EP4659158A1 - Digital computing device to achieve practical quantum machine learning capabilities - Google Patents

Abstract

The disclosure relates to a parallel computing method, dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs) and non-transitory computer readable media for simulating a quantum neural network (QNN). The method comprises simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF), by dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs).

Description

DIGITAL COMPUTING DEVICE TO ACHIEVE PRACTICAL QUANTUM MACHINE LEARNING CAPABILITIES TECHNICAL FIELD [0001] The present disclosure relates to a digital computing device to achieve practical quantum machine learning capabilities. BACKGROUND [0002] The upcoming sixth generation (6G) of wireless telecommunication networks (and beyond) will benefit from fully intelligent orchestration and management to ensure a manifold increase in the network performance and service types. New technologies are expected to provide these increasingly stringent performance requirements, among which quantum machine learning (QML) is considered a core 6G enabler. [0003] Consequently, a growing number of practitioners feel motivated to explore the possibility of harnessing the power of QML to provide advantages to machine learning (ML) algorithms. Attempts in this direction are currently based on the two assumptions/paradigms below: - Gate paradigm: Because of past success with digital computing devices, the gate paradigm of quantum computing (QC) has become the dominant approach in the QML community. Different approaches exist, mainly based on the use of gates and qubits. - Physical implementation: Today, QML capabilities are mainly achieved by exploiting physical quantum objects (such as atoms, molecules, photons, etc.). [0004] The use of simulations of these physical systems seems to be seldom contemplated. SUMMARY [0005] There is provided a parallel computing method for simulating a quantum neural network (QNN). The method comprises simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF), by dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs). [0006] There is provided a dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs), for simulating a quantum neural network (QNN). The dedicated hardware is operative to simulate nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF). [0007] There is provided a non-transitory computer readable media having stored thereon instructions for simulating a quantum neural network (QNN) on dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs). The instructions comprise simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF). [0008] The method and dedicated hardware provided herein present improvements to the way simulation of quantum machine learning operate. BRIEF DESCRIPTION OF THE DRAWINGS [0009] Figure 1 is a schematic illustration of an example layer of a quantum neural network based on Gaussian gates suggested in the CV paradigm of QC. [0010] Figure 2 is a block diagram of a two-module computing device proposed herein. The two modules consist of one cluster of connected Microcontroller Units (MCUs) with one cluster of Field Programmable Gate Arrays (FPGA) boards, respectively. [0011] Figure 3 is a schematic illustration of a high-level Inter-Integrated Circuit (I2C) connection scheme for a MCU cluster board. [0012] Figure 4 is a schematic illustration of a high-level Serial Peripheral Interface (SPI) connection scheme between a primary MCU (located on the MCU board) and various FPGA boards (one FPGA or a cluster of FPGAs). [0013] Figures 5a and b are graphs illustrating probability densities corresponding to different quasi-distribution functions of an electron moving in a linear potential. The times are t = 0 seconds for figure 5a and t = 40 femtoseconds for figure 5b. [0014] Figure 6 is a flowchart of a parallel computing method for simulating a quantum neural network (QNN). [0015] Figure 7 is a schematic illustration of a hardware in which steps and/or method described herein can be executed. [0016] Figure 8 is a schematic illustration of a virtualization environment in which the different steps and hardware components described herein can be deployed. DETAILED DESCRIPTION [0017] Various features will now be described with reference to the drawings to fully convey the scope of the disclosure to those skilled in the art. [0018] Sequences of actions or functions may be used within this disclosure. It should be recognized that some functions or actions, in some contexts, can be performed by specialized circuits, by program instructions being executed by one or more processors, or by a combination of both. [0019] Further, computer readable carrier or carrier wave may contain an appropriate set of computer instructions that would cause a processor to carry out the techniques described herein. [0020] The functions/actions described herein may occur out of the order noted in the sequence of actions or simultaneously. Furthermore, in some illustrations, some blocks, functions or actions may be optional and may or may not be executed; these are generally illustrated with dashed lines. [0021] Today, there are no known physical implementations of QML capable of computationally outperforming existing -classical- computing devices. Experimental efforts are continuing to improve QML, but are affected by intricated issues, including: - According to some estimates, the number of qubits needed for a useful quantum computer implementing the gate paradigm would be between 1,000 and 100,000. For purely QC algorithms, this would mean that at least 21,000 qubits would be necessary to solve certain kind of interesting problems better than any known classical computing device and this estimation should be increased in the case of QML applications. - This number is known to be much greater than the number of subatomic particles in the universe. Thus, logically speaking, it is difficult to see how such number of variables could stay under control (yet QC/QML theorists have managed to convince the general public that this is feasible). Similar arguments can be applied on suggested error correction methods which envisage to use even more physical qubits to achieve reliable logical qubits. - In order to scale properly, physical quantum implementations would need a total absence of decoherence effects. Without this stringent condition, only a reduced number of qubits can be built and utilized effectively while fragile effects such as quantum entanglement become extremely complicated to create and maintain. Even supposing that decoherence could be controlled in the future, the problem of measurements still remains as any measurement performed on a quantum system destroys its current configuration (wavefunction collapse). Further, to probe a solution embedded in the quantum system, an exponential number of measurements can be necessary. - In the physical world, continuous quantities (e.g., voltages or parameters defining quantum-mechanical wave functions) can be neither measured nor manipulated. In other words, no continuously variable quantity can be made to have an exact value (including zero). [0022] Ultimately these issues, mainly connected to the gate paradigm and physical implementations, are hindering the development of any concrete form of QML. [0023] The main objective of the solution proposed in this document is to provide practical and meaningful QML capabilities implemented on currently available digital technologies. The approach presented herein is based on the following assumptions: - First, it utilizes the continuous-variable (CV) paradigm of QC which, rather than using quantum bits (or qubits), encodes quantum information by means of observables which are continuous in nature (e.g., the average position of a particle). - Second, it does not aim to control any actual physical instances of a quantum system but, instead, performs very fast simulations of quantum systems based on the signed particle formulation (SPF) of quantum mechanics, and runs on a specifically tailored digital computing device. [0024] Therefore, the quantum systems exploited herein to achieve QML are simulated rather than being actual physical implementations of quantum objects. These simulations run on a suggested digital computing device which software and hardware are tailored around SPF for efficiency purposes. While slower than the physical evolution of a quantum system, it also provides a practical and constantly improvable way to achieve QML today. [0025] The approach disclosed herein might not allow to use of the full spectrum of quantum effects available in nature, however it is a practical way to access some of those effects and use them for computational purposes (for instance, quantum tunnelling and entanglement can be exploited in this suggested context). [0026] At least some of the following technological advantages may be achieved with the approach disclosed herein: - 6G networks could be better supported by using QML, since QML could help process the growing amounts of global information. - Simulated QML can be available today rather than having to wait for physically realized QML in the future. - QML use of different patterns compared with mainstream ML methods which can recognize statistical patterns in data and produce data that possess the same statistical patterns. Quantum mechanics is well-known to generate counter intuitive patterns in data and can recognize patterns that are difficult to recognize classically (i.e., by means of mainstream ML models). - The hardware design proposed herein, along with the parallelization scheme of the SPF, can be improved (in terms of execution speed) by adding more MCUs and/or FPGAs. - The machine suggested herein has a low power consumption. Compared with physical quantum systems which require expensive (and cumbersome) cryogenic facilities to avoid quantum decoherence. - The approached disclosed herein provides for controlled noise, which can be advantageous in certain situations. Having noise as a variable can provide an advantageous degree of freedom [0027] Some of the theoretical foundations are now summarized for more clarity: - Continuous-variable paradigm: Among the various QC paradigms available, the CV is the most natural for real numbers. The CV approach is based on the use of continuous observables (e.g., the position, the momentum of a particle, etc.) and, consequently, enables the treatment of problems which utilize continuous variables (e.g., the weights of an artificial neural network). This represents a relevant departure from the mainstream QML approach which is mostly based on the use of qubits. - Signed Particle Formulation: Different formulations of quantum mechanics exist today, among which the most well-known is represented by the work proposed by E. Schrödinger in 1926 in “An undulatory theory of the mechanics of atoms and molecules”, E. Schrödinger, Phys. Rev., Vol.28, No.6, (1926). In practice, this is the standard approach in quantum mechanics today. The SPF is a rather recent theory of quantum mechanics introduced by the inventor in 2015 in “A signed particle formulation of non-relativistic quantum mechanics”, J.M. Sellier, Journal of Computational Physics, Vol.297, pp.254-265, (2015). While this approach provides the same predictions as Schrodinger’s equation, its main advantage consists in the fact that it is highly parallelizable (which is not the case for the Schrödinger approach). Simulations of quantum systems performed by means of the SPF consists of two steps: 1) an ensemble of classical signed particles is constantly evolved in time and 2) a set of integrals are computed. Both these tasks are known to be highly parallelizable since they mainly consist of independent computations which do not require communication between the computing nodes. This represents an important advantage in terms of simulation speed and of needed computational resources. - Dedicated hardware: Rather than being based on the use of a physical system, simulations are performed which follow the exact same dynamics. This provides a way to immediately access to (simulated) quantum states which would be hardly accessible experimentally. To obtain the stringent performances required for such simulations, and consequently enable practical QML capabilities, the SPF runs on a dedicated -highly parallel- computing device. This work introduces a tailored computing device based on the combination of the use of clusters of (Cortex-M) microcontrollers (MCUs) -on which signed particles are simulated in parallel- and field programmable gate arrays (FPGAs) -on which various integrals are computed in parallel. This device can provide the level of parallelization required for efficient SPF simulations and, therefore, to enable practical QML capabilities. [0028] QML in the CV paradigm [0029] As discussed previously, due to the many constraints affecting qubits, the hardware quantum architecture may not be the most effective quantum framework for encoding NNs. The CV paradigm carries information in the continuous quantum states of a system, rather than on discrete qubits, which makes it a better fit in the context of computations underlying NNs. This is usually obtained by means of Bosonic particles, although Fermionic ones can be utilized as well. [0030] More specifically, continuous-variable quantum information can be encoded using different representations, for instance wavefunctions (defined over a configuration space) or quasi-distribution functions (defined over a phase-space) which are mathematically equivalent as both approaches provide the same predictions. Herein, the phase-space representation based on the SPF (described below) is utilized. In the phase-space representation, the conjugate variables x and p (with x the position and p the momentum of a particle) are treated on equal footing and quantum states are represented as real-valued functions ^^^^ = ∈ ℝ called a quasi-distribution function (“quasi” because they can have negative values as well), which is defined over the phase-space ⁽ ^^^^, ^^^^⁾ ∈ ^^^^ ⊂ ℝ² for some bidimensional domain D. [0031] In the CV paradigm, gates can be classified in two families, i.e., Gaussian and non-Gaussian. The first type of gates act on one or two quantum states. The simplest one-state Gaussian gates in use are: the rotation ^^^^( ^^^^), the displacement ^^^^( ^^^^) and the squeezing ^^^^⁽ ^^^^⁾ gates. The simplest two-state Gaussian gate is represented by the beamsplitter ^^^^ ^^^^⁽ ^^^^⁾ gate, which can be seen as the rotation between two quantum states. Mathematically, these gates read: - _{Rotation ^^^^} ⁽ _^^^^ ⁾ _:� ^{^^^^} ^_{^^^� →�} ^{cos ^^^^ ^^^^} − _{sin ^^^^ �, with ^^^^ ∈} ^[ _{0,2 ^^^^} ^] _{^^^^ . - Displacement the real and} imaginary part of α respectively. - _Squeezing - ^^^^ ∈ [0,2 ^^^^]. [0032] Concerning the non-Gaussian gates, they can be provided by any gate which acts non-linearly on the quantum information being processed (for instance, it could be a gate which exploits external noise to transform -non-linearly- a given quantum state). [0033] Mathematically speaking, a gate is represented by a potential function ^^^^ = ^^^^⁽ ^^^^⁾ which is