Quantum circuit setup for lattice gas automata simulation

Abstract

Disclosed is a quantum circuit (300) for a lattice gas automata simulation. In the quantum circuit, an initialization step (302) is performed by setting up, for the quantum circuit, a position register (l), a channel register (c) and a first ancilla register (a). To the position register (l), the channel 5 register (c) and the first ancilla register (a) are applied in a sequential order: (i) a collision step (304) constructed using a first set of multi-controlled gates (3042a-3042b), a second set of multi-controlled gates (3044a- 3044b), and four CX gates (3046a-3046d) arranged between the first set 10 and the second set, the collision step being applied to the channel register (c) and the first ancilla register (a); (ii) a mapping step (306) constructed using two Hadamard gates (3062a- 3062b) and four multi-controlled SWAP gates (3064a-3064d), the mapping step being applied to the channel register (c) and the first ancilla 15 register (a); and (iii) a propagation step (308) being applied to the position register (l) and the first ancilla register (a).

Description

[0001] QUANTUM CIRCUIT SETUP FOR LATTICE GAS AUTOMATA SIMULATION

[0002] TECHNICAL FIELD

[0003] The present disclosure relates to a method of setting up a quantum circuit for a lattice gas automata simulation. The present disclosure also relates to a quantum computer or a quantum emulator that is configured to execute the aforementioned method. The present disclosure further relates to a computer program product having computer program instructions stored thereon, the computer program instructions being executable by at least one processor in a classical computer to control a quantum computer or a quantum emulator to perform the aforementioned method.

[0004] BACKGROUND

[0005] Cellular automata have been widely explored in computational physics due to their ability to transform simple micro-dynamical rules into a specific macroscopic behaviour. A type of cellular automaton called lattice gas automaton (LGA) was devised to simulate fluid dynamics effectively; that is, LGA originated from the cellular automaton. The Hardy-Pomeau- Pazzis (HPP) and Frisch-Hasslacher-Pomeau (FHP) models are two- dimensional LGA models. LGA was largely abandoned due to the advantages presented by the lattice Boltzmann methods (LBM; which are a class of models in computational fluid dynamics), specifically the noise resilience (in contrast to the local fluctuations for the LGA) and flexibility in simulating complex multi-physics.

[0006] However, with the surge of quantum computing, LGA could be one of the key methods to realize the quantum advantage, as it provides a simple framework for encoding lattice points and can model fluid evolution even in nonlinear models. In recent years, there have been several attempts at bringing the LGA and LBM methods to quantum computing. One of the first attempts to quantize cellular automata and lattice gas automata models has been described in "From quantum cellular automata to quantum lattice gases", by D. A. Meyer, published in Journal of Statistical Physics 85 (1996) 551-574. Insisting on the exact unitarity of a collision operator to maintain consistency with standard quantum mechanics, Meyer formulated a collision operator for the simplest of one-dimensional quantum cellular automaton and quantum LGA (QLGA) models. To extend the applicability of the QLGA beyond the simulation of quantum systems, in particular to computational fluid dynamics, Yepez derived a lattice Boltzmann equation that exactly describes kinetic transport at a mesoscopic scale in a quantum lattice gas. In other words, Yepez showed that the lattice Boltzmann equation is an exact representation of the particle dynamics, including all the effects due to quantum superposition and entanglement. This has been described in "Quantum lattice-gas model for computational fluid dynamics", by J. Yepez, published in PHYSICAL REVIEW E 63 (2001) 1-18. This then lead to further work on QLGA algorithms for the simulation of the diffusion equation and Burgers equation on a one-dimensional lattice with two qubits per lattice site on a type-II quantum computer. These QLGA algorithms have been described in "Quantum lattice-gas model for the diffusion equation", by J. Yepez, published in International Journal of Modern Physics C 12 (2001) 1285-1303, and "Quantum lattice-gas model for the burgers equation", by J. Yepez, published in Journal of Statistical Physics 107 (2002) 203-224.

[0007] Although the aforementioned QLGA models make the simulation of classical physics possible on quantum computers, several drawbacks still need to be addressed before they can provide a more efficient alternative to the classical LGA. Notably, a major drawback of these QLGA models is that they require one qubit per lattice site. Furthermore, even though there are a variety of other methods for solving the fluid flow equations on quantum devices already available: for example, ranging from solving linearized differential equations using linear system solvers to using the lattice Boltzmann-based models and solving the hydrodynamic Schrodinger equation, what makes the QLGA so appealing for simulating fluid flows (or some other physical processes) on quantum devices is that all of these other methods still suffer from the same major drawback, namely, nonlinearity.

[0008] Therefore, in light of the foregoing discussion, there exists a need to overcome the aforementioned drawbacks.

[0009] SUMMARY

[0010] The present disclosure seeks to provide a method of setting up a quantum circuit for a lattice gas automata simulation. The present disclosure also seeks to provide a quantum computer or a quantum emulator that is configured to execute the aforementioned method. The present disclosure further seeks to provide a computer program product having computer program instructions stored thereon, the computer program instructions being executable by at least one processor in a classical computer to control a quantum computer or a quantum emulator to perform the aforementioned method. The aim of the present disclosure is achieved by a method, a quantum computer or a quantum emulator, and a computer program product, as defined in the appended independent claims to which reference is made to. Embodiments of the present disclosure substantially enable improvements in simulation speed, computational efficiency, and resilience to quantum noise by optimizing quantum gate complexity and resource allocation within the lattice gas automata framework. These technical benefits facilitate scalable quantum fluid dynamics simulations, making efficient use of quantum hardware capabilities for large-scale, high-fidelity applications. Advantageous features are set out in the appended dependent claims. Throughout the description and claims of this specification, the words "comprise" , "include", "have", and "contain" and variations of these words, for example "comprising" and "comprises" , mean "including but not limited to", and do not exclude other components, items, integers or steps not explicitly disclosed also to be present. Moreover, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[0011] BRIEF DESCRIPTION OF THE DRAWINGS

[0012] FIG. 1 depicts a two-dimensional lattice having a size of N x M lattice sites;

[0013] FIG. 2 depicts a schematic diagram of a quantum circuit for a lattice gas automata simulation;

[0014] FIG. 3A depicts a part of a quantum circuit for a lattice gas automata simulation, in accordance with an embodiment of the present disclosure;

[0015] FIG. 3B depicts another part of the quantum circuit, in accordance with an embodiment of the present disclosure;

[0016] FIGs. 4A, 4B and 4C depict examples of experiments in which a comparison has been made between the classical LGA model and a new QLGA model for different time steps t = 0, t = 160, t = 320, respectively, pursuant to an embodiment of the present disclosure; and

[0017] FIGs. 5A, 5B and 5C show how simulation behaves in a presence of noise for different time steps t = 2, t = 16, t = 24, respectively, pursuant to an embodiment of the present disclosure. DETAILED DESCRIPTION OF EMBODIMENTS

[0018] The following detailed description illustrates embodiments of the present disclosure and ways in which they can be implemented. Although some modes of carrying out the present disclosure have been disclosed, those skilled in the art would recognize that other embodiments for carrying out or practising the present disclosure are also possible.

