Qutrit-Inspired Fully Self-Supervised Shallow Quantum Learning Network for Brain Tumor Segmentation Debanjan Konar, MIEEE, Siddhartha Bhattacharyya, SMIEEE, Bijaya K

Qutrit-inspired Fully Self-supervised Shallow Quantum Learning Network for Brain Tumor Segmentation Debanjan Konar, MIEEE, Siddhartha Bhattacharyya, SMIEEE, Bijaya K. Panigrahi, SMIEEE, and Elizabeth Behrman

Abstract—Classical self-supervised networks suffer from con- recognition [5]–[8]. Nevertheless, quantum neural network vergence problems and reduced segmentation accuracy due models [9], [10] implemented on actual quantum processors to forceful termination. Qubits or bi-level quantum bits often are realized using a large number of quantum bits or qubits describe quantum neural network models. In this article, a novel self-supervised shallow learning network model exploiting as matrix representation and as well as linear operations on the sophisticated three-level qutrit-inspired quantum information these vector matrices. However, owing to complex and time- system referred to as Quantum Fully Self-Supervised Neural intensive quantum back-propagation algorithms involved in the Network (QFS-Net) is presented for automated segmentation supervised quantum-inspired neural network (QINN) architec- of brain MR images. The QFS-Net model comprises a trin- tures [11], [12], the computational complexity increases many- ity of a layered structure of qutrits inter-connected through parametric Hadamard gates using an 8-connected second-order fold with an increase in the number of neurons and inter-layer neighborhood-based topology. The non-linear transformation of interconnections. the qutrit states allows the underlying quantum neural network Automatic segmentation of brain lesions from Magnetic Res- model to encode the quantum states, thereby enabling a faster onance Imaging (MRI) greatly facilitates brain tumor iden- self-organized counter-propagation of these states between the tification overcoming the manual laborious tasks of human layers without supervision. The suggested QFS-Net model is tailored and extensively validated on Cancer Imaging Archive experts or radiologists [13]. It contrasts with the manual brain (TCIA) data set collected from Nature repository and also tumor diagnosis which suffers from significant variations in compared with state of the art supervised (U-Net and URes-Net shape, size, orientation, intensity inhomogeneity, overlapping architectures) and the self-supervised QIS-Net model. Results of gray-scales and inter-observer variability. Recent years have shed promising segmented outcome in detecting tumors in terms witnessed substantial attention in developing robust and effi- of dice similarity and accuracy with minimum human intervention and computational resources. cient automated MR image segmentation procedures among the researchers of the computer vision community. Index Terms—Quantum Computing, Qutrit, QIS-Net, MR The current work focuses on a novel quantum fully self- image segmentation, U-Net and URes-Net. supervised learning network (QFS-Net) characterized by qutrits for fast and accurate segmentation of brain lesions. The I.INTRODUCTION primary aim of the suggested work is to enable the QFS-Net uantum computing supremacy may be achieved through for faster convergence and making it suitable for fully auto- Q the superposition of quantum states or quantum par- mated brain lesion segmentation obviating any kind of training allelism and quantum entanglement [1]. However, owing or supervision. The proposed quantum fully self-supervised to the lack of computing resources for the implementation neural network (QFS-Net) model relies on qutrits or three- of quantum algorithms, it is an uphill task to explore the level quantum states to exploit the features of quantum cor- quantum entanglement properties for optimized computation. relation. To eliminate the complex quantum back-propagation arXiv:2009.06767v1 [quant-ph] 14 Sep 2020 Nowadays, with the advancement in quantum algorithms, the algorithms used in the supervised QINN models, the QFS-Net classical systems embedded in quantum formalism and in- resorts to a novel fully self-supervised qutrit based counter spired by qubits cannot exploit the full advantages of quantum propagation algorithm. This algorithm allows the propagation superposition and quantum entanglement [2]–[4]. Due to the of quantum states between the network layers iteratively. The intrinsic characteristics offered by quantum mechanics, the primary contributions of our manuscript are fourfold and are implementation of Quantum-inspired Artificial Neural Net- highlighted as follows: works (QANN) has been proven to be successful in solving specific computing tasks like image classification, pattern 1) Of late, the quantum neural network models and their implementation largely rely on qubits and hence, we D. Konar and B. K. Panigrahi are with the Department of Electrical have proposed a novel qudit embedded generic quantum Engineering, Indian Institute of Technology Delhi, New Delhi, India, Email: [email protected] and [email protected] neural network model applicable for any level of S. Bhattacharyya is with the Department of Computer Science and En- quantum states such as qubit, qutrit etc. gineering, CHRIST (Deemed to be University), Bangalore, India, Email: 2) An adaptive multi-class Quantum Sigmoid (QSig) acti- [email protected] India E. Behrman is with the Department of Mathematics and Physics, Wichita vation function embedded with quantum trit or qutrit is State University, Wichita, Kansas, USA, Email: [email protected] incorporated to address the wide variation of gray scales 2