applied to the electronic quasi-distribution encoding the (quantum) information to be processed. In this context, a quantum neural network can be considered as a sequence of gates applied to quantum states, such as the one shown in Fig.1, for example (which shows one layer of a quantum NN of the prior art). [0034] The CV paradigm comes with its own notion of universality, which is considered as the ability to approximate any arbitrary transformations of the form ^^^^ _^^^^ = with i the imaginary unit, t the time variable and ^^^^ = ^^^^⁽ ^^�^^, ^^̂^^⁾ (a polynomial function of ( ^^�^^, ^^̂^^) with arbitrary but fixed degree), where ^^�^^ and ^^̂^^ are the position and momentum operators respectively. Consequently, given an input state (x₁, p₁) and an output state (x₂, p₂), one can mathematically prove that a sequence of Gaussian and non-Gaussian gates always exists which transform the input state into the output state. [0035] The signed particle formulation of quantum mechanics [0036] The signed particle formulation’s numerical discretization, known as the Wigner Monte Carlo method, is a phase-space formulation and, therefore, utilizes the concept of quasi-distribution functions (of signed particles). This novel formalism has been shown to be uncommonly advantageous in terms of needed computational resources and of parallelization scheme efficiency, allowing time-dependent simulations of quantum many-body systems on relatively small machines in both the density functional theory and first-principles frameworks, as well as of systems of indistinguishable Fermions [0037] Postulates. Three postulates (or rules) which completely define the SPF are now introduced. For the sake of simplicity, only the case of a single electron in a one- dimensional configuration space is addressed herein. In this formulation, a signed particle is a mathematical object which phase-space coordinates are represented by the couple (x,p). Hereinbelow, the terms “Newtonian” and “classical” are used as synonyms, moreover “signed” particles are not physical particles but only virtual ones. - Postulate I. Physical systems can be described by means of ensembles of (virtual) Newtonian particles, i.e., provided with a position x and a momentum p simultaneously, and which carry a sign that can be positive or negative. - Postulate II. A signed particle, evolving in a given potential ^^^^ = ^^^^( ^^^^), behaves as a field-less classical point-particle which, during the time interval dt, creates a new pair of signed particles with a probability ^^^^� ^^^^⁽ ^^^^⁾� ^^^^ ^^^^ where: +^∞ l^′im⁺ � ^^^^ _^^ ⁺ ^_^( ^^^^; ^^^^∆ ^^^^^′) _{^^^^ →0} ^^^^=−∞ and where ^^^^ _^^ ⁺ ^_^( ^^^^; ^^^^) is the positive part of the following quantity: (known as the Wigner kernel). If, at the moment of creation, the parent particle has sign s, position x and momentum p, then the new particles are both located in - Postulate III. Two particles with opposite sign and same phase-space coordinates ( ^^^^, ^^^^) annihilate (i.e., they cancel each other, and they both can be removed from the simulation). [0038] It is possible to see this formulation as made of two parts: 1) one which deals with updating the signed particles and 2) one which deals with the computation of the various integrals needed (i.e., the two functions ^^^^ _^^^^ = ^^^^ _^^^^( ^^^^; ^^^^) and ^^^^ = ^^^^( ^^^^)). [0039] Below are presented several aspects of the implementation of the SPF. [0040] Initial conditions. In practice, one starts from an ensemble of signed particles distributed in the phase-space according to some specified initial quasi-distribution function corresponding to the quantum state of a given electron (i.e., the experimental conditions), and which may be obtained from some wave-function or density matrix for example. To exemplify this point, an extract from the C code implemented in this work is presented next. int i,j; int m,n; int NUM; int MAX; int number_of_particles_in_cell; // NUM stores the total number of signed particles involved NUM=0; // total number of signed particles allowed in a (phase- space) cell MAX=16; // loop over the cells in spatial domain for(i=1;i<=NX;i++){ // loop over the cells in the space of momenta for(j=0;j<2*NKX-1;j++){ // number of particles in (i,j)-th cell number_of_particles_in_cell=(int)(fabs(FW1[i][j]*MAX)); for(n=1;n<=number_of_particles_in_cell;n++){ // global index of the n-th particle m=NUM+n-1; // position of the particle if(rnd()>0.5) P[m]=(i-0.5+0.5*rnd())*DX; else P[m]=(i-0.5-0.5*rnd())*DX; // momentum of the particle K[m]=j-NKX+1; // sign of the particle if(FW1[i][j]>0) S[m]=+1; else S[m]=-1; } // update the total number of signed particles NUM+=number_of_particles_in_cell; } } [0041] In the previous code, an ensemble of signed particles is stored as a set of three one-dimensional arrays P[.], K[.] and S[.] which represent the position, the momentum (in the shape of an integer in a discretized momentum space) and the sign respectively, and where the two-dimensional array FW[.][.] is the quasi-distribution function describing the initial conditions of the system. [0042] Consequently, in this specific setting, the i-th signed particle has: - position P[i], - momentum K[i]*DKX, (with DKX is the length of the cell in the momentum space), - sign S[i], and the quasi-distribution function evaluated on the l-th spatial cell and m-th momentum cell (on a point in the discretized phase-space) is represented by the value FW[l][m]. [0043] Thus, the code provided previously can be explained as follows: 1) the maximum number of signed particles in a cell is fixed (in the variable MAX), 2) initial conditions are provided in the shape of a (normalized) quasi-distribution function FW[.][.], 3) for every cell in the discretized phase-space: 3.1) the local number of particles in the cell is computed, (in number_of_particles_in_cell), 3.2) a random position, a momentum, and a sign are assigned to every particle in the cell, 4) the total number of particles involved in the simulation is updated. [0044] Further practical details to evolve an ensemble of signed particles can now be introduced. [0045] Time-dependent evolution of the signed particles. Given a signed particle at time t with sign s, mass m, and (phase-space) coordinates (x, p), indicated as (s, m; x, p), the operator ^^̂^^ is introduced which, given one signed particle represented as (s, m; x, p), constructs a new set of three particles in the following way: - At time t, a random number ^^^^ ∈ ^[0,1^] is generated and the quantity ^^^^ ^^^^ = - At time ^^^^ + ^^^^ ^^^^, the initial particle evolves as field-less (i.e., such as ^^^^( ^^^^) = 0 f_{or any x in the spatial domain) and has new coordinates} - A pair of new signed particles is created at time ^^^^ + ^^^^ ^^^^, where the particle with the sign +s has coordinates ( ^^^^₂, ^^^^₂) ^′ ^^^^ + ^^^^� and the particle with the sign -s has coordinates ⁽ ^^^^ , ^^^^₃ ⁾ and the quantity p’ c_{omputed from the normalized probability} ^{^^^^ ^^^^( ^^^^; ^^^^)} _{� ^^^^( ^^^^) .} [0046] To exemplify this, some code is provided next which shows how signed particles are evolved and how new couples of signed particles are consequently created. // evolution and couple creation for(n=0;n<NUM;n++){ // ... if(UPDATED[n]==NO){ hmt=HBAR/M*DT; // drift n-th particle x0=P[n]; k0=K[n]*DKX; i=(int)(x0/DX)+1; // evolve position and wave vector of the n-th particle if((i>0 && i<=NX) && (-NKX<K[n] && K[n]<NKX)){ P[n]=x0+hmt*k0; // compute the probability that the wave-vector evolves // check if a couple of (+,-) have to be created if(GAMMA[i]!