The performance of variational quantum algorithms relies on the success of using quantum and classical computing resources in tandem. Here, we study how these quantum and classical components interrelate. In particular, we focus on algorithms for solving the combinatorial optimization problem MaxCut, and study how the structure of the classical optimization landscape relates to the quantum circuit used to evaluate the MaxCut objective function. In order to analytically characterize the impact of quantum features on the critical points of the landscape, we consider a family of quantum circuit ans\"atze composed of mutually commuting elements. We identify multiqubit operations as a key resource and show that overparameterization allows for obtaining favorable landscapes. Namely, we prove that an ansatz from this family containing exponentially many variational parameters yields a landscape free of local optima for generic graphs. However, we further prove that these ans\"atze do not offer superpolynomial advantages over purely classical MaxCut algorithms. We then present a series of numerical experiments illustrating that noncommutativity and entanglement are important features for improving algorithm performance.
The performance of variational quantum algorithms relies on the success of using quantum and classical computing resources in tandem. Here, we study how these quantum and classical components interrelate. In particular, we focus on algorithms for solving the combinatorial optimization problem MaxCut, and study how the structure of the classical optimization landscape relates to the quantum circuit used to evaluate the MaxCut objective function. In order to analytically characterize the impact of quantum features on the critical points of the landscape, we consider a family of quantum circuit ans"atze composed of mutually commuting elements. We identify multiqubit operations as a key resource and show that overparameterization allows for obtaining favorable landscapes. Namely, we prove that an ansatz from this family containing exponentially many variational parameters yields a landscape free of local optima for generic graphs. However, we further prove that these ans"atze do not offer superpolynomial advantages over purely classical MaxCut algorithms. We then present a series of numerical experiments illustrating that noncommutativity and entanglement are important features for improving algorithm performance.
Combinatorial optimization is a promising application for near-term quantum computers, however, identifying performant algorithms suited to noisy quantum hardware remains as an important goal to potentially realizing quantum computational advantages. To address this we present a variational quantum algorithm for solving combinatorial optimization problems with linear-depth circuits. Our algorithm uses an ansatz composed of Hamiltonian generators designed to control each term in the target combinatorial function, along with parameter updates following a modified version of quantum imaginary time evolution. We evaluate this ansatz in numerical simulations that target solutions to the MAXCUT problem. The state evolution is shown to closely mimic imaginary time evolution, and its optimal-solution convergence is further improved using adaptive transformations of the classical Hamiltonian spectrum. With these innovations, the algorithm consistently converges to optimal solutions, with interesting highly-entangled dynamics along the way. We further demonstrate the success of this approach by performing optimization of a truncated version of our ansatz with up to 32 qubits in a trapped-ion quantum computer, using measurements from the quantum computer to train the ansatz and prepare optimal solutions with high fidelity in the majority of cases we consider. The success of these large-scale quantum circuit optimizations, in the presence of realistic hardware constraints and without error mitigation, mark significant progress on the path towards solving combinatorial problems using quantum computers. We conclude our performant and resource-minimal approach is a promising candidate for potential quantum computational advantages.
The recent literature on near-term applications for quantum computers contains several examples of the applications of hybrid quantum-classical variational approaches. This methodology can be applied to a variety of optimization problems, but its practical performance is not well studied yet. This paper moves some steps in the direction of characterizing the practical performance of the methodology, in the context of finding solutions to classical combinatorial optimization problems. Our study is based on numerical results obtained applying several classical nonlinear optimization algorithms to Hamiltonians for six combinatorial optimization problems; the experiments are conducted via noise-free classical simulation of the quantum circuits implemented in Qiskit. We empirically verify that: (1) finding the ground state is harder for Hamiltonians with many Pauli terms; (2) classical global optimization methods are more successful than local methods due to their ability of avoiding the numerous local optima; (3) there does not seem to be a clear advantage in introducing entanglement in the variational form.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
Variational quantum algorithms, and in particular variants of the variational quantum eigensolver, have been proposed as approaches to combinatorial optimization (CO) problems. With only shallow ansatz circuits, these methods are considered suitable for current noisy intermediate-scale quantum hardware. Yet, the resources required to train variational circuits often scale superpolynomially with problem size. In this study, we numerically investigate the practical implications of this scaling for CO problems, using Max-Cut and random QUBO instances as benchmarks. For a fixed computational budget, we compare the average performance of training shallow variational circuits, sampling with replacement, and greedy local search. We identify the minimum problem size at which the quantum algorithms consistently outperform random sampling, and for each size, we characterize their separation from greedy local search. Beyond average-case performance, we analyze correlations between algorithms across individual problem instances. These results strengthen the case for intuitive yet meaningful benchmarks of variational quantum algorithms for CO problems under realistic resource constraints.
Abstract Variational quantum algorithms, and in particular variants of the variational quantum eigensolver, have been proposed as approaches to combinatorial optimization (CO) problems. With only shallow ansatz circuits, these methods are considered suitable for current noisy intermediate-scale quantum hardware. Yet, the resources required to train variational circuits often scale superpolynomially with problem size. In this study, we numerically investigate the practical implications of this scaling for CO problems, using Max-Cut and random QUBO instances as benchmarks. For a fixed computational budget, we compare the average performance of training shallow variational circuits, sampling with replacement, and greedy local search. We identify the minimum problem size at which the quantum algorithms consistently outperform random sampling, and for each size, we characterize their separation from greedy local search. Beyond average-case performance, we analyze correlations between algorithms across individual problem instances. These results strengthen the case for intuitive yet meaningful benchmarks of variational quantum algorithms for CO problems under realistic resource constraints.
Optimization plays a central role in variational quantum algorithms, where the objective function typically takes the form F(θ)=∑m=1Mfm(Tr[U(θ)ρmU†(θ)Om]), with U(θ) being a parameterized quantum ansatz. Understanding the optimization landscape of such objective functions is crucial for assessing the trainability and performance of these algorithms. For the special case M=1, it is known that under certain assumptions, the landscape is free of false traps (FTs), i.e., local optima that are not global. In this work, we investigate optimization landscapes of the general case M≥1 and show that the landscape becomes intrinsically more complex. First, we establish a complete framework for analyzing critical features of the optimization landscape, by deriving necessary and sufficient conditions to identify and classify all critical points under some assumptions, which is also of practical importance in designing efficient algorithms independent of whether these assumptions are satisfied. Then, we show that FTs can still emerge on landscapes for M>1, standing in stark contrast to the M=1 case and further revealing that parameter sufficiency alone is not enough to guarantee a trap-free landscape. Moreover, we uncover a close connection that the emergence of FTs is necessarily attributed to the loss of distinguishability among the states and/or operators, and fundamentally, to the loss of compatibility of the spectral ordering governed by different objective terms. Our results provide a deeper understanding of the optimization complexity and practical guidance for both algorithmic and problem-setting designs.