Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, Fernando G. S. L. Brandão
This chapter covers variational quantum algorithms, which act as a primitive ingredient for larger quantum algorithms in several application areas, including quantum chemistry, combinatorial optimization, and machine learning. Variational quantum algorithms are parameterized quantum circuits where the parameters are trained to optimize a certain cost function. They are often shallow circuits, which potentially makes them suitable for near-term devices that are not error corrected.
This chapter covers variational quantum algorithms, which act as a primitive ingredient for larger quantum algorithms in several application areas, including quantum chemistry, combinatorial optimization, and machine learning. Variational quantum algorithms are parameterized quantum circuits where the parameters are trained to optimize a certain cost function. They are often shallow circuits, which potentially makes them suitable for near-term devices that are not error corrected.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.
This chapter introduces variational quantum algorithms, a class of hybrid methods that leverage parametrized quantum circuits and classical optimization to solve problems in chemistry, optimization, and machine learning. Focusing on the variational quantum eigensolver and the quantum approximate optimization algorithm, we explore their algorithmic structure, theoretical motivation, and practical implementation on near-term devices. We discuss key challenges such as barren plateaus, ansatz design, and measurement overhead, as well as opportunities for adaptive circuits, noise-aware optimization, and emerging applications in quantum machine learning. The chapter lays the foundation for understanding variational classifiers and generative models in subsequent chapters.
Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.