Hybrid classical-quantum method for computational resource allocation

Abstract

A method and apparatus for allocating computational resources to wireless devices in a communication network. The method comprises iterating a classical combinatorial optimization solver on a combinatorial optimization problem representing the allocation of computational resources until the classical combinatorial optimization solver reaches a local minimum. The method comprises warm starting a QAOA circuit on the edge user allocation as a Boolean satisfiability, SAT, problem wherein each decision variable corresponds to a computational resource or an edge device by encoding continuous variables corresponding one-to- one to the decision variables of the SAT problem into the QAOA circuit, the continuous variables obtained from the classical combinatorial optimization solver at the local minimum. The method comprises iterating the classical combinatorial optimization solver on the optimization problem using continuous values of the qubits of the QAOA circuit. The method comprises initiating allocation of computational resources to wireless devices.

Description

[0001] HYBRID CLASSICAL-QUANTUM METHOD FOR COMPUTATIONAL RESOURCE ALLOCATION TECHNICAL FIELD The disclosure relates to methods for resource allocation in telecommunication infrastructure, and in particular to a hybrid quantum-classical method and a related apparatus, quantum circuit, computer program, and computer program product. BACKGROUND Telecommunication infrastructures are currently experiencing a substantial surge indemand for Internet services. Consequently, it is of great interest to strategicallydesign telecommunication networks to optimize energy consumption and the flow of service data. Addressing this key challenge requires an efficient algorithm to solve combinatorial optimization problems. Examples include i) maximum likelihood basedsoft multiple input multiple output (MIMO) detection, ii) low density parity check(LDPC) and polar codes decoding, iii) allocating radio resources, physical cellidentifier, iv) placing virtual network functions over the baseband resource pool in thecloud, v) offloading wireless device specific tasks to edge compute servers and vi)offloading wireless devices to micro base stations within the coverage area of amacro cell to improve the quality of service While numerous classical algorithms are available to tackle such optimization issues,their efficacy diminishes as the problem size expands. Conversely, noisyintermediate-scale quantum computers encounter challenges due to noise, resulting in low accuracy in solutions and prolonged runtime when tackling with problems with high circuit depth and substantial number of variables. Hence, quantum algorithmssuch as the standard quantum approximate optimization algorithm (QAOA) may notsingularly arise as optimal techniques for these complex problems. To address this issue, certain scholarly works proposed recursive warm-start QAOA by classical optimizers [2, 6], which enhances solution accuracy. Nonetheless, it is crucial to note that this technique entails performing QAOA iterations of an order proportional to(N / 2), with N representing the number of decision variables to maintain an advantageover purely classical methods. Furthermore, known methods employ a samplingmethod to obtain bitstring solutions. As indicated in [7], it is necessary to calculate the probability distribution across all possible bitstrings. In the case of computing the marginal distribution, as applicable to chaotic circuits, a certain number of qubits are uniformly chosen, and traced out. The probability distribution of the remaining qubits is then uniformly determined, which still demands considerable computation significantly decreasing the advantage over purely classical solutions. Existing hybrid quantum-classical algorithms for combinatorial optimization problems

