Quantum Approximation for Wireless Scheduling

04/14/2020 ∙ by Jaeho Choi, et al. ∙ 0

This paper proposes a quantum approximate optimization algorithm (QAOA) method for wireless scheduling problems. The QAOA is one of the promising hybrid quantum-classical algorithms for many applications and it provides highly accurate optimization solutions in NP-hard problems. QAOA maps the given problems into Hilbert spaces, and then it generates Hamiltonian for the given objectives and constraints. Then, QAOA finds proper parameters from classical optimization approaches in order to optimize the expectation value of generated Hamiltonian. Based on the parameters, the optimal solution to the given problem can be obtained from the optimum of the expectation value of Hamiltonian. Inspired by QAOA, a quantum approximate optimization for scheduling (QAOS) algorithm is proposed. First of all, this paper formulates a wireless scheduling problem using maximum weight independent set (MWIS). Then, for the given MWIS, the proposed QAOS designs the Hamiltonian of the problem. After that, the iterative QAOS sequence solves the wireless scheduling problem. This paper verifies the novelty of the proposed QAOS via simulations implemented by Cirq and TensorFlow-Quantum.

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I Introduction

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 called

quantum 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 Preliminaries

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:

(1)

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:

(2)

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.

(3)
(4)
(5)
(6)

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:

(7)

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:

(8)

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:

(9)
s.t. (11)
where (12)

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].

1

1

Case C: Both Scheduled

0

1

Case B: 1 Node Scheduled

0

0

Case A: Both Unscheduled

Fig. 1: The number of possible cases when a single edge exists. The scheduled and unscheduled nodes have states and . and represent arbitrary nodes, and , , and represent edges in each case.

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:

(13)

Due to quantum Fourier expansion, (13) can be mapped to Boolean Hamiltonian where and are an Identity operator and the Pauli- operator, respectively [hadfield2018representation]:

(14)

According to (13)–(14), the objective function (9) can be represented as following Hamiltonian.

(15)

where is the Pauli- operator applying on . The objective of the model is to maximize , thus the objective Hamiltonian which should be minimized is as follows:

(16)

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:

(17)

where is the number of nodes and is the number of .

According to (8)–(12), can be redefined to with symbols in Sec. III as follows:

(18)

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]:

(19)
(20)

where and are the Pauli- operators applying on and , respectively.

According to (19)–(20), the objective function (18) can be represented as following Hamiltonian:

(21)

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:

(22)

Based on the definitions of and , the problem Hamiltonian can be defined as follows:

(23)

where is the penalty rate, which indicates the rate of which affects compared to . According to (16) and (22), both and should be minimized, thus should be minimized as well.

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:

(24)

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:

