Nowadays, quantum computing and communications have received a lot of attention by academia and industry research communities. In particular, quantum computing based NP-hard problem solving is of great interest [farhi2014quantum]. Among them, quantum approximate optimization algorithm (QAOA) is one of the well-known quantum computing based optimization solvers [farhi2014quantum], and it has been verified that the QAOA outperforms the others in many combinatorial problems. Based on this nature, it is obvious that quantum computing can be used for various multi-scale communications applications [mbmc1, mbmc2, mbmc3].
In this paper, a large-scale and multi-scale scheduling problem is formulated with maximum weight independent set (MWIS) formulation where the weight is defined as the queue-backlog to be transmitted over wireless channels [ton16kim]
. According to the fact that the MWIS problem is NP-hard, heuristic algorithms are desired, and in this paper, a novel QAOA-based algorithm is designed in order to solve MWIS-based wireless scheduling problems, so calledquantum approximate optimization for scheduling (QAOS), in this paper.
The proposed QAOS works as follows. First of all, the objective function and constraint functions are formulated for MWIS. Next, corresponding objective Hamiltonian and constraint Hamiltonian are designed which map the objective function and the constraint function, respectively; and then, the problem Hamiltonian which should be optimized is formulated as the form of linear combinations of the objective Hamiltonian and constraint Hamiltonian. In addition, the mixing Hamiltonian is formulated using a Pauli- operator. Based on the definitions of the problem Hamiltonian and the mixing Hamiltonian, two corresponding unitary operators, i.e., problem operator and mixing operator, can be defined, respectively; and then parameterized state can be generated by alternately applying the two unitary operators. Then, the sample solutions can be obtained by the measurement of the expectation value of problem Hamiltonian on the parameterized state, and the parameters can be optimized in a classical optimization loop. Finally, the optimal solution of the MWIS problem can be obtained by the measurement of the expectation value of problem Hamiltonian on the state generated by optimal parameters. As verified in performance evaluation, the QAOS outperforms the other algorithms, e.g., random search and greedy search.
Ii-a Bra-ket Notation
In quantum computing, bra-ket notation is generally used to represent qubit states (or quantum states). It is also called Dirac notation as well as the notation for observable vectors in Hilbert spaces. Here, a ket and a bra can represent the column and row vectors, respectively. Thus, single qubit states, i.e.,and , are presented as follows:
where , ; and means Hermitian transpose. Accordingly, the superposition state of a single qubit is as follows where and
are probability amplitudes that are complex numbers:
Ii-B Quantum Approximate Optimization Algorithm (QAOA)
QAOA is one of the well-known noisy intermediate-scale quantum (NISQ) optimization algorithms to combat combinatorial problems [preskill2018quantum]. QAOA formulates (i.e., problem Hamiltonian) and (i.e., mixing Hamiltonian) from the optimization objective function ; and then generates the parameterized states by alternately applying the and based on initial state . Here, , , , and are defined as follows.
where , , and is the Pauli- operator applying on the qubit.
In QAOA, through iterative measurement on , the expectation value of should be taken, and then eventually, the samples of should be computed as follows:
The optimal values of the parameters and can be obtained by classical optimization methods, e.g., gradient descent. Therefore, the solution can be computed from (7) via the the parameters obtained. Eventually, it can be observed that QAOA is a hybrid quantum-classical optimizer which is needed the proper design of Hamiltonian; and the key is finding good parameters in the classical loop [hadfield2019quantum].
Iii Multi-Scale Scheduling Modeling using Maximum Weight Independent Set (MWIS)
Suppose a network consists of the set of one-hop links [ton16kim]. For the scheduling, a conflict graph is organized where the set of vertices is (the links) and two vertices are connected by an edge if the corresponding links suffer from interference. The conflict graph can be formulated by its adjacency matrix, whose are defined as follows:
For multi-scale scheduling, the objective is for finding the set of links (i.e., nodes of the conflict graph) where adjacent two connected links via edges cannot be simultaneously selected because the adjacent two connected links are interfering to each other. This is equivalent to the case which maximizes the summation of weights of all possible independent sets in a given conflict graph. Thus, it is obvious that multi-scale scheduling can be formulated with MWIS as follows:
where is a positive integer weight at . The above formulation ensures that conflicting links are not scheduled simultaneously: If (no edge between and ), then , i.e., both indicator functions can be . In contrast, if , , i.e., at most one of the two indicators can be . In wireless communication research, the where is usually considered as transmission queue-backlog at which should be processed when the link is scheduled. More details are in [ton16kim].
Iv Quantum Approximate Optimization for Scheduling (QAOS)
In this section, Hamiltonian for QAOA is designed based on the scheduling model in Sec. III; and then Quantum Approximate Optimization for Scheduling (QAOS) algorithm is proposed by applying the designed Hamiltonian to QAOA.
Iv-a Design the problem Hamiltonian,
The problem Hamiltonian is designed by a linear combination of the objective Hamiltonian and the constraint Hamiltonian . The objectives and constraints of the problem are contained by and , respectively.
Iv-A1 Objective Hamiltonian
Suppose that a basic Boolean function exists as follows:
Due to quantum Fourier expansion, (13) can be mapped to Boolean Hamiltonian where and are an Identity operator and the Pauli- operator, respectively [hadfield2018representation]:
Iv-A2 Constraint Hamiltonian
In MWIS problem, the banned condition is a case where both nodes directly connected to the edge are scheduled, as shown in Case C in Fig. 1. If the weights of the and in Case C are defined as and respectively; then the constraint function , which counts banned conditions can be represented as follows:
where is the number of nodes and is the number of .
where is a Boolean operator AND; and the reason why the coefficient is in (18) because both and represent the same edge. The AND Boolean function can be mapped to Boolean Hamiltonian as follows [hadfield2018representation]:
where and are the Pauli- operators applying on and , respectively.
where and are the Pauli- operators applying on and , respectively. The constraint of the model is to minimize , and then the constraint Hamiltonian is as follows:
Iv-B Design the mixing Hamiltonian,
The mixing Hamiltonian, denoted by , generates a variety of cases that can appear in the problem. MWIS can be formulated by a binary bit string that represents a set of nodes (e.g., ); thus various cases can be created by flip the state of each node represented by or . The bit-flip can be handled by the Pauli- operator, thus is as:
where is the Pauli- operator applying on . In other words, is a transverse-field Hamiltonian [hadfield2019quantum].
Iv-C Apply to QAOA sequence
The application of the designed Hamiltonian to QAOA sequence starts to conduct when the design of Hamiltonian, i.e., and , is completed. First, the parameterized state can be generated by applying and defined in (16), (22), (23), and (24), to (6). Here, the initial state is set to the equivalent superposition state using the Hadamard gates. The expectation value of can be measured on the generated parameterized state where the and are iteratively updated in a classical optimization loop. When the QAOA sequence terminates, the optimal parameters and are obtained; thus the solution for link scheduling can be obtained by the measurement of the expectation value of on the optimal state as follows, where is the expectation value of (9) over the returned solution samples: