NISQ Era: Noisy Intermediate-Scale Quantum Devices

• NISQ is a phase of quantum computing using processors with 50–100 qubits that exhibit partial quantum advantage amid significant noise.
• NISQ-era systems enable hybrid quantum–classical algorithms and simulations, serving as practical testbeds for exploring quantum complexity and advantage.
• Error mitigation and hardware–software co-design are pivotal in optimizing shallow circuits and benchmarking quantum supremacy in noisy environments.

The Noisy Intermediate-Scale Quantum (NISQ) era refers to the current and near-term generation of quantum computing hardware that is characterized by quantum processors containing roughly 50–100 qubits, in which quantum gates and measurements are implemented with significant noise and without full error correction. These systems are large enough that their quantum states cannot be efficiently simulated by brute-force classical computation, yet they are not reliable or scalable enough to support fault-tolerant quantum algorithms. NISQ hardware is expected to serve as a pivotal technological bridge, offering practical testbeds for quantum algorithm development, quantum simulation, and benchmarking, while helping the community identify which problems are accessible to near-term quantum advantage and guiding progress toward universal, fault-tolerant quantum computation.

1. NISQ Device Capabilities and Computational Frontier

NISQ devices, with 50–100 controllable qubits, operate beyond the reach of classical supercomputers for certain tasks due to the exponential growth of the quantum Hilbert space with system size. The total number of complex parameters to describe a quantum state grows as $2^n$ for $n$ qubits, which rapidly becomes intractable for $n \gtrsim 50$. NISQ platforms can efficiently prepare and probe highly entangled states, and simulate many-body quantum physics that are classically inaccessible. They are uniquely positioned for:

• Quantum simulation of strongly correlated and dynamical systems, including analog simulations of target Hamiltonians, lattice models, and quantum field theory phenomena.
• Executing hybrid quantum–classical algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE), leveraging quantum state preparation and measurement in tandem with classical optimization to navigate large, entangled Hilbert spaces.
• Conducting quantum sampling tasks, such as random quantum circuit sampling, which underpin ‘quantum supremacy’ benchmarks and exceed classical high-performance computing capabilities (Preskill, 2018, Villalonga et al., 2019, AbuGhanem et al., 2023).

Algorithmic complexity arguments for quantum speedup in this context are exemplified by the exponential nature of some quantum algorithms—for instance, the HHL algorithm for matrix inversion achieves a logarithmic runtime $O(\log N)$ (for input dimension $N$) under ideal conditions, contrasting with classical polynomial or worse scaling. However, algorithms like HHL generally exceed NISQ-era capabilities due to their depth requirements.

2. Sources and Impact of Noise: Fundamental Limitations

NISQ devices are inherently limited by the noise in gate operations, finite qubit coherence times, measurement errors, and restricted connectivity. Gate error rates for superconducting and trapped-ion devices are typically above $0.1\%$ for two-qubit gates, and the overall circuit reliability is sharply constrained by the product of error rates and the circuit size. The accumulated failure probability for a circuit with $G$ gates and per-gate error $\epsilon$ is approximately $P_\text{failure} \approx G \cdot \epsilon$ when $G\epsilon \ll 1$.

Physical constraints imply that only shallow circuits (typically no more than $\sim 1000$ gates in depth) can be executed reliably before the quantum signal is drowned in noise. This limitation obstructs the implementation of deep, fault-tolerant algorithms required for transformative applications (for example, Shor's algorithm for factoring large numbers or large-scale quantum simulation of chemistry).

Noise also imposes limits on computational complexity: for fundamental problems like unstructured search (Grover’s problem), NISQ algorithms cannot achieve the quadratic quantum speedup; the required number of queries remains nearly linear in the problem size due to the restriction in maintaining depth and coherence (Chen et al., 2022).

3. Algorithmic Approaches, Benchmarks, and Applications

Within the operational envelope of NISQ hardware, research focuses on algorithms and modalities amenable to shallow circuits and error resilience. Key applications and approaches include:

• Many-body quantum simulations, which take advantage of quantum entanglement to access challenging regions of quantum statistical mechanics (e.g., simulation of exotic phases and non-equilibrium dynamics). NISQ devices have provided convincing platforms for the realization and paper of phenomena such as discrete time crystals, by enabling controlled, programmable, and site-resolved experiments that were previously infeasible (Ippoliti et al., 2020).
• Hybrid quantum–classical algorithms, notably QAOA and VQE, which optimize parameterized quantum circuits steered by classical algorithms. These are tailored to the shallow depths and high-fidelity gates available in NISQ hardware, and have been investigated for chemistry, optimization, and machine learning.
• Quantum machine learning and generative modeling, such as the use of restricted Boltzmann machines for state reconstruction from noisy quantum measurement data, represent near-term algorithmic advances that integrate redundancy and error mitigation (Torlai et al., 2019).
• Quantum sampling benchmarks, most notably random circuit sampling (RCS), provide objective metrics to establish quantum computational advantage. These tasks are benchmarked by cross-entropy benchmarking (XEB) and have been simulated with high-performance classical codes (e.g., qFlex), pushing the classical boundary to hundreds of qubits and setting a clear demarcation for quantum supremacy claims (Villalonga et al., 2019).