[0019] The present disclosure provides a new method for simulating fluid flows using a new QLGA model on a quantum device (namely, a quantum computer or a quantum emulator). To begin with, there will be described some elements of the QLGA model as follows:

[0020] 1. The QLGA model is applied on a two-dimensional lattice, which is a regular lattice in which lattice sites cover a two-dimensional space. The characteristics of the two-dimensional lattice play a major role in the modelling capacity of the QLGA. Specific lattice symmetries are often necessary for the simulation of partial differential equations. In the new QLGA model, periodic lattice boundaries are assumed.

[0021] 2. A set of Boolean variables, referred to as "channels", are represented as follows: n (r, t) = { n1(r, t), n2(r, t), n3(r, t), n4(r, t) } such that n e C, wherein "C" represents a set of all channels, "r" represents a given lattice site in the two-dimensional lattice, and "t" indicates a given time step. The channels provide information about the occupation (ni = 1) or lack (ni = 0) of particles defined at each time step t and each lattice site r.

[0022] 3. A following micro-dynamical equation can be used to describe the time evolution of particles moving from a lattice site situated at a lattice site "r" to a neighbouring lattice site "r+vi" with a given velocity "vi": Di (r + Vi, t+1) = ni (r, t) + Δ1 [n (r, t)]

[0023] The function Ai ; C — > C is referred to as a "collision operator", throughout the present disclosure.

[0024] 4. In simulations and model interpretation, the physics is not captured by a microscopic evolution of any system but instead by a macroscopic behaviour that emerges from a local interaction and propagation of particles. Taking this fact into account, an average value of the channel occupation number <ni> is introduced. In other words, derivation of physical variables is managed by introducing an averaged value of the channel occupation <ni>, where a mass density is calculated as follows: p = ∑i <ni>

[0025] Moreover, a momentum density is calculated as follows: puα= ∑i <ni> Viα where "p" denotes mass density, "u" denotes a macroscopic velocity, and "v" denotes a particle velocity (namely, a velocity of a given particle). An average can be calculated over an ensemble of experiments with different initial conditions or by performing spatial averaging using sub-lattices of the two-dimensional lattice. This allows to calculate an average occupancy of a lattice site on the macroscopic scale, thereby reducing the noise.

[0026] It is also noted that the collision operator Ai is applied to every lattice site at the same time, obeying the conditions of conserving mass and momentum as follows: ∑i Δi (n) = 0 ∑i Δi Ai (n) = 0 This means that the time evolution is synchronous for each lattice site. At the same time, the fluid dynamics are homogeneous, meaning that the collision rules are equal in every lattice site.

[0027] The aforementioned elements of the QLGA model have been illustrated in FIG. 1, which depicts a two-dimensional lattice having a size of N x M lattice sites. It will be appreciated that for the sake of simplicity, the new QLGA model will be described using a square lattice (namely, N = M), but the new QLGA model is not restricted to such.

[0028] The new QLGA model is based on a two-dimensional HPP model, which originally developed as a cellular automaton to recreate fluid dynamics in 1970s by Hardy, Pomeau and de Pazzis. Similar to the HPP model, the new QLGA model involves discrete dynamics of particles moving and colliding on the two-dimensional lattice, whilst conserving the momentum and the particle count. The new QLGA model makes use of the von Neumann method, meaning that only vertical and horizontal neighbours are given four different channels, depicted as ni, n2, n3 and n4 in FIG. 1.

[0029] In this regard, there are only two simple collision rules. First, if two particles going down and up collide at a lattice site as described by a channel state | 0101>,, an output will be one particle moving left and another particle moving right, giving a channel state | 1010>. The opposite applies when two particles going left and right collide at a lattice site, giving an output of one particle moving up and another particle moving down. The collision operator can thus be seen as an identity for all the states except: | 1010> --> | 0101>, and | 0110> --> | 1010>.

[0030] In other words, all other states remain unaffected by the collision process, ensuring that the mass and momentum are conserved. This property makes the dynamics self-dual and satisfies detailed balance. During a propagation step, each particle is advanced along a prescribed direction.

[0031] Next, FIG. 2 depicts a schematic diagram of a quantum circuit for a lattice gas automata simulation. The quantum circuit comprises an initialization step, a collision step, a mapping step, a propagation step, and a measurement step, marked INITIALIZATION, COLLISION, MAPPING, PROPAGATION and MEASUREMENTS, respectively, in FIG. 2. In this regard, the quantum circuit mostly follows the structure of the classical LGA. Each of these steps requires a particular algorithm design that is tailored to quantum devices. As will be illustrated later in FIG. 3, the superposition principle for the representation of the two-dimensional lattice in LGA provides a certain level of flexibility for building the initialization and collision steps. Moreover, the mapping step (also called as superposition mapping) between the collision step and the propagation step is introduced, in order to enable use of the state-of-the-art quantum algorithms for the propagation step. Furthermore, the new QLGA model uses a binary representation of the channels, as in the classical LGA.

[0032] The initialization step is one of the most important steps of any quantum algorithm with complex initial conditions. Initializing non-structured data can lead to the number of multi-controlled gates growing exponentially with respect to the number of qubits. Several methods have been researched to reduce the complexity of the initialization step, for example, such as described in "Transformation of quantum states using uniformly controlled rotations", by M. Mottonen et al., published in Quantum Information and Computation 5 (2005) 467-473, "A divide- and-conquer algorithm for quantum state preparation", by I. F. Araujo et al., published in Scientific Reports 11 (2021); and "Efficient boolean methods for preparing uniform quantum states", by F. Mozafari et al., published in IEEE Transactions on Quantum Engineering 2 (2021) 1-12. However, for purely practical reasons, the reverse iterative procedure proposed by Shende et al. has been utilised, with focus on the implementation of the LGA itself, in an example embodiment of the present disclosure. This reverse iterative procedure has been described in "Synthesis of quantum-logic circuits", by V. Shende et al., published in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25 (2006) 1000-1010. Optionally, in this regard, the initialization step is performed using a first equation: wherein "IHPP" represents the encoding of the initial channel occupancy as an operator, so that the initial quantum state | qj>in will be as represented by the first equation above.

