### Particle swarm optimization (PSO)

Particle swarm optimization belongs to a department of the SI algorithm that was first meant for simulating social conduct after which developed for constrained and unconstrained issues and likewise utilized in discrete and steady optimization issues. It was first developed by Kennedy and Eberhart in 1995^{4}. The principle thought of the PSO algorithm is to share one of the best place of the entire swarm in each era after which transfer them towards their very own best-known place and the whole swarm’s best-known place within the search house concurrently. Then, particles are up to date based on the next equation:

$$start{aligned} V_{i,j}(t+1)= & {} wV_{i,j}(t)+c_{1} r_{1} left( p_{Best_{i,j}}- X_{i,j}(t) proper) + c_{2} r_{2} (g_{Best_{j}} – X_{i,j}(t)), X_{i,j}(t+1)= & {} X_{i,j}(t)+V_{i,j}(t+1),~qquad i=1,2, ldots , M,qquad j=1,2,ldots , D, finish{aligned}$$

the place (x_{ij}(t)) and (v_{ij}(t)) are the worth and velocity of (j{textual content {th}}) variable of the (i{textual content {th}}) particle in the course of the (t{textual content {th}}) iteration, respectively. *M* is the scale of the inhabitants, *D* is the dimension, (p_{Best_{i,j}}) is the (j{textual content {th}}) variable of the (i{textual content {th}}) finest answer thus far, (g_{Best_j}) is the (j{textual content {th}}) variable of the worldwide finest particle within the swarm, (c_1), and (c_2) are “Private Studying Coefficient” and “World Studying Coefficient”, respectively, that are predefined, *w* is a continuing, often called “Inertia Weight”, and management the worldwide and native search means of PSO and (r_1) and (r_2) are random numbers in [0, 1]. This course of iterates in order to achieve the suitable health or attain most iterations.

The pseudo-code of the PSO algorithm is illustrated in Algorithm 1.

### JAYA algorithm

The JAYA algorithm is a meta-heuristic algorithm not too long ago developed for constrained and unconstrained issues and can be utilized in discrete and steady optimization issues. It was first developed by Rao in 2016^{23}. On this algorithm, two parameters should be initialized for any optimization issues: the scale of the inhabitants, and the utmost variety of iterations. Then preliminary options are randomly generated within the possible area. Afterward, the health value of the generated options is calculated and one of the best and the worst options within the inhabitants are recognized. Then every variable of each answer is up to date based on the next equation:

$$start{aligned} X_{i,j}(t+1) = X_{i,j}(t) + r_{1} left( X_{j,finest}- vert X_{i,j}(t) vert proper) -r_{2} left( X_{j,worst}- vert X_{i,j}(t) vert proper) , finish{aligned}$$

(1)

the place (X_{i,j}(t)) is the worth of the (j{textual content {th}}) variable for the (i{textual content {th}}) candidate in the course of the (t{textual content {th}}) iteration, (X_{j,finest}) is the worth of the variable *j* for one of the best candidate and (X_{j,worst}) is the worth of the variable *j* for the worst candidate. (X_{i,j}(t+1)) is the up to date worth of (X_{i,j}(t)) and (r_{1}) and (r_{2}) are random numbers in (left[ 0,1right]). If the brand new answer is in a greater situation than the present one, it’s changed by the brand new one. This process is repeated in order to attain the suitable health or attain most iterations.

### Improved sine cosine algorithm

Sine cosine Algorithm (SCA) is a physics-based algorithm and was first developed by Mirjalili et al. in 2016 for fixing optimization issues^{7}. The principle downside of SCA is having low optimization precision and native minima trapping as a result of its exploration and exploitation mechanism. In Ref.^{24} the authors proposed a novel technique to beat the weak point of the algorithm by substituting a brand new replace mechanism. They proposed a brand new model of SCA known as Improved Sine Cosine algorithm (ISCA). Just like different meta-heuristic optimization ISCA begins the optimization course of by producing the preliminary options randomly within the possible area. Afterward, the health value of the generated options is calculated and one of the best answer within the inhabitants is recognized. Then every variable of each answer is up to date based on the sine and cosine features as follows:

$$start{aligned} X_{i,j}(t+1)=left{ start{array}{rl} X_{i,j}(t) + r_{1}(t) sin (r_{2}) displaystyle Massive vert r_{3} P_{i,j}(t)- X_{i,j}(t)Massive vert &{} if quad r_{4} geqslant 0.5, X_{i,j}(t) + r_{1}(t) cos (r_{2}) displaystyle Massive vert r_{3} P_{i,j}(t)- X_{i,j}(t)Massive vert &{} if quad r_{4} < 0.5, finish{array} proper. finish{aligned}$$