[0016] deploy quantum variational ansatzes that represent the problem structure partially and feed measurement outcomes from the quantum computer to classical heuristics to indirectly compute optimal solution. For example greedy flip together with QAOA has been used to solve MAX-CUT problem. The standard QAOA requires high circuit depth to estimate a solution with goodaccuracy, which demands high computational cost on both simulator and realquantum hardware [1].The recursive warm-start QAOA (WS-QAOA) is defined as a loop of classical andquantum algorithms. At every iteration of the loop, it runs classical optimizer to solve the relaxed problem, and randomly rounds the solution. Then, it performs several WS-QAOA, each one is initialized by different solution of randomized rounding. Regarding the probability of sampling bit-strings obtained from the WS-QAOAs, it finds a pair of variables with highest absolute correlation and eliminates one of them. The loop repeats until the number of remaining variables is small enough to be solved by exact diagonalization. This scenario necessities performing many iterations of QAOA, particularly for the problem with a substantial number of variables [2]. The sampling process used in both cases of the QAOA and the recursive WS-QAOA circuit requires substantial computational cost and is not practical as dealing with many variables in the problem. SUMMARY It is an object of the present invention to provide a method circumventing at least the above disadvantages.According to a first aspect of the disclosure, there is a method for allocatingcomputational resources to wireless devices in a communication network. Themethod is performed by a classical computer with access to a classical combinatorial optimization solver and a quantum approximate optimization algorithm, QAOA,circuit. The method comprises iterating the classical combinatorial optimizationsolver on a combinatorial optimization problem representing the allocation of computational resources until the classical combinatorial optimization solver reaches a local minimum. The method comprises warm starting the QAOA circuit on the edge user allocation as a Boolean satisfiability, SAT, problem wherein each decision variable corresponds to a computational resource or an edge device by encoding continuous variables corresponding one-to-one to the decision variables of the SAT problem, the continuous variables obtained from the classical combinatorial optimization solver at the local minimum, into the QAOA circuit. The method comprises iterating the classical combinatorial optimization solver on the optimization problem using continuous values of the qubits of the QAOA circuit. The method comprises based on an obtained solution to the combinatorial optimization problem, initiating allocation of computational resources to wireless devices. According to an embodiment of the first aspect, the combinatorial optimizationproblem is one of: an integer linear program; a SAT problem; or a quadraticunconstrained binary optimization (QUBO) problem. According to an embodiment of the first aspect, the allocation of computational resources comprises allocating computational resources to wireless devices to maximize the number of wireless devices served by the network, wherein the SAT problem comprises a Max-1-k-SAT problem, wherein each wireless device is allocated computing resources from k servers, where the Max-1-k-SAT problem is such that variables represent servers and clauses represent wireless devices. According to an embodiment of the first aspect, the continuous variables are encoded into an angle of rotation gates operating on qubits, the qubits representing binary variables corresponding one-to-one to the continuous variables of the classical optimization solver. According to an embodiment of the first aspect, the QAOA circuit is configured byobtaining an Ising Hamiltonian of the SAT problem and selecting a minimum numberof ansatz layers to obtain an approximation ratio within a first threshold of a ground state energy of the Ising Hamiltonian, wherein the ground state energy is calculated as a sum of local expectation values, each local expectation value obtained by contracting a tensor network of the QAOA circuit.According to an embodiment of the first aspect, the continuous value of a qubit of theQAOA circuit corresponding to a variable is obtained by computing a trace of a density matrix over all other variables by contracting the tensor network of the QAOA circuit. According to a second aspect of the disclosure, there is an apparatus for allocating computational resources to wireless devices in a communication network. Theapparatus comprises classical processing circuitry and a memory. The apparatushas access to a classical combinatorial optimization solver and a quantumapproximate optimization algorithm, QAOA, circuit. The apparatus is configured toiterate the classical combinatorial optimization solver on a combinatorial optimization problem representing the allocation of computational resources until the classical combinatorial optimization solver reaches a local minimum. The apparatus is configured to warm start the QAOA circuit on the edge user allocation as a Boolean satisfiability, SAT, problem wherein each decision variable corresponds to a computational resource or an edge device by encoding continuous variables corresponding one-to-one to the decision variables of the SAT problem, the continuous variables obtained from the classical combinatorial optimization solver at the local minimum, into the QAOA circuit. The apparatus is configured to iterate the classical combinatorial optimization solver on the optimization problem using continuous values of the qubits of the QAOA circuit. The apparatus is configured to, based on an obtained solution to the combinatorial optimization problem, initiate allocation of computational resources to wireless devices. According to an embodiment of the second aspect, the QAOA circuit is implemented using quantum circuitry.According to an embodiment of the second aspect, the QAOA circuit is simulated onclassical hardware. According to an embodiment of the second aspect, the allocation of computational resources comprises allocating computational resources to wireless devices to maximize the number of wireless devices served by the network, wherein the SAT problem comprises a Max-1-k-SAT problem, wherein each wireless device is allocated computing resources from k servers, where the Max-1-k-SAT problem is such that variables represent servers and clauses represent wireless devices. According to an embodiment of the second aspect, the continuous variables are encoded into an angle of rotation gates operating on qubits, the qubits representing binary variables corresponding one-to-one to the continuous variables of the classical optimization solver. According