Approximate Optimization Research Articles

Scaling whole-chip QAOA for higher-order ising spin glass models on heavy-hex graphs

We show that the quantum approximate optimization algorithm (QAOA) for higher-order, random coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from 16 up to 127 qubits for p = 1 up to p = 5, which allows for computationally efficient parameter transfer of QAOA angles. Matrix product state (MPS) simulation is used to compute noise-free QAOA performance. Hardware-compatible short-depth QAOA circuits are executed on ensembles of 100 higher-order Ising models on noisy IBM quantum superconducting processors with 16, 27, and 127 qubits using QAOA angles learned from a single 16-qubit instance using the JuliQAOA tool. We show that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution. We show that p = 1 QAOA energy landscapes remain very similar as the problem size increases using NISQ hardware gridsearches with up to a 414 qubit processor.

A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms

A key component of variational quantum algorithms (VQAs) is the choice of classical optimizer employed to update the parameterization of an ansatz. It is well recognized that quantum algorithms will, for the foreseeable future, necessarily be run on noisy devices with limited fidelities. Thus, the evaluation of an objective function (e.g., the guiding function in the quantum approximate optimization algorithm (QAOA) or the expectation of the electronic Hamiltonian in variational quantum eigensolver (VQE)) required by a classical optimizer is subject not only to stochastic error from estimating an expected value but also to error resulting from intermittent hardware noise. Model-based derivative-free optimization methods have emerged as popular choices of a classical optimizer in the noisy VQA setting, based on empirical studies. However, these optimization methods were not explicitly designed with the consideration of noise. In this work we adapt recent developments from the “noise-aware numerical optimization” literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. We study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in scikit-quant. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers and the Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing program under contract number DE-AC02-06CH11357. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0177 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0177 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Grover-QAOA for 3-SAT: quadratic speedup, fair-sampling, and parameter clustering

Abstract The SAT problem is a prototypical NP-complete problem of fundamental importance in computational complexity theory with many applications in science and engineering; as such, it has long served as an essential benchmark for classical and quantum algorithms. This study shows numerical evidence for a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random sampling for finding all solutions to 3-SAT (All-SAT) and Max-SAT problems. G-QAOA is less resource-intensive and more adaptable for these problems than Grover’s algorithm, and it surpasses conventional QAOA in its ability to sample all solutions. We show these benefits by classical simulations of many-round G-QAOA on thousands of random 3-SAT instances. We also observe G-QAOA advantages on the IonQ Aria quantum computer for small instances, finding that current hardware suffices to determine and sample all solutions. Interestingly, a single-angle-pair constraint that uses the same pair of angles at each G-QAOA round greatly reduces the classical computational overhead of optimizing the G-QAOA angles while preserving its quadratic speedup. We also find parameter clustering of the angles. The single-angle-pair protocol and parameter clustering significantly reduce obstacles to classical optimization of the G-QAOA angles.

Efficient quantum circuits based on the quantum natural gradient

Efficient preparation of arbitrary entangled quantum states is crucial for quantum computation. This is particularly important for noisy intermediate-scale quantum simulators relying on variational hybrid quantum-classical algorithms. To that end, we propose symmetry-conserving modified quantum approximate optimization algorithm (SCom-QAOA) circuits. The depths of these circuits depend not only on the desired fidelity to the target state but also on the amount of entanglement the state contains. The parameters of the SCom-QAOA circuits are optimized using the quantum natural gradient method based on the Fubini-Study metric. The SCom-QAOA circuit transforms an unentangled state into a ground state of a gapped one-dimensional Hamiltonian with a circuit depth that depends not on the system size but rather on the finite correlation length. In contrast, the circuit depth grows proportionally to the system size for preparing low-lying states of critical one-dimensional systems. Even in the latter case, SCom-QAOA circuits with depth less than the system size were sufficient to generate states with fidelity in excess of 99%, which is relevant for near-term applications. The proposed scheme enlarges the set of the initial states accessible for variational quantum algorithms and widens the scope of investigation of nonequilibrium phenomena in quantum simulators. Published by the American Physical Society 2024

An Efficient Integral Approach for Kinematic Reliability Sensitivity Analysis of Industrial Robots

