing computational probabilities

These methods can provide polynomial or exponential speedups for certain matrix operations under specific conditions — sparse matrices or when approximate solutions are acceptable. The runtime is typically O(poly(log n, κ, 1/ε)) where κ is the condition number, compared to classical O(n³) or O(n^2.81) algorithms.

IV. Quantum Transformer Inference Speedup

When a transformer processes an input sequence X with n tokens and dimension d, it performs multiple matrix multiplications per layer: QKV projections (O(nd²) each), attention scores QK^T (O(n²d)), and feedforward networks.

QKV Projections

Using quantum algorithms, we can speed up Q=XW_Q from classical O(nd²) to approximately O(√d·log²d) by encoding input vectors using amplitude encoding (O(log d) qubits per vector) and applying block-encoded weight matrices via QSVT. This gives a polynomial speedup from millions of operations down to thousands per token.

Attention Mechanism

The attention mechanism S=QK^T/√d classically costs O(n²d). Using Grover-based amplitude amplification to compute dot products, each attention score reduces from O(n) to O(√n) operations. Total quantum complexity becomes O(√nd·log(1/ε)) instead of O(n²d) — a quadratic speedup in sequence length. For n=2048, this reduces 51 billion operations to around 1 million quantum operations.

Total Speedup

V. Practical Considerations

To implement quantum matrix multiplication, we need fault-tolerant quantum computers with sufficient qubits and low gate error rates. Three main approaches exist:

• Parameterized Quantum Circuits (PQC): Suitable for NISQ devices, hardware-efficient but limited by qubit numbers and circuit depths
• Quantum Linear Algebra (QLA): Block encoding and QSVT for attention matrix calculations; requires fault-tolerant hardware but offers strong theoretical speedups
• Hybrid Classical-Quantum: The most achievable near-term strategy — classical GPUs manage the ML pipeline while quantum computers accelerate specific intensive matrix operations

VI. Classical Optimization Techniques

While quantum computers mature, classical optimization remains important:

• Sparsity: Sparse attention patterns (local, strided, block sparse), pruning, and dynamic sparsity during training
• Low-Rank Approximation: SVD, low-rank factorization, and tensor decomposition for weight matrices
• Quantization: Reducing precision from 32-bit to 8-bit or lower through post-training quantization, quantization-aware training, and mixed precision

VII. Conclusion

This paper discusses efficient matrix multiplication techniques essential for large language models and transformer architectures. While classical algorithms provide incremental improvements, quantum approaches for actual matrix multiplication — such as quantum linear algebra techniques and recent quantum transformer implementations — provide theoretical speedups through superposition and amplitude amplification. To realize these improvements, we need fault-tolerant hardware, hybrid architectures, and quantum-native algorithm design. Integration with classical optimization techniques such as sparsity exploitation, low-rank approximation, and quantization shows the most potential for near-term progress. The future of large language models may depend on successfully combining quantum acceleration with classical optimizations to overcome current computational bottlenecks.