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arXiv:2407.14178 · APL Photonics

Emulating quantum computing with optical matrix multiplication

Mwezi Koni, Hadrian Bezuidenhout, Isaac M. Nape

Published Oct 1, 2024• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0230335

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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.
This paper proposes a novel approach to accelerating matrix multiplication by exploiting the principles of quantum state superposition. The core idea centers on decomposing the matrix multiplication process into a series of quantum state superposition calculations and leveraging quantum interference effects for enhanced computational speed. Traditional parallel matrix multiplication algorithms primarily rely on classical computation methods, often failing to effectively harness the inherent parallel processing capabilities of quantum hardware. This work presents a quantum algorithm designed specifically to address this limitation. The algorithm utilizes the superposition of quantum states to perform multiple matrix multiplication operations concurrently. The mathematical framework is presented, detailing the quantum gate sequence and the overall computational flow. The potential for significant speedup is discussed, acknowledging the challenges inherent in realizing such an algorithm on current and near-term quantum devices. The key contribution lies in the direct mapping of matrix multiplication onto a quantum system, optimizing for superposition and interference. The theoretical analysis provides a foundation for future research and development in this area.
One of the main challenges in quantum computing is implementing algorithms within quantum optical free-space and integrated circuits. This research focuses on the quantum addition algorithm, derived from the Deutsch-Jozsa algorithm, to advance quantum computing using optical technologies. The developed quantum optical integrated circuit is built on a hybrid substrate with interconnected components on a silicon dioxide base. Qubits and their states are arranged according to the requirements of each quantum algorithm. In the Fourier transform section, qubits act as propagation modes within silicon waveguides, with emitted photons representing the Fourier transform after traveling a distance of π/2. The phase-shifting section consists of two parts: one shifts the phase of incoming photons by π, while the other modulates this shift based on the quantum full adder algorithm. The first employs a photonic crystal for short phase shifts, which are adjusted via impurities in the silicon substrate and controlled by an electro-optic circuit. Additionally, single-mode waveguides made of silicon dioxide and air serve as qubits. By coupling photon amplitudes, a comparator circuit is triggered based on the generated output. Through these integrations, the sum of two quantum numbers can be represented at the output. This underscores the importance of designing and simulating a full adder within quantum optical integrated circuits to address the complexities inherent in quantum computing.
Classical simulation is essential for developing and testing quantum circuits, but the exponential growth of Hilbert space challenges conventional digital processors. Here, we introduce a classical optical quantum simulation architecture that treats gates and batches of input states as independent tensor dimensions rather than separate simulation instances, while mapping eligible tensor network contractions onto the same processor. Using co-designed multidimensional encoding and readout across spatial and wavelength degrees of freedom, we experimentally realize high-order tensor-matrix multiplication in a single propagation and process all computational-basis inputs for each 4-qubit gate. Using a physics-based optical propagation model, we further numerically evaluate the architecture for the Deutsch algorithm and for the tensor network of all 128 target instances of a 7-qubit Grover algorithm. These results establish a unified framework for gate-state batch and tensor network simulation, providing a route to exploit optical concurrency across quantum simulation tasks.
A set of experimental techniques to digitally shape an optical beam are presented. In particular, we show how to digitally study the interference phenomena in different geometries and observables. We also present a novel way of generating arbitrary vector beams and controlling the spatial coherence of a classical beam using Spatial Light Modulators. These type of manipulations are further employed to control and experimentally implement quantum algorithms, more precisely the Deutsch, Deutsch-Jozsa and Grover`s search algorithms.