Abstract Assessment of the reliability and reliability sensitivity of positioning accuracy for industrial robots subject to aleatoric and epistemic uncertainties registers a challenging task. This study proposes a new optimized moment-based method for kinematic reliability analysis and its sensitivity analysis, which incorporates the sparse grid (SPGR) technique and the saddlepoint approximation (SPA) method. To start with, the positioning accuracy reliability and its sensitivity models of industrial robots are established via computational optimization techniques and kinematic criteria. The kinematic accuracy reliability and its sensitivity are then calculated. Specifically, the sparse grid technique is adopted to approach the positioning error statistical moments and moment sensitivities. On this basis, positioning accuracy reliability bounds and reliability sensitivity bounds are obtained by the saddlepoint approximation method and optimization techniques. Finally, two practical examples are implemented to demonstrate the proficiency of the currently proposed method against Monte Carlo simulation (MCS) results. The results show that the currently proposed method exhibits superior computational accuracy and efficiency in kinematic reliability and its sensitivity analyses for industrial robots.

Syndrome decoding by quantum approximate optimization

Syndrome decoding by quantum approximate optimization

Novel Application of Quantum Computing for Routing and Spectrum Assignment in Flexi-Grid Optical Networks

Flexi-grid technology has revolutionized optical networking by enabling Elastic Optical Networks (EONs) that offer greater flexibility and dynamism compared to traditional fixed-grid systems. As data traffic continues to grow exponentially, the need for efficient and scalable solutions to the routing and spectrum assignment (RSA) problem in EONs becomes increasingly critical. The RSA problem, being NP-Hard, requires solutions that can simultaneously address both spatial routing and spectrum allocation. This paper proposes a novel quantum-based approach to solving the RSA problem. By formulating the problem as a Quadratic Unconstrained Binary Optimization (QUBO) model, we employ the Quantum Approximate Optimization Algorithm (QAOA) to effectively solve it. Our approach is specifically designed to minimize end-to-end delay while satisfying the continuity and contiguity constraints of frequency slots. Simulations conducted using the Qiskit framework and IBM-QASM simulator validate the effectiveness of our method. We applied the QAOA-based RSA approach to small network topology, where the number of nodes and frequency slots was constrained by the limited qubit count on current quantum simulator. In this small network, the algorithm successfully converged to an optimal solution in less than 30 iterations, with a total runtime of approximately 10.7 s with an accuracy of 78.8%. Additionally, we conducted a comparative analysis between QAOA, integer linear programming, and deep reinforcement learning methods to evaluate the performance of the quantum-based approach relative to classical techniques. This work lays the foundation for future exploration of quantum computing in solving large-scale RSA problems in EONs, with the prospect of achieving quantum advantage as quantum technology continues to advance.

Revolutionizing Optimization: The Game-Changing Power of Quantum Algorithms and Next-Generation Computational Technologies

Quantum algorithms offer promising advancements in solving complex optimization problems that are critical in various fields, including logistics, finance, and machine learning. Classical optimization techniques, while effective, often struggle with large-scale and high-dimensional data sets, leading to inefficiencies and long processing times.Quantum algorithms, leveraging the principles of superposition, entanglement, and quantum parallelism, provide a potential solution by exponentially speeding up certain computational tasks. This paper explores key quantum algorithms for optimization, including Grover's algorithm, the Quantum Approximate Optimization Algorithm (QAOA), and the Quantum Annealing method. We analyze their theoretical foundations, implementation challenges, and practical applications. Special attention is given to the performance comparisons between quantum and classical approaches in solving combinatorial optimization problems, such as the traveling salesman problem and portfolio optimization. We also discuss the limitations of current quantum hardware and the prospects for future improvements that could make quantum optimization algorithms more practical for real-world use.Our findings suggest that, while quantum algorithms are still in their early stages, they hold immense potential for transforming optimization in a variety of domains, offering new pathways toward more efficient problem-solving in both theoretical and applied contexts. we explore how hybrid quantum-classical approaches can enhance optimization results by combining the strengths of both paradigms. Future research directions are highlighted, focusing on improving quantum error correction, scaling quantum systems, and developing more robust quantum algorithms to tackle increasingly complex optimization problems.

