Towards Finding an Optimal Flight Gate Assignment on a Digital Quantum Computer

TL;DR

This paper explores using a quantum algorithm called VQE to solve the airport gate assignment problem, showing promising results and better performance with entanglement, indicating potential for quantum optimization in complex logistical tasks.

Contribution

The paper introduces a qubit-efficient binary encoding and applies CVaR-VQE to the gate assignment problem, demonstrating improved solution quality and favorable scaling on classical simulations.

Findings

CVaR-VQE outperforms naive VQE in solution quality

Entangling gates improve the performance of the ansatz

Scaling of cost function calls is not exponential in tested regimes

Abstract

We investigate the performance of the variational quantum eigensolver (VQE) for the optimal flight gate assignment problem. This problem is a combinatorial optimization problem that aims at finding an optimal assignment of flights to the gates of an airport, in order to minimize the passenger travel time. To study the problem, we adopt a qubit-efficient binary encoding with a cyclic mapping, which is suitable for a digital quantum computer. Using this encoding in conjunction with the Conditional Value at Risk (CVaR) as an aggregation function, we systematically explore the performance of the approach by classically simulating the CVaR-VQE. Our results indicate that the method allows for finding a good solution with high probability, and the method significantly outperforms the naive VQE approach. We examine the role of entanglement for the performance, and find that ans\"atze with entangling gates allow for better results than pure product states. Studying the problem for various sizes, our numerical data show that the scaling of the number of cost function calls for obtaining a good solution is not exponential for the regimes we investigate in this work.

Equations

$$x_{i\alpha }=\{\begin{array}{ll}1,& \text{if flight i∈F is assigned to gate α∈G},\\ 0,& \text{otherwise}.\end{array}$$

$$T(x)=T^{\text{arr}}(x)+T^{\text{dep}}(x)+T^{\text{trans}}(x),$$

$$T^{\text{arr/dep}}(x)=\sum _{i\alpha }{n}_i^{\text{arr/dep}}{t}_{\alpha }^{\text{arr/dep}}{x}_{i\alpha },$$

$$T^{\text{trans}}(x)=\sum _{i,j,\alpha ,\beta }{n}_{ij}^{\text{trans}}{t}_{\alpha \beta }{x}_{i\alpha }{x}_{j\beta }.$$

$$\forall i\in F\sum _{\alpha }{x}_{i\alpha }=1.$$

$$\forall \alpha \in G\text{and}\forall (i,j)\in P{x}_{i\alpha }\times x_{j\alpha }=0,$$

$$P=\{(i,j)\in F\times F:t_i^{\text{in}}<t_j^{\text{in}}<t_i^{\text{out}}+t^{\text{buf}}\}.$$

$$C^{\text{one}}(x)=\sum _i{\left(\sum _{\alpha }{x}_{i\alpha }-1\right)}^2,$$

$$C^{\text{not}}(x)=\sum _{(i,j)\in P}\sum _{\alpha }{x}_{i\alpha }{x}_{j\alpha }.$$

$$Q(x)$$

$$=c+\sum _{i\alpha }{h}_{i\alpha }\times x_{i\alpha }+\sum _{i\alpha j\beta }{J}_{i\alpha j\beta }\times x_{i\alpha }{x}_{j\beta }.$$

$$\hat{H}$$

$$R_{\text{fea}}^{\text{one}}={\left(\frac{|G|}{2^{|G|}}\right)}^{|F|}.$$

$$|{\alpha }^{\prime }⟩\leftrightarrow \text{gate}\alpha ={\alpha }^{\prime }\mathrm{mod}|G|\in G,$$

$${\hat{P}}_i({\alpha }^{\prime })$$

$$={\left(\text{op}{z}_0{z}_0\otimes \cdots \otimes \text{op}{z}_{M-1}{z}_{M-1}\right)}_i.$$

$$\hat{H}(\hat{Z})={\hat{H}}^{\text{arr}}+{\hat{H}}^{\text{dep}}+{\hat{H}}^{\text{trans}}+{\lambda }^{\text{not}}{\hat{H}}^{\text{not}},$$