(25)
1# (Step 1) Implementation of Operators with Cirq
2# - Problem Operator
3def problem_op(mwis_graph, weight, penalty_rate, qubits, p, gamma):
4# - Objective Operator, Eqs. (16) & (26)
5    for n in mwis_graph.nodes: # n: node
6        yield cirq.rz(-(1/2)*gamma[p]*weight[n])(qubits[n])
7# - Constraint Operator, Eqs. (22) & (27)
8    for e in mwis_graph.edges: # e: edge
9        weight_sum = weight[e[0]] + weight[e[1]]
10        yield cirq.CZPowGate(exponent=(1/8)*gamma[p]*penalty_rate*weight_sum/np.pi, global_shift=0)(qubits[e[0]], qubits[e[1]])
11# - Mixing Operator, Eqs. (24) & (29)
12def mixing_op(mwis_graph, qubits, p, beta):
13    for n in mwis_graph.nodes: # n: node
14        yield cirq.rx(beta[p][n])(qubits[n])
15...
16# (Step 2) Training with TensorFlow-Quantum
17...
18# - Build the Keras model
19model = tf.keras.Sequential()
20model.add(tf.keras.layers.Input(shape=(), dtype=tf.dtypes.string))
21model.add(tfq.layers.PQC(model_circuit, model_readout)) # Parametrized Quantum Circuit
22model.compile(
23    loss=tf.keras.losses.mean_absolute_error,
24    optimizer=tf.keras.optimizers.Adam(0.03))
25model.fit(input_,optimum,epochs=1000,verbose=1)\end{lstlisting}
26    \caption{Parts of Python codes using Cirq and TensorFlow-Quantum for solving the MWIS-based scheduling problem.}
27    \label{fig:code}
28\end{figure}
29
30
31\section{Performance Evaluation}\label{sec:5}
32The proposed QAOS algorithm is implemented using Cirq and TensorFlow-Quantum developed for NISQ algorithm and quantum machine learning computation~\cite{cirq,tfq}.
33% 제안된 QAOS NISQ 알고리즘  양자 기계 학습을 위해 개발된 Cirq  TensorFlow-Quantum 사용하여 구현되었다.
34
35\subsection{Software Implementation}
36The application of the quantum gates, the basic units of the quantum circuit, is expressed by unitary operators.
37% quantum circuit 기본 단위인 양자 게이트의 적용은 unitary operator 표현된다.
38Based on the definitions of Hamiltonians in Sec.~\ref{sec:4}, the objective operator $U_O(\gamma_{\zeta})$, constraint operator $U_C(\gamma_{\zeta})$, problem operator $U_P(\gamma_{\zeta})$, and mixing operator $U_M(\beta_{\zeta})$ which are unitary operators can be defined as follows:
39
40% 섹션4의 해밀토니안 정의를 기반으로, unitary operator problem operator mixing operator 다음과 같이 정의할  있다.
41\begin{eqnarray}
42    U_O(\gamma_{\zeta})&=&e^{-i\gamma_{\zeta} H_O}, \label{eq:U_O}\\
43    U_C(\gamma_{\zeta})&=&e^{-i\gamma_{\zeta} \rho H_C}, \label{eq:U_C}\\
44    U_P(\gamma_{\zeta})&=&U_O(\gamma_{\zeta}) U_C(\gamma_{\zeta}) = e^{-i\gamma_{\zeta} (H_O+\rho H_C)}, \label{eq:U_P}\\
45    % &=& e^{-i\gamma_{\zeta} H_O}e^{-i\gamma_{\zeta} \rho H_C} \nonumber \\
46    U_M(\beta_{\zeta})&=&e^{-i\beta_{\zeta} H_M}, \label{eq:U_M}
47\end{eqnarray}
48where $\gamma_{\zeta}$ and $\beta_{\zeta}$ are in $\gamma\equiv\gamma_{1}\cdots\gamma_{p}$
49and $\beta\equiv\beta_{1}\cdots\beta_{p}$, respectively: $\zeta \in\mathbb{Z}^{+}$ and $1\leq \zeta \leq p$. %, and the others are defined in~\eqref{eq:state},~\eqref{eq:H_O},~\eqref{eq:H_C},~\eqref{eq:H_P}, and~\eqref{eq:H_M}.
50%\begin{eqnarray}
51%    U_P(\gamma_{\zeta})&=& e^{-i\gamma_{\zeta} H_P} \nonumber \\
52%    &=& e^{-i\gamma_{\zeta} (H_O+\rho H_C)} \nonumber \\
53%    &=& e^{-i\gamma_{\zeta} H_O}e^{-i\gamma_{\zeta} \rho H_C} \nonumber\\
54%    & &\text{where }\gamma_{\zeta} \text{ is in }\gamma\equiv\gamma_{1}\cdots\gamma_{p}, \\
