WO2023106978A1

WO2023106978A1 - Determining a solution to an optimization problem in radio and core networks

Info

Publication number: WO2023106978A1
Application number: PCT/SE2021/051217
Authority: WO
Inventors: Ahsan Javed AWAN; Davit Petrosyan
Original assignee: Telefonaktiebolaget Lm Ericsson (Publ)
Priority date: 2021-12-07
Filing date: 2021-12-07
Publication date: 2023-06-15

Abstract

Methods and apparatus for determining a solution to an optimization problem are provided. In an example aspect, the method comprises (i) formulating the optimization problem such that the formulated optimization problem comprises at least one single association constraint, (ii) manipulating quantum states of qubits of a quantum computing device based on the formulated optimization problem, wherein manipulating quantum states of the qubits comprises applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device, (iii) obtaining a measurement of each of the one or more qubits, and (iv) based on the obtained measurements, determining a solution to the optimization problem.

Description

DETERMINING A SOLUTION TO AN OPTIMIZATION PROBLEM IN RADIO AND CORE NETWORKS

Technical Field

Examples of the present disclosure relate to determining a solution to an optimization problem, for example using a method that uses a quantum computing device.

Background

The evolution of 5G towards increased connectivity, intelligent network platforms and edge computing to accelerate latency sensitive applications may increase the complexity of existing optimization problems in radio and core networks to the extent where these problems cannot be solved efficiently using classical methods.

One example of these existing optimization problems is the problem of determining a subset of instances of a virtual machine in a wireless network, wherein each instance is associated with a cloud of the wireless network, so as to minimize the total instances of virtual machines while satisfying all instances of virtual functions of the network. This problem may be reduced to a Set Cover problem [1], which is an NP-complete problem.

In another example, the physical cell identifier (PCI) allocation problem may be reduced to a vertex coloring problem, which is also an NP-complete problem [9],

Similarly, a problem of offloading user tasks to edge servers in a network in an efficient manner can be formulated as a variable sized vector bin packing problem, which is also a NP-hard problem.

Combinatorial optimization problems include problems of offloading UEs to micro base stations within a coverage area of a macro cell to improve the quality of service [10], the channel allocation problem described in reference [8] and variants of this problem. For these problems, there are no known efficient algorithms which may be run on a classical computer to find an optimal solution. Combinatorial optimization problems in complexity class NP are today solved approximately using different kinds of heuristic algorithms on classical computers. However, the obtained solutions may be far from optimal and/or the execution time of these algorithms may be long. For example, the reference [1] uses the Branch and Bound Approach and Simulated Annealing heuristic to optimize the Set Cover problem. Although these methods may perform well for the small size network considered in reference [1], the problem belongs to the complexity class NP. Therefore, the proposed methods cannot solve this problem for larger network architectures efficiently. The quality of these algorithms will also vary inversely to the running time of the algorithm (that is, long execution times are required to obtain high quality solutions), causing large problem instances to be practically unsolvable using classical computers. Solving these optimization problems faster and more accurately would result in major efficiency improvements.

Some of the aforementioned problems may be solvable in polynomial time on emerging quantum computers that exploit the principles of quantum mechanics (for example, superposition, entanglement, etc.), provided the problem is in the complexity class bounded-error quantum polynomial time (BQP), or features an exponential speed-up compared to current exact classical algorithms. However, near term quantum computers (NISQ) have fewer noisy qubits with shorter decoherence time. Therefore, there is a need to devise techniques that account for the limitations of these NISQ devices, while enabling these computationally intensive radio access network (RAN) workloads to be solved.

Presently, manners in which quantum computing devices could be used to improve the execution time of these combinatorial optimization problems are being researched.

One area of such research is focused on using the Quantum Adiabatic Algorithm (QAA) on quantum annealers. The QAA transfers the highest energy state from one system to the highest energy state in another system given a sufficiently large annealing time. However, improving this annealing time has proven difficult, and some optimization problems are presently impossible to solve in this manner as a result of these annealing time limits [2],

Another area of such research is focused on utilizing quantum computing devices to solve combinatorial optimization problems more efficiently. One of the algorithms being utilized for this purpose is the Quantum Approximate Optimization Algorithm (QAOA).

In 2014, Fahri et.al. [2] introduced the QAOA, which can be used to determine approximate solutions to combinatorial optimization problems. The QAOA is based on the QAA, which can determine optimal solutions to an optimization problem, whereas the QAOA can determine an approximate solution to the problem. The QAOA uses a Trotterization of the time evolution used in the QAA, in order to achieve an approximate algorithm with regards to a parameter p, which is described in greater detail below.

The QAOA begins with the preparation of quantum bits (qubits) in a quantum computing device (or a quantum circuit). A qubit differs from a classical bit in the sense that, until the qubit is measured, the qubit will be in a superposition of both the state 0, and the state 1 , whereas a classical bit can only be in the state 0, or the state 1 . Qubits may also take advantage of quantum entanglement, a quantum mechanical phenomenon that allows n number of qubits to represent 2ⁿ states.

In standard QAOA, the initial quantum state that is prepared is a superposition of all possible quantum states of the qubits of the quantum computing device. For example, in reference [2], each qubit is initially be prepared in the polar state, in which the probability to measure each qubit in either the 0 state, or the 1 state, is equal.

Following the initial state preparation, a first set of unitary operations (representing an objective function of an optimization problem that the QAOA is determining a solution for), manipulate the states of the qubits. The phase of the first set of unitary operations is dependent on a first angle y. This first set of unitary operations is also known as the “cost Hamiltonian”.

Following this, a second set of unitary operations (that represent the subspace of the possible solutions to the optimization problem) manipulate the states of the qubits. In standard QAOA, the choice of this second set of unitary operations (which are typically known as the “mixer”, or the “mixer Hamiltonian”) will typically be one or more Pauli X gates. The phase of the second set of unitary operations, in a similar manner to the first set of unitary operations, dependent on a second angle p.

In this example, when determining the first and second sets of unitary operations, the objective function of the optimization problem is encoded onto an Ising Hamiltonian [3, 4], The Ising Hamiltonian is a mathematical model consisting of lattices of sites. Each site Sj can be either in the state -1 , or the state +1. The Ising model can be transformed to a physical model, by assuming that each site is an atom with a magnetic spin moment -1 , or +1. The resulting Hamiltonian (that is, both the cost Hamiltonian and the mixer Hamiltonian) will then be the sum of the coupling strength between spin i and j, and the external magnetic field acting on spin i. The Ising model is closely related to a quadratic unconstrained binary optimization (QU BO) model [5] and is therefore suited for the encoding of optimization problems. By assuming that the magnetic spin i is aligned with the z-axis on the Bloch-sphere, each spin Sj can be represented by the Pauli Z matrix. The Ising Hamiltonian will then describe the quantum states of the qubits of the quantum computing device, that represent the binary decision variables of the optimization problem, and can be transformed by the unitary operations that are run in the QAOA.

Following the manipulation of the states of the qubits, a measurement of each qubit is performed. For each qubit, the measurement is a projection of the state of each qubit onto the computational basis 0 or 1. Depending on how the qubits were manipulated by the first and second sets of unitary operations, the probability of measuring each qubit in either the state 0, or the state 1 will vary. Thus, this algorithm requires many repetitions to ensure that the approximate solutions are reliable.

The angles y and p are the used to generate an energy landscape. An energy landscape illustrates how the expected cost of the objective function (that represents the optimization problem) varies depending on the values of y and that are used to execute the QAOA as described above. The expected cost of the objective function will correspond to the probability of the optimal solution being obtained when executing the QAOA in accordance with those corresponding values of y and p.

The generated energy landscape is then optimized over the angles y and p, to determine either the lowest expected cost (if the solution to the optimization problem aims to minimize the value of the objective function), or the maximum expected cost (if the solution to the optimization problem aims to maximize the value of the objective function). Following this determination, a probability of obtaining the optimal solution to the optimization problem when using the corresponding values of y and p when executing the QAOA may be determined.

The QAOA may then be executed using these corresponding values of y and p, which will result in the greatest probability of the QAOA returning an optimal solution to the optimization problem. Furthermore, when executing the QAOA, the first and second sets of unitary operations can be applied (that is, manipulate the states of the qubits) p times prior to the aforementioned measurements being performed. The probability of obtaining the optimal solution from the measurement will then increases with p. The variable p comes from a Trotterization of the time evolution of the Hamiltonian in the QAA, where a higher value of p approximates the time evolution more precisely [2],

It is noted that for the QAOA proposed in [2], the total subspace encoded onto the qubits of the quantum computing device may contain infeasible solutions to the optimization problem, as a result of the choice of the initial state of the qubits and the mixer. Infeasible solutions to the optimization problem include solutions that do not meet the constraints of the optimization problem, for example.

Hadfield et.al [3] investigated a set of problems, including coloring and graph problems, and proposed different initial states and mixers for solving these sets of problems using a QAOA. These different initial states and mixers reduced the subspace of possible solutions that could be obtained when executing the QAOA, resulting in a higher probability of the optimal solution to the problems being obtained.

