The era of Noisy Intermediate-Scale Quantum (NISQ) computing is characterized by quantum processors containing tens to a few hundred qubits. While these devices represent significant technological achievements, they are fundamentally limited by noise – unwanted interactions and imperfect operations that corrupt quantum information. Qubits in NISQ devices have limited coherence times, meaning they can only maintain their quantum states for short durations, and quantum gates (the operations performed on qubits) are prone to errors. Furthermore, these systems lack the capacity for full quantum error correction, which requires a substantial overhead in qubit numbers and is necessary for fault-tolerant quantum computing. These constraints necessitate the development of specialized algorithms designed explicitly to function within the limitations of NISQ hardware.
Designing algorithms for NISQ computers presents unique challenges. The primary obstacle is the pervasive noise, which accumulates rapidly and restricts the depth (number of sequential operations) of quantum circuits that can be reliably executed. Limited qubit connectivity on current hardware architectures also poses difficulties, often requiring additional operations (like SWAP gates) that increase circuit depth and introduce more noise. Therefore, effective NISQ algorithm design focuses on creating shallow circuits that minimize the number of gates, particularly the more error-prone two-qubit gates.
A dominant paradigm in NISQ algorithm design is the hybrid quantum-classical approach. These methods leverage the strengths of both quantum and classical computers. Typically, a quantum processor is used to execute a parameterized quantum circuit, preparing a quantum state and measuring certain properties. The results are then fed into a classical computer, which runs an optimization algorithm to update the parameters for the next iteration on the quantum processor. This iterative loop aims to find the optimal parameters that correspond to the solution of a given problem, attempting to offload computationally demanding parts to the quantum device while managing control and optimization classically.
Variational Quantum Algorithms (VQAs) are the most prominent category of these hybrid algorithms. They are considered strong candidates for achieving practical applicability on NISQ systems. VQAs involve preparing a quantum state using a parameterized circuit (an "ansatz") and then classically minimizing a cost function derived from measurements of that state. Notable examples include the Variational Quantum Eigensolver (VQE), often applied to problems in quantum chemistry and materials science to find ground state energies, and the Quantum Approximate Optimization Algorithm (QAOA).
QAOA is specifically designed to find approximate solutions to combinatorial optimization problems, which are relevant in fields like logistics and finance. It involves repeatedly applying layers of quantum operations corresponding to the problem's cost function and a "mixer" Hamiltonian. While promising, the performance of standard QAOA can be limited, especially with the shallow circuits feasible on NISQ devices. Consequently, significant research focuses on developing improved QAOA variants. Examples include Parity QAOA, which addresses connectivity issues and offers some error mitigation benefits, ADAPT-QAOA and its dynamic version (Dynamic-ADAPT-QAOA) which aim to build more efficient, noise-resilient circuits, and Multiscale QAOA, which combines QAOA with classical techniques.
Given the unavoidable noise in NISQ hardware, quantum error mitigation is a crucial area of research. Unlike quantum error correction, which aims to detect and fix errors directly but demands substantial resources, error mitigation techniques use various methods (often involving additional measurements and classical post-processing) to statistically reduce the impact of noise on the final computation outcome. Common techniques include Zero-Noise Extrapolation (ZNE), Dynamic Decoupling (DD), Twirled Readout Error Extraction (T-REx), Clifford Data Regression (CDR), and Probabilistic Error Cancellation. The choice of mitigation technique often depends on the specific hardware, the type of noise present, and the algorithm being run. Exploiting redundancy in how problems are encoded onto the qubits can also contribute to mitigating errors.
Effective algorithm design also requires being hardware-aware, tailoring the algorithms to the specific capabilities and constraints (like qubit connectivity and available gate sets) of the target quantum device. This hardware-efficient approach is essential for maximizing performance on current machines.
Furthermore, researchers are exploring novel techniques to enhance NISQ algorithms. This includes investigating the fundamental properties of VQA circuits, such as their expressibility (ability to represent desired states), trainability (ease of optimization, avoiding issues like "barren plateaus"), and generalization capabilities, particularly for quantum machine learning applications. Integrating classical machine learning techniques with VQAs, such as using neural networks to parameterize quantum circuits (Neural Network VQA or NNVQA) or using generative models like GPTs to help construct circuits (Generative Quantum Eigensolver or GQE), represents another exciting direction, potentially accelerating training and improving performance. Dynamic Quantum Circuits (DQC), which utilize mid-circuit measurements and classical control, are also being explored to overcome limitations in qubit count.
In summary, designing algorithms for NISQ devices is a highly active and challenging field. The central task is to develop methods that can provide useful computational results despite significant hardware noise and limitations. This relies heavily on hybrid quantum-classical strategies like VQAs (especially QAOA for optimization), sophisticated error mitigation techniques, and hardware-aware design principles. As research progresses (as of May 2025), new approaches integrating AI and dynamic circuit capabilities are emerging, pushing the boundaries of what can be achieved with near-term quantum computers while simultaneously gathering insights crucial for the eventual development of fault-tolerant quantum computing.