55%    U_M(\beta_{\zeta})&=&e^{-i\beta_{\zeta} H_M} \nonumber\\
56%    & &\text{where }\beta_{\zeta} \text{ is in %}\beta\equiv\beta_{1}\cdots\beta_{p},
57%\end{eqnarray}
58%where $\zeta \in\mathbb{Z}^{+}$ and $1\leq \zeta \leq p$, and the other symbols are defined in~\eqref{eq:state},~\eqref{eq:H_O},~\eqref{eq:H_C},~\eqref{eq:H_P}, and~\eqref{eq:H_M}.
59% 여기서 zeta 양의 정수이고, 다른 기호에 대한 정의는 (6), (16), (22), (23), (24)에 있다.
60Note that implementing $U_P(\gamma_{\zeta})$ and $U_M(\beta_{\zeta})$ is the core of QAOS implementation.
61% QAOS 구현에는 $U_P(\gamma_{\zeta})$ $U_M(\beta_{\zeta})$ 구현이 핵심이다.
62
63In Fig.~\ref{fig:code}, \texttt{cirq.rz()} and \texttt{cirq.CZPowGate()} are used for the implementation of $U_O(\gamma_{\zeta})$ and $U_C(\gamma_{\zeta})$, respectively; and based on these, $U_P(\gamma_{\zeta})$ is implemented as~\eqref{eq:U_P}.
64% 그림2에서 $U_O(\gamma_{\zeta}), U_C(\gamma_{\zeta})의 구현에는 각각 cirq.rz()와 cirq.CZPowGate()가 사용되었으며, 이를 바탕으로 U_P(\gamma_{\zeta})는  (28)과 같이 구현되었다.
65Notice that \texttt{cirq.rz()} represents the rotation-$Z$ gate, and \texttt{cirq.CZPowGate()} represents the quantum gate that applies a phase to the state $\ket{11}$.
66% cirq.rz()는 rotation-$Z$ gate, cirq.CZPowGate()는 $\ket{11}$ 위상을 적용하는 양자 게이트를 나타낸다.
67In addition, $U_M(\beta_{\zeta})$ is implemented using \texttt{cirq.rx()} which means the rotation-$X$ gate.
68% 또한, U_M(\beta_{\zeta})는 rotation-$X$ gate 사용하여 구현되었다.
69
70The part that finds the optimal parameters using Keras (one of well-known open-source deep learning computation libraries) is (Step 2), from line $16$ to line $25$, in Fig.~\ref{fig:code}.
71% Keras model 이용한 학습 파트에 대해서는 그림2의 스텝2에 나타나 있다.
72In this model, the parametrized quantum circuit (PQC) layer provides auto-management of variables in the parameterized circuit~\cite{tfq}.
73% 여기서, PQC 레이어는 parameterized circuit에서 variable auto Keras management 제공한다.
74% 원문: The PQC layer provides automated Keras management of the variables in a parameterized circuit.
75
76% In this model, Adam is used as a gradient-based optimizer~\cite{kingma2014adam}.
77% 해당 모델에서는 first-order gradient-based optimizer of stochastic objective function Adam 사용하였다.
78
79\begin{figure}[t]
80    \centering
81        \includegraphics[width =1.0\columnwidth]{sim.pdf}
82        \vspace{-3mm}
83    \caption{Performance Evaluation Results.}
84    \label{fig:sim}
85\end{figure}
86
87\subsection{Results}
88The performance of our proposed QAOS algorithm is compared with random search and greedy search. In addition, the QAOS algorithm executes with different $p$ value settings where the $p$ value means the number of alternation of $U_P(\gamma_{\zeta})$ and $U_M(\beta_{\zeta})$ in~\eqref{eq:U_P} and~\eqref{eq:U_M}, i.e., $\zeta \in\mathbb{Z}^{+}$ and $1\leq \zeta \leq p$.
89% $p$ value $U_P$ $U_M$ 교번연산 횟수를 나타낸다.
90
91For the performance evaluation, we generate random graphs with $10$ nodes, i.e., links in conflict graphs; and then random search, greedy search, and QAOS algorithms are performed for the given random graphs.
92The measurement of each QAOS is performed $1,000$ times in each simulation (i.e., in each randomly generated conflict graph), and the solution that is returned with the maximum probability is selected as the solutions of each simulation.