Summary

A first aspect of the present disclosure provides a method of determining a solution to an optimization problem. The method comprises:

(i) formulating the optimization problem such that the formulated optimization problem comprises at least one single association constraint;

(ii) manipulating quantum states of qubits of a quantum computing device based on the formulated optimization problem, wherein manipulating quantum states of the qubits comprises applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device;

(iii) obtaining a measurement of each of the one or more qubits; and

(iv) based on the obtained measurements, determining a solution to the optimization problem.

Another aspect of the present disclosure provides an apparatus for determining a solution to an optimization problem. The apparatus comprises a processor and a memory. The memory contains instructions executable by the processor such that the apparatus is operable to: (i) form the optimization problem such that the formulated optimization problem comprises at least one single association constraint;

(ii) manipulate quantum states of qubits of a quantum computing device based on the formulated optimization problem, wherein manipulating quantum states of the qubits comprises applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device;

(iii) obtain a measurement of each of the one or more qubits; and

(iv) based on the obtained measurements, determine a solution to the optimization problem.

A further aspect of the present disclosure provides an apparatus for determining a solution to an optimization problem. The apparatus is configured to:

(i) form the optimization problem such that the formulated optimization problem comprises at least one single association constraint;

(iii) obtain a measurement of each of the one or more qubits; and

Embodiments of the present disclosure aim to improve the quality of the solutions obtained from QAOAs. It is noted that particular structure present in an optimization problem, or particular properties of an optimization problem, may be exploited to improve the execution time of a QAOA, and/or improve the quality of a solution obtained from the QAOA.

For example, the quality of the solutions obtained from QAOA may be improved by expressing one of the constraints of the optimization problem (that the QAOA is determining a solution for) in the form of swap mixer [3], Such an expression improves the quality of the obtained solution under the noise of near-term quantum computers. Furthermore, observed symmetry in a generated energy landscape may be exploited by allowing the angle search space to be restricted, both improving the execution time of a QAOA and increasing the likelihood of the optimal solution being determined by the QAOA.

Brief Description of the Figures

For a better understanding of examples of the present disclosure, and to show more clearly how the examples may be carried into effect, reference will now be made, by way of example only, to the following Figures in which:

Figure 1 is a flow chart of an example of a method 100 of determining a solution to an optimization problem;

Figure 2 is a flow chart of an example of a further method 200 of determining a solution to an optimization problem;

Figure 3 shows an example of a Cloud Radio Access Network (CRAN) 300;

Figures 4a and 4b illustrate examples of energy landscapes;

Figures 5a and 5b illustrate examples of energy landscapes;

Figures 6a and 6b illustrate example quantum circuits 600a, 600b respectively;

Figures 7a and 7b show the probabilities of finding an optimal solution to the optimization problem for various problem instances;

Figure 8a shows the probabilities of determining an optimal solution to the optimization problem for a 6-qubit problem instance;

Figure 8b shows the probabilities of determining an optimal solution to the optimization problem for a 8-qubit problem instance;

Figure 9 shows an example of a network 900 comprising a mobile edge computing system; Figure 10a-10e illustrate examples of energy landscapes;

Figure 11 shows the probabilities of determining an optimal solution for a 3-qubit problem instance;

Figures 12a and 12b show the probabilities of determining an optimal solution for the four different problem instances; and

Figure 13 is a schematic of an example of an apparatus 1300 for determining a solution to an optimization problem.

Detailed Description

The following sets forth specific details, such as particular embodiments or examples for purposes of explanation and not limitation. It will be appreciated by one skilled in the art that other examples may be employed apart from these specific details. In some instances, detailed descriptions of well-known methods, nodes, interfaces, circuits, and devices are omitted so as not obscure the description with unnecessary detail. Those skilled in the art will appreciate that the functions described may be implemented in one or more nodes using hardware circuitry (e.g., analog and/or discrete logic gates interconnected to perform a specialized function, ASICs, PLAs, etc.) and/or using software programs and data in conjunction with one or more digital microprocessors or general purpose computers. Nodes that communicate using the air interface also have suitable radio communications circuitry. Moreover, where appropriate the technology can additionally be considered to be embodied entirely within any form of computer- readable memory, such as solid-state memory, magnetic disk, or optical disk containing an appropriate set of computer instructions that would cause a processor to carry out the techniques described herein.

Hardware implementation may include or encompass, without limitation, digital signal processor (DSP) hardware, a reduced instruction set processor, hardware (e.g., digital or analogue) circuitry including but not limited to application specific integrated circuit(s) (ASIC) and/or field programmable gate array(s) (FPGA(s)), and (where appropriate) state machines capable of performing such functions. It will be appreciated that the terms user equipment and device may be used interchangeably throughout the disclosure. Furthermore, the quantum circuit and quantum computing device may be used interchangeably throughout the disclosure.

For example, the quality of the solutions obtained from QAOA may be improved by expressing one of the constraints of the optimization problem (that the QAOA is determining a solution for) in the form of swap mixer [3], Such an expression improves the quality of the obtained solution under the noise of near-term quantum computers.

Furthermore, observed symmetry in a generated energy landscape may be exploited by allowing the angle search space to be restricted, both improving the execution time of a QAOA and increasing the likelihood of the optimal solution being determined by the QAOA.

In order to improve the quality of the solution obtained from a QAOA for a particular optimization problem, a single association constraint of the optimization problem may be encoded as part of the mixer Hamiltonian, through the use of a swap mixer. The swap mixer , that is comprised within a second set of one or more unitary operations that represent a subspace of possible solutions to the optimization problem, is a quantum circuit that preserves the Hamming Distance of 1 among the quantum states. An example implementation of a swap mixer is described at reference [11],

A single association constraint may be of the form:

where x^ is a binary decision variable, and x^ = 1 if an entity i within the set N is associated to an entity j within the set M. In other words, when a single association constraint is met, for each member of a first set, there will only be one association between the member of the first set, and the second set.

Therefore, for a subset of binary variables to which a single association constraint applies, the Hamming weight of the values of these binary variables (that are determined as part of a solution to the optimization problem) must be equal to one.

In some examples, by utilizing a swap mixer in the second set of unitary operations (which are applied to the qubits representing the subset of binary variables to which the single association constraint applies), the swap mixer will then preserve the Hamming weight of the states of the qubits when manipulating the states of the qubits. That is, the swap mixer will only swap between possible states of the qubits (that correspond to the subset of binary variables) that collectively obey the single association constraint, as the swap-mixer forces the Hamming weight of the states of the qubits to be conserved while the mixing is executed.

Furthermore, as the methods utilizing a swap mixer in this manner in some examples search over a smaller subspace of possible solutions to the optimization problem, these methods may be more resistant to noise present in the quantum circuit, resulting in a greater probability that the method will return the optimal solution to the optimization problem in noisy conditions (as infeasible solutions to the problem are not even being searched over).

An example method utilizing this structural property of certain optimization problems is now described.

Figure 1 is a flow chart of an example of a method 100 of determining a solution to an optimization problem. The method comprises, in step 102, formulating the optimization problem such that the formulated optimization problem comprises at least one single association constraint. Step 104 of the method 100 comprises manipulating quantum states of qubits of a quantum computing device based on the formulated optimization problem, wherein manipulating quantum states of the qubits comprises applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device. Step 106 of the method 100 comprises obtaining a measurement of each of the one or more qubits. Step 108 comprises, based on the obtained measurements, determining a solution to the optimization problem.

In some embodiments, the at least one single association constraint may be of the form:

where x_ij is a decision variable, and x_ij = 1 if an entity i within the set N is associated to an entity j within the set M.

In some embodiments, the formulated optimization problem may further comprise at least one constraint that comprises one or more variables that are represented with binary encoding. As will be explained in greater detail below, representing one or more variables using binary encoding may enable the optimization problem to be represented using fewer binary variables. In such cases, there will be fewer values of these binary variables that then need to be determined in order to determine a solution to the optimization problem. As such, fewer qubits (where each qubit represents one of the binary variables) will need to be utilized in the quantum computing device in order to determine a solution to the optimization problem on execution of the method 100. This may then reduce amount of noise introduced when solving the optimization problem using the quantum computing device.

In some embodiments, an initial state of the qubits of the quantum computing device may correspond to a solution to the optimization problem that obeys the at least one single association constraint. In some embodiments, where each qubit of the quantum computing device represents a binary variable of which a value is to be determined as part of the solution to the optimization problem, performing a measurement on each of the qubits will allow a bit string to be obtained. For each qubit, the measurement will determine the qubit as being in either a first state representing a 0, or a second state representing a 1. The obtained bit string comprised of these determined 0s and 1s will then represent a value for each of the binary variables that these qubits of the quantum computing device respectively correspond to, therefore providing a solution to the optimization problem. By manipulating the qubits of the quantum computing device such that they initially represent a solution to the optimization problem that meets the single association constraint (that is, the initial states of the qubits collectively represent values of the binary variables that obey the single association constraint), this enables the swap mixer to then switch between this initial feasible solution to the optimization problem, and the other feasible solutions to the optimization problem that also obey the single association constraint. As a result of the aforementioned Hamming weight preservation, the swap mixer will not switch between states of these qubits that do not obey the single association constraint.

In some embodiments, manipulating quantum states of the qubits may comprise applying a first set of one or more unitary operations that represent the optimization problem to one or more qubits of the quantum computing device, in a similar manner as described above.