=0.){ for(time=0.;time<DT;){ rdt=-log(rnd())/GAMMA[i]; time+=rdt; if(time<DT){ created=NO; r=rnd(); sum=0.; // random selection of the wave-vector for(j=0;((created==NO) && (j<NKX));j++){ p=fabs(VW[i][j])/GAMMA[i]; if((sum<=r) && (r<(sum+p))){ number_of_created_particles+=2; num=INUM+number_of_created_particles; // select a random time interval when the creation happens // assign position P[num-2]=P[num-1]=x0+HBAR/(MSTAR*M)*time*k0; // assign wave-vector if(VW[i][j]>=0.){ K[num-2]=K[n]+j; K[num-1]=K[n]-j; } else { K[num-2]=K[n]-j; K[num-1]=K[n]+j; } // assign the sign if(W[n]==1){ W[num-2]=+1; W[num-1]=-1; } else { W[num-2]=-1; W[num-1]=+1; } // assign flag to evolve the particles at the next loop UPDATED[num-2]=UPDATED[num-1]=NO; // ... created=YES; } sum+=p; } } } } } UPDATED[n]=YES; } } In the previous code: - DT is the time step, - GAMMA[i] is the value ^^^^( ^^^^ _^^^^), - VW[i][j] is the Wigner kernel ^^^^ _^^^^� ^^^^ _^^^^ , ^^^^ _^^^^�, - NX is the number of cells in the discretized spatial domain, - NKX is the number of cells in the discretized momentum domain, - DX is the length of the spatial cell, - DKX is the length of the cell in the momentum space, - M is the mass of an electron, and - HBAR is the reduced Planck constant. [0047] The following explains how the predictions are computed within this theoretical framework. [0048] Predictions and macroscopic variables. Predictions, in the context of SPF, are obtained by averaging microscopic variables (i.e., variables directly describing one or more facet of a signed particle, e.g., its position) which, in turn, provides the value of macroscopic variables (e.g., the position of an electron). In more mathematical details, a macroscopic variable defined over the phase-space A = A(x; p) is computed as the average of this value for every single signed particle, taking into account their sign, i.e.: where s_i, x_i and p_i are the sign, the position and the momentum, respectively, of the i- th signed particle. An example is provided by the probability density which corresponds to A(x; p) = 1. [0049] Using the theoretical framework introduced so far, it is now possible to provide an adapted definition of quantum neural network (QNN). [0050] Quantum neural networks [0051] QNNs may take many forms and shapes, like common ANNs, but herein a layered QNN having an input layer, one or more hidden layers and an output layer is considered, as it represents a simple starting point to introduce the new concepts presented herein. [0052] Information representation. On QNNs, information is encoded through quasi-distributions of electrons. In practice, a complex number c is encoded in the macroscopic averages of the position ^̅^^^ and momentum ^̅^^^ of an electron so that the position corresponds to the real part of the complex number (i.e., ^^^^ = ^^^^ ^^^^( ^^^^)) and the momentum corresponds its imaginary part (i.e., ^^^^ = ^^^^ ^^^^⁽ ^^^^⁾, i.e., ^^^^ = ^̅^^^ + ^^^^ ^̅^^^, where: with the spatial domain being the interval [0,L] and the function ^^^ _^^_^^^ = ^^^ _^^_^^^( ^^^^; ^^^^; ^^^^) being a quasi-distribution function corresponding to some given electron (note that a real number is a number which imaginary part is equal to 0, and, for convenience, the numbers x and p could be represented in arbitrary units). Consequently, a QNN can be seen as a series of (Gaussian and/or non-Gaussian) gates which are applied on the quasi-distribution to process the information encoded in the layers. [0053] It should be noted that the quantum systems selected to store quantum information can be limited to one-dimensional ones, while the various gates necessary to process that information can be one-body or two-body gates. This greatly simplifies the complexity of the simulations required to achieve practical QML, since it avoids the need for multi-dimensional, many-body simulations which, in turn, would be a daunting task. [0054] Input layer. The input layer consists of a signal made of N components/numbers which enter through the first layer of the network (just like in common ANNs), expressed as N complex numbers, say ( ^^^^₁, ^^^^₂, … , ^^^^ _^^^^) ∈ ℂ ^{^^^^}. In turn, these numbers are internally encoded into N quasi-distribution functions with the corresponding average positions and momenta (see above). The quasi-distribution functions are, then, migrated to the next (hidden) layer by providing them as initial conditions (i.e., a quantum state) to be transformed by a given gate (just like an activation function coming from the hidden layer). [0055] Hidden layer(s). In the same way activation functions would be applied to the information coming from the previous layer in ANNs, the hidden layers of a QNN can be seen as a series of Gaussian and/or non-Gaussian gates applied to the electronic quasi-distribution functions encoding the information to be processed. Every gate comes with one or more hyper-parameters that needs to be tuned to perform the transformation needed, in other words they represent the weights of the network just as in mainstream ANNs. [0056] Note that although one object is to obtain QML, because the actual computations are performed in a simulated environment, it is possible to introduce gates which would not be possible to implement in an actual physical system. In fact, it is possible to define and utilize gates which are not necessarily unitary (i.e., a surjective bounded operator on a Hilbert space that preserves the inner product). This represents an advantage over the use of physical systems since it provides a further (relevant and influent) degree of freedom in the choice of internal operations achievable in the network layers. For instance, one could introduce and use gates which purposes are simply to copy a quantum state from one layer to the next one. [0057] Output layer. Finally, once processed by the previous hidden layers, the information arrives at the output layer still in the shape of electronic quasi-distribution functions. Supposing that M quasi-distribution functions are transferred to the output layer, the output will consist of the averages extractable from such distributions, i.e.: where the function ^{^^^^} = ^^^ _^^_^^^ ⁽ ^^^^; ^^^^⁾ is the quasi-distribution of the i-th electron, with i=1,…,M. Therefore, the output of the network is represented by the following _numbers [0058] This concludes the part describing QNNs. The description of the computing machinery, and how the software is implemented on it to obtain practical QML capabilities is described next. [0059] Example hardware implementation [0060] From a purely hardware perspective, the QML machine 200 described herein be considered as the interaction of two modules, one dedicated to the evolution of ensembles of signed particles (one or more clusters 205 of MCUs 215) and one dedicated to the computation of integrals (one or more clusters 210 of FPGA boards 220). Every MCU cluster 205 is connected to one cluster of FPGAs 210 through its primary node (which, in turn, becomes the primary node of the network consisting of the primary MCU and the connected FPGA boards). Every communication between the primary MCU and a cluster of FPGA boards is made through the SPI protocol while communication on the cluster of MCUs is provided through the I2C protocol. Optionally, the primary MCU node of a cluster of MCUs can be connected to a computer (desktop or laptop) through the Universal Asynchronous Receiver- Transmitter (UART)/ Universal Synchronous/Asynchronous Receiver-Transmitter USART protocol. [0061] For clarity purposes, Fig.2 graphically presents an example device 200 at a high level while Fig.3 shows more details