$${\hat{H}}^{\text{arr/dep}}$$

$${\hat{H}}^{\text{trans}}$$

$${\hat{H}}^{\text{not}}$$

$$\text{op}{z}_k{z_k}_i=\left(\hat{I}+(-1{)}^{z_k}{Z}_{i\times M+k}\right)/2,$$

$${\text{CVaR}}_{\xi }(X)=\mathbb{E}\left[X|X\le F_X^{-1}(\xi )\right],$$

$${\text{CVaR}}_{\xi }=\frac{1}{\lceil \xi K\rceil }\sum _{i=1}^{\lceil \xi K\rceil }{E}_i.$$

$$\text{Qcircuit@C}=1em@R=.7em$$

$$\text{lstick}|0⟩$$

$$c$$

$$h_{i\alpha }$$

$$J_{i\alpha j\beta }$$

$${\delta }_{ij}^P=\{\begin{array}{ll}1,& \text{if (i,j)∈P},\\ 0,& \text{otherwise}.\end{array}$$

$$H$$

$$=c+\frac{1}{2}\sum _{i\alpha }{h}_{i\alpha }+\frac{1}{4}\sum _{i\alpha j\beta }{J}_{i\alpha j\beta }$$

$$+\sum _{i\alpha }\left[-\frac{1}{2}{h}_{i\alpha }-\frac{1}{4}\sum _{j\beta }(J_{i\alpha j\beta }+J_{j\beta i\alpha })+\frac{1}{4}{J}_{i\alpha i\alpha }\right]{\hat{Z}}_{i\alpha }$$

$$+\frac{1}{4}\sum _{i\alpha \ne j\beta }{J}_{i\alpha j\beta }{\hat{Z}}_{i\alpha }{\hat{Z}}_{j\beta }.$$

$$\hat{H}$$

$$c^{\prime }$$

Full text

††preprint: MIT-CTP/5484

I Introduction

In recent years, variational quantum algorithms (VQAs) Peruzzo et al. (2014); McClean et al. (2016); Cerezo et al. (2021) have become increasingly relevant due to substantial progress in quantum hardware development. Such algorithms typically do not require deep quantum circuits that could only be faithfully executed on fully error-corrected quantum computers. Instead, they are amenable to noisy intermediate-scale quantum (NISQ) devices (see, e.g., Refs. Peruzzo et al. (2014); Kandala et al. (2017); Kokail et al. (2019); Hartung and Jansen (2019); Hempel et al. (2018); Barkoutsos et al. (2020); Atas et al. (2021); Mohammadbagherpoor et al. (2021) for various proof-of-principle demonstrations). While such algorithms are typically heuristics without proven performance guarantees, there are indications that VQAs can outperform classical algorithms for certain computationally hard problems. Besides various applications for quantum simulations, the solution of combinatorial optimization problems are further candidates of widespread applications that can be tackled with VQAs Farhi et al. (2014).

In order to assess the potential of VQA approaches for real-world applications, it can be useful to investigate applications beyond purely academic problems and to focus on certain industrial use-cases, as they typically exhibit additional complexity. One such example is the flight-gate assignment (FGA) problem Kim et al. (2017); Stollenwerk et al. (2019). The FGA problem is a quadratic assignment problem Finke et al. (1987) with additional constraints, as typical for real-world applications. Previous works mainly investigated the solution of the FGA problem Stollenwerk et al. (2019) and related problems Venturelli et al. (2016); Stollenwerk et al. (2020a, 2021) with quantum annealers. Here, the constraints are incorporated into an unconstrained cost function by penalty terms. These approaches have a number of disadvantages, one of which is the typically exponentially small subspace of valid solutions in the entire Hilbert space Stollenwerk et al. (2021). One method for mitigating this issue is to constrain the algorithm to only search the feasible subspace. This idea was originally proposed for quantum annealing Hen and Spedalieri (2016) and later adapted to variational algorithms Hadfield et al. (2019); Mohammadbagherpoor et al. (2021). The applicability of the latter approach for FGA was investigated by deriving suitable algorithmic primitives for constraint invariance Stollenwerk et al. (2020b). Reference Mohammadbagherpoor et al. (2021) implemented a proof-of-principle VQE for the FGA problem using an encoding incorporating some of the constraints on IBM’s quantum hardware, thus demonstrating the suitability of problem for digital quantum devices.