In some embodiments, applying the first set of one or more unitary operations may manipulate the quantum states of the qubits based on one or more constraints of the optimization problem. For example, the first set of one or more unitary operations may correspond to a cost Hamiltonian for the optimization problem. In some embodiments, the first set of unitary operations may be based on a first angle. For example, the first set of unitary operations may be based on the angle y, in a similar manner as described above.

In some embodiments, the swap mixer may be comprised within a second set of one or more unitary operations that represent a subspace of possible solutions to the optimization problem. For example, the second set of one or more unitary operations may correspond to a mixer Hamiltonian for the optimization problem. In some embodiments, the application of the swap mixer to the one or more of the qubits of the quantum computing device may restrict the subspace of possible solutions to the optimization problem that may be determined using the quantum computing device to a subspace of possible solutions that meet the at least one single association constraint. That is, utilizing the swap mixer in this manner may reduce the subspace over which the method 100 searches over to in order to determine a solution to the optimization problem. In some embodiments, the second set of unitary operations may be based on a second angle. For example, the second set of unitary operations may be based on the angle p, in a similar manner as described above.

In some embodiments, the step of manipulating quantum states of the qubits may comprise:

(a) applying the first set of unitary operations to one or more of the qubits of the quantum computing device,

(b) applying the second first set of unitary operations to one or more of the qubits of the quantum computing device, and

(c) repeating steps (a) and (b) at least once.

As noted above, repeating the steps (a) and (b) in this manner (in other words, increasing the value of the variable p when executing the method 100) may increase the probability of determining an optimal solution to the optimization problem when executing the method 100.

As will be explained in greater detail below, in some embodiments, the optimization problem may represent a problem of determining a subset of instances of a virtual machine in a wireless network, wherein each instance is associated with a cloud of the wireless network to which one or more base stations in the wireless network can connect, so as to minimize a latency between the one or more base stations and the clouds of the wireless network.

As will be explained in greater detail below, in some embodiments, the optimization problem may represent a problem of determining a subset of edge servers in a wireless network to which one or more user equipments, UEs, in the wireless network can connect, so as to maximize a measure of efficiency of the wireless network. In some embodiments, the measure of efficiency represents the number of UEs in the network assigned to the subset of edge servers over the total number of edge servers within the subset.

It will be appreciated that in some examples both the optimization problem that represents a problem of determining a subset of instances of a virtual machine in a wireless network, and an optimization problem that represents a problem of determining a subset of edge servers in a wireless network, can be formulated such that they comprise at least one single association constraint.

Examples of further optimization problems that may be solved according to the method 100 include: user association in 5g heterogenous networks, physical cell ID allocation, and sub channel allocation in device-to-device communication.

Figure 2 is a flow chart of an example of a further method 200 of determining a solution to an optimization problem.

The method comprises, in step 202, formulating a binary integer linear programming (BILP) problem that represents the optimization problem.

Generally, an optimal solution to a BILP problem can be found by determining the values of binary variables that maximize or minimize (depending on the optimization problem that is to be solved) the value of an objective function of the BILP problem, where the objective function comprises the binary variables.

In such an embodiment, the qubits of the quantum computing device (as described with reference to method 100) represent these binary variables, and the measured states of these qubits correspond to the values of these binary variables for the obtained solution to the optimization problem.

In some embodiments, the BILP problem may comprise a problem of determining a solution to the optimization problem that maximizes or minimizes a value of an objective function, wherein the objective function comprises at least one single association constraint. As noted above, whether the problem aims to determine a solution to optimization problem that maximizes or minimizes the value of the objective function will depend on the optimization problem itself, and the formulation of the objective function.

It will be appreciated that, in embodiments in which a BILP problem that represents the optimization problem has been formulated, the first set of unitary operations may represent the objective function of the BILP problem. In some embodiments, the formulated optimization problem may further comprise at least one constraint comprising one or more variables that are represented with binary encoding, in a similar manner as described above.

At step 204, the method 200 comprises determining whether the optimization problem (in this example the BILP problem) comprises a single association constraint. In response to, at step 204, determining that the optimization problem does comprise a single association constraint, the method proceeds to step 214.

Some examples of single association constraints in optimization problems relating to telecommunication networks, and of constraints that may be reformulated to form a single association constraint in optimization problems relating to telecommunication networks, include: enforcing that, during virtual network function placement in cloud radio network, each RRH is handled by a baseband unit pool over the cloud [1], enforcing that, in edge user allocation, each user is to be assigned to a server [6], enforcing that, in joint optimization of user association and cell activation in 5G heterogenous networks, each user in the macro cell can be served by one cell [10], enforcing that, in sub channel allocation, a transmitter is allocated at most one subchannel [8], enforcing that, in physical cell ID allocation, a cell cannot be allocated with more than one PCI [9], enforcing that, while placing microservices in mobile edge computing systems, each microservice must be assigned to at least one edge node.

In response to, at step 204, determining that the optimization problem does not comprise a single association constraint, the method 200 proceeds to step 206. At step 206, the method 200 comprises determining whether a constraint of the optimization problem can be reformulated as a single association constraint. In response to, at step 206, determining that a constraint of the optimization problem can be reformulated as a single association constraint, the method 200 proceeds to step 208.

At step 208, the method 200 comprises reformulating the constraint as a single association constraint. For example, the constraint of enforcing that, in sub channel allocation, a transmitter is allocated at most one sub-channel may be reformulated as a constraint of enforcing that a transmitter is allocated to one sub-channel. In another example, the constraint of enforcing that, while placing microservices in mobile edge computing systems, each microservice must be assigned to at least one edge node, may be reformulated as a constraint of enforcing that each microservice must be assigned to one edge node.

In some embodiments, step 206 may comprise recognizing that the optimization problem comprises a constraint that defines that a total weight of a mapping between two sets of variables may be: equal to or greater than 1 , or less than or equal to 1. In these circumstances, a constraint of this form may then be reformulated to limit that constraint to one which defines that the total weight of the mapping must be equal to 1 . It will be appreciated that a solution that meets this reformulated constraint also corresponds to a solution that meets the original constraint.

In response to, at step 206, determining that no constraint of the optimization problem can be reformulated as a single association constraint, the method 200 proceeds to step 210. At step 210, the BILP problem is a transformed to form a quadratic unconstrained binary optimization, QllBO, model that represents the objective function, or an Ising model that represents the objective function.

In some embodiments, the step 210 comprises the steps 210a and 210b. In step 210a, the method 200 comprises converting the inequality constraints of the BILP problem to equality constraints using binary slack variables.

In step 210b, the method 200 comprises scaling the quadratic form of each equality constraint with a penalty, and, for each scaled constraint, adding the scaled constraint to the objective function to form a QllBO model. Step 210b then further comprises encoding the formed QllBO model onto a Hamiltonian using an Ising model, in a similar manner as described above.

At step 212, the transformed optimization problem is solved. For example, where the transformed optimization model comprises a QllBO model, the transformed optimization model may be encoded onto and solved using a quantum annealer. In another example, where the transformed optimization model comprises an Ising model, the transformed optimization model may be encoded onto and solved using a quantum computing device. In some embodiments, the QllBO model may also be solved using an Ising Processing Unit (IPU), or an FPGA. Referring again to step 208, following the execution of step 208, the method 200 proceeds to step 214. At step 214, the method 200 comprises transforming the updated BILP problem to form a quadratic unconstrained binary optimization, QllBO, model that represents the objective function, or an Ising model that represents the objective function. In some embodiments, the step 214 comprises the steps 210a and 210b, as described above.

At step 216, the method 200 comprises encoding the one or more single association constraints in the mixer Hamiltonian using one or more swap-mixers. The swap mixers may then be applied to the qubits of the quantum computing device that represent the subset of binary variables to which the single association constraint applies. Therefore, step 216 may comprise applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device.

At step 218, the method 200 comprises determining whether there is a symmetry present in an energy landscape.

In some embodiments, generating the energy landscape may comprise: determining a set of values of the first angle; determining a set of values of the second angle; and for each value of the set of values of the first angle: for each value of the set of values of the second angle: performing steps 104 to 108 of the method 100 to determine a solution to the optimization problem; and based on the determined solution, determine a value of the objective function.

As noted above, the energy landscape illustrates how the expected cost of the objective function (that represents the optimization problem) varies depending on the values of y and β that are used when executing the method 100 or 200. A symmetry may be present in the energy landscape in the sense a subsection of the energy landscape may repeat within the total generated energy landscape. It will be appreciated that these repetitions may be rotated and/or translated throughout the energy landscape. In other words, in some embodiments, determining that a symmetry exists in the energy landscape may comprise analyzing the energy landscape, for example by performing spatial analysis on the energy landscape.

In some embodiments, it may be determined that a symmetry is present in the energy landscape based on the properties of the operators that are to be applied to the qubits of the quantum computing device.

For example, in some embodiments, the step 216 may comprise, when the first set of one or more unitary operations represent an Ising model that represents the objective function: determining whether the Ising model contains real integers, determining whether the first set of unitary operations and the second set of unitary operations are commutative, and in response to a determination that the Ising model contains real integers, and a determination that the first set of unitary operations and the second set of unitary operations are commutative: determining that there is a symmetry of the determined values of the objective function within the space of the determined values.