on the connections between the MCUs (I2C) 215. Fig.3 also shows a power supply 305 and a connection to an external computer 310. Fig.4 shows the SPI connection scheme between the primary MCU 215’ of an MCU cluster 205 and various FPGAs 220 (acting as digital integrators). [0062] Example software implementation [0063] The evolution of quantum systems in the signed particle formulation is represented by two independent steps which are repeated in a loop until a final state if reached. These two steps consist of 1) the evolution of the signed particles and 2) the computation of two integrals to obtain the Wigner kernel V_W=V_W(x;p) and the function γ= γ(x). These two steps can be parallelized, with great computational benefits, as follows: - evolution of the signed particles. The signed particles in use in the SPF are independent, Newtonian, field-less particles. During a simulation, the code needs to update the position and momentum of every signed particle and, eventually, create new pairs of signed particles (if the right conditions are reached, see the postulates above). From a computational perspective, this is equivalent to say that every single signed particle can be evolved on its own and if subsets of particles are sent to a certain number of computing devices, those devices will not need to communicate with each other for that purpose. In practice, the evolution of the signed particles is a highly parallelizable process. Therefore, the execution time for this process grows only linearly with the number of signed particles involved, i.e., it scales very well. - computation of the integrals. It is well known that, from a purely numerical perspective, integrals can be computed as the sum of different terms. In this context, the integral of a function ^^^^ = ^^^^( ^^^^) over the domain [0, L] becomes: ^{provided by the} user. In the specific case at hand, the dimension of both integrals can be fixed, with a value of 1 for the function γ=γ(x) and a value of 2 for the Wigner kernel VW=VW(x; p). Therefore, the formula introduced to compute these integrals can work properly and accurately since the dimensionality is very small. Moreover, such task can be parallelized since it can be computed in independent chunks. [0064] The two components constituting the computing hardware and tailored around SPF to achieve relatively fast simulations of quantum systems, are discussed next. [0065] The cluster of microcontrollers [0066] As mentioned previously, the machine suggested herein consists of two main components. Hereinbelow, the MCU cluster 205 is presented. [0067] The MCU cluster is a parallel computing device which is constituted of Cortex-M7 MCUs connected with each other in a primary-secondary fashion, through the I2C communication protocol (thus the MCUs are hard wired with each other). A standard software framework, light and small in dimensions, is implemented to handle the communications between the secondary and the primary nodes, along with the computations to be performed. Consequently, this component of the machine does not necessitate any operating system and/or communication libraries. [0068] The choice of the Cortex-M family is not a restriction and other types of MCUs could be utilized as well. The same applies for the I2C communication protocol (the SPI protocol could equivalently be used). The reason for selecting this specific type of connection/protocol is twofold: - on one hand, it does not require the implementation of any software (as being directly embedded in the hardware of the MCUs), - on the other hand, it requires less wiring than other communication protocols (for instance, SPI requires 4 wires between 2 MCUs to work properly, while I2C requires only 3 wires) while still providing fast enough communication capabilities. [0069] Finally, the connection between the primary node of the prototype and a personal computer is easily achieved by connecting the primary MCU 215’ of the board to the USB port of the PC 310 and using the UART/USART protocol. [0070] To validate the approach suggested in this work, an example implementation was made using 12 MCUs (1 primary and 11 secondary, this number is arbitrary and other machines with a different number -larger or smaller- of nodes can be readily obtained by using the same techniques). More specifically, 12 Cortex-M7 embedded on development boards known as Teensy 4.0 have been arranged on 1 printed circuit board. Therefore, a cluster of MCUs consisted of 1 board containing 12 MCUs, where one MCU was utilized as the primary while the others were used as computing nodes (in other words, the secondary nodes). The communication bus consisted of simple wires with pull-up resistors (when needed to strengthen the signal over the bus). Only one power supply was necessary which consisted of a converter from 110 Volt to 12 Volt (0.5 Ampere in output) which was, in turn, transformed into 5 Volt. [0071] The evolution of the signed particles was performed by spreading the ensemble of particles over all available computing (secondary) MCUs which, then, performed this task independently. [0072] The FPGA-based integrator [0073] A field-programmable gate array (FPGA) is an integrated circuit designed to be configured/programmed by a user after manufacturing. A typical FPGA contains an array of programmable logic blocks, and a hierarchy of reconfigurable interconnections which allow blocks to be wired together. Those logic blocks also include memory elements, which can range from simple flip-flops to more complete blocks of memory. Thus, FPGAs can be configured/programmed to perform complex functions, allowing flexible reconfigurable computing as performed in computer software. In the example implementation, an ARTY S7-50 board with a Spartan-7 FPGA was programmed by using the Verilog Hardware Description Language (HDL). [0074] It should be noted that FPGAs come with the advantage of providing significantly fast computational capabilities, compared with other computing devices, because of their truly parallel nature and optimality in terms of the number of gates used for computational purposes (as a matter of fact, an average FPGA can go beyond 800 times faster than an average CPU). Therefore, FPGAs are very suitable devices for tasks such as computing discretized integrals since, in turn, this can be efficiently performed in parallel. [0075] Performing a numerical integration in a system as the one proposed herein, consists in computing a sum which is made of various terms computed in parallel on the FPGAs. These terms can be chosen arbitrarily, and, in this work, 10 terms were selected in the following way (the following formula does not represent a restriction for the method suggested in this work): where the terms Tj are computed contemporaneously in parallel on the FPGA. [0076] This summation is performed, within each FPGA, with the following steps: 1. A series of real numbers is first shared with an FPGA, which need to be summed to obtain the final integral, e.g., the numbers f(x1), f(x2), …, f(xN), and the cell length Δx. In this specific case, the primary MCU sends the series of numbers to the FPGA through the SPI communication protocol. 2. While receiving this series of numbers, the FPGA stores them in memory. 