[0033] In the initialization step, a two-dimensional lattice is encoded in the superposition of basis states of a position register "I" with n (= log? N) qubits, where N is the number of lattice sites in the lattice. Additionally, a first ancilla register "a" having three ancilla qubits is used for the switch from the binary representation of the channels to the superposition. The first ancilla register is not needed for the encoding itself, but it is used later for the collision step, the mapping step and the propagation step. In this regard, the propagation step may be implemented using well- known state shift techniques, for example, such as described in "Efficient parallelization of quantum basis state shift", by Ljubomir Budinski et al., published in Quantum Science and Technology 8 (2023) 045031, and "Efficient and scalable quantum walk algorithms via the quantum fourier transform", by A. Shakeel, published in Quantum Information Processing 19 (2020). In such a case, an additional variable ancilla register having a single qubit may be introduced during the propagation step, if needed. In the first equation, "d" represents binary strings that are composed from the qubits (h, I2, ...In) of the position register "I" encoding positions of lattice sites and the four qubits (ci, C2, C3, C4) of the channel register "c" encoding occupancies of four channels per lattice site. For example, a binary string |00010101> would refer to the occupancy of 10101 > at the lattice site positioned at |0001>. It will be appreciated that the contents of the active channel |ciC2C3C4>i varies depending on the position "i", and each qubit encodes a particular channel. A particle with a direction "n" is encoded with a corresponding qubit "c". It will be appreciated that these binary strings represent basis states of a sub-system only without ancilla qubits, and do not represent basis states of the entire system. With the first ancilla register "a", these binary strings repeat as substrings of complete binary strings corresponding to the basis states of the entire system, according to the tensor products with the ancilla qubits. The initialization step using a binary encoding scheme for the position register "I" and the channel register "c" optimizes memory efficiency within the quantum circuit, addressing a key limitation of the existing QLGA models (namely, the prior art) that require one qubit per lattice site. By encoding lattice site positions and channel occupancies as binary strings, this method minimizes the required qubit resources, allowing the quantum circuit to handle larger lattice configurations with fewer qubits. This approach significantly reduces both memory overhead and initialization complexity, enabling efficient scalability for quantum lattice gas automata simulations.

[0034] In all the equations, that follow, there will be considered a sum over all the lattice sites i ∈ {0,..., 2n-1}. In the initiation step, the Xij and yij coordinates of the lattice sites are mapped into one-dimensional space using the following relation: rk = Yi,j + Xi,j (N - 1), with i, j = 0,..., N-1 and k = 0,..., (N2- 1 ) It will be appreciated here that a square lattice configuration has been assumed with N = M, for the sake of simplicity only.

[0035] Next, in the collision step, a switch is performed between two particular occupancy sub-states as mentioned in the collision rules earlier. Since the occupancy of the channels is stored only in the channel register "c", the collision step involves only those four qubits of the channel register. Optionally, in this regard, the collision step is applied using a second equation: wherein δ represents a channel map that maps channel states, and "CHPP" represents the gate sequence as an operator, thereby obtaining the new state | ψ >col. Moreover, the collided channel is denoted by |c'ic'2c'3C4>. The use of the second equation for the collision step enables a precise mapping of the channel states, crucial for accurate particle collision simulation. This method enhances the fidelity of state transitions by systematically mapping particle interactions within lattice sites, thus reducing computational errors and ensuring consistent performance across simulation steps. Unlike the existing QLGA models, this method and the new QLGA model utilize the binary encoding scheme that yields robustness to errors, while still benefitting from the quantum speedup provided by the logarithmic scaling of the quantum circuit.

[0036] Optionally, in this regard, the channel map maps the channel states according to following collision rules: a collision between one particle moving down and another particle moving up at a lattice site and represented by a channel state | 0101 > is mapped to an output in which one particle moves left and another particle moves right as represented by a channel state | 1010>; a collision between one particle moving left and another particle moving right at a lattice site and represented by the channel state 11010> is mapped to an output in which one particle moves up and another particle moves down as represented by the channel state 10101 > ; and any other channel state remains unchanged.

[0037] This channel map can be represented as follows:

[0038] Defining specific collision rules in the channel map improves the accuracy of particle movement representation by maintaining consistency in particle direction and state transitions. These predefined rules enhance computational efficiency by reducing the complexity of gate operations required for collision processing and ensuring accurate state transitions. This approach reduces errors in fluid dynamics simulations, which are critical for applications requiring high-resolution modelling of particle interactions, and lays a robust foundation for efficient propagation and mapping steps.

[0039] Subsequently, the occupancy of each channel is propagated spatially according to a prescribed velocity of a particular LGA model. For the new QLGA model, the particles residing on the channel pointing to the right / left direction are propagated one step at a time in the right / left direction; likewise, the particles residing on the channel pointing to the up / down direction are propagated one step at a time in the up / down direction. The practical implementation of this movement is straightforward classically, but the quantum implementation is more involved. According to an example embodiment, a state shift method as detailed in "Efficient parallelization of quantum basis state shift", by Ljubomir Budinski et al., published in Quantum Science and Technology 8 (2023) 045031 is used. This requires a rearrangement of the qubit encoding. Particularly, in order to propagate each qubit state independently in the channel register "c", the occupancy sub-state needs to be decomposed. In this regard, each qubit from the channel register "c" is mapped into a superposition of states using the first ancilla register "a" entangled with the lattice sites. This is achieved by first bringing the ancilla qubits "a2" and "a3" from the first ancilla register "a" into a superposition state using Hadamard gates followed by four multi-control SWAP gates between the channel register "c" and the first ancilla register "a", as depicted in FIG. 3A later.

[0040] Optionally, in this regard, the mapping step is applied using a third equation:

[0041] Using the third equation, each channel of the channel register "c" is mapped into the sub-states marked with the corresponding state of the ancilla qubits "a2" and "as". As a result, an equal superposition of the lattice sites (represented by the position register "I") and each channel (represented by the channel register "c") is obtained. This allows to apply one-step shift in different directions independently for each channel. By mapping each channel of the channel register into the sub-states marked with the corresponding state of the ancilla qubits, this method achieves the equal superposition of lattice sites and channels. This configuration allows for independent one-step shifts in multiple directions per channel, enhancing the flexibility and control of particle movement within the lattice. The technical benefit includes increased efficiency in managing particle propagation across complex lattice structures, for example, such as those with high-dimensional configurations or non-uniform boundaries. This increased efficiency reduces the required gate operations, optimizes resource usage, and facilitates scalable simulations of fluid dynamics, even on hardware with limited qubit availability.

[0042] Finally, the propagation of the particles along the corresponding directions can be implemented with the increment and decrement operations on the basis states. In the new QLGA model, along a horizontal direction, the increment moves the particle nl one step to the right, while the decrement moves the particle n3 one step to the left (see FIG. 1). Along a vertical direction, the increment moves the particle n2 one step to the up direction, while the decrement moves the particle n4 one step to the down direction (see FIG. 1).

[0043] It will be appreciated that several quantum algorithms (namely, quantum circuits) exist for these shift operators, as they are crucial in several applications, such as the quantum random walk, matrix block encoding, and the quantum lattice Boltzmann method. For an efficient implementation, a shift based on the quantum Fourier transformation has been proposed by Shakeel in "Efficient and scalable quantum walk algorithms via the quantum fourier transform", published in Quantum Information Processing 19 (2020), and a parallel shift method by Budinski et al. in "Efficient parallelization of quantum basis state shift", published in Quantum Science and Technology 8 (2023) 045031.

[0044] An implementation of the parallel shift for the new QLGA model in the case of 256 lattice sites (namely, eight qubits in the position register"!") is illustrated in conjunction with FIG. 3B, with an additional variable ancilla register having a single qubit. In this regard, propagation in four different orthogonal directions on the two-dimensional lattice can be constructed by sandwiching a quantum circuit for the left / right propagation between two symmetrical sets of controlled SWAP gates. This gives a permutation of the sub-states for the particles propagating in the left and right directions, effectively parallelizing the four directions.

