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 $O(N^2\log_2N)$ , 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 $O(N^{2.371339})$ . Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V ${\scriptstyle 2}$ V), Quantum Vector-Matrix Multiplication (V ${\scriptstyle 2}$ M), QKMM (M ${\scriptstyle 2}$ M), Quantum One-to-Many Matrix Multiplication(O ${\scriptstyle 2}$ M) 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 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 $O(N^2\log_2N)$, 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 $O(N^{2.371339})$. Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V${\scriptstyle 2}$V), Quantum Vector-Matrix Multiplication (V${\scriptstyle 2}$M), QKMM (M${\scriptstyle 2}$M), Quantum One-to-Many Matrix Multiplication(O${\scriptstyle 2}$M) 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.

quant-pharXiv:2602.05541Feb 6, 20260 citations

Emulating quantum computing with optical matrix multiplication

Mwezi Koni, Hadrian Bezuidenhout, Isaac M. Nape

Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well-known algorithm, namely, Deutsch–Jozsa’s algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light, which is the main resource for optical vector-matrix multiplication. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore, our work advances the use of structured light for quantum information processing.Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well-known algorithm, namely, Deutsch–Jozsa’s algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light, which is the main resource for optical vector-matrix multiplication. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore, our work advances the use of structured light for quantum information processing.

arXiv:2407.14178APL PhotonicsOct 1, 20247 citations

Towards realistic architectures for linear optical quantum computing

Sam Morley-Short

Jan 1, 20190 citations

A Programmable and Reconfigurable Photonic Simulator for Classical XY Models

Jiayi Ouyang, Yuxuan Liao, Xue Feng +6 more

In this work, we proposed and experimentally demonstrated a photonic simulator for XY models, which is a typical kind of classical spin models. By encoding the XY spins on the phase term of the input light field, the corresponding XY Hamiltonian could be performed on the output light intensities. The simulator is mainly based on a programmable and reconfigurable optical vector-matrix multiplication system, which can map arbitrary XY models within the dimensionality limit. Here, we demonstrated the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional XY model, in which the expectation values of some observables are calculated and consistent with the theory. Besides, we performed the ground state search of two 25-spin XY models with different spin connections and coupling strengths. Our proposal paves a new way to investigate the XY spin system.In this work, we proposed and experimentally demonstrated a photonic simulator for XY models, which is a typical kind of classical spin models. By encoding the XY spins on the phase term of the input light field, the corresponding XY Hamiltonian could be performed on the output light intensities. The simulator is mainly based on a programmable and reconfigurable optical vector-matrix multiplication system, which can map arbitrary XY models within the dimensionality limit. Here, we demonstrated the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional XY model, in which the expectation values of some observables are calculated and consistent with the theory. Besides, we performed the ground state search of two 25-spin XY models with different spin connections and coupling strengths. Our proposal paves a new way to investigate the XY spin system.

arXiv:2401.08055Jan 16, 20240 citations

Towards Optical Quantum Computing

Sverre Gullikstad Johnsen

Jan 1, 20020 citations

O(N^2\log_2N)

, 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(N^{2.371339})

. Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V

{\scriptstyle 2}

V), Quantum Vector-Matrix Multiplication (V

{\scriptstyle 2}

M), QKMM (M

{\scriptstyle 2}

M), Quantum One-to-Many Matrix Multiplication(O

{\scriptstyle 2}

M) 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.