Benchmarking strategies and their standardization (e.g., RCS/XEB) enable transparent comparison between hardware platforms and facilitate progress in both device calibration and algorithm development.

4. Error Mitigation and Circuit Optimization

Error mitigation is at the core of algorithmic and architectural advances in the NISQ era. Strategies include:

• Careful engineering of shallow, noise-resilient circuit structures and the heuristic reduction of gate count and circuit depth (e.g., optimizing basis encoding, minimizing SWAP networks, and employing parallelization via circuit “shifting”).
• Dynamically allocating logical qubits to physical qubits based on device connectivity and varying error rates, performed through intelligent transpilation and mapping techniques that minimize the necessity for fidelity-costly gate insertions.
• Incorporation of robust classical post-processing steps, as in measurement error mitigation by calibration matrix inversion $t = C^{-1} m$, where $m$ is the raw measurement vector, $C$ is the calibration matrix, and $t$ is the corrected probability vector.

While the pursuit of full quantum error correction is ongoing, present methods focus on noise-aware compilation, primitive error mitigation (e.g., randomized compiling, zero-noise extrapolation), and precise resource management to maximize computational yield (Preskill, 2018, Leymann et al., 2020, Bandic et al., 2022).

5. Hardware and Software Stacks: Integration and Co-Design

Contemporary NISQ systems feature a deeply intertwined integration of hardware and software layers—termed full-stack quantum computing. Unlike the abstraction-prone stacks of classical systems, NISQ platforms exploit vertical co-design and cross-layer feedback:

• Hardware-aware compilation optimizes algorithm mapping subject to physical connectivity and calibration data. This includes tailoring quantum circuit scheduling, qubit mapping, SWAP insertion, and employing performance models (e.g., circuit fidelity as a product of per-gate fidelities $F \approx \prod_i F_i$).
• Software frameworks are developed with modularity and back-end agnosticism (e.g., TensorCircuit integrates tensor network contraction, automatic differentiation, and JIT compilation for fast, scalable simulation and hybrid quantum-classical workflows) (Zhang et al., 2022).
• Resource estimation and management now emphasize dynamic, real-time querying of the device state, integrating continuous monitoring of qubit properties, error rates, and device topology before job execution (Lammers et al., 6 Aug 2025). The proposed “middleware” layers dynamically mediate between job requirements and physical capability.

These co-designed approaches are essential for optimizing the limited and fluctuating resources available during experimental runs and for accelerating algorithm iterations.

6. Impact, Limitations, and Outlook

The NISQ era enables practical exploration of quantum computational tasks that are elusive or impossible for classical systems, but does so under clear and quantifiable constraints. NISQ devices expand the frontier of quantum simulation, optimization, sampling, and benchmarking but are not anticipated to impact conventional applications—such as cryptography—until scalability and error rates improve substantially.

The landscape of NISQ computational complexity is now more clearly articulated: natural problems exist where quantum advantage is provable (e.g., Bernstein–Vazirani under noise), but others (e.g., shadow tomography) remain out of reach due to exponential scaling in noise sensitivity. Theoretical statements rigorously place NISQ below the full computational reach of fault-tolerant quantum computers (BQP) but above that of classical probabilistic computation (BPP) when modeled with realistic noise (Chen et al., 2022).

The NISQ era further catalyzes hardware advances, algorithmic innovation, and experimental feedback. Progress toward scalable, error-corrected architectures will draw directly upon the experimental and computational foundations established in this period.

7. Path Forward: From NISQ to Fault-Tolerance

While NISQ devices will not enable the transformative quantum applications foreseen in fault-tolerant computing, they are essential as an experimental and theoretical proving ground. Future directions require:

• Continual reduction of gate and measurement error rates to enable deeper circuits and fault tolerance.
• Development of error correction protocols suitable for scaling to thousands or millions of physical qubits, with efficient decoders and practical syndrome extraction.
• Expansion of middleware and resource management systems to adapt to dynamic and heterogeneous quantum architectures.
• Standardization of benchmarking and reproducibility to systematically assess quantum advantage and inform community-wide progress.

This evolutionary process situates the NISQ era as a necessary and productive stepping stone towards reliable, universal, and large-scale quantum computation (Preskill, 2018, Cheng et al., 2023, AbuGhanem et al., 2023).