[0045] It will be appreciated that expansion of the aforementioned parallel shift to the larger number of lattice sites is straightforward. It will be appreciated that any other shift algorithm can alternatively be utilised, instead of the aforementioned parallel shift. Applying the propagation operator "PHPP" on state results in a new state,

[0046] Optionally, in this regard, the propagation step is applied using a fourth equation:

[0047] wherein p is equal to a number of lattice sites along a given dimension. It will be appreciated that in a case where the two-dimensional lattice is assumed as a square lattice, the number of lattice sites along both directions are equal.

[0048] In the above equation, π+(i) and πv-(i) account for periodic single-step shifts in a first dimension as follows: and and π+p(i) and πr-p(i) account for periodic shifts in a second dimension as follows: and

[0049] It will be appreciated here that periodic lattice boundaries are assumed. Namely, the lattice indexing is assumed to be periodic. The binary strings "d" representing the position and the channels have to split, as the channel information is moved along the directions on the lattice. With this propagation step, one time step of the new QLGA model is completed.

[0050] The position registers "'I" and the ancilla register "a" are measured at the end of each time step. As the encoding in this new QLGA model is binary, the states do not carry any information in the amplitudes or the phases. Information about the occupancy of the channel is stored in the ancilla qubit "ai", and the particular channel is identified with the states of the ancilla qubits "a2" and "a3". In other words, when the state I 00>a2a3is observed, the ancilla qubit "a1" denotes the right moving channel, while an observation of the state | 10>a2a3means that the ancilla qubit "a1" denotes the up moving channel, and so on. These measurement results are then used to prepare the initial state for the next time step. The use of the fourth equation to implement periodic shifts enables seamless particle movement across lattice boundaries, creating a continuous flow that accurately represents periodic boundary conditions. This feature is essential for simulations that require boundary continuity, such as fluid dynamics models involving closed-loop systems or large-scale environmental simulations, for example, such as climate modelling, ocean current modelling, or aerodynamic flow analysis. By minimizing edge effects and reducing errors associated with boundary transitions, this approach improves simulation stability and fidelity. As a result1 it ensures consistent performance across large-scale simulations, as demonstrated in FIG. 3B.

[0051] Referring next to FIG. 3A, depicted is a detailed diagram of a part of a quantum circuit 300 for a lattice gas automata simulation, in accordance with an embodiment of the present disclosure. In this regard, an embodiment of the present disclosure provides a method of setting up the quantum circuit 300 for a lattice gas automata simulation, the method comprising: performing an initialization step 302 by setting up, for the quantum circuit 300, a position register I, a channel register c and a first ancilla register a, wherein the position register I comprises a plurality of qubits (l1, 21, ...In), the channel register c comprises four qubits Ci, C2, C3, and C4, and the first ancilla register a comprises three ancilla qubits a1 a2 and as; and applying to the position register I, the channel register c and the first ancilla register a in a sequential order:

[0052] (i) a collision step 304 constructed using a first set of multicontrolled gates 3042a-3042b, a second set of multi-controlled gates 3044a-3044b, and four CX gates 3046a-3046d arranged between the first set and the second set, the collision step 304 being applied to the channel register c and the first ancilla register a; (ii) a mapping step 306 constructed using two Hadamard gates 3062a-3062b and four multi-controlled SWAP gates 3064a- 3064d, the mapping step 306 being applied to the channel register c and the first ancilla register a; and

[0053] (iii) a propagation step 308 being applied to the position register I and the first ancilla register a.

[0054] Optionally, in the method, a two-dimensional lattice is represented by a grid of lattice sites (for example, an N x M grid as shown in FIG. 1), wherein in the initialization step 302, the qubits of the position register I are used to encode positions of the lattice sites in said grid. In this regard, the position register I encodes lattice sites using bitstrings that are defined by qubit tensor products. As an example, assuming a square lattice, with 8 qubits in the position register I, there would be 256 ( = 2A8) lattice sites. It will be appreciated that the size (namely, the number of qubits) of the position register I can be arbitrary; a conceivable minimum being 2 qubits for a 2 x 2 lattice, and in theory, having no upper limit. It will also be appreciated here that for the sake of simplicity, the two-dimensional lattice can be assumed as a square lattice, but the method and the quantum circuit 300 are not restricted to such. Moreover, it is assumed that the two-dimensional lattice is finite, and is completely indexed by the position register I. Periodicity is assumed at the boundaries of the lattice, so that it loops around itself. Using the position register I to encode lattice sites in a two-dimensional grid allows for efficient spatial mapping within the quantum circuit. This encoding minimizes the qubit footprint by representing complex lattice structures within fewer qubits, thus optimizing memory and computational resources. Unlike the existing QLGA models requiring one qubit per lattice site, this approach significantly reduces overhead, making it particularly relevant in scaling simulations for larger lattice configurations. This enables the quantum circuit to handle complex spatial models with minimal quantum resource consumption.

[0055] Moreover, the position register I is used to index the lattice sites, over which the occupancies encoded in the channel states are initialized. Optionally, in this regard, in the initialization step 302, the four qubits C1, C2, C3 and C4 of the channel register c are initialized to encode respective occupancies of four channels per lattice site. The four channels correspond to particle movement along respective ones of four directions in the two-dimensional lattice (which directions are shown as ni, nz, ns and n 4 in FIG. 1). The initialization of the four qubits C1, C2, C3 and C4 of the channel register c representing occupancies enables accurate tracking of particle movements within each lattice site. This configuration enhances the fidelity of the lattice gas automata simulation by accurately encoding directional particle movement, critical for modelling fluid dynamics. The technical advantage includes greater precision in state tracking and reducing simulation errors, which contributes to more reliable simulation outcomes in fluid dynamics applications.

[0056] It will be appreciated that the new quantum circuit 300 demonstrates LGA as a quantum-native method for simulating fluid flow. In particular, the new QLGA model demonstrated by the new quantum circuit 300 uses the quantum superposition property to encode the structure of the two- dimensional lattice. Moreover, the drawback of lattice representation in existing QLGA models (namely, requiring one qubit per lattice site) has been addressed by exploiting the superposition property of all lattice sites. As a result, the quantum circuit 300 scales logarithmically with the number of states and is capable of simulating LGA with logarithmic complexity in the number of CX gates, thereby exhibiting an exponential speedup for one time step and demonstrating a potential for significant computational advantage in systems with a large number of states. This is in contrast to the classical LGA and other existing QLGA models, which have linear complexity with respect to the number of lattice sites. The quantum circuit 300 is based on a two-dimensional HPP model, and is interchangeably referred to as the "new QLGA model", throughout the present disclosure.

[0057] Additionally, the quantum circuit 300 (namely, the new QLGA model) is more resilient to quantum noise than the closely-related quantum lattice Boltzmann method, which is very sensitive to changes in the state amplitudes. Moreover, the quantum circuit 300 (namely, the new QLGA model) also solves the issue of simulating nonlinearities, as they are introduced as a result of local interactions between particles, exhibiting a complex behaviour.

