Reducing the Complexity of Matrix Multiplication to $O(N^2log_2N)$ by an Asymptotically Optimal Quantum Algorithm

Matrix multiplication is a fundamental classical computing operation whose efficiency becomes a major challenge at scale, especially for machine learning applications. Quantum computing, with its inherent parallelism and exponential storage capacity, offers a potential solution to these limitations. This work presents a quantum kernel-based matrix multiplication algorithm (QKMM) that achieves an asymptotically optimal computational complexity of $ O(N^2 \log_2 N) $, outperforming the classical optimal complexity of $ O(N^{2.371552}) $, where $N$ denotes the matrix dimension. Through noiseless and noisy quantum simulation experiments, we demonstrate that the proposed algorithm not only exhibits superior theoretical efficiency but also shows practical advantages in runtime performance and stability.

💡 Research Summary

The paper introduces a quantum‑kernel‑based matrix multiplication algorithm (QKMM) that achieves a gate‑count complexity of O(N² log N), which is asymptotically optimal and improves upon the best known classical exponent O(N²·371552). The authors begin by reviewing five decades of progress in classical fast matrix multiplication—from Strassen’s O(N²·81) algorithm to the most recent laser‑hash hybrid achieving O(N²·371552). They point out that classical complexity is measured by the number of scalar multiplications, whereas quantum algorithms lack a universally accepted metric; they therefore adopt the number of elementary quantum gates (single‑ and two‑qubit operations) as the performance measure.

QKMM is built on a “quantum kernel inner‑product” primitive. Four hierarchical modules are defined:

1. V2V (vector‑to‑vector) computes the inner product of two normalized vectors directly from the ground state without ancillary qubits, outperforming the Swap Test and Hadamard Test in both gate count and ancilla overhead.

2. V2M (vector‑to‑matrix) extends V2V to perform a matrix‑vector multiplication in a single circuit, exploiting quantum superposition to process all rows in parallel.

3. M2M (matrix‑to‑matrix) further extends V2M to full matrix multiplication, allowing the entire product to be obtained from one quantum circuit without repeated inner‑product subroutines.

4. M‑MM (multi‑matrix) generalizes M2M so that a single circuit simultaneously multiplies one matrix with many others, demonstrating the extreme parallelism possible on a quantum processor.

The authors analytically derive the gate‑count for each module. V2V requires O(N log N) gates, V2M O(N² log N), and both M2M and M‑MM retain the O(N² log N) bound because the additional parallelism does not increase the asymptotic gate count. A mathematical proof shows that O(N² log N) = N² + o(N²), confirming that the algorithm asymptotically approaches the theoretical lower bound O(N²).

Experimental validation is performed with both noiseless and noisy simulations using the pyQ‑Panda library on the Origin Quantum platform. In the noiseless case, matrix sizes from 4×4 to 16×16 are tested. QKMM consistently outperforms Swap Test and Hadamard Test implementations, achieving speed‑ups of 2–5× that grow with N. In the noisy simulations, realistic superconducting hardware parameters (T₁≈50 µs, T₂≈30 µs, single‑qubit fidelity 99.8 %, two‑qubit ECR fidelity 97.5 %) are applied. Fidelity remains above 0.96 for V2V and above 0.90 for M2M, while mean error scales linearly with increasing noise, providing a quantitative picture of robustness on near‑term devices.

Table 1 compares QKMM with prior quantum matrix multiplication approaches (Swap Test, Hadamard Test) in terms of qubit requirements and gate complexity. QKMM uses the fewest gates (O(N² log N)) and comparable or fewer qubits. Compared to the best classical algorithms, the only extra factor is a logarithmic term, suggesting that for sufficiently large N the quantum method could deliver practical speed‑ups, assuming hardware advances.

The paper acknowledges several limitations. Input vectors and matrix rows must be normalized quantum states, which introduces a preprocessing cost. Circuit depth grows with log N, so for very large N the coherence time of current NISQ devices may become a bottleneck. Scaling to truly large matrices (e.g., 1024×1024) would require a substantial increase in qubit count and a reduction in gate error rates beyond today’s capabilities. The authors propose future work on more efficient state‑encoding schemes, error‑correction integration, hybrid quantum‑classical strategies, and the development of large‑scale quantum memory architectures.

In conclusion, the work demonstrates that a quantum‑kernel‑based approach can achieve asymptotically optimal matrix multiplication complexity, provides thorough theoretical analysis, and validates the concept through extensive simulations under realistic noise models. It offers a compelling pathway toward leveraging quantum parallelism for linear‑algebraic workloads in the NISQ era and beyond.