Optimized Noise Suppression for Quantum Circuits

Quantum computation promises to advance a wide range of computational tasks. However, current quantum hardware suffers from noise and is too small for error correction. Thus, accurately utilizing noisy quantum computers strongly relies on noise characterization, mitigation, and suppression. Crucially, these methods must also be efficient in terms of their classical and quantum overhead. Here, we efficiently characterize and mitigate crosstalk noise, which is a severe error source in, for example, cross-resonance based superconducting quantum processors. For crosstalk characterization, we develop a simplified measurement experiment. Furthermore, we analyze the problem of optimal experiment scheduling and solve it for common hardware architectures. After characterization, we mitigate noise in quantum circuits by a noise-aware qubit routing algorithm. Our integer programming algorithm extends previous work on optimized qubit routing by swap insertion. We incorporate the measured crosstalk errors in addition to other, more easily accessible noise data in the objective function. Furthermore, we strengthen the underlying integer linear model by proving a convex hull result about an associated class of polytopes, which has applications beyond this work. We evaluate the proposed method by characterizing crosstalk noise for two chips with up to 127 qubits and leverage the resulting data to improve the approximation ratio of the Quantum Approximate Optimization Algorithm by up to 10% compared with other established noise-aware routing methods. Our work clearly demonstrates the gains of including noise data when mapping abstract quantum circuits to hardware native ones. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue on Quantum Computing. Funding: This work was supported by Bavarian state government; Bayerische Staatsministerium für Wirtschaft, Landesentwicklung und Energie. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0551 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0551 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Approximate optimality and the risk/reward tradeoff given repeated gambles

Approximate optimality and the risk/reward tradeoff given repeated gambles

Applying quantum approximate optimization to the heterogeneous vehicle routing problem

Quantum computing offers new heuristics for combinatorial problems. With small- and intermediate-scale quantum devices becoming available, it is possible to implement and test these heuristics on small-size problems. A candidate for such combinatorial problems is the heterogeneous vehicle routing problem (HVRP): the problem of finding the optimal set of routes, given a heterogeneous fleet of vehicles with varying loading capacities, to deliver goods to a given set of customers. In this work, we investigate the potential use of a quantum computer to find approximate solutions to the HVRP using the quantum approximate optimization algorithm (QAOA). For this purpose we formulate a mapping of the HVRP to an Ising Hamiltonian and simulate the algorithm on problem instances of up to 21 qubits. We show that the number of qubits needed for this mapping scales quadratically with the number of customers. We compare the performance of different classical optimizers in the QAOA for varying problem size of the HVRP, finding a trade-off between optimizer performance and runtime.

Bayan algorithm: Detecting communities in networks through exact and approximate optimization of modularity

Bayan algorithm: Detecting communities in networks through exact and approximate optimization of modularity

Quantum-Based Maximum Likelihood Detection in MIMO-NOMA Systems for 6G Networks

As wireless networks advance toward the Sixth Generation (6G), which will support highly heterogeneous scenarios and massive data traffic, conventional computing methods may struggle to meet the immense processing demands in a resource-efficient manner. This paper explores the potential of quantum computing (QC) to address these challenges, specifically by enhancing the efficiency of Maximum-Likelihood detection in Multiple-Input Multiple-Output (MIMO) Non-Orthogonal Multiple Access (NOMA) communication systems, an essential technology anticipated for 6G. The study proposes the use of the Quantum Approximate Optimization Algorithm (QAOA), a variational quantum algorithm known for providing quantum advantages in certain combinatorial optimization problems. While current quantum systems are not yet capable of managing millions of physical qubits or performing high-fidelity, long gate sequences, the results indicate that QAOA is a promising QC approach for radio signal processing tasks. This research provides valuable insights into the potential transformative impact of QC on future wireless networks. This sets the stage for discussions on practical implementation challenges, such as constrained problem sizes and sensitivity to noise, and opens pathways for future research aimed at fully harnessing the potential of QC for 6G and beyond.

Mapping quantum circuits to shallow-depth measurement patterns based on graph states

Abstract The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions—a so-called standard form—which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus, enabling a significant decrease in circuit depth for certain applications. We show that groups of mutually-commuting operators can be implemented using fully-parallel, i.e. non-adaptive, measurements within our protocol. In addition, we discuss how groups of mutually commuting observables can be simulatenously measured by adjusting the resource state, rather than performing a costly basis transformation prior to the measurement as it is done in the circuit model. Finally, we demonstrate the utility of our technique on two examples of high practical relevance—the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver (VQE) for the ground-state energy estimation of the water molecule. For the VQE, we find a reduction of the depth by a factor of 4 to 5 using measurement patterns vs. the standard circuit model. At the same time, since we incorporate the simultaneous measurements, our patterns allow us to save shots by a factor of at least 3.5 compared to measuring Pauli strings individually in the circuit model.