to an embodiment of the second aspect, the QAOA circuit is configured byobtaining an Ising Hamiltonian of the SAT problem and selecting a minimum numberof ansatz layers to obtain an approximation ratio within a first threshold of a ground state energy of the Ising Hamiltonian, wherein the ground state energy is calculated as a sum of local expectation values, each local expectation value obtained by contracting a tensor network of the QAOA circuit. According to an embodiment of the second aspect, the continuous value of a qubit of the QAOA circuit corresponding to a variable is obtained by computing a trace of a density matrix over all other variables by contracting the tensor network of the QAOA circuit.According to a third aspect of the disclosure, there is a computer programcomprising computer-readable instructions which, when executed by processing circuitry of an apparatus, cause the apparatus to perform the method according to any embodiment of the first aspect.According to a fourth aspect of the disclosure, there is a computer program productcomprising a non-transient computer-readable storage medium on which a computerprogram according to the third aspect is stored.BRIEF DESCRIPTION OF THE DRAWINGS Fig.1 is a flowchart of a method according to the disclosure. Fig. 2 is a random network illustrating an embodiment of methods of thedisclosure. Fig.3 is a quantum circuit according to embodiments of the disclosure. Fig.4 is a line graph of an outcome of a simulation of methods of the disclosure. Fig.5 is a simplified quantum circuit according to embodiments of the disclosure. Fig.6 is a bar chart of outcomes of a simulation of methods according to the disclosure. Fig.7 is a bar chart of outcomes of a simulation of methods according to the disclosure. Fig. 8 is a bar chart of outcomes of a simulation of methods according to thedisclosure. Fig. 9 is a bar chart of outcomes of a simulation of methods according to thedisclosure. Fig.10 is a simplified box plot of outcomes of a simulation of methods according to the disclosure. Fig.11 is a schematic illustration of an apparatus performing methods according to the disclosure. Fig.12 is a schematic overview of a subset of a network in which methods according to the disclosure may be implemented. DETAILED DESCRIPTION OF THE DRAWINGS Fig.1 is a flowchart of a method 100 for allocating computational resources to wireless devices. The method 100 is a hybrid quantum-classical method performed by a classical computing device with access to a classical combinatorial optimizationsolver and a quantum approximate optimization algorithm, QAOA, circuit.The method 100 comprises iterating the classical combinatorial optimization solver on a combinatorial optimization problem representing the allocation of computationalresources to wireless devices until the classical optimization solver reaches a localminimum.Allocation of computational resources to wireless devices is a fundamental problemin, among other applications, telecommunications and cloud computing. In particular,the problem comprises determining, given a set number of available computational resource providers (radio access nodes, servers), how to distribute wireless devicesamong the resources to enable the largest number of devices to access thetelecommunications network / access the cloud network.For example, in mobile edge computing, a large number of edge servers aredistributed in close proximity to wireless devices. An optimal deployment aims tomaximize the number of allocated wireless devices and minimize the number ofemployed edge servers, while maintaining the required quality of service for users.Such optimization problems can be formulated as Boolean satisfiability (SAT)problems. SAT problems involve determining the optimal values for a set of variablesdenoted as {^^}, collectively known as a configuration ^, to either maximize or minimize a given objective function. As a specific category of multivariate problems,SAT problems are characterized by N Boolean variables and M constraints(clauses). An illustrative example of such a problem instance is a 2-SAT problem which can be expressed in conjunctive normal form (CNF) as follows: ^(^) ≔ (^^ ⋁ ^^) ⋀ (^^ ⋁ ^^)⋀ … ⋀(^^ ⋁ ^^),where ⋁ and ⋀ denote Boolean expressions of OR (disjunction) and AND(conjunction) respectively. The CNF is a logical conjunction of clauses, with eachclause being a logical disjunction of precisely two literals, where a literal is a variableor a negation of a variable. If a clause is evaluated to TRUE, it is SAT, otherwise it isunsatisfied, UNSAT. The Max-2-SAT problem, as a variant of 2-SAT problem,identifies a variable assignment which maximizes the count of SAT clauses in theCNF of the 2-SAT problem. Each clause is associated to a correspondingmathematical expression, ensuring that the minimal value of the mathematicalexpression is achieved when the clause is true, as follows: Using conventional Ising embedding methodologies [3], the Max-2-SAT problem canbe transformed into Ising Hamiltonians represented in the following: where ^ is the number of clauses,binary matrix with ^^^ = 1 if the variable ^^ is included in the clause ^^ as positiveliteral, ^^^ = −1 if the variable ^^ is included in the clause ^^ as negative literal, and^^^ = 0 otherwise. Using Eqs. 1 and 2, the Max-2-SAT problem is reduced intominimizing a combination of linear and quadratic equations. The solutions of theIsing model will be a set of ^^ = {1, −1}. To convert the Ising solutions into the binaryones, the following equation is applied:^^ = (1 − ^^) / 2. (Eq. 3)The 1-k-SAT problem represents a natural variation of the SAT problem, where eachclause satisfies the condition that exactly one literal must be assigned the value 1.For ease of notation, the Max-1-3-SAT problem will be used as an illuminatingexample in the detailed description, but the method 100 applies to the general Max-1-k-SAT problem. A clause in the Max-1-3-SAT problem with all positive literals canbe expressed as: ^^^⋁^^⋁^^^ → (^^ + ^^ + ^^ − 1)^. (Eq.4) The corresponding Hamiltonian can be formulated as: where ^, ℎ, and ^ are defined as in Eq. 2.The method 100 applies where the combinatorial optimization problem exhibits abackbone structure, wherein a subset of the decision variables called the backbonetake the same value in all solutions. More concretely, the backbone corresponds towireless devices which are allocated resources by the same server / radio accessnode in all optimal solutions. All Max-2-SAT and Max-1-k-SAT problems exhibit thebackbone structure [8, 9], as does the problem of allocating computational resourcesto wireless devices. The resource allocation problem exhibits a backbone structuredue to geographical proximity and devices requiring computational support which can only be satisfied by specific servers / combinations of servers. A simplified example