in MR images. computing [31], [32]. Recent advances in both hardware and 3) The convergence analysis of the QFS-Net model is theoretical development may enable their implementation on provided, and its super-linearity is also demonstrated the noisy intermediate scale (NISQ) computers that will soon experimentally. The proposed qutrit based quantum be available [24], [28], [29], [33]. Konar et al. [7], [8], neural network model tries to explore the superposition Schutzhold et al. [34], Trugenberger et al. [35] and Masuyama and entanglement properties of quantum computing in et al. [36] suggested quantum neural networks for pattern classical simulations resulting in faster convergence of recognition tasks which deserve special mention for their the network architecture yielding optimal segmentation. contribution on QNN. 4) The suggested QFS-Net model is validated extensively The classical self-supervised neural network architectures using Cancer Imaging Archive (TCIA) data set collected employed for binary image segmentation suffer from slow from Nature repository [14]. Experimental results show convergence problems [37], [38]. To overcome these chal- the efficacy of the proposed QFS-Net model in terms lenges, the authors proposed the quantum version of the of dice similarity, thus promoting self-supervised proce- classical self-supervised neural network architecture relying on dures for medical image segmentation. qubits for faster convergence and accurate image segmentation The remaining sections of the article are organized as and implemented on classical systems [6]–[8]. Furthermore, follows. Section II reviews various supervised artificial neural the recently modified versions of the network architectures networks and deep neural network models useful for brain relying on qubits and characterized by multi-level activation MR image segmentation. A brief introduction of qutrits and function [39]–[41], are also validated on MR images for brain generalized D-level quantum states (qudits) is provided along lesion segmentation and reported promising outcome while with the preliminaries of quantum computing in Section III. compared with current deep learning architectures. However, A novel quantum neural network model characterized by the implementation of these quantum neural network models qudits is illustrated in Section IV. A vivid description of the on classical systems is centred on the bi-level abstraction suggested QFS-Net and its operations characterized by qutrit of qubits. In most physical implementations, the quantum has been provided in Section V. Results and discussions shed states are not inherently binary [42]; thus, the qubit model light on the experimental outcomes of the proposed neural is only an approximation that suppresses higher-level states. network model in Section VI. Concluding remarks and future The qubit model can lead to slow and untimely convergence directions of research are confabulated in Section VII. and distorted outcomes. Here, three-level quantum states or qutrits (generally D-level qudits) are introduced to improve the convergence of the self-supervised quantum network models. II.LITERATURE REVIEW Recent years have witnessed various machine learning clas- sifiers [15], [16] and deep learning technologies [17]–[20] A. Motivation for automated brain lesion segmentation for tumor detec- The motivation behind the proposed QFS-Net over the deep tion. Examples include U-Net [18] and UResNet [20], which learning based brain tumor segmentation [17], [18], [21], [22] have achieved remarkable dice score in auto-segmentation are as follows: of medical images. Of late, Pereira et al. [21] suggested a modified Convolutional Neural Network (CNN) introducing 1) Huge volumes of annotated medical images are required small size kernels to obviate over-fitting. Moreover, CNN for suitable training of a convolutional neural network, based architectures suffers due to lack of manually segmented and it is also a paramount task to acquire. or annotated MR images, intensive pre-processing and ex- 2) The extensive and time-consuming training of deep pert image analysts. In these circumstances, self-supervised neural network-based MR image segmentation requires or semi-supervised medical image segmentation is becoming high computational capabilities (GPU) and memory re- popular in the computer vision research community. Wang et sources. al. [22] contributed an interactive method using deep learning 3) In contrast to automatic brain lesion segmentation, the with image-specific tuning for medical image segmentation. slow convergence and over-fitting problems often affect Zhuang et al. [23] suggested a Rubik’s cube recovery based the outcome, and hence extra efforts are required for self-supervised procedure for medical image segmentation. suitable tuning of hyper-parameters of the underlying However, the interactive learning frameworks are not fully deep neural network architecture. self-supervised and suffer from the complex orientation and 4) Moreover, the lack of image-specific adaptability of the time-intensive operations. convolutional neural network leads to a fall in accuracy Quantum Artificial Neural Networks (QANN) were first pro- for unseen medical image classes. posed in the 1990s [24]–[26], as a means of obviating some A potential solution to devoid the requirement of training data of the most recalcitrant problems that stand in the way of the and the problems faced by intensely supervised convolutional implementation of large scale quantum computing: algorithm neural networks prevalent to medical image segmentation design [27], noise and decoherence [28], [29], and scaleup is a fully self-supervised neural network architecture with [30]. Amalgamating artificial neural networks with intrinsic minimum human intervention. The novel qutrit-inspired fully properties of quantum computing enables the QANN models self-supervised quantum learning model incorporated in the to evolve as promising alternatives to quantum algorithmic QFS-Net architecture presented in this article is a formidable 3

2π th contribution in exploiting the information of the brain lesions where θ = e D is the D complex root of unity. That is, the and poses a new horizon of research and challenges. operator X shifts a computational basis state |ki to the next state, and the Z operator multiplies a computational basis state III.FUNDAMENTALS OF QUANTUM COMPUTING by the appropriate phase factor. Note that these two operators generate the generalized Pauli group. Quantum computing offers the inherent features of superpo- The Hadamard gate is one of the basic constituents of sition, coherence, decoherence, and entanglement of quantum quantum algorithms, as its action creates superposition of the mechanics in computational devices and enables implementa- basis states. On qutrits it is defined as [43] tion of quantum computing algorithms [26]. Physical hardware in classical systems uses binary logic; however, most quantum  1 1 1  1  2π − 2π systems have multiple (D) possible levels. States of these H = √  1 e 3 e 3  (6) 3 − 2π  2π systems are referred to as qudits. 1 e 3 e 3 The special case of the generalized Hadamard gate for qudits A. Concept of Qudits is given by In contrast to a two-state quantum system, described by a D−1 X i D 2iπ 2iπ 2iπ qubit, a (D > 2) multilevel quantum system is represented H|ki = θ |ii, where θ = cos( )+ sin( ) = e D k i D D by D basis states. We choose, as is usual the so-called i=0 “computational” basis: |0i, |1i, |2i,... |D −1i. A general pure (7) th state of the system is a superposition of these basis states Here  is the imaginary unit and the angle θ is the i root of 1. represented as ω   We define a rotation gate R(ω) = e 3 , which transforms a α0 0 0 0 qutrit in state (α0, α1, α2) to the (rotated) state (α , α , α ), α 0 1 2 |ψi = α |0i+α |1i+α |2i+...+α |D−1i =  1  as follows: 0 1 2 D−1  ...  √    0      α 1 + cos ω − 2 sin ω 1 − cos ω α0 αD−1 0 √ √ α0 = 1 × α (1)  1  2  sin ω √2 cos ω − 2 sin ω   1  2 2 α0 α subject to the normalization criterion |α0| + |α1| + ... + 2 1 − cos ω 2 sin ω 1 + cos ω 2 2 (8) |αD−1| = 1 where, α0, α1, . . . αD−1 are complex quantities, Note that the rotation gate defined above is a unitary operator. i.e., {αi} ∈ C. Physically, the absolute magnitude squared of each coefficient αi represents the probability of the system being measured to be in the corresponding basis state |ii. IV. QUANTUM NEURAL NETWORK MODELBASEDON In this article, we use a three-level system (D = 3), i.e., a QUDITS (QNNM) basis of {|0i, |1i and |2i} for each quantum trit or qutrit. One A quantum neural network dealing with discrete data is physical example of a qutrit is a spin-1 particle. A general realized on a classical system using quantum algorithms and pure (coherent) state of a qutrit [42] is a superposition of all acts on quantum states through a layered architecture. In this the three basis states, which can be represented as proposed qudit embedded quantum neural network model,   the classical network inputs are converted into D-dimensional α0 2π th quantum states [0, D ] or qudits. Let the k input be given by |ψi = α0|0i + α1|1i + α2|2i =  α1  (2) xk. We apply a standard classical sigmoid activation function α2 fQNNM (xk), which yields binary classical outcome [0, 1]. 2 2 2 subject to the normalization criterion |α0| +|α1| +|α2| = 1. 1 For example, the state fQNNM (xk) = (9) 1 + e−xk √ √ 2 3 3 We then map that quantity onto the amplitude for the kth basis |ψ3i = √ |0i + √ |1i + √ |2i (3) 10 10 10 state as 2π has a probability of the being measured to be in the basis state |α i = ( f (x )) (10) |2i of k D QNNM k 2 3 The suggested QNNM model comprises multiple D- |h2|ψ3i| = (4) 10 T dimensional qudits, ZD = {(|z1i, |z2i, |z3i,... |zDi) : Similarly, the probabilities of the quantum state |ψ3i being |zki ∈ Z(k = 1, 2,...D)}. The inner product between the T measured to be in each of the other two basis states |0i and input quantum states |ψDi = (|α1i, |α2i, |α3i,..., |αDi) 4 3 T |1i are 10 and 10 respectively. and the quantum weights |WDi = (|θ1i, |θ2i, |θ3i,..., |θDi) is defined as D B. Quantum Operators X hψ |W i = hα |θ i = T |α iH |θ i = We define generalized Pauli operators on qudits as D D k k D k D k k=1 (11) D−1 D−1 D D X X X X 2kπ 2kπ X = |k + 1(mod3)ihk|,Z = θk|kihk| (5) α |ki( cos( ) +  sin( )) k D D k=0 k=0 k=1 k=1 4