At step 220, the method 200 comprises, in response to determining that a symmetry is present in the energy landscape, restricting the angle search space in the method 200. It will be appreciated that, in embodiments in which the angle search space is restricted to the subset of the energy landscape that is determined to repeat over the total energy landscape, such a restriction will prevent the repeating elements of the energy landscape from being repeatedly searched over.

In some embodiments, the step 220 may comprise, in response to a determining that there is a symmetry of the determined values of the objective function within the space of the determined values: determining the set of values of the first angle based on the determination; and determining the set of values of the second angle based on the determination. In other words, the set of values of the first angle, and the set of values of the second angle, may be restricted such to only correspond to a repeating portion of the energy landscape.

The method 200 then moves from step 220 to step 222. In response to determining that there is no symmetry present in the energy landscape at step 216, the method 200 will also move to step 222.

At step 222, the method 200 comprises determining the ground state of the formed Ising model (that is represented by the operations that are acting on the qubits of the quantum computing device), in a similar manner as described above. By determining the ground state, and performing the measurements described with reference to method 100, a bit string representing a solution to the optimization problem may be determined.

For example, in some embodiments, the step 222 comprises based on the objective function, determining a maximum or minimum value of the determined values, and performing steps 104 to 108 of the method 100, based on the value of the first angle and the value of the second angle corresponding to the determined value, to determine a solution to the optimization problem.

In some embodiments, an initial state of the qubits of the quantum computing device corresponds to a solution to the optimization problem that obeys the at least one single association constraint.

At step 224, the method 200 comprises decoding an obtained bit string to obtain a solution to the optimization problem.

As noted above, each bit of the obtained bit string represents a value of the binary variables which the qubit (that was measured to obtain the value of the bit) represents, and therefore, the bit string provides a solution to the optimization problem. That is, step 224 comprises, based on the obtained measurements, determining a set of values of a plurality of binary variables of the BILP problem, wherein the set of values represent a solution to the optimization problem. A method of generally optimizing a binary integer linear program on a quantum computer is now described:

Firstly, a relevant optimization problem is formed as a BILP, and following this, it is determined if the BILP comprises a single association constraint of the form:

where is a decision variable, and x^ = 1 if an entity i is associated to entity j. That is, the single association constraint restricts that every entity i in N is associated to exactly one entity j in M.

If no single association constraints exists, it is determined whether any of the constraints of the optimization problem can be transformed to a single association constraint. Following this, the BILP formulation is updated accordingly.

The inequality constraints of the BILP formulation are then converted to equality constraints using slack variables represented with binary encoding (rather than one-hot encoding) to reduce the number of qubits required to solve the optimization problem. The constraints are then scaled with appropriate Lagrangian multipliers and added as penalty terms to the objective function. The updated objective function is then mapped to a QUBO or Ising model.

Following this, the single association constraint is encoded in the mixer Hamiltonian using one or more swap-mixers.

Following this, it is determined if the generated energy landscape comprises any potential symmetries. If a symmetry exists, it is then exploited by restricting the angle search space.

The ground state to the Ising model is then found using a quantum computing device, executing a QAOA with restricted angle search space.

The ground state is returned as a bit string on measuring the qubits of the quantum computing device, and the bit string is then decoded to return a solution to the optimization problem. The methods 100 and 200 may be utilized to determine solutions to the following optimization problems described with reference to Figures 3 and 9.

Figure 3 shows an example of a Cloud Radio Access Network (CRAN) 300.

In the CRAN 300, low energy base stations (BSs) 302a, 302b, 302c, 302d are deployed over a geographical area. Each BS 302a, 302b, 302c, 302d is respectively connected to a cloud 304a, 304b, 304c via a finite capacity backhaul link. These BSs 302a, 302b, 302c, 302d then serve one or more user equipments, UEs, in the wireless network 300.

Baseband processing unit (BBU) functions are implemented on virtual machines (VMs) in the cloud 304a, 304b, 304c over commodity hardware. These BBU functions (which are built-in software) are termed virtual functions (VFs). By placing the VFs in the CRAN 300 in an optimal manner, delays experienced by the end-users of the CRAN 300 will be reduced. These delays may depend on the Remote Radio Head (RRH) to cloud 304a, 304b, 304c allocation in the CRAN 300, and/or may depend on the service placement over the resource pool, for example.

It will be appreciated that a non-optimal allocation of the VFs in the CRAN 300 may result in unacceptable delays to the end-users of the CRAN 300, and/or may result in QoS and cost violations, and/or may not fully take advantages of the CRAN 300 architecture. There is therefore a need for efficient algorithms that are capable of mapping BBU service requirements in the CRAN 300, to the available virtual resources in the CRAN 300, which as a result minimize the end-to-end delays to the end users of the CRAN 300.

Existing algorithms presently allocate service demands at base stations 302a, 302b, 302c, 302d, in the form of VFs, to VMs across multiple clouds 304a, 304b, 304c in the network 300, in order to minimize either the response time or the latency to users. These allocations also satisfy cost constraints, capacity constraints and placement constraints [1] in the CRAN 300. For example, the maximum number of instances of a VM which may be deployed in each cloud 304a, 304b, 304c is bounded by the capacity of that cloud 304a, 304b, 304c and demands of the VM. In another example, the minimum number of VMs that need to be deployed on a particular cloud 304a, 304b, 304c for a particular service is bounded by the fraction of the total client traffic from all the sites assigned to that cloud 304a, 304b, 304c. That is, in this example, the optimization problem may feature several constraints relating to cloud capacity, VM capacity, link delays, queuing constraints, cost threshold, SLAs for response time, and an integrity constraint.

Therefore, in this embodiment, the optimization problem represents a problem of determining a subset of instances of a virtual machine in a wireless network, wherein each instance is associated with a cloud of the wireless network to which one or more base stations in the wireless network can connect, so as to minimize a latency between the one or more base stations and the clouds of the wireless network.

In this embodiment, the optimization problem is formulated as a binary integer linear programming, BILP, problem that represents the optimization problem, wherein the BILP problem comprises a problem of determining a solution to the optimization problem that minimizes a value of an objective function, and wherein the objective function comprises at least one single association constraint.

The objective function may be formulated as follows:

where the binary variable represents if a BS i is being handled by a cloud j, where xtj = 1 if a BS j is being handled by a cloud j, and x^ = 0 otherwise, n is the number of BSs in the network, m is the number of clouds in the network,

T_tj is the communication delay, and and Cj is the computational delay.

The first constraint of the aforementioned optimization problem enforces that, for the determined solution, every BS 302a, 302b, 302c, 302d is to be handled by a cloud 304a, 304b, 304c. The first constraint may be formulated as follows:

It will be appreciated that this first constraint is a single association constraint.

The second constraint of the optimization problem is a cloud capacity constraint. The cloud capacity constraint ensures that a cloud 304a, 304b, 304c in the network 300 will not be overloaded in the determined solution, by restricting the number of VMs that can be available on that cloud 304a, 304b, 304c. The second constraint may be formulated as follows:

where:

I^l _j is a decision variable representing how many VMs of type 1 are available on cloud j, δ₁ is the demand of VM 1, and

K_j is the resource capacity of cloud j.

It will be appreciated that the second constraint is an inequality constraint. Prior to adding the second constraint to the objective function, the second constraint may be reformulated as an equality constraint through the introduction of slack variables as follows:

where y^p is a binary decision variable indicating how much of the capacity of cloud j is being used.

Encoding these binary variables in this manner is known as one-hot encoding in some examples. It will be appreciated that, for this type of encoding and for large values of Kj, an equivalent large number of binary variables will need to be included within the BILP formulation of the optimization problem.

Therefore, when such an encoding is utilized when executing a QAOA, a large number of qubits will be required to represent the large number of binary variables. This may result in an increase in the computation time required to execute the QAOA on the quantum computing device comprising these qubits.

It will also be appreciated that, when executing a QAOA that utilizes such an encoding on a real (as opposed to a simulated) quantum computing device (that is, a quantum computing device that features noise), the increased number of qubits will also increase the overall noise in the circuit.

Therefore, in this example, in order to reduce the number of binary variables from Kj to log₂ Kj, the cloud capacity is instead encoded using binary encoding in the reformulated second constraint [4]:

The third constraint of the optimization problem is the VM capacity constraint. The VM capacity constraint ensures that the data traffic from the BSs 302a, 302b, 302c, 302d does not overload the VMs. The third constraint may be formulated as follows:

where:

Δj is the demand for BS i per byte of traffic,

Wj is the traffic generated by BS i in number of packets, and

Kj is the capacity of VM 1.

It will be appreciated that the third constraint is an inequality constraint. In this example, the third constraint is added to the objective function by reformulating the third constraint as an equality constraint, through the introduction of slack variables. In this example, binary encoding (as described above) is also used when reformulating the third constraint, to reduce the number of binary decision variables comprised within the BILP formulation of the optimization problem. The third equality constraint may then be reformulated as follows:

The first constraint, the reformulated second constraint and the reformulated third constraint are then added to the objective function:

where P_1; P₂, P₃, P₄ are penalty terms that respectively scale the components of the objective function. In this example, the penalty terms were selected following multiple simulations to determine a solution to the optimization problem.