will now be discussed in more detail. In the simplified example, every server has the capacity to connect with up to six wireless devices, and everywireless device can be connected to maximum three servers.The aim is to optimize the number of wireless devices served by a specific number ofservers. The problem is modelled as a Max-1-3-SAT problem, where the decisionvariables correspond to servers and the clauses represent wireless devices. To generate corresponding networks for the Max-1-3-SAT problem, we generate abi-regular graph where every vertex in a first set of vertices of the bi-regular graphrepresents a server and has exactly a degree of 6, and each vertex in a second setof vertices represents a wireless device and has exactly a degree of 3. Each vertexcorresponds to a variable, and an edge connects two vertices if the two vertices bothappear in a common clause. Alternatively, a computational resource allocation problem may be modelled as a Max-2-SAT problem. By way of a second simplified example, we use the followingprocedure to create a Max-2-SAT problem corresponding to a network resourceallocation problem: 1. Generate a cycle consisting of N vertices and N edges, where N representsthe number of variables. This ensures that the graph obtained from thisprocedure is always connected, and we are dealing with hard instances. 2. Add additional edges randomly to the variables until the total number of edgesreaches the pre-defined value of ^^, where ^ is the proportion of clauses tovariables. In this example, ^ = 3.3. Flip positive literals to negatives randomly, with a probability of p=0.25.Fig. 2 schematically represents a random network constructed by the aboveinstruction where the number of variables and the number of clauses are N=20, M=60, respectively.Returning to Fig. 1, the method comprise optimizer for variational ansatz s iterating101 the classical combinatorial optimization solver on a combinatorial optimizationproblem representing the allocation of computational resources until the classicalcombinatorial optimization solver reaches a local minimum.The combinatorial optimization problem may be written in any suitable way which iscompatible with the classical combinatorial optimization solver. By way of example, the combinatorial optimization problem may be written as aninteger linear optimization problem and the classical combinatorial optimizationsolver may implement a variation of the Simplex algorithm. Alternatively, thecombinatorial optimization problem may be formulated as a quadratic unconstrainedbinary optimization (QUBO) problem. Alternatively, the classical combinatorialoptimization problem may be formulated as a SAT problem, as in the simplifiedexample. The classical combinatorial optimization problem may implementoptimization solvers such as Gurobi or CEPLEX or any state-of the-art optimizationsolver. After formulating the combinatorial optimization problem, the Ising Hamiltonian ^^ofthe combinatorial optimization problem may be derived, where each decisionvariable in the combinatorial optimization problem corresponds to a site in the lattice^^ and each site / decision variable can be in either state -1 or +1.To pinpoint the scenario in which the classical combinatorial optimization solver isdeemed to be caught in a local minimum, a parameter ^^^^^ is used in someembodiments. The parameter ^^^^^ signifies a number of successive iterationsduring which the classical combinatorial optimization solver yields objective valuesdiffering by at most some pre-determined tolerance ^. When the number of iterations^^^^^ of identical subsequent objective values is reached, the classical combinatorialoptimization solver is deemed to have found a local minimum and the current configuration of continuous variables obtained by the classical combinatorialoptimization solver at the discovered local minimum is used in the to warm start theQAOA circuit.The method comprises warm starting 102 the QAOA circuit on the edge userallocation as a SAT problem, wherein each decision value corresponds to acomputational resource or an edge device, by encoding a configuration of continuous variables corresponding one-to-one to the decision variables of the SAT problem, the continuous variables obtained from the classical combinatorialoptimization solver at the local minimum, into the QAOA circuit.The QAOA, developed by Farhi et al. [1], represents a hybrid classical-quantum variational algorithm designed for the purpose of estimating the minimum energy associated with a Hamiltonian problem ^^. To achieve this, the algorithm parameterizes the wave function in terms of angle parameters ^⃗, ^⃗, with the objective of minimizing the following mathematical expression: QAOA, themixer Hamiltonian is denoted as ^ = ^(^) ^ ^, and the initial state is defined as|Ψ^^ = | +^, which corresponds to the superposition of all computational basis states.Furthermore, the parameters ^⃗, ^⃗ are iteratively adjusted within a classical learningloop to minimize the objective function ^^. One of the generalizations of the QAOA incorporates a warm-start strategy for quantum optimization [2]. This method initiates the quantum algorithm with a state aligning with the solution obtained by the relaxation of a combinatorial optimization problem, which can be expressed as: So, the uniform superposition of states in the standard QAOA is substituted by: in which ^^∗values denote the solutions obtained by the classical optimizer for therelaxed version of the problem. Moreover, ℇ is a regularization parameter, taking thevalues of [0, 0.5]. Since this new initial state is required to be the ground state of the mixer Hamiltonian, the modified mixer Hamiltonian is defined as follows: (Eq. 9) Therefore, the warm-start QAOA circuit is designed as in Fig. 3, where ^ is the circuitdepth. A technical effect of the warm-starting technique is to enable the quantumalgorithm to integrate the performance guarantees of the classical algorithm. In some embodiments, the continuous variables are encoded into an angle ofrotation gates of the QAOA circuit operating on qubits, where the qubits representbinary variables corresponding one-to-one to the continuous variables of theclassical optimization solver. The mixer Hamiltonian of the gates is adjusted suchthat the ground state of the mixer Hamiltonian is equal to the new initial stateobtained from the continuous variables. Each time the initial state of QAOA isupdated, the parameters ^⃗, ^⃗ are iteratively adjusted with an optimizer for variationalansatz. Returning to the simplified example, we set the parameter ^^^^^as 10, the circuitdepth ^ as 2, and the circuit parameter ℇ in Eq. 8 as 0.1. Fig.4 illustrates theoutcome of a simulation for the simple example of a Max-2-SAT problem with 10variables and ^^^^^ = 10. The graph illustrates a number of iterations an energy ofthe corresponding