where TD and HD are the transformation (realization map- Hence, each quantum neuron constitutes a qutrit state desig- ¯ ping) and the Hadamard gate, respectively. ψD is the complex nated as ψij. conjugate of ψD and is deﬁned as Each layer of the quantum self-supervised neural network

† architecture is organized by combining the qutrit neurons hψD| = |ψDi = (|α¯1i, |α¯2i, |α¯3i,..., |α¯Di) (12) in a fully-connected fashion with intra-connection strength 2π qutrit In the suggested quantum neural network model, let us con- as 3 ( state). The main characteristic of the network architecture lies in the organization of the 8-connected second- sider the set of all quantum states be denoted as QD(Z) and order neighborhood subsets of each quantum neuron in the the D-dimensional realization transformation, TD : QD(Z) → 2D layers of the underlying architecture and propagation to the R is deﬁned as subsequent layers for further processing. The input, intermedi- T |ψDi = ate/hidden and output layers are inter-connected through self- T (Re|α1i, Im|α1i, Re|α2i, Im|α2i, . . . Re|αDi, Im|αDi) forward propagation of the qutrit states in the 8-connected (13) neighborhood fashion. On the contrary, the inter-connections are established from the output layer to intermediate layer en- for all |ψ i = (|α i, |α i,... |α i)T ∈ Q ( ) and ∀i ∈ D 1 2 D D Z tailing self-counter-propagation obviating the quantum back- D, |α i = cos ω |0i + j sin ω |1i. The input-output association i i i propagation algorithm and thereby reducing time complexity. of a D-dimensional basic quantum neuron in the proposed Finally, a quantum observation process allows the qutrit states QNNM model at a particular epoch (t) is modeled as to collapse to one of the basis states (0 or 1 as 2 is con- t t 2π t t sidered as a temporary state). We obtain true outcome at the |Oki = TD(|hki) = T ( δDk − arg(|yki)) (14) D output layer of the QFS-Net once the network converges, else where, quantum states undergo further processing. N t X t t−1 t |yki = HD(|θi,ki)TD(|Ok i) − HD(|ξi i) (15) A. Qutrit-inspired Fully Self-supervised Quantum Neural i=1 Network Model Here, the quantum phase transformation parameter (weight) The novel quantum fully self-supervised neural network between the kth output neuron and the ith input neuron is model based on qutrits adopts twofold schemes. The qutrit |θ i and the activation is |ξti. The D-dimensional Hadamard i,k i neurons of each layer are realized using a T transformation gate parameters are designated by δ . Considering the basis Dk gate (realization mapping) and the inter-connection weights state |D − 1i, the true outcome of the quantum neuron k at are mapped using the phase Hadamard gates (H) applicable the output layer is obtained through quantum measurement of on qutrits. The angle of rotation is set as relative difference D-dimensional quantum state |Ot i as k of quantum information (marked by pink arrow in Figure 1) k t 2 qutrit MQNMM = |Im(|Oki)| (16) between each candidate neuron and the neighborhood qutrit neuron of the same layer employed in the rotation t t where, the imaginary section of Ok is referred to as Im(Ok). gate for updating the inter-layer interconnections. The rotation It is worth noting that the realization mapping T trans- angle for the inter-connection weights and the threshold are forms quantum states to probability amplitudes and hence set as ω and γ, respectively. The inter-connection weights the quantum state is destroyed on implementation in classical between the qutrit neurons (denoted as k and i) of two adjacent systems. However, the suggested quantum neural network layers are depicted as |θiki and measured as the relative model is not a quantum neural network in the true sense of difference between the ith candidate qutrit neuron and the the term. It is a quantum mechanics-inspired hybrid neural 8-connected neighborhood quantum neuron k. The realization network model implementable on classical systems. of the network weights are mapped using the Hadamard gate (H) inspired by the proposed QNNM model by suppressing V. QUANTUM FULLY SELF-SUPERVISED NEURAL the highest basic level (|2i) of qutrit as a temporary storage NETWORK (QFS-NET) as The suggested quantum fully self-supervised neural network 2π 2π 2π cos( 3 ωi,k) architecture comprises trinity layers of qutrit neurons arranged H(|θiki) = cos( ωi,k) +  sin( ωi,k) = 2π 3 3 sin( 3 ωi,k) as input, intermediate and output layers. A schematic outline of (17) the QFS-Net architecture as a quantum neural network model where,  is an imaginary unit. The role of relative measure of is illustrated Figure 1. The information processing unit of the quantum fuzzy information lies in the fact that the distinc- the QFS-Net architecture is depicted using quantum neurons tion between the foreground and background image pixels is (qutrits) reﬂected in the trinity layers using the combined clearly visible on adapting the relative measures. Assuming matrix notation. the quantum fuzzy grade information at the ith candidate   |ψ11i |ψ12i |ψ13i ... |ψ1mi neuron and its 8-connected second order neighborhood neuron  ......  as µi and µi,k respectively, the angle of the Hadamard gate is    ......  determined as    ......  ωi,k = 1 − (µi − µi,k); k ∈ {1, 2,... 8} (18) |ψn1i |ψn2i |ψn3i ... |ψnmi 5