In this paper, we systematically assess the performance of VQE for the FGA problem, by numerically studying its performance using the Conditional Value at Risk (CVaR) Barkoutsos et al. (2020) as an aggregation function. We adopt an encoding that avoids a dominant subspace of invalid solutions, which is similar to the one of Ref. Mohammadbagherpoor et al. (2021), with the addition of a cyclic mapping. Our study demonstrates that utilizing this encoding, the CVaR-VQE performs significantly better than the naive encoding used in previous works. From classically simulating the CVaR-VQE for various problem sizes up to 18 qubits, our results indicate that the number of cost function calls to obtain a reasonably large contribution of the optimal solution in the final state does not scale exponentially with the problem size. Furthermore, we examine the role of entangling gates in the ansatz. Our results demonstrate that ansätze creating entanglement between qubits show a significantly better performance than circuits preparing only product states.

The paper is organized as follows. In Sec. II, we first introduce the FGA problem, before discussing the one-hot encoding and the binary encoding of the problem. Subsequently, we discuss the CVaR-VQE method and the types of ansätze we use in our simulations in Sec. III. Section IV shows our numerical results for classically simulating the CVar-VQE for various problem sizes, and a comparison between entangling ansätze and ansätze that only produce product states. Finally, we conclude in Sec. IV.

II The flight-gate assignment problem and its encoding into quantum states

In this section, we first introduce the FGA problem and then proceed with discussing two ways of encoding the problem into quantum states: the one-hot encoding, which does not incorporate any of the constraints, and a binary encoding that integrates some of the constraints.

II.1 The flight-gate assignment problem

The FGA problem aims at minimizing the total transit time of passengers in an airport by finding an optimal gate assignment of the flights. Although there are multiple scenarios for optimizing the gate assignment of flights at an airport, we choose the one where we seek to minimize the total transfer time of passengers at the airport Kim et al. (2017). In this scenario, we have three kinds of passengers in an airport: arriving passengers, departing passengers, and transfer passengers. The arriving passengers land at the airport with an inbound flight and need to walk from the arrival gate to the baggage claim before leaving the airport. Departing passengers enter the airport through the security checkpoint and leave with an outbound flight. Transfer passengers arrive at the airport with an inbound flight, have to walk to the gate of their connecting flight, and leave with an outbound flight. To model the problem mathematically given a set of flights $F$ and a set of gates $G$, we consider a set of binary decision variables $x_{i\alpha}$ that represent whether a flight $i$ is assigned to a gate $\alpha$ or not:

[TABLE]

Throughout the paper, we refer to gates with Greek indices, to flights with Latin indices, and $x=(x_{i\alpha})\in\{0,1\}^{|F|\times|G|}$ is a binary vector collecting all of the $|F|\times|G|$ decision variables. The total passenger travel time can then be expressed as a function of $x$ and is given by

[TABLE]

where the three parts arise from the contributions of the different types of passengers. The time $T^{\text{arr/dep}}$ represents the total transit time of arriving/departing passengers and is given by the partial sums

[TABLE]

where $n_{i}^{\text{arr/dep}}$ is the number of passengers arriving/departing with flight $i$, and $t_{\alpha}^{\text{arr/dep}}$ is the time it takes to walk from/to gate $\alpha$. The total time $T^{\text{trans}}$ of the transfer passengers is given by the sum of the times $t_{\alpha\beta}$ that it takes to go from gate $\alpha$ to gate $\beta$ for each of the $n_{ij}^{\text{trans}}$ passengers who transfer from flight $i$ to flight $j$ (or vice versa), given that flight $i$ is assigned to gate $\alpha$ and flight $j$ is assigned to gate $\beta$,

[TABLE]

Note that $T^{\text{trans}}(x)$ contains a term quadratic in the decision variables. Thus, minimizing the total time in Eq. (2) is an instance of a quadratic assignment problem, which are in general NP-hard Garey and Johnson (1979).