[0058] In accordance with an embodiment of the present disclosure, in the collision step 304, the first set of multi-controlled gates 3042a-3042b comprise in a sequential order: a first multi-controlled gate 3042a that uses a first qubit Ci and a third qubit C3 of the channel register c as controls corresponding to a state of | 1>, a second qubit cz and a fourth qubit C4 of the channel register c as controls corresponding to a state of |0>, and a first ancilla qubit a1 of the first ancilla register a as a target; and a second multi-controlled gate 3042b that uses the first qubit C1 and the third qubit C3 of the channel register c as controls corresponding to the state of |0>, the second qubit C2 and the fourth qubit C4 of the channel register c as controls corresponding to the state of 11 >, and the first ancilla qubit a1 of the first ancilla register a as a target.

[0059] In the collision step 304, the four CX gates 3046a-3046d are arranged after the first set of multi-controlled gates 3042a-3042b and before the second set of multi-controlled gates 3044a-3044b, wherein the four CX gates 3046a, 3046b, 3046c and 3046d use in a sequential order, respectively: the first qubit Ci, the second qubit C2, the third qubit C3 and the fourth qubit C4 of the channel register c as respective targets. Each of the four CX gates 3046a-3046d use the first ancilla qubit ai of the first ancilla register a as a control corresponding to the state of | 1>.

[0060] Moreover, in the collision step 304, the second set of multi-controlled gates 3044a-3044b comprises in a sequential order: a third multi-controlled gate 3044a that uses the first qubit Ci and the third qubit C3 of the channel register c as controls corresponding to the state of |0>, the second qubit C2 and the fourth qubit C4 of the channel register c as controls corresponding to the state of 11 >, and the first ancilla qubit ai of the first ancilla register a as a target; and a fourth multi-controlled gate 3044b that uses the first qubit Ci and the third qubit C3 of the channel register c as controls corresponding to the state of | 1>, the second qubit C2 and the fourth qubit C4 of the channel register c as controls corresponding to the state of |0>, and the first ancilla qubit ai of the first ancilla register a as a target.

[0061] Utilizing the aforementioned multi-controlled gates in the collision step 304 allows for precise manipulation of quantum states corresponding to particle interactions at each lattice site. This approach ensures accurate quantum state transitions that reflect particle collisions, reducing error rates and improving coherence across simulation steps. This precision significantly enhances the accuracy of the fluid dynamics model, especially under noisy conditions. The robustness to errors that are typical in current quantum hardware makes the quantum circuit with this particular encoding well-suited for practical, large-scale quantum simulations.

[0062] In accordance with an embodiment of the present disclosure, in the mapping step 306, one of the two Hadamard gates (namely a Hadamard gate 3062a) is applied to a second ancilla qubit a2 of the first ancilla register a, while another of the two Hadamard gates (namely a Hadamard gate 3062b) is applied to a third ancilla qubit a3 of the first ancilla register a, the two Hadamard gates 3062a-3062b being applied before the four multi-controlled SWAP gates 3064a-3064d.

[0063] Moreover, in the mapping step 306, the four multi-controlled SWAP gates 3064a-3064d comprise in a sequential order: a first multi-controlled SWAP gate 3064a that is applied between the first qubit Ci of the channel register c and the first ancilla qubit ai of the first ancilla register a, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of |0> and the third ancilla qubit a3 of the first ancilla register a as a control corresponding to the state of | 1>; a second multi-controlled SWAP gate 3064b that is applied between the second qubit cz of the channel register c and the first ancilla qubit a1 of the first ancilla register a, and that uses the second ancilla qubit a2 of the first ancilla register a as a control corresponding to the state of 11> and the third ancilla qubit as of the first ancilla register a as a control corresponding to the state of |0>; a third multi-controlled SWAP gate 3064c that is applied between the third qubit C3 of the channel register c and the first ancilla qubit a1 of the first ancilla register a, and that uses the second ancilla qubit a2 and the third ancilla qubit a3 of the first ancilla register a as controls corresponding to the state of | 1>; and a fourth multi-controlled SWAP gate 3064d that is applied between the fourth qubit C4 of the channel register c and the first ancilla qubit a1 of the first ancilla register a, and that uses the second ancilla qubit a2 and the third ancilla qubit a3 of the first ancilla register a as controls corresponding to the state of |0>.

[0064] The mapping step 306, which incorporates superposition through the aforementioned controlled SWAP gates, effectively reduces computational complexity by encoding multiple particle states simultaneously. This design significantly increases computational efficiency, as fewer operations are needed for particle state transitions. Consequently, this approach provides a notable speed advantage for high-resolution simulations, enabling faster processing and lower power consumption on quantum devices. Such improvements are crucial for scalable quantum simulations.

[0065] With reference to FIG. 3A, there is shown only a part of the quantum circuit 300, which part comprises the initialization step 302, the collision step 304 and the mapping step 306. It will be appreciated that the propagation step 308 can be performed using various different quantum sub-circuits. One example quantum sub-circuit that can be utilised to perform the propagation step 308 will now be illustrated in conjunction with FIG. 3B.

[0066] With reference to FIG. 3B, the two-dimensional lattice is assumed to be a square lattice having a size of 256 lattice sites. Thus, all the lattice sites of this lattice can be encoded using 8 qubits only. Accordingly, in such a case, the position register I comprises 8 qubits h, h, I3, k, Is, k, I7 and Is-

[0067] Moreover, in the example quantum sub-circuit of the propagation step 308, a variable ancilla register having a single ancilla qubit anc is also introduced. In accordance with an embodiment of the present disclosure, the propagation step 308 comprises: a first set of SWAP gates 3082a-3082f; a second set of SWAP gates 3084a-3084f; and a cascade of gates 3086a-3086n, arranged between the first set of SWAP gates 3082a-3082f and the second set of SWAP gates 3084a- 3084f.

[0068] The first set of SWAP gates 3082a-3082f comprises in a sequential order: a first SWAP gate 3082a that is applied between the first ancilla qubit ai and the second ancilla qubit az of the first ancilla register a; a second SWAP gate 3082b that is applied between the second ancilla qubit az and the third ancilla qubit as of the first ancilla register a; a first controlled SWAP gate 3082c that is applied between a first qubit li and a fifth qubit I5of the position register I, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of | 1>; a second controlled SWAP gate 3082d that is applied between a second qubit l2and a sixth qubit k of the position register I, and that uses the second ancilla qubit a2 of the first ancilla register a as a control corresponding to the state of | 1>; a third controlled SWAP gate 3082e that is applied between a third qubit I3 and a seventh qubit I7 of the position register I, and that uses the second ancilla qubit a2 of the first ancilla register a as a control corresponding to the state of | 1>; and a fourth controlled SWAP gate 3082f that is applied between a fourth qubit I4 and an eighth qubit Is of the position register I, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of | 1>. The second set of SWAP gates 3084a-3084f comprises in a sequential order: a fifth controlled SWAP gate 3084a that is applied between the fourth qubit k and the eighth qubit l8 of the position register I, and that uses the second ancilla qubit a2 of the first ancilla register a as a control corresponding to the state of | 1>; a sixth controlled SWAP gate 3084b that is applied between the third qubit I3 and the seventh qubit I7 of the position register I, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of | 1>; a seventh controlled SWAP gate 3084c that is applied between the second qubit l2 and the sixth qubit k of the position register I, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of | 1>; an eighth controlled SWAP gate 3084d that is applied between the first qubit h and the fifth qubit I5 of the position register I, and that uses the second ancilla qubit az of the first ancilla register a as a control corresponding to the state of | 1>; a third SWAP gate 3084e that is applied between the second ancilla qubit az and the third ancilla qubit as of the first ancilla register a; and a fourth SWAP gate 3084f that is applied between the first ancilla qubit ai and the second ancilla qubit az of the first ancilla register a.