the place (X_{i,j}(t)) is the worth of the (j{textual content {th}}) variable for the (i{textual content {th}}) candidate in the course of the (t{textual content {th}}) iteration, (P_{j}(t)) is the (j{textual content {th}}) variable of one of the best answer in the course of the (t{textual content {th}}) iteration. (X_{i,j}(t+1)) is the up to date worth of (X_{i,j}(t)) and (r_{1}(t)) is described as follows:

$$start{aligned}r_{1}(t)=a left( 1- left( frac{t}{T} proper) ^alpha proper) ^beta ,finish{aligned}$$

through which (alpha) and (beta), are constructive actual numbers. (r_{2}), (r_{3}), and (r_{4}) are random numbers within the vary (left[ 0,1right]). If the brand new answer is in a greater situation than the present one is changed by the brand new one. This process is repeated in order to attain the suitable health or attain most iterations.

### Quantum particle swarm optimization (QPSO)

Nature is basically based mostly on quantum mechanical guidelines, though the quantum results are extra vital in micro-scale methods. One of many basic ideas of quantum mechanics is wave-particle duality, through which all of the details about a particle (state of affairs, place, velocity, vitality, and many others.) is described as a wave operate, (psi (r,t)), similar to a normalized quantum eigenstate. From the quantum standpoint, a wave packet, which is a superposition of too many waves, can characterize a localized particle in house beneath a bodily potential, however it’s broadened to some extent because of the uncertainty precept. In keeping with the Heisenberg uncertainty precept, the precise place of a quantum particle and its velocity can’t be concurrently decided. That’s true for its vitality and its quantum state lifetime as nicely. Subsequently, the expectation worth of every amount in quantum mechanics is represented as a probabilistic worth, which is set by the likelihood density operate, (|psi (r,t)|^2), which describes the likelihood of a particle to be present in a given quantum state (a given place, momentum, vitality, and many others). Within the final years, many researchers have tried to introduce quantum ideas by way of varied mathematical frameworks and succeeded in using them in optimization algorithms. The quantum particle swarm optimization (QPSO) algorithm is impressed by the quantum conduct of nature. The principle thought behind the QPSO is to discover a correct wave operate, related to a quantum particle in a possible subject. To seek out the optimum answer, QPSO exploits the quantum likelihood density operate to steer particles to the probably positions (or to probably the most doable states in a extra common sense). As a result of probabilistic nature of quantum mechanics, the correlation between quantum particles, and the mutual affect of their eigenstates, it’s anticipated that the answer lies in probably the most possible area of the search extent. Though there may be not an express relation between quantum options and the time complexity, it’s anticipated that the answer lies in probably the most possible area of the search extent, leading to higher looking efficiency. The QPSO algorithm, based mostly on some quantum potential fields, corresponding to sq. nicely, 1D potential nicely, the 1D-quantum easy harmonic oscillator, Coulomb-like sq. root subject, Lorentz potential subject, and Rosen–Morse has been already used and developed by a number of authors^{14,21,22}. The quantum wave features utilized in all the talked about research fulfill the standard linear Schrödinger equation,

$$start{aligned} ihbar frac{partial psi }{partial t}+frac{hbar ^2}{2m}frac{partial ^2psi }{partial x^2}-V(x,t)psi =0. finish{aligned}$$

(2)

Then again, there are a lot of pure phenomena in physics and engineering, described by nonlinear equations. The Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger (NLS) equation, and the sine-Gordon equation are some well-known nonlinear equations which were used extensively in reference to many bodily phenomena. They’re precisely solvable equations with soliton-like options.

Within the sequel, we current the likelihood density operate of such issues, particularly, quantum solitons with a self-consistent answer to the NLS equation.

### Solitons

Solitons or solitary waves are primarily the analytical options of bodily integrable nonlinear partial differential equations. They’re ubiquitous phenomena each in classical and quantum points with an unlimited variety of functions Classical solitons in physics are non-dispersive pulses, touring lengthy distances (Fig. 1a). Some of the fascinating and distinctive options of solitons is having neither deformation nor attenuation throughout propagation in a nonlinear dispersive medium. Making the most of the nonlinearity of the medium, solitons can reconstruct themselves, regardless of dispersion results. Quantum solitons are certainly the quantum states of classical options (Fig. 1b). They’re handled as particle-like wave packets with their very own coherent eigenstates and vitality eigenvalues (Fig. 1c).