3. The terms Tj, for j=1, …, 10, are computed in parallel by using Verilog non- blocking commands. Once these terms are computed, they are summed to obtain the final value for the numerical integral at hand. 4. The points above are organized in the shape of a state machine. By default, at the beginning, the state machine is in the “idle” state. This continues for every clock cycle until information is sent to the FPGA. A “go” bit is set to 1 once the transmission of the data is completed. Then the actual computation is triggered, and the state machine goes to the state of “computing”. Finally, when the (parallel) computation of the integral is performed, the value is stored in memory and the state of the machine switches to “completed”. When requested, the FPGA sends the result back to the primary MCU by means of SPI (state “sending”). Once this happens, the state machine switches back to “idle”. At that point, the FPGA waits until a new series of numbers is provided. [0077] The same process can be generalized for a cluster of FPGAs where each FPGA computes one terms T_j of the integral provided previously and every term is computed, in turn, as a sum of terms computed in parallel. Thus, the specific implementation described in this document does not represent a limitation for the method suggested. [0078] A validation test was done for the QML computing machinery proposed in this work. [0079] Applications [0080] Hereinbelow, the results of simulations performed on a prototype, for validation purposes, is presented. An archetypal quantum system has been chosen for this purpose which consists of one electron which moves in a linear potential miming the presence of a displacement gate applied to the position of the electron. In the CV paradigm, this represents one of the fundamental operations which are utilized to obtain QNNs. The validation test clearly shows that the computing device proposed in this work can perform the operations required to achieve practical QML. The example prototype machine which was used for such test consisted of one MCU board made of 12 parallel Cortex-M7 MCUs and one FPGA board with one Spartan-7 chip. [0081] Displacement [0082] This test consisted of one electron moving in a finite one-dimensional domain of total length L_X equal to 200 nanometers. The initial conditions for the electron were represented by an average position x0 = 68.5 nanometers, and momentum p0 = 6π⁄50 nanometers^-1. A linear potential, mathematically expressed as with V_P = -0.3 Volt, was applied to the electron for 40 femtoseconds. To show that relatively high execution speeds can be obtained by the approach suggested in this work, a set of different hyperparameters was selected and the corresponding execution times are reported accordingly (see table 1 below). In particular, the hyperparameters that have been tuned for fast execution times, consisted of: - The number of cells in the discretized one-dimensional domain (NX): it ranged from a maximum of 200 cells to a minimum of 50 cells. From the table, it can clearly be seen that, to keep the accuracy of the simulation high enough (i.e., to keep the prediction ^̅^^^ accurate enough), it was not convenient to use a resolution lower than a hundred cells. - The initial number of signed particles representing the electron (NP): it ranged from a maximum of 1,000,000 to a minimum of 100,000. From the table below, it is clear that the accuracy of the simulation can be kept even with a number of signed particles as low as a 100,000. - Total number of time steps and the time step (DT): in every simulation performed, the total number of time steps and the step itself were chosen so that they always corresponded to a final time equal to 40 fs. The time step ranged from a maximum of 0.32 fs to a minimum of 0.01 fs. Consequently, the total number of time steps ranged from a maximum of 4,000 to a minimum of 125. - Annihilation frequency: in order to keep under control, the number of signed particles involved in a simulation, an annihilation process was applied at regular intervals. It ranged from a maximum of one annihilation every 100 iterations down to the case of no annihilation applied at all. [0083] Typical plots of the results, corresponding to the first row of the table below, are presented in Fig.5a and b, for the times t = 0 seconds (5a) and t = 40 femtoseconds (5b). The graphs show the probability densities. [0084] The table below shows the value obtained for the average position of the electron after 40 fs (last right-hand side column) for every set of hyperparameters. Moreover, the corresponding execution time is reported in the second last right-hand side column. The execution time for the simulation of a displacement gate applied to the information encoded in the quasi-distribution function of an electron can be drastically reduced while keeping a good accuracy. [0085] The computational times reported below are measured on a rather simple and limited prototype, and improvements are possible. For instance, the proposed computing device can be improved by adding more MCUs and FPGAs to work in parallel since the mathematical algorithms necessary to make the system run do not require any modifications. This clearly shows that QML capabilities are at reach by using the machine suggested in this work. NX NP # of DT Annihilation Exec. ^�^^^ iterations freq. time 200 1,000,008 4000 0.01 fs 100 0.977 sec 94.55 nm 100 1,002,493 2000 0.02 fs 100 0.590 sec 94.55 nm 50 979,773 1000 0.04 fs 100 0.272 sec 88.93 nm 200 499,952 4000 0.01 fs 100 0.509 sec 94.55 nm 100 501,253 2000 0.02 fs 100 0.386 sec 94.55 nm 100 250,608 2000 0.02 fs 100 0.136 sec 94.55 nm 100 125,302 2000 0.02 fs 100 0.090 sec 94.55 nm 100 125,302 2000 0.02 fs 1000 0.090 sec 94.55 nm 100 125,302 2000 0.02 fs none 0.068 sec 94.55 nm 100 125,302 1000 0.04 fs none 0.045 sec 94.55 nm 100 125,302 500 0.08 fs none 0.022 sec 94.55 nm 100 100,239 125 0.32 fs none 0.003 sec 94.55 nm Table 1 – execution results [0086] The worst and best results are in bold character in the table above. [0087] Turning to figure 6, there is provided a parallel computing method 600 for simulating a quantum neural network (QNN). The method comprises simulating, step 605, nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF), by dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs). [0088] The virtual gates may each be represented by a potential function ^^^^ = ^^^^⁽ ^^^^⁾ and the signed particles may each be provided with a position x, a momentum p and a sign s that can be positive or negative. [0089] The signed particles may represent, or may be used to represent, fermions. The signed particles may more specifically represent, or be used to represent, electrons. [0090] Evolving the signed particles may further comprise, for each signed particle, step 610: - computing a new pair of signed particles by computing two integrals, for computing a Wigner kernel ^^^^ _^^^^( ^^^^; ^^^^), and a gamma function ^^^^( ^^^^), the new pair of signed particles having a probability of ^^^^� ^^^^( ^^^^)� ^^^^ ^^^^, and - updating the position of the signed particle according to the momentum of the signed particle. [0091] The two integrals may be defined as: +^∞ l ^{+ ′}i →m 0⁺ � V_W(x; M∆p^′) M=−∞ a^nd where: D^{p' indicates to compute VW using a momentum integral,} V_{+ W(x; p) is the positive part of VW(x; p), p' is a normalized probability computed with} ^{^^^^ ^^^^( ^^^^; ^^^^)} _{� ^^^^(} M is the mass of an electron, ∆p^′ represents a discretization in the momentum space, ℏ is the reduced Planck constant, i is the imaginary unit, and x' is a variable used for integration. [0092] The evolved signed particles may be obtained by averaging positions, momentums and signs of new pairs of signed particles that are not cancelled, cancelled signed particles being particles that have opposite signs and same phase- space coordinates (x, p). Prior to evolving, the signed particles may be distributed in a