[0069] It will be appreciated that the number of controlled SWAP gates in the first set 3082a-3082f and the second set 3084a-3084f depend on the size of the two-dimensional lattice and its lattice representation. For the example case of 256 lattice sites (that are encoded using only eight qubits in the position register I), there are only four controlled SWAP gates in each of the first set 3082a-3082f and the second set 3084a-3084f. If N is equal to a total number of qubits in the position register I, the first controlled SWAP gate of the first set can be considered to be applied between the first qubit h and an (N / 2+l)thqubit IN / 2+I of the position register I; a last controlled SWAP gate of the first set can be considered to be applied between an N / 2thqubit IN / 2 and an Nthqubit IN of the position register I. Moreover, it will be appreciated that the second set of SWAP gates 3084a-3084f is merely a reverse of the first set of SWAP gates 3082a-3082f, and can be devised accordingly.

[0070] The cascade of gates 3086a-3086n (arranged between the first set 3082a-3082f and the second set 3084a-3084f) comprises in a sequential order: a first CX gate 3086a that uses the single ancilla qubit anc of the variable ancilla register as a target and uses the first qubit h of the position register I as a control corresponding to the state of 11>; a second CX gate 3086b that uses the single ancilla qubit anc of the variable ancilla register as a target and uses the first ancilla qubit ai of the first ancilla register a as a control corresponding to the state of 11>; a third CX gate 3086c that uses the second qubit h of the position register I as a target and uses the single ancilla qubit anc of the variable ancilla register as a control corresponding to the state of 11>; a first two-controlled gate 3086d that uses the first ancilla qubit ai of the first ancilla register a as a control corresponding to the state of |0>, the single ancilla qubit anc of the variable ancilla register as a control corresponding to the state of 11>, and the second qubit I2 of the position register I as a target; a second two-controlled gate 3086e that uses the first ancilla qubit a1 of the first ancilla register a and the single ancilla qubit anc of the variable ancilla register as controls corresponding to the state of | 1>, and the third qubit h of the position register I as a target; a first multi-controlled gate 3086f that uses the second qubit I2 and the third qubit I3 of the position register I and the single ancilla qubit anc of the variable ancilla register as controls corresponding to the state of 11>, and the fourth qubit I4 of the position register I as a target; a third two-controlled gate 3086g that uses the first ancilla qubit a1 of the first ancilla register a and the single ancilla qubit anc of the variable ancilla register as controls corresponding to the state of | 1>, and the third qubit I3 of the position register I as a target; a fourth two-controlled gate 3086h that uses the first ancilla qubit ai of the first ancilla register a as a control corresponding to the state of |0>, the single ancilla qubit anc of the variable ancilla register as a control corresponding to the state of 11>, and the second qubit I2of the position register I as a target; a second multi-controlled gate 3086i that uses the second qubit I2of the position register I and the first ancilla qubit ai of the first ancilla register a as controls corresponding to the state of |0> and the single ancilla qubit anc of the variable ancilla register as a control corresponding to the state of | 1>, and the third qubit I3 of the position register I as a target; a third multi-controlled gate 3086j that uses the second qubit I2 of the position register I, the first ancilla qubit a1 of the first ancilla register a and the single ancilla qubit anc of the variable ancilla register as controls corresponding to the state of | 1>, and the third qubit I3 of the position register I as a target; a fourth CX gate 3086k that uses the single ancilla qubit anc of the variable ancilla register as a target and uses the first ancilla qubit ai of the first ancilla register a as a control corresponding to the state of | 1>; and a fifth CX gate 30861 that uses the single ancilla qubit anc of the variable ancilla register as a target and uses the first qubit l1of the position register I as a control corresponding to the state of | 1>.

[0071] The cascade of gates 3086a-3086n further comprises two X gates, wherein a first X gate 3086m is applied between the first CX gate 3086a and the fifth CX gate 30861, and a second X gate 3086n is applied after the fifth CX gate 30861.

[0072] It will be appreciated that the number and design of the two-controlled gates and the design of the multi-controlled gates in the cascade of gates 3086a-3086n also depend on the size of the two-dimensional lattice and its lattice representation. Symmetries are well observable and evident in the arrangement of various gates in the cascade of gates 3086a-3086n. As an example, the first CX gate 3086a and the second CX gate 3086b are arranged symmetrically opposite to the fifth CX gate 30861 and the fourth CX gate 3086k, respectively. As another example, the first two- controlled gate 3086d and a second two-controlled gate 3086e are arranged symmetrically opposite to the fourth two-controlled gate 3086h and the third two-controlled gate 3086g, respectively.

[0073] It is evident that the first set of SWAP gates 3082a-3082f, the second set of SWAP gates 3084a-3084f and the cascade of gates 3086a- 3086n allow for the propagation step 308 to scale logarithmically.

[0074] In an embodiment of the present disclosure, the quantum circuit 300, after set up, is used for performing the lattice gas automata simulation. After the propagation step 308 is performed, measurements can be taken for one time step, and the quantum circuit 300 can be repeated from the initialization step 302 for a next time step. This enhances operational efficiency by allowing the quantum circuit 300, once set up, to proceed directly with the lattice gas automata simulation without reinitializing complex parameters. This continuous execution capability minimizes time lost to setup phases and enhances throughput, especially in applications requiring iterative simulations, for example, such as fluid flow modelling and real-time environmental monitoring. The result is a more responsive simulation process, enabling higher data processing speeds and improved scalability for real-time or multi-step quantum fluid dynamics simulations.

[0075] In another aspect, an embodiment of the present disclosure provides a quantum computer or a quantum emulator configured to execute the aforementioned method. The quantum computer or the quantum emulator configured to implement this method benefits from optimized resource management and parallel processing capabilities, significantly improving computational efficiency for high-complexity simulations. This configuration is particularly advantageous for executing lattice gas automata models, as it supports rapid processing with reduced gate operations and enhanced accuracy under quantum noise. By leveraging these capabilities, the result is a robust and scalable platform capable of running scalable quantum simulations with improved resilience and precision, even in noisy environments.