In addition, there are two constraints in the FGA problem. Firstly, each flight can only be assigned to one gate, so there can only be a single non-zero decision variable among those belonging to the same flight. This constraint can be enforced by imposing

[TABLE]

Secondly, there can be at most a single flight at a gate at the same time, because flights departing at the same time from the airport cannot be assigned to the same gate. This can be expressed as

[TABLE]

where $P$ is the set of forbidden flight pairs,

[TABLE]

In the expression above, $t_{i}^{\text{in/out}}$ is the time of arrival/departure of flight $i$, and $t^{\text{buf}}$ is a buffer time between two flights at the same gate. In the following, we refer to an assignment of the decision variables fulfilling the two constraints above as a feasible assignment.

The encoding presented above requires $|G|$ decision variables $x_{i1}\dots x_{i|G|}$ for each flight $i\in F$, which can be interpreted as a bit string. The constraint in Eq. (5) then implies that only a single entry in such a bit string can be nonzero. Hence, we call the encoding presented above the one-hot encoding. Since for each flight only $|G|$ assignments of the corresponding decision variables are compliant with the constraint in Eq. (5), the total number of feasible assignments is upper bounded by $|G|^{|F|}$.

II.2 Hamiltonian formulation using the one-hot encoding

In order to treat the problem on a quantum computer, we have to formulate the problem as a (quantum) Hamiltonian. In order to minimize the objective function $T(x)$ subject to the constraints in Eqs. (5) and (6), we want to incorporate the constraints in the objective function. To this end, we translate them to positive semidefinite penalty terms whose kernel corresponds to valid solutions fulfilling the constraints. These penalty terms can then simply be added to the objective function with a large positive constant in front, thus ensuring that the global minimum is the optimal solution fulfilling the constraints.

Equation (5) can be represented as a penalty term,

[TABLE]

while the second constraint in Eq. (6) can be formulated as

[TABLE]

Considering both the objective function and the penalty terms, the total cost function can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem:

[TABLE]

In the equation above, $c$, $h_{i\alpha}$, and $J_{i\alpha j\beta}$ are the coefficients of the corresponding terms, which depend on $t_{\alpha}^{\text{arr/dep}}$, $t_{\alpha\beta}$, $n_{i}^{\text{arr/dep}}$, and $n_{ij}^{\text{trans}}$. The explicit formulas of these coefficients are shown in Eq. (20) of Appendix A. The parameters $\lambda^{\text{one}}$ and $\lambda^{\text{not}}$ are constants that have to be chosen large enough to ensure the solution of the QUBO problem above satisfies the constraints. For practical purposes, the values of these parameters might have to be set carefully to make the optimization procedure efficient Stollenwerk et al. (2019).

In order to solve this problem using a quantum device, the QUBO problem has to be mapped to a Hamiltonian acting on qubits. This can be easily realized by replacing the binary decision variables $x_{i\alpha}$ in $Q(x)$ with the operators $(I-\hat{Z}_{k})/2$, where $I$ is the identity and $\hat{Z}_{k}$ is the Pauli $Z$-matrix acting on the qubit that encodes the decision variable $x_{i\alpha}$. Substituting this transformation into the QUBO problem in Eq. (10), we obtain the (quantum) Hamiltonian

[TABLE]

where $N=|F|\times|G|$ and $c^{\prime}$, $h_{p}^{\prime}$, and $J_{pq}^{\prime}$ are coefficients related to the ones of the original QUBO problem (see Eq. (22) in Appendix A for details). The bit strings $x$ are now encoded by a computational basis state $\ket{x}$, and we call $\ket{x}$ a feasible state if $x$ represents a feasible assignment. The optimal solution of the FGA problem subject to the constraints corresponds to the ground state of the Hamiltonian above. By construction, the ground state will be a computational basis state since $\hat{H}$ is diagonal in the $Z$-basis.