Fig. 1: qutrit-inspired Quantum Fully Self-Supervised Neural Network (QFS-Net) architecture where H represents Hadamard gate and T is realization gate (only three inter-layer connections are shown for clarity).

l th The 8-fully intra-connected spatially arranged neighborhood where, |ψki is the output of the k constituent qutrit neuron qutrit neurons contribute to the candidate quantum neuron (say at the lth layer and the contribution of each 8-connected i0) of the adjacent layer through the transformation gate (T ) neighborhood qutrit neurons of the kth candidate neuron is l−1 and the realization mapping defined as expressed as |ψk i i.e. , X " 8 # |ψi0 i = T (|µi,ki)H(|θiki0 i) = 2π X |ψl i = T δl − arg{ H(|θl i)T (|ψl−1i) − H(|ξl i)} k k 3 k k,i k,i k (19) i=1 X 2π 2π [µ {cos( ω ) +  sin( ω )}] (23) i,k 3 i,k 3 i,k k Quantum observation on a qutrit neuron transforms a quantum In addition, the contribution of the 8-fully intra-connected spa- state into a basis state and a true outcome (|1i) is obtained on tially arranged neighborhood qutrit neurons are accumulated measurement from the qutrit neuron considering the imaginary l at the candidate qutrit neuron as the quantum fuzzy context section of |ψk) as sensitive activation (ξi) and is presented using the Hadamard l l 2 gate as Ok = |Im(|ψki)| (24) 2π 2π cos( 2π γ ) i.e H(|ξ i) = cos( γ ) +  sin( γ ) = 3 i (20) i i i sin( 2π γ ) 8 3 3 3 i X 2π Ol = QSig( T (ψl−1) cos( (ωl − γl ))+ k k,i 3 k,i k where, the angle of the Hadamard gate is defined as i=1 (25) X 2π γ = ( µ )  sin( (ωl − γl ))) i i,k (21) 3 k,i k k where, the quantum phase transmission parameter from the The self-supervised forward and counter propagation of input qutrit neuron i (the neighborhood of kth qutrit neuron the QFS-Net are guided by a novel qutrit based adaptive at the layer l−1 is depicted as i) to intermediate qutrit neuron multi-class Quantum Sigmoid (QSig) activation function with l l k with activation ξk, is ωk,i. The rotation gate parameters are quantum fuzzy context sensitive thresholding as discussed in l l expressed as δk with the parameters of activation as γk at the the following subsection V-C. The basis of network dynamics layer l. The activation function employed in the proposed QFS- of the QFS-Net is centred on the bi-directional self-organized Net model is a novel adaptive multi-class qutrit embedded propagation of the qutrit states between the intermediate and sigmoidal (QSig) activation function which is illustrated in output layers via updating of inter-connection links. the following subsection V-C. The network basic input-output relation is presented through the composition of a sequence using the transformation gate (T ) and the realization mapping defined as B. Qutrit-Inspired Self-supervised Learning of QFS-Net Let us consider, the interconnection weights in terms of 8 l X l−1 l l qutrit between the input and the hidden or intermediate layer |ψki = QSig( T (ψk,i )H(hθi|ξii)) (22) l are expressed as |θ 0 i (here any candidate qutrit neuron at i=1 ipi 6 the input layer is i, its corresponding candidate neuron at the levels with various schemes of activation. The QSig activation next subsequent intermediate layer is i0 and its corresponding function, employed in the QFS-Net model is defined as 8-connected neighborhood neurons are described by p) and l 1 for the intermediate layer to the output layer are |θjqj0 i (here QSig(x) = −λ(xh−η) (28) any candidate qutrit neuron at the intermediate layer is j, its κϑ + e corresponding candidate neuron at the next subsequent output where, QSig(x) represents the adaptive multi-class quantum layer is j0 and its corresponding 8-connected neighborhood sigmoidal (QSig) activation function with steepness parameter neurons are described by q) at the lth iteration. The activation l λ, step size h and activation η described by qutrits. The multi- at the intermediate layer and output layer are expressed as |ξji l level class output, κϑ as qutrit is defined as and |ξki, respectively. The self-supervised counter-propagation of the quantum states from output to intermediate layer is QN l performed through the interconnection weight |θ 0 i (here κϑ = (29) krk τϑ − τϑ−1 any candidate qutrit neuron at the output layer is k, its corresponding candidate neuron at the next subsequent intermediate The gray-scale intensity index is expressed as κϑ (1 ≤ κϑ ≤ layer is k0 and its corresponding 8-connected neighborhood L) where ϑ is the class index. The ϑth and ϑ − 1th class neurons are described by r). The outcome of a qutrit neuron responses are denoted as τϑ and τϑ−1, respectively and the l (|ψki) at the output layer can be expressed as sum of the containment of 8-connected neighborhood qutrit 8 8 neurons representing gray-scale pixels is denoted by QN . The l X l−1 l l X |ψki = QSig( T (|ψjq i)H(hθjqj0 |ξki) = QSig( q=1 q=1 00 00 0 0 8 2π X 2π l−1 l−1 l l 0 0 T ( × QSig( ( xip)H(hθ 0 |ξj i)))H(hθjqj0 |ξki) 3 3 ipi p=1 00 00 (26) 0 0 00 00

0 0 i.e., 000 000 0 0 8 8 X 2π X 2π |ψl i = QSig( T ( × (QSig( ( x ) (a) L = 3 (b) L = 4 k 3 3 ip q=1 p=1 00 00 0 0 2π 2π l l l l (27) 0 0 cos( (ω 0 − γ )) cos( ((ω 0 − γ ))+ 3 ipi j 3 jqj k 00 00 2π l l 2π l l 0 0  sin( (ω 0 − γ )) sin( (ω 0 − γ ))))) 3 ipi j 3 jqj k 00 00 0 0