[0076] The present disclosure also relates to the quantum computer or the quantum emulator that is configured to execute the aforementioned method as described above. This tailored configuration of the quantum computer or the quantum emulator for lattice gas automata simulations provides enhanced control over quantum resources, ensuring precise state management and consistency throughout simulation steps. By minimizing latency and improving stability, this setup supports high- precision modelling even on noisy quantum hardware, thus enabling accurate, large-scale fluid dynamics simulations that are well-suited for real-world applications, for example, such as climate modelling or aerodynamic flow analysis. Various embodiments and variants disclosed above, with respect to the aforementioned method and the quantum circuit 300, apply mutatis mutandis to the quantum computer or the quantum emulator. In an embodiment, the quantum computer or the quantum emulator is used to perform the lattice gas automata simulation, as described above.

[0077] It will be appreciated here that the term "quantum emulator" refers to an emulator for a quantum computer. A quantum emulator can be a classical computer that is configured to emulate the behaviour of a quantum computer.

[0078] In a third aspect, an embodiment of the present disclosure provides a computer program product having computer program instructions stored thereon, the computer program instructions being executable by at least one processor in a classical computer to control a quantum computer or a quantum emulator to perform the aforementioned method. The present disclosure also relates to such a computer program product. The computer program product, designed to control the quantum computer or the quantum emulator for this simulation method, facilitates seamless integration between classical and quantum computing systems for lattice gas automata simulations. By enabling a classical computer to input physical data parameters into a quantum computer for simulation, and subsequently process the quantum simulation results back into physical data results, this computer program product creates an efficient and interactive feedback loop. This design optimizes resource utilization by offloading computationally intensive tasks to the quantum computer, thereby saving classical computing resources such as processing capacity and memory. The dynamic interaction allows for real-time updates to input parameters, efficient visualization of simulation results, and reduced computational overhead, making the computer program product adaptable for diverse applications such as fluid dynamics modelling and optimization problems. The product's adaptability ensures broad applicability across different platforms, facilitating scalable and efficient quantum simulations. Various embodiments and variants disclosed above, with respect to the aforementioned method, apply mutatis mutandis to the computer program product. The computer program product can be in a form of a non-transitory computer-readable medium, a software application, or a cloud-based service. The software application may, for example, be downloadable from an App store, or a website that servers as a distribution platform for the software application. The cloudbased service may be implemented as a web-based platform that provides software functionality according to the aforementioned method over a communication network (for example, such as the Internet).

[0079] EXPERIMENTAL PART

[0080] There will now be considered an analysis of the gate complexity of the quantum circuit 300 of the new QLGA model as compared to the prior art. The focus of this analysis will be on three main steps that make the core of the QLGA model, namely: the collision step 304, the mapping step 306, and the propagation step 308.

[0081] The collision step 304 and the mapping step 306 between binary encoding to superposition scales as O(1), as the number of channels is constant. However, the propagation step 308 depends critically on multicontrolled gates and will be the source for most of the complexity in the quantum algorithm. In the illustrated example embodiment, the propagation step 308 is implemented using the basis state shift, for example, as described in "Efficient parallelization of quantum basis state shift", by Ljubomir Budinski et al., published in Quantum Science and Technology 8 (2023) 045031. This document provides an extensive complexity analysis for the canonical shift, the Quantum Fourier Transform (QFT) variation, and the parallel shift, together with an outline of an ancilla decomposition of the multi-controlled gates. The results focus on a transpilation with the IBM basis set of gates CX, I, R.Z, SX and X using standard compilation techniques. It can be concluded that the best choice for conducting the shift step is the parallel shift algorithm, as it provides a linear scaling ncx(n) = 15(n - 6) + 149 in terms of CX gates, when the number of working qubits "n" is at least six.

[0082] To test the new QLGA model, two benchmarks have been carried out. First, the results of the new QLGA model are compared against a classical version. For this purpose, sub-lattices were used for spatial-averaging. As the initial condition, a sharp profile was set up in the middle of the domain as 95 % channel occupancy and 5 % of probability for the rest of the model. Several simulations were conducted using the different variants of QLGA, and validated against classical simulations, yielding the same results. One example of these experiments is illustrated in FIGs. 4A, 4B and 4C, wherein the comparison has been made between the classical LGA model and the new QLGA model for different time steps t = 0, t = 160, t = 320, respectively. The simulation was performed using the Aer simulator from Qiskit SDK, without noise, for example, as described in Qiskit contributors, "Qiskit: An open-source framework for quantum computing", 2023. Besides the sub-lattice averaging, it was found that a statistical ensemble, with different initial conditions for each trial, is necessary to reduce the noise, due to the limited number of qubits in current simulators and quantum computers (more qubits are needed to increase the sub-lattice sites). The results clearly show a diffusion-like behaviour for both of the classical LGA model and the new QLGA model, and the results clearly overlap for each time step. This confirms that the new QLGA model is capable of recreating the behaviour of the classical LGA. Quantum computing is advancing fast and new devices and architectures with more qubits and less noise are continuously being presented. Noise is unavoidable in quantum devices of today, and it is relevant to find quantum algorithms that can be used under noisy conditions. Therefore, there was also explored how the new QLGA model behaves under different error rates. Due to the high cost of quantum computers and limited access to machines with varying error rates, the new QLGA model was tested with the Qiskit Aer simulator and constructed toy models for the noise. The error rates of the device were manually included. Depolarizing errors are the major contributors to noise in real quantum devices, for example, as also shown in "Simple mitigation of global depolarizing errors in quantum simulations", by J. Vovrosh et al., published in Phys. Rev. E 104 (2021) 035309. Other variables to consider are the number of shots to extract results, the connectivity of the device, and the set of native gates used. To simplify the analysis as much as possible, the IBM native gate set, which is composed of RZ, X, SX and CX gates, was selected, and the new QLGA model was tested under three noise levels. Each noise model has been created using single-qubit errors, two-qubit errors, a depolarizing error, and a readout error provided by Qiskit.

[0083] Table 2 provides the different error rates used for the low, mid, and high noise conditions, as follows:

[0084] Increasing the number of shots should lead to more accurate simulations. To test this assumption, simulations were run for every noise level with the minimum number of shots required to observe propagation and diffusion up to time step t = 24. FIGs. 5A, 5B and 5C show how the simulation behaves in the presence of noise for different time steps t = 2, t = 16, t = 24, respectively. The number of shots was adjusted to the minimum necessary for accurate simulations. An ensemble average of fifteen was used. Additionally, the noiseless simulation was performed with 350 shots, which is found to measure all the states with high probability for this simulation.

[0085] The higher the noise, the more shots are required to replicate similar results, as was expected. With increasing noise, it becomes harder to differentiate between the junk states and the desired states in the measurement and post-processing. When the noise is high enough, no valid information can be extracted, and the model freezes or has no propagation using up to 100k shots. Despite using the IBM native gates here, the simulation with mid-level noise (which includes the same quantum depolarizing error rates from Quantinuum H2 and all-to-all connectivity) indicates that useful simulations could be performed with current quantum computers. These results indicate that while high noise levels make distinguishing between the results and computational junk states challenging, accurate simulations could still be achievable with current noisy devices despite the need for increasing shots.