P₄ = 2.

In this example, the penalty term P₂ was selected such that it is more expensive to set every binary variable to be 0 in a potential solution, than it is to have every BS 302a, 302b, 302c, 302d connected to a cloud 304a, 304b, 304c in a potential solution.

In this example, P₃ was selected such that it is favored to use more clouds in a potential solution, over overloading too few clouds in a potential solution.

In this example, P₄ was selected so as to slightly increase the expense of overloading the VMs in a potential solution to the optimization problem.

This objective function is then transformed to a quadratic unconstrained binary optimization, QUBO, problem or model.

A QUBO problem is defined for example using an upper triangular matrix Q of size NxN with N being the number of logical qubits and a binary vector as

minimizing the following energy function E:

where x_i ∈ {0,1} and Qij ∈ R, i,j = 1,2, . . , , N. Each output decision variable x_; represents the measured value (classical) of a logical qubit. The main diagonal elements Qj; of the matrix Q are the linear coefficients in the function E which represents the qubit biases, and the off-diagonal elements Qjj of the matrix Q are the quadratic coefficients in the function E which represents the coupling strengths between neighboring qubits. Note that,

This can be expressed more concisely in some examples as:

The QU BO model was formed through evaluating the squares and calculating the matrix Q. The evaluations of the squared terms are provided below. The formed QllBO model comprises the following constraint, which ensures that every BS 302a, 302b, 302c, 302d in the network 300 is connected to a cloud 304a, 304b, 304c:

The formed QU BO model further comprises the following cloud capacity constraint:

The formed QU BO model further comprises the following VM capacity constraint:

The formed QllBO model further comprises the following binary variables:

In this example, the QllBO model is then transformed and encoded onto a Hamiltonian using the Ising model, in a similar manner as described above [5],

The ground state of the Ising model, which contains an optimal solution to the optimization problem, is then approximated using a QAOA (as described in [2, 3]).

In this example, the QAOA was implemented (for the formed Ising model) for four problem instances, respectively using 4, 6, 8 and 12 qubits. Each of these problem instances were simulated for p = 1, 2, 3, and optimized over the resulting energy landscapes to determine the optimal angles of p and y, and calculated the probability of determining the optimal solution associated with these determined angles. ,or the generated energy landscapes, differential evolution was used to determine the optimal angles, using a maximum of 1000 iterations. This maximum corresponds to 60000 function evaluations (that is, the number of times that the quantum circuit is executed).

In accordance with the method 100 of Figure 1 , a swap mixer may be utilized when executing the QAOA, to restrict the subspace of possible solutions to the optimization problem that may be determined using the quantum computing device to a subspace of possible solutions that meet the at least one single association constraint.

In other words, in this example, the search space of the QAOA may be limited by the use of a swap mixer for each set of i in the constraint

As noted above, in standard QAOA, the σ_x-mixer is used to search over the entire space of possible solutions. In other words, the entire space of possible bit strings that may be returned by the quantum computing device is searched over, including those which do not meet the single association constraint.

However, if only possible solutions to the optimization problem that meet the single association constraint are considered, the search space of possible solutions may be limited to j possible states, rather than of 2^j for each i.

Considering the single association constraint for an instance of the optimization problem with one BS and three clouds, the three binary variables X11,X12 X13 may be used to represent which of the three clouds the BS is connected to. The three possible solutions that then meet the single association constraint are as follows:

_X11 = l,x₁₂ = 0,x₁₃ = 0,

_X11 = o,x₁₂ = l,x₁₃ = 0,

_X11 = 0,x₁₂ = 0,x₁₃ = 1, That is, only 3 states, out of the 2³ states that these binary variables could encode, provide solutions to the optimization problem that meet the single association constraint.

In this example, when the swap mixer is utilized as part of executing the QAOA, a feasible initial state of the qubits (that represent the binary variables) chosen that corresponds to one of the solutions to the optimization problem that meets the single association constraint (for example x₁₁ = 1,x₁₂ = 0, x₁₃ = 0). Following this initialization, on executing the QAOA, the swap mixer will swap between the possible states that correspond to the solutions that meet the single association constraint, as the swap mixer will conserve the Hamming weight of the states which are being swapped over.

It is noted that it also possible to encode the binary variable Xjj for each i with binary encoding, which would result, in this example, in the same number of states being searched over as when utilizing the swap mixer as described above. However, it will be appreciated that it is more difficult the encode the interactions between the binary variables on the qubits of the quantum computing device when utilizing binary encoding in this manner.

Simulations using the swap mixer for the aforementioned four problem instances were then repeated and compared to corresponding simulations with a σ_x-mixer. Following these simulations, properties including circuit depth, and the number of gates in the circuits, were compared. Simulations with noise were then also repeated for the 6- and 8-qubit problems instances. The simulations with noise were executed using a backend that model an IBM quantum computer, and the obtained results were compared with the results from the ideal simulations.

Figures 4a and 4b illustrate examples of the energy landscapes (that is, the expected cost of the objective function corresponding to each value of p and y obtained following simulations of the 6-qubit problem with the σ_x- and swap mixers respectively, for p = 1.

Figures 5a and 5b illustrate examples of the energy landscapes (that is, the expected cost of the objective function corresponding to each value of p and y) obtained following simulations of the 8-qubit problem with the σ_x- and swap mixers respectively, for p = 1. It is noted that for p > 1, the energy landscapes cannot be visualized in this manner, but the symmetry described below also applies to these instances.

As shown in Figures 4a, 4b, 5a and 5b, each of the illustrated energy landscapes are symmetric.

As a result of the symmetries present in the energy landscapes, when optimizing over each energy landscape, the optimization may be performed over a smaller part of the energy landscape. As a result, the computation time for the optimization process may be reduced, as the redundancy of optimizing over a repeating part of the energy landscape is removed.

As illustrated in Figures 4b and 5b, the energy landscapes obtained when performing simulations that utilize the swap mixer feature more sudden changes in gradient over the landscape. This larger change in gradient present in the energy landscape may make the optimization process (that is aiming to find the global minimum/maximum of the expected cost) more difficult. For example, to compensate for the larger change in gradient, the step size utilized in the optimization process may need to be reduced, which would make the process more computationally expensive. It is also noted that, generating an energy landscape with a smaller step size between the values of p and y would allow improved conclusions to be drawn over the energy landscape, at the tradeoff of computational expense.

However, as shown in Figure 5b, the expected cost of the objective function also reaches a lower global minimum when the swap mixer is utilized. As a result, the values of p and Y that correspond to this lower global minimum will result in an increased probability of the optimal solution to the optimization problem being obtained, when the QAOA is executed in using these corresponding angles.

Figures 6a and 6b illustrate example quantum circuits 600a, 600b for implementing the 6-qubit problem instance as described above, where the quantum circuit 600a utilizes a o^-mixer, and the quantum circuit 600b utilizes a swap mixer, respectively. As shown in Figure 6a, the qubits representing the binary decision variables of the optimization problem are initialized in superposition using Hadamard (H) gates. The initialized quantum state is then transformed by the cost Hamiltonian, and then manipulated by the mixer Hamiltonian. Following this, the quantum state is measured by projecting the state onto the computational basis. In the cost Hamiltonian, interactions between two qubits (for example, ) are captured by the application of the ZZ

gates, and single qubit operations are captured by the application of the R_z gates. The σx-mixer is then implemented by the application of the R_x gates.

In this illustrated example, the problem instance relates to a network comprising 1 BS and 2 clouds, and the binary variables represented by the qubits of the quantum computing devices illustrated in Figures 6a and 6b correspond to this problem instance.

As shown in Figure 6b, the initial state of the qubits corresponds to a solution in which the BS is assigned to the first cloud. This is initialized through the bitflip gate 602b acting on qubit q₀. It is noted that this initial state represents a feasible solution to the optimization problem, in the sense that it meets the single association constraint.

Hadamard (H) gates are then applied to qubits q2, qs, q4 and qs. Following this, the cost Hamiltonian transforms the initialized quantum state. In the cost Hamiltonian, interactions between two qubits (for example,

are captured by the application of the ZZ gates, and single qubit operations are captured by the application of the R_z gates. Following this, the swap mixer (which in this example comprises by the Rxx gate 604b, and the RYY gate 606b) keeps the rotations in the {10,01} subspace for the qubits qo and qi (that is, the qubits whose space is to be restricted). This ensures that single association constraint is met. R_x gates are then applied to the remaining qubits q2, q3, q4 and q5,, and the quantum state is measured by projecting the state onto the computational basis.

T able 1 shows the circuit properties of the quantum circuit 600a, for a plurality of different problem instances that require varying numbers of qubits (to represent the varying number of binary variables required to determine a solution for that problem instance).

T able 2 shows the circuit properties of the quantum circuit 600b, for a plurality of different problem instances that require varying numbers of qubits (to represent the varying number of binary variables required to determine a solution for that problem instance).

Table 1

Table 2 The results presented in Table 1 and Table 2 were generated by executing, for each problem instances, ideal simulations (with all to all qubit connectivity) of the corresponding quantum circuit.