solution. The larger graph illustrates the situation where the classical combinatorial optimization solver is allowed to run for over 200 iterations to obtain a solution. The smaller graph is an enhancement of a subsection of the largergraph, illustrating the point where the iterations would stop. The arrow on the smallergraph indicates the point at which the classical combinatorial optimization solverstops, i.e.10 subsequent iterations have yielded minimal or no improvement in the energy of the solution.Returning to Fig. 1, to warm start the QAOA, the solutions obtained by the classicaloptimizer at the local minimum are utilized as the parameters ^^ in the QAOA circuit(see Eq. 8). The shallow depth QAOA circuit is obtained by choosing the minimumnumber of ansatz layers ^ to obtain sufficiently good ground state energy of the IsingHamiltonian, wherein a tolerance for an approximation ratio for the ground stateenergy may be tuned by the implementer. The circuit parameters ^⃗, ^⃗ are iterativelyadjusted with an optimizer for variational ansatz, and the ground state energy iscalculated by adding the local (single qubit and two qubit) expectation values, eachexpectation value obtained by contracting the tensor network of the reduced QAOAcircuit corresponding to the local expectation value. The number of ansatz layers ^remains fixed when scaling to large problem instances.By executing the QAOA, optimal values for the parameters ^⃗, ^⃗ may be determined.After providing the quantum circuit with the optimal parameters ^^⃗^^, ^^⃗^^, a single-qubit reduced density matrix is computed. The value of Ising solution ^^ corresponding to variable ^^ representing thecontinuous value of a qubit of the QAOA circuit can be obtained by computing thetrace of the density matrix over all ^^ , ^ ≠ ^ using, for example, tensor networksimulation. The values ^ represent the Ising solutions are in the range of [-1, 1], andcan be converted into the solutions of the relaxed binary problem in the range of [0, 1] (Eq. 7) using the mapping mentioned in Eq. 3.For reasons of clarity and completeness, an overview of the concept of the reduceddensity matrix is presented. Consider a quantum system whose state can bedescribed by the density matrix , and split the quantumsystem into two non-overlapping subsystems ^ and ^. The reduced density matrixof the subsystem ^ is defined as: (Eq.10) the wavefunction of the subsystem ^. To illustrate the computation ofthe reduced density matrix by tensor network contraction, we provide the following schematic figure: In Fig.5, the circles represent the initial wavefunction tensor network, and the gates correspond to the unitary operators in Eq.6. When simulating the reduced density matrix of Eq. 10 using tensor network, a significant fraction of the unitary gatescancel, as they do not contribute to the reduced density matrix. This phenomenon, known as light cone or casual cone rule, has been implemented in Quimb [4]. To obtain the optimal solutions, we are interested in the reduced density matrix of each individual qubit. For the specific qubit i, we compute ^^by retaining qubit i and tracing out the rest. The final configuration of continuous decision variables in the Ising space will be obtained by: (Eq.11) where Z is the Pauli-matrix in the Z-direction.The obtained solutions to the relaxed binary problem are then be used to initializethe classical combinatorial optimizer. The classical combinatorial optimizer mayiterate 103 ^^^^^,^^^^^^^ times (^^^^^,^^^^^^^ may be set to any natural number, by wayof example e.g. 1, 5, or 10). If the solutions of the classical combinatorial optimizerafter ^^^^^,^^^^^^^ iterations yield lower energy compared to the one obtained afterstopping the classical optimizer at the local minimum, the classical optimizer isinitiated with these solutions, iterating until the energy of the outcomes exceeds thatof the classical combinatorial optimization solver. Fig. 6 presents the average energydifference obtained from the final outcomes of the classical and the hybrid classical-quantum optimizers in 10 different simulations of this example implementation.In some embodiments, method steps 101, 102, and 103, are further iterated whenadditional local minima are encountered in step 103.Once a sufficiently low-energy solution is obtained, the method comprises initiating104 allocation of computational resources to wireless devices based on the obtainedsolution. A sufficiently low-energy solution may comprise a solution where thedistribution of computational resources is sufficiently good for the application. Alternatively or in addition, a sufficiently good solution may comprise a solution obtained after a predetermined amount of computational power has been used toobtain the solution. In particular, initiating allocation of computational resources maycomprise transmitting, to an orchestrator node, a determined low-energy allocation of servers to wireless devices.Given the general objective of minimizing energy when solving optimizationproblems, the energy difference ^^^^^^^^ − ^^^^^^^^^^^^^ represented on the y-axis ofFig. 6 serves as a metric for evaluating the performance of the method on thesimplified example. Notably, the consistently positive mean energy differencesignifies the outperformance of the method of the disclosure against the pureclassical optimizer.Fig. 7 depicts the number of QAOA iterations required to solve a combinatorialoptimization problem as a function of the number of variables in the problem. As canbe observed in Fig. 6, the method 100 outperforms a classical method using only a small number of iterations of the QAOA circuit. Additionally, the circuit depth used forthese simulations was ^ = 2, indicating that the method 100 significantly reduces therequirements on quantum hardware and / or quantum simulations compared to state- of-the-art hybrid methods.Fig. 8 depicts an energy difference between the continuous-value solutions obtainedby pure classical and hybrid classical-quantum algorithms used to solve thesimplified example. As can be inferred from Fig. 8, the energy difference grows asthe number of users increases. The method therefore results in increasingimprovements over the prior art as the problem complexity increases.Fig. 9 is a chart illustrating a number of iterations of the method 100 compared withan energy of the obtained solution. The line chart demonstrates that performing asmall number of iterations of the method 100 leads to a significant improvement over a purely classical combinatorial optimization solver, and, in particular, that scaling the number of users / wireless devices by a factor of four only requires on average doubling the number of iterations of the method 100.Fig. 10 illustrates a comparison between the method 100 and Loandra [13, 14] whichis one of the well-known MaxSAT-solvers in the “MaxSAT Evaluation 2022”