000 000 0 0 where, xip represents the classical input to the neighborhood neuron p with respect to a candidate neuron i at the input (c) L = 6 (d) L = 8 layer which is subsequently transformed to a qutrit state 2π Fig. 2: Multi-level class outcome of QSig activation function (|φipi = 3 xip) and  is an imaginary unit. An adaptive multi- class qutrit embedded sigmoidal (QSig) activation function for λ = 15, 20, 25 and h = 1 with segmentation levels employed in this self-supervised network model governs the activation at the intermediate and output layers and also the generalized version of the QSig activation function defined subsequent processing of the quantum states guided by various in Equation 28 can be modified leveraging κϑ with various thresholding schemes. 2π subnormal responses σκϑ as qutrit where 0 ≤ σκϑ ≤ 3 . The multi-level class output is obtained on superposition of the C. Adaptive Multi-class Quantum Sigmoidal (QSig) activation subnormal responses and the generic QSig activation function function can be expressed as In this paper, we have introduced an adaptive multi-class 1 QSig(x; κϑ, τϑ) = L+1 (30) sigmoidal activation function in quantum formalism suitable −λ(x−(ϑ− )τϑ−1−η) κϑ + e 2 for pixel wise multi-class segmentation of medical images varying with multi-intensity gray-scales. The proposed QSig In order to ensure that the number of distinct κϑ parameters activation function is the modification on the recently de- is to be equal to the number of multi-level classes (L − 1), veloped quantum multi-level sigmoid activation function em- Equation 31 depicts the closed form of the resultant QSig ployed in authors’ previous work [39], [40]. An optimized function as version of similar function is also introduced in [41]. How- L ever, the requirement of finding optimal thresholding of the X L + 1 Qsig (x) = Qsig(x − (ϑ − )τ ); images in the activation function is computationally exhaustive R 2 ϑ−1 ϑ=1 (31) and time dependent. The proposed QSig relies on an adaptive L + 1 step length incorporating the total number of segmentation (ϑ − )τϑ−1 ≤ x ≤ ϑτϑ 2 7

Substituting Equation 30 in Equation 31, the updated form is that the convergence of the QFS-Net is faster than that of the expressed as QIS-Net and also follows super-linearity. This claim is also

L substantiated by the number of iterations required to converge X 1 for each image slice in QFS-Net and QIS-Net as illustrated in QSigR(x; κϑ, τω) = L+1 (32) −λ(x−(ϑ− 2 )τϑ−1−η) ϑ=1 κϑ + e Figure 4. Different forms of the QSig activation function with different Convergence of QIS-Net for class level S1 Convergence of QIS-Net for class level S2 0.8 0.8 values of the steepness parameters are illustrated in Figure2. 0.7 0.7

0.6 0.6

0.5 0.5 D. Updating Inter-connection Weight using Hadamard Gate 0.4 0.4 0.3 0.3 The interconnection weights and the activation of QFS-Net 0.2 0.2 architecture are updated using a Hadamard gate (H) working 0.1 0.1 0 0 on qutrit as follows. 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50   (a) QIS-Net, S1 (b) QIS-Net, S2 1 1 1 Convergence of QIS-Net for class level S3 Convergence of QIS-Net for class level S4 0.8 0.8 ι+1 1  2π 4ω − 2π 4ω ι H(|θ i) = √  1 e 3 e 3  H(|θ i) 0.7 0.7 2π 2π 3 − 4ω  4ω 0.6 0.6 1 e 3 e 3 0.5 0.5

(33) 0.4 0.4  1 1 1  0.3 0.3 ι+1 1  2π 4γ − 2π 4γ ι 0.2 0.2 √ 3 3 H(|ξ i) =  1 e e  H(|ξ i) 0.1 0.1 2π 2π 3 − 4γ  4γ 0 0 1 e 3 e 3 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 (34) where (c) QIS-Net, S3 (d) QIS-Net, S4 Convergence of QFS-Net for class level S1 Convergence of QFS-Net for class level S2 ι+1 ι ι 0.8 0.8 ω = ω + 4ω (35) 0.7 0.7

0.6 0.6 and 0.5 0.5 γι+1 = γι + 4γι (36) 0.4 0.4 0.3 0.3

0.2 0.2

The suitable tailoring of the phase angle in the Hadamard 0.1 0.1

0 0 gate advocates the stability of the QFS-Net or its convergence 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 which is very crucial for self-supervised networks where (e) QFS-Net, S1 (f) QFS-Net, S2 the loss function (here error function) is dependent on the Convergence of QFS-Net for class level S3 Convergence of QFS-Net for class level S4 0.8 0.8 interconnection weights. Hence, the phase angles are evalu- 0.7 0.7 ated using 4ωι and 4γι as given in Equations 18 and 21, 0.6 0.6 0.5 0.5 respectively. It is worth noting that the qutrit based quantum 0.4 0.4 neural network provides faster convergence compared to the 0.3 0.3 0.2 0.2 classical neural networks. This is due to the fact that whereas 0.1 0.1

0 0 the classical neural networks are formed using the multipli- 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 cation of input vector and the weight vector guided by an (g) QFS-Net, S3 (h) QFS-Net, S4 activation function, the quantum-based networks incorporate the frequency components of the weights and their inputs Fig. 3: Convergence analyses of the suggested qutrit-inspired thereby enabling faster convergence of the network states. This QFS-Net and qubits embedded QIS-Net [39] for four different inherent novel feature of the quantum neural networks facili- activation schemes tates the qutrit based fully self-organized quantum algorithm to be employed in QFS-Net to converge super-linearly, as shown in Figure 3. The loss function cum QFS-Net network error VI.RESULTS AND DISCUSSION function is deﬁned on quantum measurement in the following way. A. Data Set

N 8 Cancer Imaging Archive (TCIA) data are available from 1 X X 2 ζ(ω, γ) = Θ (ω , γ )ι+1 − Θ (ω , γ )ι the Nature repository [14] and the experiments have been N ik ik i ik ik i i k=1 performed on the same data sets using the suggested QFS- (37) Net model characterized by qutrits and an adaptive multi- ι where, Θik(ωik, γi) represents the true interconnection class quantum sigmoidal (QSig) activation function. In contrast ι weight terms of the inter-connection weights |θiji as expressed with the automatic brain lesion segmentation, four distinct using the Hadamard gate (H) at an instance (ι). ζ(ω, γ) is a activation schemes have been tested, and experiments are also coherent error function of ω and γ. Convergence analysis of performed using Quantum-Inspired Self-supervised Network the proposed qutrit-inspired QFS-Net is provided in Appendix (QIS-Net) [39], U-Net [18] and URes-Net [20] architectures. Section A and demonstrated experimentally with qubit embed- The U-Net [18] and URes-Net [20] architectures are trained ded QIS-Net [39] as shown in Figure 3. It can be summarized with 2000 MR images and validated and tested on 120 and 880 8