Note that in the encoding presented above, each decision variable is mapped to a single qubit. Hence, a total number of $|F|\times|G|$ qubits are required to address the problem on a quantum computer. However, only $|G|^{|F|}$ of the $2^{|G|\times|F|}$ basis states correspond to an assignment for which Eq. (8) is zero. Hence, the fraction of states in the Hilbert space fulfilling the first constraint, and correspondingly the number of feasible states, will decay exponentially with the problem size:

[TABLE]

As a result, searching for the optimal solution will become increasingly challenging for increasing numbers of flights and gates.

II.3 Hamiltonian formulation using a binary encoding

In order to avoid this exponential decay of the feasible subspace, we use a binary encoding for the FGA problem and derive the corresponding Hamiltonian, which is similar to the efficient embedding in Ref. Mohammadbagherpoor et al. (2021). In addition, we use a more efficient cyclic mapping as shown below.

As we have discussed in Sec. II.1, there are $|G|$ assignments compliant with the first constraint in Eq. (5) for the decision variables corresponding to each flight. These assignments can be represented with $M=\lceil\log(|G|)\rceil$ (qu)bits using a binary encoding. Since $|G|$ is in general not a power of 2, we choose to map the elements in $G$ to the $2^{M}$ basis states $\ket{\alpha^{\prime}}$ cyclically as

[TABLE]

where $\alpha^{\prime}=0,\dots,2^{M}-1$. In contrast, the previous work in Ref. Mohammadbagherpoor et al. (2021) added a penalty term for the additional states $\{|\alpha^{\prime}\rangle:|G|\leq\alpha^{\prime}<2^{M}\}$, in case $G$ is not a power of 2. However, this will lead to an exponential decay with $|F|$ for the fraction of feasible states, as these are given by $\left(G/2^{M}\right)^{|F|}$. The cyclic mapping used in this work can avoid this exponential decay of feasible states and will usually lead to many degenerate ground states, rendering it easier to find an optimal solution. All in all, for a total of $|F|$ flights, this encoding allows us to represent all possible assignments with $|F|\times M$ qubits, a lot less than that required for the one-hot encoding. Moreover, by construction, all solutions in this encoding automatically fulfill the constraint in Eq. (5).

In order to be able to solve the problem on a quantum computer using a VQA, we have to translate the Hamiltonian in Eq. (11) to this encoding. To this end, we define a set of projection operators $P_{i}(\alpha^{\prime})$, $i=0,\cdots,F-1$ given by

[TABLE]

In the expression above, $z_{0}\cdots z_{M-1}$ is the bit string for the binary representation of $\alpha^{\prime}$ and the index $i$ indicates the set of qubits related to flight $i$, on which the projection operators are acting on. Applying $\hat{P}_{i}(\alpha^{\prime})$ to one of the basis states encoding the solutions compliant with the first constraint for flight $i$ results in a $1$, if and only if flight $i$ is assigned to gate $\alpha$, $\hat{P}_{i}(\alpha^{\prime})\ket{\beta^{\prime}}_{i}=\delta_{\alpha^{\prime}\beta^{\prime}}$. Using these projection operators, the Hamiltonian can be expressed as

[TABLE]

where the individual terms are given by

[TABLE]

In the expression above, $\alpha$ and $\beta$ refer to the gate indices after applying the mapping from Eq. (13). Note that we no longer have to impose the first constraint from Eq. (5) with a penalty term anymore, as it is fulfilled by construction. Moreover, the Hamiltonian can be easily decomposed into Pauli operators using the relation

[TABLE]

where we have chosen a linear ordering of the qubits.

The binary encoding with cyclic mapping still allows for unfeasible states, as the second constraint from Eq. (6) is not automatically fulfilled. Compared to the exponential decay observed for the one-hot encoding, the ratio of feasible solutions for the binary encoding is a lot larger, and it decays only very slowly with problem size, as shown in Fig. 1.

In conjunction with its reduced qubit requirements, the binary encoding with cyclic mapping is significantly more amenable for NISQ devices, which provide only limited resources. The Hamiltonian corresponding to the binary encoding consists of $\mathcal{O}(|F|^{2}\times|G|^{2})$ Pauli $Z$-terms with order $2\times\lceil\log(G)\rceil$ or less, meaning that each Pauli $Z$-term only acts nontrivially on at most $2\times\lceil\log(G)\rceil$ qubits. Thus, the expectation value of the Hamiltonian can be evaluated efficiently on a quantum computer.