[0015] . To do this comparison, we firstly need to convert the continuous-value solutions obtained by solving relaxed problem (Eq.7). This can be achieved by defining a threshold, determined as the mean value of all continuous-value solutions: As can be seen in Fig. 10, the method 100 produces higher quality solutionscompared to both Loandra and the classical combinatorial optimization solver alone.Fig. 11 is a schematic overview of an apparatus 1100 performing methods accordingto the disclosure. The apparatus may, for example, comprise a separate computingapparatus comprised in a data center or a communication network, such as a telecommunications network or a cloud computing network. Alternatively, the apparatus may be comprised in an apparatus comprised in for example a data center or a communication network. For example, the apparatus may be comprised in a more general computing apparatus in a data center or comprised in an orchestration node of a cloud network or comprised in a network node of atelecommunications network. The apparatus comprises classical processing circuitry1101 and a memory 1102. The memory may further comprise a computer-readablestorage medium 1103 on which computer readable instructions 1104 are storedwhich, on execution by the classical processing circuitry, cause the apparatus to perform embodiments of the method 100. The apparatus has access to a classical combinatorial optimization solver 1105. In some embodiments, the classical combinatorial optimization solver is comprised inthe apparatus, for example by being implemented on a logical block of the classicalprocessing circuitry and memory. In other embodiments, the classical combinatorial optimization solver is implemented in a physical distinct computing device, or a virtual computing device separate from the apparatus.The apparatus further has access to a QAOA circuit 1106. In some embodiments,the QAOA circuit is comprised in the apparatus. In other embodiments, the QAOA circuit is comprised in a separate quantum computing device. In some embodiments,the QAOA circuit is implemented using quantum processing circuitry. In otherembodiments, the QAOA circuit is implemented as a simulation on running on a classical computer. The apparatus is comprised in a communication network. A subsection of a network in which the apparatus may be comprised is depicted in Fig.12. Fig.12 depicts a subset of a communication network 1200 comprising wireless devices 1201a, 1201b,1201c, 1202d, 1202e and servers 1202a, 1202b, 1202c. The communicationnetwork further comprises the apparatus 1100. The apparatus performsembodiments of the method 100 according to the disclosure to determine theallocation of servers 1202a, 1202b, 1202c to wireless devices 1201a, 1201b, 1201c, 1201d, 1201e. An example allocation is illustrated in Fig.12 as dashed linesconnecting servers and wireless devices. In some embodiments, the apparatus isconnected to an orchestration node 1203, which performs the allocations ofcomputational resources to the wireless devices according to the solution obtainedby the apparatus. The communication network may be a telecommunicationsnetwork according to a current telecommunications standard such as 4G Long TermEvolution, 5G New Radio, or any future standard as determined by the 3rd Generation Partnership Project, such as 6G, or any other telecommunication standardizing body, such as European Telecommunications Standards institute.