Convergence of QIS-Net and QFS-Net for class level S1 Convergence of QIS-Net and QFS-Net for class level S2 50 50 The suggested qutrit-inspired fully self-supervised shallow 45 45 quantum learning model is experimented with the multi-level 40 40

35 35 gray-scale images using distinct classes L = 4, 5, 6, 7 and 8 30 30 characterized by an adaptive multi-class quantum sigmoidal 25 25

20 20 (QSig) activation function. In this experiment, the steepness 15 15 in the QSig activation, λ is varied in the range 0.23 to 0.24 10 10 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 with step size 0.001. It has been observed that in majority

(a) ηβ (b) ηχ cases, λ = 0.239 yields optimal performance. The empirical

Convergence of QIS-Net and QFS-Net for class level S3 Convergence of QIS-Net and QFS-Net for class level S4 40 45 goodness measures [Positive Predictive Value (PPV ), Sensi-

40 35 tivity (SS), Accuracy (ACC) and Dice Similarity(DS) [44]]

35 30 are assessed to evaluate the experimental outcome using four 30 25 25 thresholding schemes (ηβ, ηχ, ηξ, ην ) [39], [45] as discussed

20 20 in the supplementary materials section for different level sets.

15 15 The dice score is often used to measure the similarity of the 10 10 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 segmented brain lesions and regions of interest (ROIs).

(c) η (d) ην ξ C. Experimental Results Fig. 4: Average number of iterations of each brain slice using Extensive experiments have been performed in the cur- QFS-Net based on qutrit and QIS-Net [39] based on qubits for rent setup, and experimental outcomes are reported with the four various thresholding schemes (a)ηβ, (b)ηχ, (c)ηξ, (d)ην demonstration of numerical and statistical analyses using the using class level S2 [39] proposed QFS-Net, QIS-Net [39], convolutional U-Net [18] and Residual U-Net (URes-Net) architectures [20]. The hu- contrast-enhanced Dynamic Susceptibility Contrast (DSC) MR man expert segmented skull-tripped contrast enhanced DSC 512 × 512 images, respectively. The QIS-Net and the proposed QFS-Net brain MR input image slices of size and ROIs are also tested on the same number of 880 contrast-enhanced are provided in Figure 5 as samples. The demonstration of DSC MR images. QFS-Net segmented images followed by the essential post- processed outcome on the slice no. 37 for class level L = 8 with four distinct activation schemes (ηβ, ηχ, ηξ, ην ) are shown B. Experimental Setup in Figure 6. It is evident from the experimental data provided In this current work, extensive experiments have been in Table I that the proposed QFS-Net performs optimally carried out on 3000 Dynamic Susceptibility Contrast (DSC) for the 8-connected quantum fuzzy pixel information het- brain MR images of Glioma patients from TCIA data sets of erogeneity assisted activation (ηξ) with L = 8 and gray size 512×512 using Nvidia RTX 2070 GPU System with high- scale set S2 in comparison with other thresholding schemes performance systems with MATLAB 2020a and Python 3.6. and gray scale sets under the four evaluation parameters The 2D segmented images are processed through a 2D binary (ACC, DS, P P V, SS) [44]. The segmented tumors obtained circular mask to obtain the brain lesion in the suggested QFS- using the proposed self-supervised procedure under L = 8 Net framework. The lesion or brain tumor detection mask is class transition levels with four different thresholding schemes binarized using a threshold of 0.5, and in the case of QFS- ηβ, ηχ, ηξ and ην are demonstrated in Figures 7- 8 for the class Net and QIS-Net [39], it is seen that with a radius of 5 boundary sets S1 and S2 [39], respectively. The segmented pixels, the segmented ROIs perform optimally while compared images using the remaining two class boundary sets (S3 and with the human expert segmented images. Experiments are S4) [39] are provided in the supplementary materials section. also performed on two recently developed CNN architectures The segmented ROIs describing the whole tumor region after suitable for medical image segmentation viz., convolutional the masking procedure using QIS-Net, U-Net and URes-Net U-Net [18] and Residual U-Net (URes-Net) [20] available in are also reported in Figure 9. GitHub. The U-Net and URes-Net networks are rigorously trained using the stochastic gradient descent algorithm with learning rate 0.001 and batch size 32 allowing maximum 50 epochs to converge. The segmented output images resemble in size with the dimensions of the binary mask and the outcome 1 is considered as tumor region and 0 as background in detecting complete tumor. The pixel by pixel comparison with (a) Input Slice (b) Input Slice (c) ROI Slice (d) ROI Slice the manually segmented regions of interest or lesion mask #37 #69 #37 #69 allows evaluating the dice similarity, which is considered as Fig. 5: Dynamic Susceptibility Contrast (DSC) skull stripped a standard evaluation procedure in automatic medical image brain MR images with size 512×512 and manually segmented segmentation. The evaluation process involves the manually ROI slices [14] segmented lesion mask as ground truth, and each 2D pixel is predicted as either True Positive (TRP ) or True Negative Table II presents the numerical results obtained using the (TRN ) or False Positive (TRN ) or False Negative (FLN ). proposed QFS-Net and QIS-Net [39] on evaluating the av- 9 erage accuracy (ACC), dice similarity score (DS), positive prediction value (PPV ), and sensitivity (SS) as reported under L = 8 class transition levels (S1,S2,S3,S4) [39] with four distinct thresholding schemes (ηβ, ηχ, ηξ and ην ). The average number of iterations required to converge for each class boundary set is also reported in Table II. In addition, (a) ηβ (b) ηχ (c) ηξ (d) ην Table III summarises the results obtained using convolutional U-Net [18] and Residual U-Net (URes-Net) [20] architectures for two distinct convolutional masks with size 3 × 3 and 5 × 5 with stride sizes of 1 and 2. However, the convolutional based architectures (U-Net and URes-Net) marginally outperform our proposed qutrit-inspired fully self-supervised quantum (e) ηβ (f) ηχ (g) ηξ (h) ην neural network model QFS-Net and the previously developed QIS-Net [39] based on qubits. The box plots are also demonstrated in the supplementary materials section citing the outcome reported in Tables II and III, respectively. Moreover, to show the effectiveness of our proposed QFS-Net over QIS- Net, U-net and URes-Net, we have conducted one-sided two- (i) ηβ (j) ηχ (k) ηξ (l) ην sample Kolmogorov-Smirnov (KS) [46] test with signiﬁcance Fig. 7: Segmented ROIs describing the complete tumor region level α = 0.05. It is interesting to note that in spite being after the post-processing using the proposed QFS-Net on slice a fully self-supervised quantum learning inspired by qutrits, #69 [14] using L = 8 transition levels with four different the QFS-Net has shown similar accuracy (ACC) and dice thresholding schemes (ηβ, ηχ, ηξ, ην ) (a − e) with class-level similarity (DS) compared with U-Net [18] and URes-Net [20]. S1 [39] Hence, it can be concluded, that the performance of the QFS- Net model on Dynamic Susceptibility Contrast (DSC) brain MR images is statistically signiﬁcant and offers a potential alternative to the solution of deep learning technologies.