As shown in Table 2, utilizing a quantum circuit with a swap mixer for a particular problem instance results in the quantum circuit comprising a greater number of 2-qubit gates, but fewer single qubit gates (in comparison to utilizing a quantum computing device with a σ_x mixer)

It is also noted that the circuit properties, for circuits that utilize a swap mixer, are heavily dependent on the number of binary variables the swap mixer is swapping between.

For example, a 20-qubit problem instance that comprises only 1 binary variable of type x, and a 20-qubit problem instance that comprises 10 binary variables of type x, will have largely different circuit properties when a swap mixer is utilized for the problem instances. It is also noted that, when utilizing a swap mixer for these aforementioned problem instances on a real quantum computer (as opposed to an ideal simulation), the more qubits the swap mixer has to swap between (that is, the more qubits in the quantum circuit that relate to each other as a result of the double qubit gates utilized by the swap mixer), the greater the noise in the quantum circuit.

Table 3 shows the circuit properties of a noisy quantum circuit that utilizes a o_x mixer, for a plurality of different problem instances that require varying numbers of qubits (to represent the varying number of binary variables required to determine a solution for that problem instance).

Table 3

Table 4 shows the circuit properties of a noisy quantum circuit that utilizes a swap mixer, for a plurality of different problem instances that require varying numbers of qubits (to represent the varying number of binary variables required to determine a solution for that problem instance).

Table 4 Figures 7a and 7b show the probabilities of finding an optimal solution to the optimization problem for the four aforementioned problem instances, using the o_x mixer and the swap mixer respectively, for p=1 ,2,3.

The data presented in Figures 7a and 7b was obtained using an ideal simulator, and simulations were executed once for each problem instance.

As shown in Figure 7b, utilizing a swap mixer when executing the QAOA for these problem instances results in a higher probability of finding an optimal solution to the optimization problem. This is a result of the feasible subspace produced by the swap mixer, that the QAOA then searches over, being smaller. Theses probabilities also tend to increase faster when p is increased when the swap mixer is utilized, as also shown in Figure 7b.

Figure 8a shows the probabilities of determining an optimal solution to the optimization problem for the 6-qubit problem instance, using the o_x mixer and the swap mixer respectively, for p=1 ,2,3. The data presented in Figure 8a was obtained using a noisy simulator which mimics the properties of IBMs Tokyo backend. As shown in Figure 8a, the probability of determining the optimal solution for the 6-qubit instance increases as p increases when the swap mixer is utilized. For example, for p = 3, utilising the swap mixer improves the probability of obtaining the optimal solution by approximately 3 times, in comparison to utilizing the σ_x-mixer.

Figure 8b shows the probabilities of determining an optimal solution to the optimization problem for the 8-qubit problem instance when using the swap mixer, for p=1 ,2,3.

As shown in Figure 8b, for the 8-qubit problem instance (when simulated with noise), the probability of determining the optimal solution in fact decreases with p.

It is noted that the circuit depth for the 8-qubit problem instance is approximately 35% higher than the circuit depth for the 6-qubit problem instance (as shown in Table 4). As a result of this increase, the overall noise in the circuit also increases. The probability of determining the optimal solution then decreases as p increases, as the circuit depth and the noise increases with p. As the noise increases, decoherence is more likely to occur in the quantum circuit, and as a result, the chance of obtaining the optimal solution decreases.

It will be appreciated that, an increase in overall noise in the circuit (as a result of increased circuit depth) may for example, be reduced by utilizing binary encoding to determine the binary variables for the optimization problem, therefore reducing the number of qubits required to represent these binary variables and reducing the circuit depth.

Figure 9 shows an example of a network 900 comprising a mobile edge computing system.

In the network 900, edge servers 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h are installed at the base stations 904a, 904b, 904c, 904d in order to meet the application demands of the UEs of the network 900. The UEs within the network 900 may request different computation tasks, where each task requires a specified amount of resources, be executed on any suitable edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h of the network 900.

As shown in Figure 9, each UE falls in range of a set of edge servers 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h with the network 900.

There is therefore a need for efficient algorithms that are capable of determining which edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h each UE should connect to, in order to maximize the number of tasks that can be processed by the network 900. Such allocation problems are known as Edge User Allocation (EUA) problems [6],

The definition of the Optimal Edge User Allocation (EUA) problem presented in [6] is as follows: Given m edge servers ⁿ users

and d

different resource types (for example, RAM, bandwidth, the number of CPU cores, etc.), each edge server has a maximum capacity for each resource type Each user also has a resource requirement for each resource type Each

server also has a given coverage area denoted by cov(sj), and each user has a coordinate defined by the distance to each edge server denoted by dj_j. The objective of the EUA problem is to assign as many users as possible to the edge servers 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h. while also minimizing the total number of utilized edge servers 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h.

Therefore, in this embodiment, the optimization problem represents a problem of determining a subset of edge servers in a wireless network to which one or more user equipments, UEs, in the wireless network can connect, so as to maximize a measure of efficiency of the wireless network. In this embodiment, the measure of efficiency represents the number of UEs in the network assigned to the subset of edge servers over the total number of edge servers within the subset.

In this example, an objective function may be formulated as follows for the EUA problem:

where:

The binary variable x_ij, represents if a user uj is connected to a server s_i, where x_ij = 1 if the user u_j is connected to a server s_i, and 0 otherwise, and

The binary variable yi represents if a server st is utilized or not. where y_i = 1 if the server st is utilized, and 0 otherwise. The aforementioned optimization problem also comprises the following three constraints:

The first constraint of the optimization problem enforces that, for each edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h, the edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h may only be assigned to users of the network 900 if the total resource requirements of the assigned users do not exceed the edge server’s maximum capacity for any resource type. The first constraint may be formulated as follows:

The second constraint of the optimization problem enforces that each user can be connected to at most one edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h. The second constraint may be formulated as follows:

It will be appreciated that the second constraint may be relaxed to form a single association constraint, in a similar manner as described above.

The third constraint of the optimization problem enforces that each user u₇ assigned to an edge server st must be in the range of that edge server 902a, 902b, 902c, 902d, 902e, 902f, 902g, 902h. The third constraint may be formulated as follows:

In this example, the aforementioned constraints are converted to penalty terms to be added to the objective function. The first constraint is converted to an equality constraint using binary slack variables, in a similar manner as described above.

Although the second constraint may be converted to an equality constraint by adding another server, s_m+1 with corresponding (such that if user u_j is assigned to server s_m+1, user uj is not assigned to any server), it will be appreciated that encoding this reformulated constraint would require n extra qubits (to represent the additional binary variables). Furthermore, the server s_m+1 would not appear in, nor modify the objective function. Instead, the second constraint may be reformulated by changing the inequality sign to an equality sign, and therefore requiring each user to be assigned to a server.

That is, the second constraint has been reformulated as a single association constraint.

The third constraint of the optimization problem can be met by not allowing any x^- to be equal to 1 if the user u₇ is outside of range of the server s_t. Therefore, the third constraint can be met by enforcing those x_tj to always be zero. Alternatively, depending on the given problem instance, those x_tj may be removed from the objective function altogether, allowing a reduction in the required number of qubits to implement the problem.

The reformulated constraints are then added to the objective function as follows:

In this example, the penalty terms were selected as A = B + 1, B = 1, and D = B + 1 to favor a solution that utilizes more servers, over a solution that leaves users unassigned, and to favor a solution that utilizes more servers, over a solution that does not meet the server capacity constraints.

By expanding the parenthesis of the reformulated objective function, the reformulated objective function may be mapped to an Ising model, in a similar manner as described above. As noted above, an Ising model represents a QllBO model featuring a change of variables. The mapping of the reformulated objective function to an Ising model may be performed by identifying the relevant terms through the utilization of for-loops.

The obtained Ising model will then represent a Hamiltonian, and a ground state of the Hamiltonian will correspond to an optimal solution of the EUA problem. The ground state of the Ising model is then approximately estimated by using a QAOA as described in [2, 3].

In accordance with the method 100 of Figure 1 , a swap mixer may be utilised to restrict the subspace of possible solutions to the optimization problem that may be determined using the quantum computing device to a subspace of possible solutions that meet the at least one single association constraint.

In other words, in this example, the search space of the QAOA is limited though the use of swap-mixers for

with fixed j, as a result of the single association constraint:

\ if an extra server for unassigned users is added, as discussed above I Vj. /

In a similar manner as described above, utilizing swap-mixers allows the number of combinations contributed from each x^ for a fixed i from 2¹ to i, as only i states are required to represent which server s; user uj is assigned to.

As noted above, the swap-mixers force the Hamming weight of the states (that represent the relevant binary variables that are to obey the single association constraint) to be conserved when the mixing is executed.

Alternatively, the binary variable x^ for each j may be represented in the reformulated objective function with binary encoding. In this example, the number of states that are searched over when executing the QAOA would then be the same as the case when the swap-mixers are utilized. However, it is difficult to encode the interactions between these binary variables (represented with binary encoding) in a low depth circuit. In a similar manner as described above, the convergence rate of executing the QAOA to determine a solution to the optimization problem may be increased by reducing the angle search space by utilizing observed symmetries in the generated energy landscapes.

In this example, the observed symmetries arise from the formulated Ising model only containing real integers, alongside the formed mixer Hamiltonian being fully commutable with the formed cost Hamiltonian.