[0002] REFERENCES 1. Farhi E, Goldstone J, Gutmann S. A quantum approximate optimizationalgorithm. arXiv preprint arXiv:1411.4028.2014 Nov 14. 2. Egger DJ, Mareček J, Woerner S. Warm-starting quantum optimization.Quantum.2021 Jun 17;5:479. 3. Zhang B, Sone A, Zhuang Q. Quantum computational phase transition incombinatorial problems. npj Quantum Information.2022 Jul 22;8(1):87. 4. (Retrieved 20-09-2024)5. Ogryczak W, Pióro M, Tomaszewski A. Telecommunications network designand max-min optimization problem. Journal of telecommunications and information technology.2005:43-56. 6. Beaulieu D, Pham A. Max-cut clustering utilizing warm-start QAOA and IBMruntime. arXiv preprint arXiv:2108.13464.2021 Aug 30. 7. Gray J, Kourtis S. Hyper-optimized tensor network contraction. Quantum.2021 Mar 15;5:410. 8. Li BF, Wei W, Liu CQ. Research on the Solution Space of 2-SAT and Max-2-SAT. InITM Web of Conferences 2016 (Vol.7, p.01016). EDP Sciences. 9. Zhang W. Phase transitions and backbones of 3-SAT and maximum 3-SAT.In Principles and Practice of Constraint Programming—CP 2001: 7th International Conference, CP 2001 Paphos, Cyprus, November 26–December 1, 2001 Proceedings 72001 (pp.153-167). Springer Berlin Heidelberg. 10. Gao F, Han L. Implementing the Nelder-Mead simplex algorithm with adaptiveparameters. Computational Optimization and Applications.2012 Jan;51(1):259-77. 11. https: / / docs.scipy.org / doc / scipy / reference / optimize.minimize-neldermead.html(Retrieved 20-09-2024) 12. Lai P, He Q, Abdelrazek M, Chen F, Hosking J, Grundy J, Yang Y. Optimaledge user allocation in edge computing with variable sized vector bin packing. InService-Oriented Computing: 16th International Conference, ICSOC 2018, Hangzhou, China, November 12-15, 2018, Proceedings 162018 (pp. 230-245). Springer International Publishing. 13. Berg J, Demirović E, Stuckey PJ. Core-boosted linear search for incompleteMaxSAT. InIntegration of Constraint Programming, Artificial Intelligence, and Operations Research: 16th International Conference, CPAIOR 2019, Thessaloniki, Greece, June 4–7, 2019, Proceedings 162019 (pp.39-56). Springer International Publishing. 14. https: / / github.com / jezberg / loandra / blob / master / INSTALL. (Retrieved 20-09-2024)https: / / maxsat-evaluations.github.io / 2022. (Retrieved 20-09-2024)Wurtz, J., Sack, S. and Wang, S.T., 2024. Solving non-native combinatorialoptimization problems using hybrid quantum-classical algorithms. arXivpreprint arXiv:2403.03153.

