We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level $p$: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-$1$ QAOA and level-$1$ RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-$1$ QAOA and RQAOA with up to $300$ qutrits applied to ensembles of randomly generated $3$-colorable constant-degree graphs. We find that level-$1$ RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices.
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal sol…
We develop a heuristic graph coloring approximation algorithm that uses the D-Wave 2X as an independent set sampler and evaluate its performance against a fully classical implementation. A randomly generated set of small but hard graph instances serves as our test set. Our performance analysis suggests limited quantum advantage in the hybrid quantum-classical algorithm. The quantum edge holds over multiple metrics and suggests that graph problem applications are a good fit for quantum annealers.
The Quantum Approximate Optimization Algorithm (QAOA) has been one of the leading candidates for near-term quantum advantage in gate-model quantum computers. From its inception, this algorithm has sparked the desire for comparison between gate-model and annealing platforms. When preparing problem statements for these algorithms, the predominant inclination has been to formulate a quadratic Hamiltonian. This paper gives an example of a graph coloring problem that, depending on its variable encoding scheme, optionally admits higher order terms. This paper presents evidence that the higher order …
In this work we start from a phenomenological Hamiltonian built from two known systems: the Hamiltonian of a pumped optomechanical system and the Jaynes Cummings Hamiltonian. Using algebraic techniques we construct an approximate time evolution operator $\hat U_{opt}$ for the forced optomechanical system (as a product of exponentials) and take the JC Hamiltonian as an interaction. We transform the later with $\hat U_{opt}$ to obtain a generalized interaction picture Hamiltonian which can be linearized and whose time evolution operator is written in a product form. The analytic results are comp…
Algorithmic discrepancy theory seeks efficient algorithms to find those two-colorings of a set that minimize a given measure of coloring imbalance in the set, its {\it discrepancy}. The {\it Euclidean discrepancy} problem and the problem of balancing covariates in randomized trials have efficient randomized algorithms based on the Gram-Schmidt walk (GSW). We frame these problems as quantum Ising models, for which variational quantum algorithms (VQA) are particularly useful. Simulating an example of covariate-balancing on an IBM quantum simulator, we find that the variational quantum eigensolve…
The quantum-classical hybrid algorithm is an algorithm that holds promise in demonstrating the quantum advantage in NISQ devices. When running such algorithms, effects from quantum noise are inevitable. In our work, we consider a well-known hybrid algorithm, the quantum approximate optimization algorithm (QAOA). We study the effects on QAOA from typical quantum noise channels and produce several numerical results. Our research indicates that the output state fidelity, the cost function, and its gradient obtained from QAOA decrease exponentially with respect to the number of gates and noise str…
In this paper we elaborate a hybrid classical-quantum framework which allows one to model and solve heat and mass transfer problems occurring in electric contacts. We utilize special functions and Harrow-Hassidim-Lloyd (HHL) quantum algorithm for finding temperature and arc flux functions exactly and approximately for the Stefan type problems. The Stefan type problems we are considering are based on the Generalized Heat Equation with free boundaries. As examples we consider exact and approximate solutions of inverse one-phase and two-phase Stefan problems. An Inverse Generalized One-Phase Stef…
Quantum Approximate Optimization Algorithm (QAOA) is studied primarily to find approximate solutions to combinatorial optimization problems. For a graph with $n$ vertices and $m$ edges, a depth $p$ QAOA for the Max-cut problem requires $2\cdot m \cdot p$ CNOT gates. CNOT is one of the primary sources of error in modern quantum computers. In this paper, we propose two hardware independent methods to reduce the number of CNOT gates in the circuit. First, we present a method based on Edge Coloring of the input graph that minimizes the the number of cycles (termed as depth of the circuit), and red…
Constrained Portfolio Optimization via Quantum Approximate Optimization Algorithm (QAOA) with XY-Mixers and Trotterized Initialization: A Hybrid Approach for Direct Indexing
SquareOne Capital（SquareOne资本）；School of Physics, Faculty of Sciences, Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学理学院物理系）