93Then, the performance of each algorithm is quantitatively measured as $\eta\triangleq \frac{a}{b}$ where $a$ and $b$ are the summation of weights of the scheduled nodes by the used algorithms and the summations of weights of the scheduled nodes by brute-force full search (i.e., exhaustive search), respectively, for the given randomly generated graphs.
94Then, the cumulative distribution functions (CDF) of $\eta$ for each algorithm is computed and illustrated in Fig.~\ref{fig:sim}.
95
96% 우리는  시뮬레이션 마다(그래프 마다), QAOS measurement 1000회 실시하였으며,  시뮬레이션에서 최대 확률로 반환되는 solution  QAOS solution으로 채택하였다.
97% Among the set of solutions that are obtained after measurement, there may be solutions that have larger summations of weights than the highest probability solution.
98% % 실제로 QAOS  경우의 measurement  반환된 solution 집합들 중에서, 최빈 솔루션의 the summation of weights 값보다  값이   솔루션도 존재하기도 한다.
99% However, for a fair evaluation of the QAOS algorithm, when the highest probability solution and the solution which has the maximum summation of weights are not equal, the highest probability solution is selected.
100% %  공정한 QAOS 평가를 위해, 최대 확률을 가진 solution 최대 가중치를 가진 solution 같지 않으면 가장 높은 확률의 solution 택하였다.
101
102As presented in Fig.~\ref{fig:sim}, QAOS algorithms with $p\geq 8$ present better performance than random search and greedy search, in any kinds of randomly generated conflict graphs.
103% 그림 3에 표기된 것과 같이, $p=8$ 이상의 QAOS 알고리즘은 random search greedy 비해 훨씬 나은 성능을 보였다.
104In these repeated simulations, the performances of QAOS algorithms are improved as $p$ value increases.
105% 또한 이번 반복 시뮬레이션에서, QAOS $p 증가함에 따라  좋은 성능을 보였다.
106In particular, the performance of QAOS algorithm with $p=10$ is much better than the QAOS algorithms with $p=8 $ and $p=9$.
107% 특히 QAOS with $p=10$ QAOS with $p=8$ QAOS with $p=9$ 비해 훨씬 좋은 성능을 보였다.
108As shown in Table~\ref{tab:tab1}, the QAOS algorithm with $p=10$ returns optimal solutions (i.e., equivalent to the solutions obtained by brute-force full search) with the ratio of $69.50\%$.
109% 표1을 보면, $p=10$에서는 전수조사의 경우와 같은 솔루션을 69.50%의 비율로 반환하였다.
110Through these performance evaluation results, it has verified that our proposed QAOS algorithm presents desired results in terms of the accuracy of the solutions.
111% 이러한 성능 평가 결과를 통해, 솔루션의 정확성이라는 측면에서 QAOS 보장된 성능을 확인할  있었다.
112
113
114\begin{table}[h]%\\
115\caption{Percentage of Optimal Solution Computation}
116\label{tab:tab1}
117\footnotesize
118\begin{center}
119  \centering
120  \begin{tabular}{c|c|c|c|c}
121    \toprule[1.0pt]
122    \centering
123    QAOS, $p=10$ & QAOS, $p=9$ & QAOS, $p=8$ & Greedy & Random \\
124    \midrule[1.0pt]
125    $69.50$\% & $49.67$\% & $42.83$\% & $33.83$\% & $15.17$\% \\
126    \bottomrule[1.0pt]
127  \end{tabular}
128\end{center}
129\end{table}
130
131
132\section{Concluding Remarks}\label{sec:6}
133In wireless network research, the large-scale and multi-scale scheduling can be modeled with the MWIS problem which is one of well-known NP-hard problems. In order to solve the MWIS problem, a QAOA-based scheduling algorithm, so called \textit{quantum approximate optimization for scheduling (QAOS)}, is proposed. The performance of our proposed QAOS is evaluated via data-intensive simulations using Cirq and TensorFlow-Quantum. As a result, we confirm that our proposed QAOS outperforms the other methods for the MWIS-based multi-scale scheduling problem.
134
135
136\bibliographystyle{IEEEtran}
137\bibliography{main}
138
139\end{document}