The recent development of logical quantum processors marks a pivotal transition from the noisy intermediate-scale quantum (NISQ) era to the fault-tolerant quantum computing (FTQC) era. These devices have the potential to address classically challenging problems with polynomial computational time using quantum properties. However, they remain susceptible to noise, necessitating noise resilient algorithms. We introduce Quantum Zeno Monte Carlo (QZMC), a classical-quantum hybrid algorithm that demonstrates resilience to device noise and Trotter errors while showing polynomial computational cost for a gapped system. QZMC computes static and dynamic properties without requiring initial state overlap or variational parameters, offering reduced quantum circuit depth.
The recent development of logical quantum processors marks a pivotal transition from the noisy intermediate-scale quantum (NISQ) era to the fault-tolerant quantum computing (FTQC) era. These devices have the potential to address classically challenging problems with polynomial computational time using quantum properties. However, they remain susceptible to noise, necessitating noise resilient algorithms. We introduce Quantum Zeno Monte Carlo (QZMC), a classical-quantum hybrid algorithm that demonstrates resilience to device noise and Trotter errors while showing polynomial computational cost for a gapped system. QZMC computes static and dynamic properties without requiring initial state overlap or variational parameters, offering reduced quantum circuit depth.
Quantum Monte Carlo and quantum simulation are both important tools for
understanding quantum many-body systems. As a classical algorithm, quantum
Monte Carlo suffers from the sign problem, preventing its applications to most
fermion systems and real-time dynamics. In contrast, on a quantum computer, we
can simulate time evolution of general systems with local interactions, yet
such algorithms usually require the development of fault-tolerant quantum
technologies. In this paper, we introduce a novel non-variational algorithm
using quantum computing as a subroutine to accelerate quantum Monte Carlo. On a
fault-tolerant quantum computer, the complexity of our algorithm is polynomial
as the same as conventional quantum simulation algorithms. On a noisy quantum
computer, our algorithm incorporating error mitigation eases the sign problem.
Even when the circuit noise is significant, our algorithm can reduce the Monte
Carlo variance by orders of magnitude.
The transition from classical to quantum computation marks a shift from deterministic binary logic to a probabilistic, entanglement‑driven framework able to address problems beyond classical reach. Today’s Noisy Intermediate ‑ Scale Quantum era features devices that lack full error correction yet already point toward larger, modular systems designed for fault‑tolerant operation. This paper examines the evolving paradigm shift from classical and NISQ ‑ era machines toward practical fault‑tolerant quantum computing, focusing on how advances in physical qubit technologies, control architectures, and error‑correction strategies are converging to bridge noisy prototypes and scalable, useful machines. We synthesize insights from historical semiconductor scaling trends, contemporary roadmaps from leading institutions and industry, and emerging modular designs with real‑time error decoding. Our analysis shows that while challenges around decoherence, connectivity, and classical co‑processing remain, coordinated progress in hardware, control, and software is laying a coherent foundation for future quantum platforms. The work concludes that quantum computing is maturing into a strategically significant paradigm, poised to redefine the limits of computation in simulation, optimization, and secure communication.
The Noisy Intermediate-Scale Quantum (NISQ) era, defined by qubit counts in the hundreds and gate error rates that preclude long coherence times, imposes severe constraints on purely quantum computation. The central problem is that achieving practical quantum advantage for real-world tasks must overcome these hardware limitations, yet fault-tolerant quantum computers with full error correction remain years away. To address this, we present a systematic survey of hybrid classical-quantum computing models, in which classical high-performance resources orchestrate the workflow, perform iterative optimization, and apply error mitigation to noisy quantum processing units. Our review encompasses variational quantum algorithms, such as the variational quantum eigensolver and quantum approximate optimization algorithm; classical optimizers tailored for noisy quantum landscapes; error mitigation techniques including zero-noise extrapolation and probabilistic error cancellation; and circuit knitting that partitions large circuits across small QPUs. We find that these hybrid architectures have already demonstrated domain-specific utility in quantum chemistry for molecular ground-state estimation, in logistics and finance for combinatorial optimization, and in machine learning for generative and recommendation models. Notably, they often surpass purely classical or purely quantum approaches on carefully chosen benchmarks. The evidence indicates that hybrid classical-quantum systems are not a temporary stopgap but a durable paradigm. Their role will persist into the early fault-tolerant era, where small, logical qubits will similarly require classical coordination. Consequently, the field urgently demands the establishment of standardized hybrid performance benchmarks, the maturation of robust quantum-classical middleware capable of efficient transpiration and execution management, and holistic co-design frameworks that seamlessly integrate hardware constraints, software stack layers, and application-specific requirements from the outset.
Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-time evolution operator with a random evolution time according to a modified Cauchy-Lorentz distribution, we can compute the expected value of an observable in imaginary-time evolution. Our algorithm approaches the exact solution given a circuit depth increasing polylogarithmically with the desired accuracy. Compared with quantum phase estimation, the Trotter step number, i.e. the circuit depth, can be thousands of times smaller to achieve the same accuracy in the ground-state energy. We verify the resilience to Trotterisation errors caused by the finite circuit depth in the numerical simulation of various models. The results show that Monte Carlo quantum simulation is promising even without a fully fault-tolerant quantum computer.
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.
Fault-tolerant quantum computing (FTQC) is a form of universal quantum computing that suppresses physical errors via quantum error correction (QEC). On the other hand, it is expected that the available code distance and the T-gate count will be restricted in the early years of FTQC. Meanwhile, quantum error mitigation (QEM) was recently introduced for suppressing errors in noisy intermediate-scale quantum (NISQ) devices; it improves the computation accuracy of near-term quantum algorithms with its overhead being a greater number of samples. In this work, we integrate QEC and QEM into an efficient FTQC architecture that effectively increases the code distance and T-gate count. Our scheme will dramatically alleviate the required qubit count by tens of percent at the cost of a constant multiplicative sampling overhead within 102 in a wide range of regimes from the post-NISQ era to early FTQC regime, which necessitates the order of from 104 to 1010 logical operations. While it has been widely believed that there do not exist useful applications in post-NISQ regimes due to a chasm between the NISQ and FTQC regimes such as overheads for encoding and T-gate distillations, our scheme can continuously bridge the two distinct regimes.