phase-space according to an initial quasi-distribution function. [0093] The MCUs may simulate the evolution of the signed particles. The FPGAs may compute the integrals. Each MCU of the plurality of MCUs may be connected to a plurality of FPGAs. [0094] The parallel computing method may further comprise any of the steps described herein. [0095] It should be noted that method and steps described herein are, generally, computer implemented. The term computer may be interpreted as having different meanings, such as explained next, for example. [0096] Referring to figure 7, there is provided an alternate apparatus (HW) 701, in which some functions and steps described herein may be implemented. The preferred implementation is made in the dedicated hardware presented previously, but a person skilled in the art would understand that other hardware could implement the method, such apparatus 701 or virtualization environment 800 described further below. [0097] The apparatus 701 (which may go beyond what is illustrated in figure 7), may be a user device, a server, network node, or other computing device which may be part of a cloud computing system, edge computing system, or which may be a standalone device. [0098] The apparatus 701 comprises processing circuitry 703 and memory 705. The memory 705 can contain instructions executable by the processing circuitry 703 whereby functions and steps described herein may be executed to provide any of the relevant features and benefits disclosed herein. [0099] The apparatus 701 may also include non-transitory, persistent, machine- readable storage media 707 having stored therein software and/or instruction 709 executable by the processing circuitry 703 to execute functions and steps described herein. The apparatus may also include network interface(s) and a power source. [00100] The instructions 709 may include a computer program for configuring the processing circuitry 703. The computer program may be stored in a physical memory local to the device, which can be removable, or it could alternatively, or in part, be stored in the cloud. The computer program may also be embodied in a carrier such as an electronic signal, optical signal, radio signal, or computer readable storage medium. [00101] Referring to figure 8, there is provided a virtualization environment 800 in which functions and steps described herein can be implemented. [00102] The virtualization environment 800 (which may go beyond what is illustrated in figure 8), may comprise systems, networks, servers, nodes, devices, etc., that are in communication with each other either through wire or wirelessly, e.g. through a network interface component (NIC) comprising physical network interface(s). Some or all of the functions and steps described herein may be implemented as one or more virtual components (e.g., via one or more applications, components, functions, virtual machines, containers, etc.) executing on one or more physical apparatus in one or more networks, systems, environment, etc. [00103] A virtualization environment provides hardware 801 comprising processing circuitry 803 and memory 805. The memory 805 can contain instructions executable by the processing circuitry 803 whereby functions and steps described herein may be executed to provide any of the relevant features and benefits disclosed herein. [00104] The hardware 801 may also include non-transitory, persistent, machine-readable storage media 807 having stored therein software and/or instruction 809 executable by the processing circuitry 803 to execute functions and steps described herein. [00105] The instructions 809 may include a computer program for configuring the processing circuitry 803. The computer program may be stored in a removable memory, such as a portable compact disc, portable digital video disc, or other removable media. The computer program may be stored in a physical memory local to the hardware 801, which can be removable, or it could alternatively, or in part, be stored in the cloud. The computer program may also be embodied in a carrier such as an electronic signal, optical signal, radio signal, or computer readable storage medium. [00106] Referring again to figures 2-4, there is provided a dedicated hardware 200 composed of a plurality of Microcontroller Units (MCUs) 215 and Field Programmable Gate Arrays (FPGAs) 220 for simulating a quantum neural network (QNN). The dedicated hardware is operative to simulate nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF). [00107] The virtual gates may each be represented by a potential function ^^^^ = ^^^^( ^^^^) and the signed particles may each be provided with a position x, a momentum p and a sign s that can be positive or negative. The signed particles may represent fermions. The signed particles may more specifically represent electrons. [00108] Evolving the signed particles may further comprise, for each signed particle: - computing a new pair of signed particles by computing two integrals, for c_{omputing a Wigner kernel ^^^^ ^^^^} ⁽ _{^^^^; ^^^^} ⁾ _{, and a gamma function ^^^^} ⁽ _^^^^ ⁾ _{, the new pair of signed particles having a probability of ^^^^� ^^^^} ⁽ _^^^^ ⁾ _{� ^^^^ ^^^^, and} - updating the position of the signed particle according to the momentum of the signed particle. [00109] The two integrals may be defined as: +^∞ l ^{+ ′ ′}im⁺� V_W(x; M∆p ) _→0 M=−∞ a^nd where D^{p' indicates to compute VW using a momentum integral,} V_{+ W(x; p) is the positive part of VW(x; p), p' is a normalized probability computed with} ^{^^^^ ^^^^( ^^^^; ^^^^)} _� M is the mass of an electron, ∆p^′ represents a discretization in the momentum space, ℏ is the reduced Planck constant, i is the imaginary unit, and x' is a variable used for integration. [00110] The evolved signed particles may be obtained by averaging positions, momentums and signs of new pairs of signed particles that are not cancelled, cancelled signed particles being particles that have opposite signs and same phase- space coordinates (x, p). [00111] Prior to evolving, the signed particles may be distributed in a phase- space according to an initial quasi-distribution function. [00112] The MCUs may simulate the evolution of the signed particles. The FPGAs may compute the integrals. Each MCU of the plurality of MCUs may be connected to a plurality of FPGAs. [00113] The dedicated hardware is further operative to execute any of the steps described herein. [00114] Referring to figures 7 and 8, there is provided a non-transitory computer readable media 707, 807 having stored thereon instructions 709, 809 for simulating a quantum neural network (QNN) on dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs). The instructions comprise simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF). The non-transitory computer readable media may further store instructions to execute any of the steps described herein. [00115] Modifications will come to mind to one skilled in the art having the benefit of the teachings presented in the foregoing description and the associated drawings. Therefore, it is to be understood that modifications, such as specific forms other than those described above, are intended to be included within the scope of this disclosure. The previous description is merely illustrative and should not be considered restrictive in any way. The scope sought is given by the appended claims, rather than the preceding description, and all variations and equivalents that fall within the range of the claims are intended to be embraced therein. Although specific terms may be employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.

