Quantum computing has emerged as a transformative paradigm for tackling complex matrix operations that underpin applications in machine learning and quantum information. In this paper, we synthesize recent advancements in quantum computing by integrating the quantum subroutines for matrix geometric means from Liu et al. (2025), which enable efficient solutions to nonlinear matrix equations like algebraic Riccati equations, with the Chebyshev-accelerated quantum matrix multiplication framework from Li et al. (2025), offering quadratic speedups for iterative chain operations on complex matrices. Through detailed reviews of each approach, we highlight shared primitives such as block-encoding, quantum singular value transformation, Hermitianization, and polynomial approximations, demonstrating their complementarity in constructing hybrid algorithms for nonlinear iterative problems. These hybrids achieve polylogarithmic complexities in matrix dimensions while optimizing precision and conditioning dependencies, exemplified in frameworks for Riccati iterations and kernel enhancements. Applications are explored in machine learning, including geometric mean metric learning for anomaly detection in classical and quantum data, and in quantum information, such as optimal fidelity and Rényi entropy estimation with accelerated power computations for error correction. Also available on Academia.edu: https://www.academia.edu/attachments/125700844/download_file?s=portfolio
Matrix multiplication is a fundamental operation in compute-intensive tasks and a key component of modern quantum acceleration frameworks. Here we present a quantum matrix multiplication algorithm based on quantum kernels (QKMM), achieving an elementary gate complexity of
O(N2log2N), with amplitude encoding overhead explicitly included and without assuming a QRAM oracle. This scaling is asymptotically lower than that of the best-known classical matrix multiplication algorithm
O(N2.371339). Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V
2V), Quantum Vector-Matrix Multiplication (V
2M), QKMM (M
2M), Quantum One-to-Many Matrix Multiplication(O
2M) and Quantum Sequential Matrix Multiplication (SMM), providing a unified framework from vector operations to parallel and sequential matrix transformations. Through noiseless simulations, realistic noise modelling and experiments on a superconducting quantum processor, we systematically characterize the numerical accuracy, resource requirements and hardware execution limits of this operator framework. Furthermore, we integrate SMM into deep neural-network inference, enabling intermediate features to propagate coherently across layers without repeated measurement and re-encoding. These results establish a pathway from quantum circuit-level algorithm design to end-to-end coherent computation, providing a quantum computing framework for matrix-centric compute-intensive applications.
Matrix multiplication is a fundamental operation in various fields, from linear algebra to computer graphics. However, traditional methods often suffer from computational bottlenecks when dealing with large matrices, limiting their applicability. This paper introduces a new algorithm employing quantum fields to significantly enhance matrix multiplication efficiency. The core of this approach lies in applying quantum fields to the matrix operations, utilizing their inherent properties to optimize the computation process. We explore a method that dynamically adjusts the quantum field parameters based on the matrix being multiplied, leading to improved convergence and computational speed. The research presented herein demonstrates the effectiveness of this technique through extensive simulations and preliminary analysis. This advancement promises to unlock new possibilities in matrix calculations, particularly for large-scale applications, and offers a viable alternative to conventional methods.
Matrix chain multiplication -- computing
W=M(0)⋯M(K−1) where
M(k)∈RPk×Pk+1 -- arises in scientific computing, machine learning, and graph analysis. Despite the importance of this problem, for chains of distinct matrices, the classical number of operations grows linearly with the chain length
K and polynomially in the matrix dimensions. We present \emph{Two-Tower Matrix Multiplication}, a quantum subroutine that encodes the product
W of the
K matrices into a quantum state in circuit depth
O(maxkpolylog(PkPk+1)), which is independent of~
K within the QRAM-based state-preparation model, whereas the qubit count is
O(∑klogPk); the total gate count remains linear in
K, so the gain is in the circuit depth. The construction interleaves state-preparation operators across two layers; within each layer, all operators act on disjoint registers and execute in parallel. This subroutine can be specialized for the chain-vector case, which computes the product of
K−1 matrices applied to a vector. We prove the correctness of the subroutine for all
K and provide two implementations using the Qiskit and QCLAB frameworks. The subroutine is applicable to any downstream quantum algorithm that operates on a matrix encoded in the statevector, including norm estimation, graph-matrix powers, linear system solving, and quantum machine learning kernels.
As the most central and computationally intensive component of deep neural networks, the execution efficiency of matrix multiplication directly determines the training and inference performance of models. Harnessing the parallel processing capabilities afforded by quantum superposition and entanglement to reshape matrix multiplication implementations has become a promising entry point for optimising underlying quantum arithmetic logic and improving the operational efficiency of quantum circuits. This paper proposes a universal quantum matrix multiplication (QMM) framework designed to achieve substantial computational acceleration through an optimised quantum arithmetic logic unit. To circumvent the limitations of multi-register and multi-control gates in conventional quantum arithmetic circuits, we encode classical data directly into parameterised
Rz rotation gates using the quantum Fourier transform (QFT), thereby reducing the base gate complexity of the quantum adder to
O(n). In addition, by adopting the column-wise multiplication principle from classical arithmetic, we optimize the gate complexity of the quantum multiplier to
O(n2). We further extend this approach to a quantum version of the Strassen algorithm, and experimentally quantify the trade-off between reduced multiplication time and increased overhead in addition resources. This work establishes a reliable technical pathway for constructing general-purpose quantum matrix operations, with the potential to unlock substantial computational power for training modern machine learning models.
This work explores the potential of Quantum Matrix Multiplication (QMM) to accelerate several computational tasks, demonstrating substantial speedups. We present three distinct applications showcasing QMM’s versatility and efficiency. We introduce a novel Hybrid-Quantum Angle Based Outlier Detection (H-QABOD) algorithm, leveraging QMM to efficiently identify outliers in numerical datasets. We evaluate H-QABOD’s performance on both synthetic and real-world datasets, benchmarking it against its classical counterpart. Furthermore, we investigate QMM’s application within quantum linear algebra, specifically for multitrace estimation and Frobenius norm computation.