(a) ηβ (b) ηχ (c) ηξ (d) ην

(e) ηβ (f) ηχ (g) ηξ (h) ην

(i) ηβ (j) ηχ (k) ηξ (l) ην Fig. 8: Segmented ROIs describing the complete tumor region after the post-processing using the proposed QFS-Net on slice #69 [14] using L = 8 transition levels with four different (i) ηβ (j) ηχ (k) ηξ (l) ην thresholding schemes (ηβ, ηχ, ηξ, ην ) (a − e) with class-level S2 [39]

(m) ηβ (n) ηχ (o) ηξ (p) ην Fig. 6: Demonstration of QFS-Net segmented images followed by essential post-processed outcome on the slice no. 37 [14] (a) QIS-Net (b) U-Net (c) URes-Net for class level L = 8 with four distinct activation schemes #69 (ηβ, ηχ, ηξ, ην ) with class-levels (a − d) for S1 , (e − h) for Fig. 9: ROI segmented output slice [14] masking followed (a) b c S2, (i − l) for S3,and (m − p) for S4 [39] by post processing using QIS-Net [39] ( ) U-Net [18] ( ) URes-Net [20] 10

TABLE I: Segmented accuracy, dice similarity score, PPV and sensitivity for the slice #37 [14] using QFS-Net ACC = DS = PPV = SS = Level Set TRP +TRN 2TRP TRP TRP TRP +FLP +TRN +FLN 2TRP +FLP +FLN TRP +FLP TRP +FLN ηβ ηχ ηξ ην ηβ ηχ ηξ ην ηβ ηχ ηξ ην ηβ ηχ ηξ ην S1 0.99 0.99 0.99 0.99 0.82 0.79 0.79 0.71 0.71 0.65 0.65 0.65 0.98 0.99 0.99 0.99 S 0.99 0.99 0.99 0.99 0.75 0.82 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.98 0.99 0.98 L = 4 2 S3 0.99 0.99 0.99 0.99 0.82 0.83 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.97 0.99 0.98 S4 0.99 0.99 0.99 0.99 0.82 0.83 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.97 0.99 0.98 S1 0.99 0.99 0.99 0.99 0.82 0.83 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.97 0.99 0.98 S 0.99 0.99 0.99 0.99 0.82 0.80 0.81 0.82 0.71 0.72 0.74 0.71 0.98 0.89 0.89 0.98 L = 6 2 S3 0.99 0.99 0.99 0.99 0.82 0.83 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.97 0.99 0.98 S4 0.99 0.99 0.99 0.99 0.82 0.83 0.78 0.82 0.71 0.72 0.65 0.71 0.98 0.97 0.99 0.98 S1 0.99 0.99 0.99 0.99 0.83 0.83 0.82 0.82 0.74 0.74 0.73 0.74 0.93 0.93 0.94 0.93 S 0.99 0.99 0.99 0.99 0.83 0.83 0.83 0.83 0.73 0.73 0.73 0.73 0.95 0.96 0.96 0.96 L = 8 2 S3 0.99 0.99 0.99 0.99 0.83 0.82 0.82 0.82 0.77 0.77 0.77 0.77 0.89 0.89 0.89 0.89 S4 0.99 0.99 0.99 0.99 0.82 0.82 0.82 0.83 0.73 0.73 0.73 0.74 0.94 0.94 0.94 0.94

TABLE II: Average performance analyses of QFS-Net and QIS-Net [39] for four distinct class levels and activation [One sided non-parametric two sample KS test [46] with α = 0.05 signiﬁcance level has been conducted and marked in bold.] ACC DS PPV SS Avg. Network Set #Iteration ηβ ηχ ηξ ην ηβ ηχ ηξ ην ηβ ηχ ηξ ην ηβ ηχ ηξ ην S1 0.990 0.987 0.987 0.988 0.799 0.783 0.782 0.788 0.713 0.695 0.691 0.698 0.955 0.954 0.957 0.957 10.78 S 0.989 0.989 0.988 0.987 0.790 0.790 0.776 0.773 0.697 0.696 0.679 0.679 0.957 0.958 0.960 0.959 11.06 QFS-Net 2 S3 0.989 0.989 0.990 0.989 0.783 0.798 0.795 0.782 0.690 0.710 0.718 0.687 0.955 0.957 0.935 0.959 10.98 S4 0.989 0.986 0.988 0.989 0.781 0.767 0.783 0.800 0.694 0.676 0.693 0.713 0.954 0.955 0.954 0.957 12.12 S1 0.986 0.987 0.986 0.986 0.784 0.771 0.767 0.766 0.698 0.688 0.680 0.672 0.956 0.947 0.951 0.960 11.77 S 0.987 0.987 0.988 0.988 0.764 0.761 0.766 0.766 0.665 0.663 0.667 0.666 0.960 0.959 0.961 0.961 12.65 QIS-Net 2 S3 0.986 0.986 0.986 0.987 0.768 0.781 0.755 0.764 0.676 0.666 0.659 0.665 0.955 0.957 0.957 0.959 12.15 S4 0.987 0.986 0.986 0.986 0.773 0.764 0.761 0.768 0.679 0.674 0.668 0.676 0.959 0.954 0.955 0.957 13.16