The quantum solitons are probably the most widespread options of the quantum NLS equation which governs many quantum phenomena in physics and engineering. The same old Schrödinger equation, (Eq. (2)), turns into nonlinear if the potential *V*(*x*, *t*) is determined by (psi), itself. The overall commonplace dimensionless type of the NLS equation so reads,

$$start{aligned} i frac{partial psi }{partial t}+pfrac{partial ^2psi }{partial x^2}+q|psi |^2psi =0, finish{aligned}$$

(3)

through which (i=sqrt{-1}), and *p* and *q* are the coefficients with a particular bodily significance. Its primary soliton-like answer is,

$$start{aligned} psi (x,t)=frac{1}{2}sqrt{frac{2A}{q}}frac{exp (iAt)}{cosh (frac{A}{2p}x)}, finish{aligned}$$

(4)

the place *A* is an arbitrary fixed associated to the wave packet’s properties (amplitude, width, and frequency). A extra common model of the NLS equation is written by way of the Hamiltonian operator ({hat{H}}), as follows:

$$start{aligned} i frac{partial psi }{partial t}={hat{H}}psi + Vpsi +q|psi |^2psi . finish{aligned}$$

(5)

On this equation, ({hat{H}}) corresponds to the associated kinetic and potential energies of the system, and is a quadratic operate of the momentum operator, ({hat{p}}=-ipartial /partial x) and ({hat{x}}), particularly, It has a common soliton-like answer as the next kind,

$$start{aligned} psi (x,t)=textsf {A}(x,t)F(z)e^{iS(x,t)}, finish{aligned}$$

(6)

which describes a touring soliton with the profile operate F(z) by way of a wave-type argument *z*^{25}. The quantum NLS in its common kind often is written as,

$$start{aligned} ihbar frac{partial psi }{partial t}=-frac{hbar ^2}{2m}frac{partial ^2psi }{partial x^2}+2c|psi |^2psi , finish{aligned}$$

(7)

the place, c, because the coupling parameter is an actual quantity. It’s constructive within the repulsive NLS and adverse within the engaging NLS, which arises from the Hamiltonian,

$$start{aligned} {hat{H}}=int ({psi _x}psi _x+cpsi ^{*}psi ^{*}psi psi )dx. finish{aligned}$$

(8)

The quantum solitons are the options of the engaging NLSs. Taking a fast have a look at the answer of some issues, one can discover the stationary quantum solitons to be as the next commonplace dimensionless kind,

$$start{aligned} psi = alpha frac{e^{iS}}{cosh (beta x)}, finish{aligned}$$

(9)

through which, (1/beta) denotes the attribute size outlined in the issue. It’s price noting that the wave operate, (psi), must be normalized in house, particularly,

$$start{aligned} int _{-infty }^{+infty }|psi |^2dx=1. finish{aligned}$$

(10)

The corresponding likelihood density operate, (|psi |^2), is given by,

$$start{aligned} |psi |^2=frac{alpha ^2}{cosh ^2(beta x)}, finish{aligned}$$

(11)

plotted in Fig. 2.

As one can see in Supplementary Appendix, though the wave features may be diverse based on the given potential, its common kind stays unchanged.

Within the subsequent half, we intend to make use of the overall soliton-solutions of the NLS equation within the QPSO.

### Quantum soliton-inspired optimization algorithms

The principle technique right here is strictly the identical because the QPSO state of affairs proposed by many authors^{14,19,21,22}, however with an *M* variety of particle-like quantum soliton wave packets as a substitute of quantum particles.

In Ref.^{26} the authors have proven that, geometrically, one of the best place of the particle probably is positioned on the linear convex mixture of one of the best native and international place often called native attractor (K_{i}(t)= left( K_{i,1}(t), K_{i,2}(t), ldots , K_{i,D}(t) proper)) outlined as follows

$$start{aligned} K_{i,j}(t)= & {} lambda ~ p_{Best_{i,j}}(t) + left( 1- lambda proper) g_{Best_{j}}(t), quad i=1,2,ldots ,M, quad j=1,2,ldots ,D, nonumber lambda= & {} frac{c_{1}r_{1}}{c_{1}r_{1}+c_{2}r_{2}}. finish{aligned}$$