In this example, these symmetries allow the angle search space to be reduced from y_k ∈ [0,2π ] and β_k ∈ [0, π ] , to y_k ∈ [0, π ] and β_k ∈ [0, π /2] respectively.

Figure 10a illustrates the energy landscape for y_k ∈ [0,2-π ] and β_k ∈ [0, π] for a 3-qubit problem instance for the optimization problem, consisting of one edge server and one user.

Figure 10b illustrates the energy landscape for y_k ∈ [0,π] and β_k ∈ [0,π/2], for a 3-qubit problem instance for the optimization problem, consisting of one edge server and one user.

It can be seen that the energy landscape as illustrated in Figure 10b is repeated four times within the energy landscape of Figure 10a (where the landscape of Figure 10b is rotated 180° in the yβ-plane when shifted π/2 along the β-axis of Figure 10a). For the aforementioned optimization problem, similar symmetries arise in the energy landscapes generated for each the following problem instances: a 3 qubit problem instance consisting of one edge server and one user, a 6 qubit problem instance consisting of one server and two users, an 8 qubit problem instance consisting of two servers and one user, and a 13 qubit problem instance consisting of two servers and four users.

Figure 10c illustrates the energy landscape for y_k ∈ [0, π ] and p_k ∈ [0, π /2] , for the 6- qubit problem instance. Figure 10d illustrates the energy landscape for y_k ∈ [0, π ] and β_k ∈ [0, π /2], for the 8- qubit problem instance.

Figure 10e illustrates the energy landscape for y_k ∈ [0, π ] and β_k ∈ [0, π /2] , for the 13- qubit problem instance.

It will be appreciated that the number of qubits used for each problem instance not only depends on the number of edge servers and the number of users for the particular problem instance, but also on the resource capacities of the edge servers, and how many servers each user is in range of.

Figure 11 shows the probabilities of determining the optimal solution for the 3-qubit problem instance consisting of one user in range of one edge server, for a number of different mixer and angle search space configurations, for p=1 ,2,3.

The probabilities were determined by executing the QAOA to find the optimal angles in terms of expected cost of the objective function they correspond to. The probability of determining the optimal solution to the optimization problem when executing the QAOA in accordance with these optimal angles was then computed.

Each simulation was executed 10 times for an ideal quantum computing device, and the average probability was plotted with the corresponding standard deviation. The first set of simulations were executed for o_x-mixers while searching the full angle search space. The second set of simulations were executed for swap-mixers while searching the full angle search space. The third set of simulations were executed for swap-mixers while only searching the restricted angle search space (as defined by the problem symmetries).

As shown in Figure 11 , the success probability (of determining the optimal solution to the optimization problem) increases with p for all cases. Furthermore, the error bars shown in Figure 11 decrease between the first set of simulations and the second set of simulations, and again between the second set of simulations and the third set of simulations. This is a result of the optimization over the energy landscape being increasingly likely to find the true global minimum of the energy landscape, and the angles associated with this true global minimum being more likely to be used and return the optimal solution. Table 5 shows the average number of function evaluations required to find optimal angles given a 3-qubit EUA problem instance.

Table 5

As shown in Table 5, for the third set of simulations, the QAOA features fewer average function evaluations. The reduction in number of function evaluations between the full and restricted angle search space simulations may be explained by the fact that the QAOA only has to search over a smaller space to determine the optimal angles. The difference in the number of function evaluations is also greater for increasing values of p between the three sets of simulations, as the restricted angle search space is reduced by a factor of 2^2p.

Figures 12a and 12b show the probabilities of determining an optimal solution for the aforementioned four problem instances, for p=1 ,2,3. The probabilities shown in Figures 12a and 12b were generated by executing a QAOA for the respective problem instances, determining the optimal angles in terms of expected cost for each problem instance, and then determining the probability of determining the optimal solution to the optimization problem for these optimal angles.

For each problem instance, a simulation for an ideal quantum computing device was executed 10 times, and the average probability (of determining the optimal solution to the optimization problem) was plotted with the corresponding standard deviation.

Figure 12a illustrates the probabilities determined for the simulations with o^-mixers while searching the aforementioned restricted angle search space. Figure 12b illustrates the probabilities determined for the simulations with swap-mixers while searching the aforementioned restricted angle search space. As shown in Figures 12a and 12b, the probability of determining the optimal solution to the optimization problem increases for each of the 3-, 6- and 8-qubit problems instances, when p increases for both simulations utilizing the o_x-mixers, and simulations utilizing the swap-mixers. However, this does not occur for the 13-qubit problem instance.

This may be as a result of the optimization over the generated energy landscapes being more likely to determine a local minimum of the energy landscape as the global minimum, as opposed to being able to identify the true global minimum. The angles returned by these local minimums then have a much lower chance of returning the optimal solution (when using these angles to execute the QAOA).

As shown in Figure 12b, utilizing the swap mixer in these aforementioned simulations of the 3-, 6- and 8-qubit problems instances will result in an increased probability of the optimal solution of the optimization problem being returned (in comparison to utilizing the σ_x mixer).

Table 6 shows the average number of function evaluations required to find optimal angles when using o-_z-mixers and a restricted angle search space for the four aforementioned EUA problem instances.

Table 7 shows the average number of function evaluations required to find optimal angles when using swap-mixers and a restricted angle search space for the four aforementioned EUA problem instances.

As shown in Tables 6 and 7, the average number of function evaluations for simulations with the restricted angle search spaces are relatively similar.

Table 6

Table 7

Table 8 shows the properties of a quantum computing device utilizing (σ_x-mixers for the aforementioned four EUA problem instances, where p=1.

Table 9 shows the properties of a quantum computing device utilizing swap mixers for the aforementioned four EUA problem instances, where p=1.

Table 8

Table 9 The circuit properties shown in Tables 8 and 9 were determined in accordance with the specifications of an existing quantum computing device, the IBM Tokyo.

As shown in Tables 8 and 9, for both circuits comprising σ_x mixers, and circuits comprising swap mixers, the circuit properties are largely similar. That is, for small problems, utilizing swap mixers (rather than σ_x mixers) may not result in a notable improvement in computation time, despite utilizing two-qubit gates rather than singlequbit gates as the σ_x-mixers utilize.

The general advantages of the proposed solutions in some examples are that larger problem instances of the aforementioned optimization problem may be able to be solved faster and more accurately with quantum computers than with classical computers.

Furthermore, by encoding slack variables with binary encoding in order to implement a QAOA, the number of qubits required to execute the QAOA may be reduced, which lowers the overall noise in the quantum circuit, as the total circuit depth and the number of gates decrease.

Furthermore, utilizing a swap mixer reduces the feasible subspace of solutions to the QAOA, and lets the mixer Hamiltonian handle the single association constraints, rather than the cost Hamiltonian. The swap mixer also significantly improves the probability of optimal solution, even under the presence of noise, allowing the aforementioned methods to be suitable for execution of near-term quantum computers that are characterized by a shorter decoherence time and noise. For example, in the case study for the problem of VNF placement in a cloud radio access network, the proposed methods improve the probability of finding the optimal solution to the optimization problem by 2.88x, in comparison to the standard QAOA approach.

Figure 13 is a schematic of an example of an apparatus 1300 for determining a solution to an optimization problem. The apparatus 1300 comprises processing circuitry 1302 (e.g. one or more processors) and a memory 1304 in communication with the processing circuitry 1302. The memory 1304 contains instructions executable by the processing circuitry 1302. The apparatus 1300 also comprises an interface 1306 in communication with the processing circuitry 1302. Although the interface 1306, processing circuitry 1302 and memory 1304 are shown connected in series, these may alternatively be interconnected in any other way, for example via a bus. In one embodiment, the memory 1304 contains instructions executable by the processing circuitry 1302 such that the apparatus 1300 is operable to form the optimization problem such that the formulated optimization problem comprises at least one single association constraint, manipulate quantum states of qubits of a quantum computing device based on the formulated optimization problem, wherein manipulating quantum states of the qubits comprises applying a swap mixer that represents the single association constraint to one or more of the qubits of the quantum computing device, obtain a measurement of each of the one or more qubits, and based on the obtained measurements, determine a solution to the optimization problem. In some examples, the apparatus 1300 is operable to carry out the methods 100 and 200 described above with reference to Figures 1 and 2 respectively.

It should be noted that the above-mentioned examples illustrate rather than limit the invention, and that those skilled in the art will be able to design many alternative examples without departing from the scope of the appended statements. The word “comprising” does not exclude the presence of elements or steps other than those listed in a claim, “a” or “an” does not exclude a plurality, and a single processor or other unit may fulfil the functions of several units recited in the statements below. Where the terms, “first”, “second” etc. are used they are to be understood merely as labels for the convenient identification of a particular feature. In particular, they are not to be interpreted as describing the first or the second feature of a plurality of such features (i.e., the first or second of such features to occur in time or space) unless explicitly stated otherwise. Steps in the methods disclosed herein may be carried out in any order unless expressly otherwise stated. Any reference signs in the statements shall not be construed so as to limit their scope.

