International Journal of Reconfigurable Computing

Volume 2019, Article ID 1949121, 14 pages

https://doi.org/10.1155/2019/1949121

## Dimension Reduction Using Quantum Wavelet Transform on a High-Performance Reconfigurable Computer

Correspondence should be addressed to Naveed Mahmud; ude.uk@329_deevan

Received 4 May 2019; Revised 16 August 2019; Accepted 1 September 2019; Published 11 November 2019

Academic Editor: Wim Vanderbauwhede

Copyright © 2019 Naveed Mahmud and Esam El-Araby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The high resolution of multidimensional space-time measurements and enormity of data readout counts in applications such as particle tracking in high-energy physics (HEP) is becoming nowadays a major challenge. In this work, we propose combining dimension reduction techniques with quantum information processing for application in domains that generate large volumes of data such as HEP. More specifically, we propose using quantum wavelet transform (QWT) to reduce the dimensionality of high spatial resolution data. The quantum wavelet transform takes advantage of the principles of quantum mechanics to achieve reductions in computation time while processing exponentially larger amount of information. We develop simpler and optimized emulation architectures than what has been previously reported, to perform quantum wavelet transform on high-resolution data. We also implement the inverse quantum wavelet transform (IQWT) to accurately reconstruct the data without any losses. The algorithms are prototyped on an FPGA-based quantum emulator that supports double-precision floating-point computations. Experimental work has been performed using high-resolution image data on a state-of-the-art multinode high-performance reconfigurable computer. The experimental results show that the proposed concepts represent a feasible approach to reducing dimensionality of high spatial resolution data generated by applications such as particle tracking in high-energy physics.

#### 1. Introduction

High-energy physics deal with advanced instruments such as particle accelerators and detectors. These machines use electromagnetic fields to accelerate charged particles to high speeds and create collisions. By studying particle collisions and tracking collision trajectories, physicists can test the predictions of many theories of particle physics such as properties of the Higgs boson [1], discovering new particle families [2] as well as many high-energy physics problems [3]. There are a number of high-energy physics (HEP) research centers [4]. The largest particle accelerator is the Large Hadron Collider (LHC) in Geneva, Switzerland. Large-scale general-purpose particle detectors have been developed at the LHC. The ATLAS [5] and Compact Muon Solenoid (CMS) [6] are two examples which are used for studying the properties of the Higgs boson and investigating new physics. The ATLAS has an inner detector that has been used to observe the decay products of collisions. The pixel detector [7] is one of the main components of the inner detector, having over 80 million readout channels [8] (pixels), which contribute to half the total readout channels of the entire experiment. Reconstruction of high-energy particles from the pixel detector is considered a critical design and engineering challenge [9], due to its large readout count, high spatial resolution, and 3D space-time measurements. There have been efforts to improve the tracking performance of the ATLAS Inner Detector [9, 10], which involved insertion of additional pixel detector layer (Insertable B-Layer). Another approach that has been considered in the ATLAS FTK (Fast Track Trigger) upgrade [11] is using variable resolution patterns, where the data from the detector is compared to generated pattern banks of particle tracks and non-intersecting data is filtered. In high-dimensional datasets, e.g., the pixel detector readout data, not all the measured data variables are relevant in understanding the underlying regions of interest (RoI). Generally, statistical predictive models are applied to multidimensional datasets for detection and pattern matching, which is a computationally expensive process. Thus, an effective method is needed to reduce the dimensionality [12] of the data in such high-dimensional spatial sets, for faster detection and matching.

As a feasible solution to this problem, we here propose combining wavelet-based dimension reduction techniques [13–15] with quantum information processing (QIP) [16] for applications in domains that generate high-dimensional data volumes such as high-energy physics (HEP). More specifically, we propose using quantum wavelet transform (QWT) to reduce the dimensionality and high spatial resolution of data in HEP particle tracking. Wavelet-based dimension reduction has been shown to be an effective technique in image preprocessing, reducing computation time, reducing interprocessor overhead, and improving classification accuracy [13–15]. Even so, the large volume of data from domains such as high-energy particle physics, present a challenge for a classical wavelet-based method. The QWT has been demonstrated in previous works to be very useful in quantum image processing and quantum data compression [16–19]. Quantum information processing uses qubits as the basic units of information storage, compared to classical binary forms, and can exploit quantum mechanical properties such as entanglement and superposition [20]. Therefore, applying QIP techniques such as QWT for dimension reduction of HEP data will bring substantial improvements in storage and computation compared to classical signal processing techniques. To the best of our knowledge, this work is the first to investigate QWT-based dimension reduction for HEP applications. We develop simple and effective algorithms for QWT and inverse-QWT (IQWT) that are best suited for dimension reduction and present corresponding emulation hardware architectures for QWT and IQWT.

The objectives and focus of our work are to demonstrate the feasibility of QWT for dimension reduction, through emulation, and to evaluate the performance of the emulation architectures. Our proposed algorithms are prototyped on an FPGA-based quantum emulator that has been developed based on our previous works [21, 22] and has been shown to emulate full quantum algorithms such as quantum Fourier transform (QFT) [23] and Grover’s search algorithm [24]. An FPGA platform was chosen because of its reconfigurability and flexibility in emulating multiple quantum algorithms. The emulator is based on the hardware system of DirectStream [25], which is a state-of-the-art reconfigurable computing platform. This emulation platform can be conveniently used to verify and benchmark future implementations of the proposed system in HEP applications. In the next section, we discuss fundamental concepts of quantum computing, QWT, and the related work done on QWT. In Section 3, we elaborate our proposed methods and emulation architectures. In Section 4, the experimental results and analysis are presented. Section 5 is our conclusion and future directions of this work.

#### 2. Background and Related Work

In this section, we discuss background concepts of quantum computing and the quantum wavelet transform. We also discuss current and related work on QWT and high-energy particle detection.

##### 2.1. Qubits, Superposition, and Entanglement

The qubit is the smallest unit of quantum information that describes a two-level quantum mechanical system. Physical implementations of the qubit can be electron/atomic/nuclear spin, where spin directions of the particle represent the two qubit levels. Other physical representations of the qubit can be photon polarization, superconducting Josephson junction, etc. [26]. The qubit is represented theoretically using the Bloch sphere [20], as shown in Figure 1. The basis states of the qubit, and are denoted by poles of the sphere. The property that distinguishes the qubit from the classical bit is superposition. The qubit can exist in a mixed or superposition state that is any other point on the surface of the sphere other than the poles. The overall state of the qubit can be defined using a linear superposition equation , where *α* and *β* are complex numbers determined from *φ* and *θ* as shown in Figure 1 and satisfying . Another distinguishing property of qubits is entanglement [20]. Two or more qubits can be entangled together, which means each entangled qubit becomes strongly correlated to the other along all possible combinations of the qubits. Outcome of measurement of one qubit is dependent on the other measurement, but individually they exhibit completely random behavior. In quantum computing, most algorithms assume that the qubits are fully entangled [21]. A system of *n* entangled qubits can be represented in vector space as complex basis state coefficients.