Claims

CLAIMS 1. A parallel computing method for simulating a quantum neural network (QNN), comprising simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF), by dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs).

2. The method of claim 1, wherein the virtual gates are each represented by a potential function ^^^^ = ^^^^⁽ ^^^^⁾ and wherein the signed particles are each provided with a position x, a momentum p and a sign s that can be positive or negative.

3. The method of claim 1, wherein the signed particles represent fermions.

4. The method of claim 3, wherein the signed particles represent electrons.

5. The method of claim 2, wherein evolving the signed particles further comprises, for each signed particle: - computing a new pair of signed particles by computing two integrals, for computing a Wigner kernel ^^^^ _^^^^( ^^^^; ^^^^), and a gamma function ^^^^( ^^^^), the new pair of signed particles having a probability of ^^^^� ^^^^( ^^^^)� ^^^^ ^^^^, and - updating the position of the signed particle according to the momentum of the signed particle.

6. The method of claim 5, wherein the two integrals are defined as: l ^{+ ′}i →m⁺ _{� VW} ⁽ _{x; M∆p} ^′) ₀ M=−∞ a^nd V_{W(x; p) =} ^{i +∞ ′ −2i} ℏ^x′∙p π_{ℏ2� dx e [V(x + x} ^′ _{) − V(x − x} ^′ _{)] −∞} where: Dp' indicates to compute VW using a momentum integral, _VW ⁺ _{(x; p) is the positive part of VW(x; p), p' is a normalized probability computed with} ^{^^^^ ^^^^( ^^^^; ^^^^)} _{� ^^^^( ^^^^) ,} M is the mass of an electron, ∆p^′ represents a discretization in the momentum space, ℏ is the reduced Planck constant, i is the imaginary unit, and x' is a variable used for integration.

7. The method of claim 5 or 6, wherein evolved signed particles are obtained by averaging positions, momentums and signs of new pairs of signed particles that are not cancelled, cancelled signed particles being particles that have opposite s_{igns and same phase-space coordinates} ⁽ _{x, p} ⁾ _.

8. The method of any one of claims 1 to 7, wherein, prior to evolving, the signed particles are distributed in a phase-space according to an initial quasi-distribution function.

9. The method of claim 1, wherein the MCUs simulate the evolution of the signed particles.

10. The method of claim 5 or 6, wherein the FPGAs compute the integrals.

11. The method of any one of claims 1 to 10, wherein each MCU of the plurality of MCUs is connected to a plurality of FPGAs.

12. A dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs), for simulating a quantum neural network (QNN), operative to simulate nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF).

13. The dedicated hardware of claim 12, wherein the virtual gates are each represented by a potential function ^^^^ = ^^^^⁽ ^^^^⁾ and wherein the signed particles are each provided with a position x, a momentum p and a sign s that can be positive or negative.

14. The dedicated hardware of claim 12, wherein the signed particles represent fermions.

15. The dedicated hardware of claim 14, wherein the signed particles represent electrons.

16. The dedicated hardware of claim 13, wherein evolving the signed particles further comprises, for each signed particle: - computing a new pair of signed particles by computing two integrals, for c_{omputing a Wigner kernel ^^^^ ^^^^} ⁽ _{^^^^; ^^^^} ⁾ _{, and a gamma function ^^^^} ⁽ _^^^^ ⁾ _{, the new pair of signed particles having a probability of ^^^^� ^^^^} ⁽ _^^^^ ⁾ _{� ^^^^ ^^^^, and} - updating the position of the signed particle according to the momentum of the signed particle.

17. The dedicated hardware of claim 16, wherein the two integrals are defined as: +^∞ lim � V_W ⁺(x; M∆p^′) M=−∞ a^nd where: D^{p' indicates to compute VW using a momentum integral,} V_{+ W(x; p) is the positive part of VW(x; p), p' is a normalized probability computed with} ^{^^^^ ^^^^( ^^^^; ^^^^)} _� M is the mass of an electron, ∆p^′ represents a discretization in the momentum space, ℏ is the reduced Planck constant, i is the imaginary unit, and x' is a variable used for integration.

18. The dedicated hardware of claim 16 or 17, wherein evolved signed particles are obtained by averaging positions, momentums and signs of new pairs of signed particles that are not cancelled, cancelled signed particles being particles that have opposite signs and same phase-space coordinates (x, p).

19. The dedicated hardware of any one of claims 12 to 18, wherein, prior to evolving, the signed particles are distributed in a phase-space according to an initial quasi- distribution function.

20. The dedicated hardware of claim 12, wherein the MCUs simulate the evolution of the signed particles.

21. The dedicated hardware of claim 16 or 17, wherein the FPGAs compute the integrals.

22. The dedicated hardware of any one of claims 12 to 21, wherein each MCU of the plurality of MCUs is connected to a plurality of FPGAs.

23. A non-transitory computer readable media having stored thereon instructions for simulating a quantum neural network (QNN) on dedicated hardware composed of a plurality of Microcontroller Units (MCUs) and Field Programmable Gate Arrays (FPGAs), the instructions comprising simulating nodes of the QNN by evolving signed particles in virtual gates defined based on the continuous variable (CV) paradigm, the evolving of the signed particles in the virtual gates being computed in parallel, using the signed particle formulation (SPF).