Abbreviations

ILP Integer linear problem

BS Base station

VM Virtual machine

QAOA Quantum approximate optimization algorithm

QAA Quantum Adiabatic Algorithm

Qubit Quantum bit QUBO Quadratic unconstrained binary optimization

QAOA Quantum Approximation Optimization Algorithm or Quantum Alternating Operator Ansatz

P Complexity class P, problems solvable in polynomial time using a deterministic Turing machine

NP Complexity class NP, problems solvable in polynomial time using a non- deterministic Turing machine

NP-hard Complexity class NP-hard, contains all problems in NP which are not in P

BQP Complexity class BQP, bounded-error quantum polynomial time

LGP Lexicographic Goal Programming

MIP Mixed Integer Programming

References

1 . Efficient Virtual Network Function Placement Strategies for Cloud Radio Access Networks, Bhamare, Deval and Erbad, Aiman and Jain, Raj and Zolanvari, Maede and Samaka, Mohammed, Computer Communications (2018), DOI: 10.1016/j.comcom.2018.05.004, ISSN: 0140-3664. Relevant pages: 4-6, 9,11.

2. A Quantum Approximate Optimization Algorithm, Edward Farhi and Jeffrey Goldstone and Sam Gutmann, arXiv preprint arXiv: 1411.4028 (2014).

3. From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz, Hadfield, Stuart and Wang, Zhihui and O’Gorman, Bryan and Rieffel, Eleanor and Venturelli, Davide and Biswas, Rupak, Algorithms (2019), DOI: 10.3390/a12020034, ISSN: 1999-4893. Relevant pages: 5-10, 11-13

4. Ising formulations of many NP problems, Lucas, Andrew, Frontiers in Physics (2014), DOI: 10.3389/fphy.2014.00005, ISSN: 2296-424X. Relevant pages: 2-3, 8-9, 13-15

5. Efficient Combinatorial Optimization Using Quantum Annealing, Hristo N. Djidjev and Guillaume Chapuis and Georg Hahn and Guillaume Rizk, arXiv: 1801.08653 (2018). Relevant pages: 1-4, 7-8. 6. Lei, P., He, Q., Abdelrazek, M., Chen, F., Hosking, J., Grundy, J., & Yang, Y. (2018). Optimal edge user allocation in edge computing with variable sized vector bin packing. In International Conference on Service-Oriented Computing (pp. 230-245). Springer, Cham. 7. Rieffel, E., Dominy, J. M., Rubin, N., & Wang, Z. (2020). XY-mixers: analytical and numerical results for QAOA. Phys. Rev. A, 101, 01232

8. Seung Geun Hong, Jinhyun Park, Saewoong Bahk. "Subchannel and Power Allocation for D2D Communication in mmWave Cellular Networks. Journal of Communications and Networks, Vol. 22, No. 2, April 2020." Journal of Communications and Networks (2020).

9. Nyberg, S., 2016. Physical cell id allocation in cellular networks.

10. Tran GK, Shimodaira H, Rezagah RE, Sakaguchi K, Araki K. Dynamic cell activation and user association for green 5G heterogeneous cellular networks. In 2015 IEEE 26th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC) 2015 Aug (pp. 2364-2368). IEEE

11 . https://qiskit.org/documentation/stubs/qiskit.circuit.library.SwapGate.html

Claims

1. A method of determining a solution to an optimization problem, the method comprising:

(iii) obtaining a measurement of each of the one or more qubits; and

2. The method according to claim 1 , wherein the at least one single association constraint is of the form:

where x^ is a decision variable, and x^ = 1 if an entity i within the set N is associated to an entity j within the set M.

3. The method according to claim 1 or 2, wherein the step of formulating the optimization problem comprises: determining whether a constraint of the optimization problem can be reformulated as a single association constraint; and in response to determining a constraint of the optimization problem can be reformulated as a single association constraint, reformulating the constraint as a single association constraint.

4. The method according to any preceding claim, wherein the formulated optimization problem further comprises at least one constraint comprising one or more variables that are represented with binary encoding.

5. The method according to any preceding claim, wherein an initial state of the qubits of the quantum computing device correspond to a solution to the optimization problem that obeys the at least one single association constraint.

6. The method according to any preceding claim, manipulating quantum states of the qubits comprises applying a first set of one or more unitary operations that represent the optimization problem to one or more qubits of the quantum computing device.

7. The method according to claim 6, wherein applying the first set of one or more unitary operations manipulates the quantum states of the qubits based on one or more constraints of the optimization problem.

8. The method according to claim 6 or 7, wherein the first set of unitary operations are based on a first angle.

9. The method according to any preceding claim, wherein the swap mixer is comprised within a second set of one or more unitary operations that represent a subspace of possible solutions to the optimization problem.

10. The method according to claim 9, wherein the application of the swap mixer to the one or more of the qubits of the quantum computing device restricts the subspace of possible solutions to the optimization problem that may be determined using the quantum computing device to a subspace of possible solutions that meet the at least one single association constraint.

11. The method according to claim 9 or 10, wherein the second set of unitary operations are based on a second angle.

12. The method according to any of claims 9-11 , when dependent on any of claims 6- 8, wherein the step of manipulating quantum states of the qubits comprises:

(a) applying the first set of unitary operations to one or more of the qubits of the quantum computing device;

(b) applying the second first set of unitary operations to one or more of the qubits of the quantum computing device; and

(c) repeating steps (a) and (b) at least once.

13. The method according to any preceding claim, wherein the formulated optimization problem comprises a binary integer linear programming, BILP, problem that represents the optimization problem.

14. The method according to claim 13, wherein the step of determining a solution to the optimization problem comprises: based on the obtained measurements, determining a set of values of a plurality of binary variables of the BILP problem, wherein the set of values represent a solution to the optimization problem.

15. The method according to claim 13 or 14, wherein the BILP problem comprises a problem of determining a solution to the optimization problem that maximizes or minimizes a value of an objective function, wherein the objective function comprises at least one single association constraint.

16. The method according to any of claims 13-15, when dependent on claim 6 or 7, wherein the first set of unitary operations represent the objective function.

17. The method according to claim 16, wherein the first set of unitary operations represent either: a quadratic unconstrained binary optimization, QUBO, model that represents the objective function, or an Ising model that represents the objective function.

18. The method according to any of claims 15-17, when dependent on 8 and 11 , wherein the method further comprises: determining a set of values of the first angle; determining a set of values of the second angle; and for each value of the set of values of the first angle: for each value of the set of values of the second angle: performing steps (ii) to (iv) of the method of claim 1 to determine a solution to the optimization problem; and based on the determined solution, determine a value of the objective function.

19. The method according to claim 18, wherein determining the set of values of the first angle, and determining the set of values of the second angle, comprises: determining whether there is a symmetry of the determined values of the objective function within the space of the determined values; and in response to a determining that there is a symmetry of the determined values of the objective function within the space of the determined values: determining the set of values of the first angle based on the determination; and determining the set of values of the second angle based on the determination.

20. The method according to claim 19, wherein the step of whether there is a symmetry of the determined values of the objective function within the space of the determined values comprises, when the first set of one or more unitary operations represent an Ising model that represents the objective function: determining whether the Ising model contains real integers; determining whether the first set of unitary operations and the second set of unitary operations are commutative; and in response to a determination that the Ising model contains real integers, and a determination that the first set of unitary operations and the second set of unitary operations are commutative: determining that there is a symmetry of the determined values of the objective function within the space of the determined values.

21. The method according to claim 18-20, the method further comprising: based on the objective function, determining a maximum or minimum value of the determined values; and performing steps (ii) to (iv) of claim 1 , based on the value of the first angle and the value of the second angle corresponding to the determined value, to determine a solution to the optimization problem.

22. The method according to any proceeding claim, wherein the optimization problem represents a problem of determining a subset of instances of a virtual machine in a wireless network, wherein each instance is associated with a cloud of the wireless network to which one or more base stations in the wireless network can connect, so as to maximize a measure of efficiency of the wireless network.

23. The method according to any of claims 1-21 , wherein the optimization problem represents a problem of determining a subset of edge servers in a wireless network to which one or more user equipments, UEs, in the wireless network can connect, so as to maximize a measure of efficiency of the wireless network.

24. The method according to claim 23, where the measure of efficiency represents the number of UEs in the network assigned to the subset of edge servers over the total number of edge servers within the subset.

25. A computer program comprising instructions which, when executed on at least one processor, cause the at least one processor to carry out a method according to any of claims 1 to 24.

26. A carrier containing a computer program according to claim 25, wherein the carrier comprises one of an electronic signal, optical signal, radio signal or computer readable storage medium.

27. A computer program product comprising non transitory computer readable media having stored thereon a computer program according to claim 25.

28. An apparatus for determining a solution to an optimization problem, the apparatus comprising a processor and a memory, the memory containing instructions executable by the processor such that the apparatus is operable to:

(iii) obtain a measurement of each of the one or more qubits; and

29. The apparatus of claim 28, wherein the memory contains instructions executable by the processor such that the apparatus is operable to perform the method of any of claims 2 to 24.

30. An apparatus for determining a solution to an optimization problem, wherein the apparatus is configured to:

(iii) obtain a measurement of each of the one or more qubits; and

(iv) based on the obtained measurements, determine a solution to the optimization problem

31. The apparatus of claim 30, wherein the apparatus is further configured to perform the method of any of claims 2 to 24.