III Variational quantum eigensolver using the conditional value at risk

The VQE is a hybrid quantum-classical algorithm for finding an approximation to the ground state of a given Hamiltonian $\hat{H}$ by minimzing $\langle\psi(\bm{\theta})|\hat{H}|\psi(\bm{\theta})\rangle$. Here, $|\psi(\bm{\theta})\rangle$ is a normalized ansatz state, which is parametrized by real numbers $\bm{\theta}$. To find an optimal set of parameters, the VQE utilizes of a feedback loop between a quantum device and a classical computer. The former is used to realize a variational ansatz $\ket{\psi(\bm{\theta})}$ in form of a parametric quantum circuit, and to measure the expectation value of the Hamiltonian. The classical computer is running a minimization algorithm suggesting a new set of parameters $\bm{\theta}^{\prime}$ based on the measurement outcome of the quantum device. Running the feedback loop until convergence, the parametric circuit encodes an approximation of the ground state of the given Hamiltonian. Due to its modest quantum hardware requirements, and its partial resilience to noise, the VQE is one of the most promising candidates for applications on NISQ devices. While the VQE was originally proposed for finding the ground state of a molecule Peruzzo et al. (2014), it can be readily applied to many other fields (see, e.g., Refs. Kokail et al. (2019); Paulson et al. (2021); Avkhadiev et al. (2020); Mazzola et al. (2021); Tilly et al. (2022)).

In particular, the VQE has been proposed to solve combinatorial optimization problems Amaro et al. (2022); Nannicini (2019); Mugel et al. (2022). Contrary to strongly-correlated quantum many-body systems, for combinatorial optimization problems the problem Hamiltonian is diagonal and the possible solutions correspond to basis states. Since we are only interested in obtaining a good candidate for the solution of the combinatorial optimization problem, the resulting state at the end of the VQE does not necessarily have to be dominated by the state encoding this solution. As long as it produces a state that has a reasonably large component of such a solution, the projective measurements at the end will reveal it, provided enough measurements are taken. Due to this property, Ref. Barkoutsos et al. (2020) argued that the CVaR is better suited as a cost function for combinatorial optimization problems than the expectation value of the Hamiltonian. The CVaR for a random variable $X$ with the cumulative density function $F_{X}$ is defined as the conditional expectation over the left $\xi$-tail of the distribution,

[TABLE]

where $\xi\in(0,1]$. This can be applied to VQE by considering only a subset of the samples obtained during the measurement process. Suppose we perform $K$ measurements resulting in the bit strings $\{z_{1},z_{2},\cdots,z_{K}\}$ and the corresponding energy values $\{E_{1},E_{2},\cdots,E_{K}\}$. Assuming the energy values are sorted in ascending order, the CVaR can be calculated as

[TABLE]

Note that for $\xi=1$ the $\text{CVaR}_{\xi}$ is nothing but the usual estimate for the expectation value with $K$ samples. In the opposite limit, $\xi\to 0$, the $\text{CVaR}_{\xi}$ corresponds to selecting the measurement that produced the lowest energy. Moreover, the definition in Eq. (19) shows that the $\text{CVaR}_{\xi}$ does essentially not reward increasing the fidelity of the VQE solution with the ground state beyond $\xi$, as we only consider the subset of the $\lceil\xi K\rceil$ measurements with the lowest energy.

In the following, we use VQE with the $\text{CVaR}_{\xi}$ as a cost function to address the FGA problem. In particular, we explore the performance for various choices of $\xi$ as a function of problem size.

IV Simulation results

In order to explore the performance of the VQE using the CVaR for the FGA problem, we perform classical simulations using the Qiskit tA-v et al. (2021) framework, assuming a perfect quantum device without shot noise, which means we evaluate the cost function exactly. For our experiments, we use the EfficentSU2 ansatz from Qiskit consisting of parametric $R_{Y}(\theta)=\exp(-i\theta Y/2)$ rotation gates and linear entangling layers of CNOT gates (see Fig. IV for an illustration).