Managing Resources for Shared Micromobility: Approximate Optimality in Large-Scale Systems

We consider the problem of managing resources in shared micromobility systems (bike sharing and scooter sharing). An important task in managing such systems is periodic repositioning/recharging/sourcing of units to avoid stockouts or excess inventory at nodes with unbalanced flows. We consider a discrete-time model; each period begins with an initial inventory at each node in the network, and then, customers (demand) materialize at the nodes. Each customer picks up a unit at the origin node and drops it off at a randomly sampled destination node with an origin-specific probability distribution. We model the above network inventory management problem as an infinite horizon discrete-time discounted Markov decision process (MDP) and prove the asymptotic optimality of a novel mean-field approximation to the original MDP as the number of stations becomes large. To compute an approximately optimal policy for the mean-field dynamics, we provide an algorithm with a running time that is logarithmic in the desired optimality gap. Lastly, we compare the performance of our mean field-based policy with state-of-the-art heuristics via numerical experiments, including experiments using Austin scooter-sharing data. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02023 .

Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements.

The Turnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. Turnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the Turnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy Turnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve Turnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the Beltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at https://github.com/Kingsford-Group/turnpikesolvermm.

Solving industrial fault diagnosis problems with quantum computers

In this article, we investigate in how far quantum computers can be leveraged to solve NP-complete fault diagnosis problems within the area of industrial cyber-physical systems. Therefore, two approaches are proposed which exploit quantum computing to solve diagnosis problems: The first method employs Grover’s algorithm, and the second is based on the Quantum Approximate Optimization Algorithm. To show the industrial application, we present an integrated approach to learn the diagnosis model from process data, check whether the model is suitable, and use it for diagnosis. The result is a method for quantum industrial fault diagnosis. For this approach, the diagnostic capabilities and the runtime have been evaluated on an IBM Falcon processor using three publicly available benchmarks from the process industry. Further, the scaling between quantum computers and classical PCs has been analyzed.

BHT-QAOA: The Generalization of Quantum Approximate Optimization Algorithm to Solve Arbitrary Boolean Problems as Hamiltonians.

A new methodology is introduced to solve classical Boolean problems as Hamiltonians, using the quantum approximate optimization algorithm (QAOA). This methodology is termed the "Boolean-Hamiltonians Transform for QAOA" (BHT-QAOA). Because a great deal of research and studies are mainly focused on solving combinatorial optimization problems using QAOA, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for Boolean problems, by transforming such problems from Boolean oracles (in different structures) into Phase oracles, and then into the Hamiltonians of QAOA. From such a transformation, we noticed that the total utilized numbers of qubits and quantum gates are dramatically minimized for the generated Hamiltonians of QAOA. In this article, arbitrary Boolean problems are examined by successfully solving them with our BHT-QAOA, using different structures based on various logic synthesis methods, an IBM quantum computer, and a classical optimization minimizer. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, and machine learning, just to name a few, in the hybrid classical-quantum domain.

A robust force identification method based on L1+2-norm approximation theory

Obtaining the real force of the structure is the basic condition for efficient vibration reduction design. However, the uneasiness and instability of the inverse problem make the force identification still challenging. On the basis of L2-norm regularization, we propose a new method for force identification based on L1+2-norm approximation theory. By introducing the L1-norm penalty operator, a joint norm approximation optimization model is constructed to limit the non-zero force and punish the force value which approaching to zero, so that it can adapt to the identification of various types of forces. Then the optimal force solution is obtained by using the Primal-dual interior point method. Finally, the effectiveness and accuracy of the proposed method are verified by numerical simulation and experiment of asymmetric plate and double-layer plate structures. Different from previous studies, the L1+2-norm method in this work has high robustness, considers the characteristics of sparse regularization and L2-norm regularization, and can adapt to the identification of continuous forces and sparse forces at the same time, which has potential engineering application value.

Inaccurate search based sequential approximate constrained optimization method for Solid Rocket Motor design

Surrogate based constrained optimization methods are widely utilized in practical solid rocket motor (SRM) design problems. To improve the design performance, this paper proposes an inaccurate search based sequential approximate constrained optimization (ISSACO) method consisting of an efficient recursive evolution Latin hypercube design (RELHD) method and a three-stage constrained sampling method. The RELHD method divides large-sample experiment design problem into several small-sample design problems to ensure the computational efficiency and the design uniformity at the same time. The three-stage constrained sampling method employs the elite archives mechanism to balance the feasibility and the optimality in optimization. Results of the numerical cases indicate that the proposed ISSACO algorithm is competitive with other state-of-art constrained optimization method in efficiency and effectiveness. The finocyl grain design and the impulse mass ratio optimization are also conducted to make further verifications, and the ISSACO method can efficiently obtain feasible optimum and proves to be a potential method for computationally intensive solid rocket motor design problems.