Claims

CLAIMS 1. A method for allocating computational resources to wireless devices in a communication network, method performed by a classical computer with access to a classical combinatorial optimization solver and a quantum approximate optimization algorithm, QAOA, circuit, the method comprising: iterating (101) the classical combinatorial optimization solver on a combinatorial optimization problem representing the allocation of computational resources until the classical combinatorial optimization solver reaches a local minimum; warm starting (102) the QAOA circuit on the edge user allocation as a Boolean satisfiability, SAT, problem wherein each decision variable corresponds to a computational resource or an edge device by encoding continuous variables corresponding one-to-one to the decision variables of the SAT problem, the continuous variables obtained from the classical combinatorial optimization solver at the local minimum, into the QAOA circuit; iterating (103) the classical combinatorial optimization solver on the optimization problem using continuous values of the qubits of the QAOA circuit; and based on an obtained solution to the combinatorial optimization problem,initiating (104) allocation of computational resources to wireless devices.

2. The method (1009 according to claim 1, wherein the combinatorialoptimization problem is one of: an integer linear program; aSAT problem; ora quadratic unconstrained binary optimization (QUBO) problem.

3. The method (100) according to claim 1 or 2, wherein the allocation ofcomputational resources comprises allocating computational resources to wireless devices to maximize the number of wireless devices served by the network, whereinthe SAT problem comprises a Max-1-k-SAT problem, wherein each wireless deviceis allocated computing resources from k servers, where the Max-1-k-SAT problem is such that variables represent servers and clauses represent wireless devices.

4. The method (100) according to any one of claims 1-3, wherein thecontinuous variables are encoded into an angle of rotation gates operating on qubits, the qubits representing binary variables corresponding one-to-one to the continuous variables of the classical optimization solver.

5. The method (100) according to any one of claims 1-4, wherein the QAOAcircuit is configured by: obtaining an Ising Hamiltonian of the combinatorial optimization problem; andselecting a minimum number of ansatz layers to obtain an approximation ratio within a first threshold of a ground state energy of the Ising Hamiltonian, wherein the ground state energy is calculated as a sum of local expectation values, each local expectation value obtained by contracting a tensor network of the QAOA circuit.

6. The method (100) according to any one of claims 1-5, wherein thecontinuous value of a qubit of the QAOA circuit corresponding to a variable is obtained by computing a trace of a density matrix over all other variables by contracting the tensor network of the QAOA circuit.

7. An apparatus (1100) for allocating computational resources to wirelessdevices in a communication network, the apparatus comprising classical processingcircuitry (1101) and a memory (1102), the apparatus having access to a classicalcombinatorial optimization solver (1105) and a quantum approximate optimizationalgorithm, QAOA, circuit (1106), the apparatus configured to:iterate (101) the classical combinatorial optimization solver on a combinatorial optimization problem representing the allocation of computational resources until the classical combinatorial optimization solver reaches a local minimum; warm start (102) the QAOA circuit on the edge user allocation as a Boolean satisfiability, SAT, problem wherein each decision variable corresponds to a computational resource or an edge device by encoding continuous variables corresponding one-to-one to the decision variables of the SAT problem, the continuous variables obtained from the classical combinatorial optimization solver at the local minimum, into the QAOA circuit;iterate (103) the classical combinatorial optimization solver on the optimization problem using continuous values of the qubits of the QAOA circuit; and based on an obtained solution to the combinatorial optimization problem, initiating (104) allocation of computational resources to wireless devices.

8. The apparatus (1100) according to claim 7, wherein the QAOA circuit isimplemented using quantum circuitry.

9. The apparatus (1100) according to claim 7, wherein the QAOA circuit is simulated on classical hardware.

10. The apparatus (1100) according to any one of claims 7-9, wherein the allocation of computational resources comprises allocating computational resources to wireless devices to maximize the number of wireless devices served by the network, wherein the SAT problem comprises a Max-1-k-SAT problem, wherein each wireless device is allocated computing resources from k servers, where the Max-1-k- SAT problem is such that variables represent servers and clauses represent wireless devices.

11. The apparatus (1100) according to any one of claims 7-10, wherein the continuous variables are encoded into an angle of rotation gates operating on qubits, the qubits representing binary variables corresponding one-to-one to the continuousvariables of the classical optimization solver.

12. The apparatus (1100) according to any one of claims 7-11, wherein theQAOA circuit is configured by: obtaining an Ising Hamiltonian of the SAT problem; andselecting a minimum number of ansatz layers to obtain an approximation ratio within a first threshold of a ground state energy of the Ising Hamiltonian, wherein the ground state energy is calculated as a sum of local expectation values, each local expectation value obtained by contracting a tensor network of the QAOA circuit.

13. The apparatus (1100) according to any one of claims 7-12, wherein thecontinuous value of a qubit of the QAOA circuit corresponding to a variable is obtained by computing a trace of a density matrix over all other variables by contracting the tensor network of the QAOA circuit.

14. A computer program (1104) comprising computer-readable instructions which, when executed by processing circuitry of an apparatus, cause the apparatus to perform the method according to any one of claims 1-6.

15. A computer program product (1103) comprising a non-transient computer- readable storage medium on which a computer program (1104) according to claim 14 is stored.

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