Quantum Algorithm Speeds up Complex Calculations to N²log₂N, a New Record

Scientists have long sought to improve the efficiency of matrix multiplication, a cornerstone of modern computation that presents significant challenges as data scales. Jiaqi Yao and Ding Liu, both from the School of Computer Science and Technology at Tiangong University, alongside Yao et al., now demonstrate a quantum kernel-based matrix multiplication algorithm (QKMM) that achieves a computational complexity of O(N^2log_2N). This represents a substantial advancement over the classical optimal complexity of O(N^3), where N denotes the matrix dimension, and signifies a potential breakthrough for applications reliant on large-scale matrix operations. Through both noiseless and noisy simulations, the researchers prove that their algorithm not only possesses superior theoretical efficiency but also exhibits promising runtime performance and stability in practical scenarios.

This represents a significant advancement over the classical optimal complexity of O(N².371552), where N denotes the matrix dimension.

The research demonstrates that this new algorithm not only possesses superior theoretical efficiency but also exhibits practical advantages in runtime performance and stability through rigorous simulation experiments. This breakthrough addresses a critical bottleneck in modern computing, particularly within the rapidly evolving field of machine learning where matrix multiplication forms the core of many computationally intensive processes.
The newly proposed algorithm leverages the inherent parallelism and exponential storage capacity of quantum computing to overcome limitations inherent in classical approaches. By reducing the computational complexity, the QKMM algorithm promises to substantially decrease the resources required for training complex artificial intelligence models.

Detailed simulations, conducted under both ideal and noisy conditions, confirm the algorithm’s scalability and resilience. These simulations reveal significant parallel acceleration as the matrix dimension increases, indicating a pathway towards practical quantum advantage. This work provides complete circuit implementations and explicit simulations, utilising gate complexity as the primary performance metric.

This allows for a direct and compelling comparison between QKMM and both classical and previously proposed quantum matrix multiplication methods. Analysis reveals that the QKMM algorithm achieves a level of asymptotic optimality, marking a crucial step forward in the pursuit of efficient quantum computation.

The research systematically characterises the decay of fidelity under noisy conditions, providing valuable insights into the algorithm’s robustness in realistic hardware environments. Furthermore, the study addresses a lack of standardised metrics for evaluating quantum matrix multiplication. By adopting gate complexity as a consistent measure, the research offers a clear benchmark for comparing different quantum matrix multiplication algorithms.

Circuit implementation and performance evaluation of quantum kernel matrix multiplication

A Quantum Kernel-based Matrix Multiplication (QKMM) algorithm forms the basis of this research, achieving an asymptotically optimal computational complexity of O(N² log N), where N denotes the matrix dimension. This new approach surpasses the classical optimal complexity of O(N²⋅371552) by leveraging the principles of quantum computation to address limitations in scaling matrix multiplication.

The methodology centres on a kernel-based implementation, designed to minimise quantum gate count as the primary performance metric, facilitating direct comparison with both classical and prior quantum methods. The study meticulously details complete circuit implementations and simulations of QKMM, conducted under both ideal noiseless conditions and those mimicking noisy superconducting hardware.

Noiseless simulations demonstrated substantial parallel acceleration as the matrix dimension increased, highlighting the potential for significant speedups. To assess robustness, researchers systematically characterised the decay of fidelity with varying noise parameters, providing insights into the algorithm’s performance in realistic hardware environments.

Comparative analysis was performed against established quantum matrix multiplication techniques, including those based on the Swap Test and the Hadamard Test. The Swap Test required 2 log₂ N + 1 qubits and exhibited a gate complexity of O(N log₂ N), while the Hadamard Test demanded log₂ N + 1 qubits and a gate complexity ranging from O(N log₂ N) to O(N³ log₂ N).

In contrast, QKMM operates with log₂ N qubits and achieves a gate complexity of O(N² log₂ N), demonstrating a clear advantage in resource efficiency and scalability. Table 1 summarises these resource requirements, providing a quantitative assessment of QKMM’s performance relative to alternative methods.

Asymptotic complexity and performance of the quantum kernel matrix multiplication algorithm

Scientists have developed a quantum kernel-based matrix multiplication (QKMM) algorithm exhibiting an asymptotically optimal computational complexity of O(N2log2N), where N denotes the matrix dimension. This surpasses the classical optimal complexity of O(N3) for matrix multiplication. The research demonstrates both theoretical efficiency and practical advantages in runtime performance and stability through noiseless and noisy simulation experiments.

The complexity of the QKMM algorithm approaches O(N2) as the matrix dimension increases, a feat not yet achieved by the best-known classical algorithms. Analysis reveals that for any N greater than 1, a function x exists, defined as ln(log2N)/ln(N), such that log N equals N multiplied by x. Consequently, the complexity O(N2log2N) can be represented asymptotically as N2 + o(1), rigorously proving the algorithm’s approach to the theoretical lower bound of O(N2).

Large-scale simulations were performed using the pyQ-Panda library on the Origin Quantum platform, utilising a server with an Intel Core i7-12700H processor (2.30GHz, 14 cores) and 24 GB of RAM. Vector inner products (V2V), serving as the fundamental building block of QKMM, were compared against the Swap Test and Hadamard Test.

Results, detailed in Fig.2(a), show V2V demonstrates a significantly lower time overhead for computing pure inner products, with this advantage increasing as dimensionality increases. Specifically, V2V eliminates the need for ancillary qubits to establish vector relations and avoids reliance on measurements to extract correlation terms.

The design allows for direct readout of the inner product term from the ground state. Vector-matrix multiplication (V2M), a single-level parallel extension of V2V, was also evaluated. The Swap Test was excluded from primary benchmarking due to its non-reusable and non-parallelizable nature, as it modifies data qubits during inner product computation. Resource requirements for the quantum comparison methods are summarised in Table 1, showing QKMM requires log2 N qubits and 2log2 N gates for V2V, 3log2 N gates for V2M, and O(Nlog2N) gates for M2M.

Quantum kernel multiplication exhibits enhanced resilience to noise and scalability

A new quantum kernel-based matrix multiplication algorithm, termed QKMM, achieves a computational complexity of, representing an improvement over the classical optimal complexity of, where denotes the matrix dimension. Demonstrations using both noiseless and noisy simulations confirm that this algorithm not only possesses superior theoretical efficiency but also exhibits practical advantages in runtime performance and stability.

These findings suggest a pathway towards more efficient handling of large-scale matrix operations using quantum computing. Evaluations under various noise conditions, including T1 relaxation, T2 dephasing, and gate noise, reveal that QKMM demonstrates greater resilience compared to other quantum inner product algorithms, maintaining more gradual fidelity reduction and slower error growth with increasing dimensionality.

While all algorithms experience fidelity reduction as dimensionality increases, QKMM’s performance deteriorates less rapidly, particularly in combined noise environments where multiple error sources interact. The algorithm maintains high fidelity above 0.95 for low matrix dimensions (2 to 4) even with concurrent noise, though gate noise becomes the dominant factor limiting fidelity at dimension 8.

The study acknowledges that performance degradation at higher dimensions arises from the nonlinear interaction of multiple noise sources, leading to rapid computational accuracy deterioration. Future research should focus on exploring the algorithm’s optimization capabilities across diverse quantum hardware noise models to further enhance its practical implementation and scalability. These results indicate the potential for QKMM to be a valuable tool for high-dimensional applications in noisy quantum systems, provided further work addresses the challenges posed by compounded noise.