TABLE III: Performance analyses of U-Net [18] and URes- and it can also be implemented using quantum gates along Net [20] for four distinct class levels and activation [One with its classical counterparts. The proposed QFS-Net model sided non-parametric two sample KS test [46] with α = 0.05 offers the possibilities of entanglement and superposition in signiﬁcance level has been conducted and marked in bold.] the network architecture, which are often missing in the Networks Conv-Mask Stride ACC DS PPV SS classical implementations. However, it is also worth noting that 3 × 3 1 0.993 0.795 0.717 0.939 the suggested qutrit-inspired fully self-supervised quantum 3 × 3 2 0.991 0.794 0.715 0.937 U-Net 5 × 5 1 0.996 0.795 0.726 0.938 neural network model is computed and experimented on a 5 × 5 2 0.990 0.797 0.718 0.940 classical system. Hence, the proposed model architecture is 3 × 3 1 0.999 0.806 0.734 0.932 not quantum in a real sense, instead it is quantum-inspired. It 3 × 3 2 0.997 0.809 0.727 0.936 URes-Net is also worth noting that the QFS-Net is validated solely for 5 × 5 1 0.998 0.805 0.729 0.939 5 × 5 2 0.991 0.796 0.717 0.937 complete tumor and the network has potential for multi-level segmentation which is evident from the segmented brain MR lesions. Nevertheless, it remains an uphill task to optimize the hyper-parameters for obtaining optimal multi-class segmentation. Authors are currently engaged in this direction. VII.CONCLUSION

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APPENDIX Also, ||γι+1 − γ|| A. Convergence analysis of QFS-Net lim ≤ 1 (51) ι→∞ ||γι − γ|| Let us consider the optimal phase angles for the weighted and matrix and the activation are denoted as ω and γ, respectively ι+1 ι and deﬁned as follows: ||µ || = O||ρ || (52) In order to prove the convergence of the sequences of {ωι} υι = ωι − ω (38) and {γι}, according to Thaler theorem, we obtain µι = γι − γ (39) ζ(ωι+1, γι+1) − ζ(ωι, γι) = (53) " ∂ζ(ω,γ)ι # and ι ι ι+1 ι ι+1 ι ι ι ∂ωik ι ι δ = ω − ω = υ − υ (40) 4ωik 4γi ∂ζ(ω,γ)ι + O ||4ωik 4γi || ∂γι ι ι+1 ι ι+1 ι ik ρ = γ − γ = µ − µ (41) ι ι ∂ζ(ω, γ) ∂ζ(ω, γ) 1 2 2 ι ι t ≈ {−σik ι } + {−σi ι } {ζ(ω , γ )} Also, the derivative of the loss function ζ(ω, γ) with respect ∂ωik ∂γik to ω, γ is depicted as follows. (54) Hence, (ζ(ωι+1, γι+1) − ζ(ωι, γι)) ≤ 0 and it is clearly N 8 ι ι ∂ζ(ω, γ) 2 X X ι evident that the sequences of {ω } and {γ } are monotonically = 4Θik(ωik, γik) ∂ωik N decreasing. The coherent nature of these two sequence leads i k=1 (42) ι+1 ι to the following. ∂Θik(ωik, γik) ∂Θik(ωik, γik) − ι ι ∂ωik ∂ωik lim ζ(ω , γ ) = (ω, γ) (55) ι→∞ ∂ζ(ω, γ) ι = The rapid convergence of the iteration sequences {ω } and ι ∂γi {γ } are due to N (43) ι+1 ι ι+1 ι+1 2 X ι ∂Θi(ωi, γi) ∂Θi(ωi, γi) ||ζ(ω , γ ) − (ω, γ)|| 4Θi(ωi, γi) − lim ≤ 1 (56) N ∂γ ∂γ ι→∞ ι ι i i i ||ζ(ω , γ ) − (ω, γ)|| where The super-linear convergence of the sequences can be shown as follows. 4Θ (ω , γ )ι = |Θ (ω , γ )ι+1 − Θ (ω , γ )ι| (44) ∂ζ(ω,γ)ι ik i ik ik ik i ik ik i Let Gω = ι , then ∂ωik and ||ωι+1|| ||ωι+1 − ω|| ||ωι+1 − ω|| = ≥ ι ι ι 2 ι ∂ζ(ω,γ)ι 1 1 Θ (ω , γ ) = [Im(H{hθ |ξ i})] = ||δ || ι ι t ik ik i ik i || − σik{ ι ζ(ω, γ) } t || σikGω{ζ(ω, γ) } (45) ∂ωik ι ι 2 [Im(cos(ωik − γi) +  sin(ωik − γi) )] (57) Hence, The change in phase or angles (4ω and 4γ) in the Hadamard ι+1 ι 1 ||ω − ω|| = O({ζ(ω, γ) } t }) (58) gate are evaluated using the following equations. ι Consequently, ∂ζ(ω, γ) 1 ι ι t ι+1 ι 4ωik = −σik{ ι ζ(ω, γ) } (46) ||ω || = O(||δ ||) (59) ∂ωik which proves that the convergence behavior of the iteration ι ∂ζ(ω, γ) ι 1 ι 4γ = −σ { ζ(ω, γ) } t sequence {ω } is super-linearly convergent. i i ι (47) ι ∂γi ∂ζ(ω,γ) Similarly, let Gγ = ∂γι , then where, the learning rate for the self-supervised updating of the ik ||γι+1|| ||γι+1 − γ|| ||γι+1 − γ|||| weights in QFS-Net is denoted as σik. It is computed using the = ≥ ι ∂ζ(ω,γ)ι 1 1 ||ρ || ι ι t relative difference between the candidate and its neighborhood || − σi{ ι ζ(ω, γ) } t || σiGγ {ζ(ω, γ) } ∂γi qutrit neurons (intensities) with t > 2 as (60) Hence, σik = µi − µik∀k = 1, 2 ... 8 (48) ι+1 ι 1 ||γ − γ|| = O({ζ(ω, γ) } t }) (61) Similarly, the learning rate for updating the activation is Consequently, denoted as σi and is equal to the quantum fuzzy contribution ||γι+1|| = O(||ρι||) (62) of the candidate neuron (µi). The conditions for the super- linear convergence of the sequences of {ωι} and {γι} can be which proves that the convergence behavior of the iteration formulated as [1] sequence {γι} is super-linearly convergent. ||ωι+1 − ω|| lim ≤ 1 (49) REFERENCES ι→∞ ||ωι − ω|| [1] L. J. Zhen, X. G. He, and D. S. Huang,“Super-linearly convergent BP and learning algorithm for feed forward neural networks,” Journal of Software, ||υι+1|| = O||δι|| (50) in Chinese, vol. 11, no. 8, pp. 1094–1096, 2000.