(12)

In keeping with this trajectory evaluation, now we have a brand new motion technique concerning one of the best native, (p_{Best_i}(t)), and international place of the entire swarm, (g_{Finest}(t)). The brand new place on this mannequin may be up to date as follows^{14,26},

$$start{aligned} x_{i}(t+1)= K_{i}(t)+L ( X_{i}(t), u), finish{aligned}$$

(13)

the place *L* denotes a displacement operate relying on the attribute size of the issue and a non-uniform distribution operate, *F*.

$$start{aligned} L=frac{1}{beta } F. finish{aligned}$$

(14)

The attribute size, representing a bodily significance, outlined as absolutely the distinction of common place of all particles’ private finest positions in swarm and the present place, that’s

$$start{aligned} frac{1}{beta }= Massive vert X_{i}(t)-frac{1}{M}sum _{i=1}^{M} p_{Best_{i}}(t)Massive vert . finish{aligned}$$

(15)

The distribution operate, *F*, which is implicitly associated to the likelihood density operate, proposes the probably place across the native attractor level and is set as (G^{-1}(u)), through which *u* is a random quantity in [0, 1], and G as a random quantity generator simulated by the likelihood density operate, is assigned to *u*,

$$start{aligned} displaystyle G(L)=frac{vert psi (L) vert ^{2} }{max (vert psi (L) vert ^{2})}:=u. finish{aligned}$$

(16)

By substituting the likelihood density operate of quantum solitons, (displaystyle |psi |^2=alpha ^2/cosh ^2(beta L)), and fixing for *L*, now we have

$$start{aligned}L=pm frac{1}{beta }cosh ^{-1}bigg (frac{1}{sqrt{u}}bigg ).finish{aligned}$$

Lastly, substituting *L* into Eq. (14), *F*(*u*) is derived as follows,

$$start{aligned} F(u)=pm cosh ^{-1}bigg (frac{1}{sqrt{u}}bigg ). finish{aligned}$$

(17)

Now, by substituting *L* in Eq. (13), the brand new place may be measured by making use of both of the next two equations:

$$start{aligned} X(t+1)= Ok(t) + frac{1}{beta }cosh ^{-1}bigg (frac{1}{sqrt{u}}bigg ), finish{aligned}$$

(a)

or

$$start{aligned} X(t+1)= Ok(t) – frac{1}{beta }cosh ^{-1}bigg (frac{1}{sqrt{u}}bigg ). finish{aligned}$$

(b)

In keeping with the quantum mechanical ideas, the quantum state (place) is undetermined till a measurement takes place. So (X(t+1)) may be up to date both by (a) or (b). Within the absence of statement, it’s certainly a superposition of each. In computation, the random operate has the identical function as that of the observer in experiments. To take action, let

$$start{aligned} X(t+1)=left{ start{array}{rl} (a) &{} if quad 0.5 leqslant upsilon leqslant 1, (b) &{} if quad 0 leqslant upsilon < 0.5, finish{array} proper. finish{aligned}$$

the place (displaystyle upsilon =rand[0,1]), then, the search radius decreases linearly by multiplying with an element *w* outlined as follows,

$$start{aligned} displaystyle w = w_{1} + frac{ (w_{0} – w_{1}) ,occasions, left( MaxIt – t proper) }{ MaxIt}, finish{aligned}$$

(18)

the place (w_0) and (w_1) are the preliminary and last values of *w* respectively. *MaxIt* is the utmost variety of iterations and *t* is the present search iteration quantity. Subsequently, Eq. (13) can rewritten as follows,

$$start{aligned} X_{i}(t+1)=left{ start{array}{rl} K_{i}(t) + w displaystyle Massive vert frac{1}{M}sum _{i=1}^{M} p_{Best_{i}}(t)- X_{i}(t)Massive vert cosh ^{-1}left( frac{1}{sqrt{u}}proper) &{} if quad upsilon geqslant 0.5 K_{i}(t) – w displaystyle Massive vert frac{1}{M} sum _{i=1}^{M} p_{Best_{i}}(t)- X_{i}(t)Massive vert cosh ^{-1}left( frac{1}{sqrt{u}}proper) &{} if quad upsilon <0.5 finish{array}. proper. finish{aligned}$$

The pseudo-code of the quantum soliton-inspired particle swarm optimization (QSPSO) algorithm is illustrated in Algorithm 2.

https://www.nature.com/articles/s41598-022-18351-0