Claims

CLAIMS1. A method of setting up a quantum circuit (300) for a lattice gas automata simulation, the method comprising: performing an initialization step (302) by setting up, for the quantum circuit, a position register (I), a channel register (c) and a first ancilla register (a), wherein the position register (I) comprises a plurality of qubits (li, I2, ...In), the channel register (c) comprises four qubits (ci, C2, C3, C4), and the first ancilla register (a) comprises three ancilla qubits (ai, a2, as); and applying to the position register (I), the channel register (c) and the first ancilla register (a) in a sequential order:(i) a collision step (304) constructed using a first set of multicontrolled gates (3042a-3042b), a second set of multi-controlled gates (3044a-3044b), and four CX gates (3046a-3046d) arranged between the first set and the second set, the collision step being applied to the channel register (c) and the first ancilla register (a);(ii) a mapping step (306) constructed using two Hadamard gates (3062a-3062b) and four multi-controlled SWAP gates (3064a- 3064d), the mapping step being applied to the channel register (c) and the first ancilla register (a); and(iii) a propagation step (308) being applied to the position register (I) and the first ancilla register (a).

2. The method according to claim 1, wherein a two-dimensional lattice is represented by a grid of lattice sites, wherein in the initialization step (302), the qubits of the position register (I) are used to encode positions of the lattice sites in said grid.

3. The method according to any of the preceding claims, wherein in the initialization step (302), the four qubits (ci, C2, C3, C4) of the channel register (c) are initialized to encode respective occupancies of four channels per lattice site.

4. The method according to any of the preceding claims, wherein in the collision step (304), the first set of multi-controlled gates (3042a- 3042b) comprises in a sequential order: a first multi-controlled gate (3042a) that uses a first qubit (ci) and a third qubit (C3) of the channel register (c) as controls corresponding to a state of 11>, a second qubit (C2) and a fourth qubit (C4) of the channel register (c) as controls corresponding to a state of |0>, and a first ancilla qubit (ai) of the first ancilla register (a) as a target; and a second multi-controlled gate (3042b) that uses the first qubit (ci) and the third qubit (C3) of the channel register (c) as controls corresponding to the state of |0>, the second qubit (C2) and the fourth qubit (C4) of the channel register (c) as controls corresponding to the state of 11>, and the first ancilla qubit (ai) of the first ancilla register (a) as a target; wherein the four CX gates (3046a-3046d) are arranged after the first set of multi-controlled gates (3042a-3042b) and before the second set of multi-controlled gates (3044a-3044b), wherein the four CX gates (3046a, 3046b, 3046c, 3046d) use in a sequential order: the first qubit (ci), the second qubit (C2), the third qubit (C3) and the fourth qubit (C4) of the channel register (c) as respective targets, each of the four CX gates using the first ancilla qubit (ai) of the first ancilla register (a) as a control corresponding to the state of | 1>; wherein the second set of multi-controlled gates (3044a-3044b) comprises in a sequential order:a third multi-controlled gate (3044a) that uses the first qubit (ci) and the third qubit (C3) of the channel register (c) as controls corresponding to the state of |0>, the second qubit (C2) and the fourth qubit (C4) of the channel register (c) as controls corresponding to the state of 11>, and the first ancilla qubit (ai) of the first ancilla register (a) as a target; and a fourth multi-controlled gate (3044b) that uses the first qubit (ci) and the third qubit (C3) of the channel register (c) as controls corresponding to the state of | 1>, the second qubit (C2) and the fourth qubit (C4) of the channel register (c) as controls corresponding to the state of 10>, and the first ancilla qubit (ai) of the first ancilla register (a) as a target.

5. The method according to any of the preceding claims, wherein in the mapping step (306), one of the two Hadamard gates (3062a) is applied to a second ancilla qubit (32) of the first ancilla register (a), while another of the two Hadamard gates (3062b) is applied to a third ancilla qubit (as) of the first ancilla register (a), the two Hadamard gates (3062a- 3062b) being applied before the four multi-controlled SWAP gates (3064a-3064d), wherein the four multi-controlled SWAP gates (3064a-3064d) comprise in a sequential order: a first multi-controlled SWAP gate (3064a) that is applied between a first qubit (ci) of the channel register (c) and a first ancilla qubit (ai) of the first ancilla register (a), and that uses the second ancilla qubit (as) of the first ancilla register (a) as a control corresponding to a state of |0> and the third ancilla qubit (as) of the first ancilla register (a) as a control corresponding to a state of | 1>;a second multi-controlled SWAP gate (3064b) that is applied between a second qubit (C2) of the channel register (c) and the first ancilla qubit (a1) of the first ancilla register (a), and that uses the second ancilla qubit (a2) of the first ancilla register (a) as a control corresponding to the state of | 1> and the third ancilla qubit (as) of the first ancilla register (a) as a control corresponding to the state of |0>; a third multi-controlled SWAP gate (3064c) that is applied between a third qubit (cs) of the channel register (c) and the first ancilla qubit (a1) of the first ancilla register (a), and that uses the second ancilla qubit (a2) and the third ancilla qubit (as) of the first ancilla register (a) as controls corresponding to the state of | 1>; and a fourth multi-controlled SWAP gate (3064d) that is applied between a fourth qubit (C4) of the channel register (c) and the first ancilla qubit (ai) of the first ancilla register (a), and that uses the second ancilla qubit (a?) and the third ancilla qubit (as) of the first ancilla register (a) as controls corresponding to the state of |0>.

6. The method according to any of the preceding claims, wherein the initialization step (302) is performed using a first equation:wherein ch represents binary strings that are composed from the qubits (li, I2, ...In) of the position register (I) encoding positions of lattice sites and the four qubits (C1, C2, C3, C4) of the channel register (C) encoding occupancies of four channels per lattice site.

7. The method according to any of the preceding claims, wherein the collision step (304) is applied using a second equation:wherein δ represents a channel map that maps channel states.

8. The method according to claim 7, wherein the channel map maps the channel states according to following collision rules: a collision between one particle moving down and another particle moving up at a lattice site and represented by a channel state | 0101 > is mapped to an output in which one particle moves left and another particle moves right as represented by a channel state | 1010>; a collision between one particle moving left and another particle moving right at a lattice site and represented by the channel state | 1010> is mapped to an output in which one particle moves up and another particle moves down as represented by the channel state | 0101 > ; and any other channel state remains unchanged.

9. The method according to any of the preceding claims, wherein the mapping step (306) is applied using a third equation:

10. The method according to any of the preceding claims, wherein the propagation step (308) is applied using a fourth equation:wherein p is equal to a number of lattice sites along a given dimension, wherein πr+(i) and πv-(i) account for periodic single-step shifts in a first dimension as follows:andand wherein π+p(i) and π-p(i) account for periodic shifts in a second dimension as follows:and11. The method according to any of the preceding claims, wherein the quantum circuit (300), after set up, is used for performing the lattice gas automata simulation.

12. A quantum computer or a quantum emulator configured to execute a method according to any of the preceding claims.

13. The quantum computer or the quantum emulator according to claim12, wherein the quantum computer or the quantum emulator is used to perform a lattice gas automata simulation.

14. A computer program product having computer program instructions stored thereon, the computer program instructions being executable by at least one processor in a classical computer to control a quantum computer or a quantum emulator to perform a method according to any of claims 1 to 11.

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