Current quantum computing research is divided into two eras: near-term intermediate scaling quantum (NISQ) and fault-tolerant quantum computing (FTQC). There is a significant gap in these both in terms of hardware and algorithms. An important question is “how to transition from NISQ to FTQC”? Some research looks at an intermediate third era of quantum computing, sometimes referred to as “early” fault-tolerant quantum computing (EFTQC). The idea is to establish a trade-off between hardware and algorithmic parameters such that we have limited size circuits, albeit large enough to allow some amount of error-correction at the cost of more number of samples and increased run-time. This also requires adjustment to algorithms for various tasks. One way to do this is to allow the algorithm to tolerate some level of noise in the input. These algorithms are called “robust” quantum algorithms. In this chapter, we will understand what EFTQC means and how we can distinguish between the three eras of quantum computing. Furthermore, we will look at the ideas behind algorithms suitable for EFTQC devices. Lastly, we will look at two examples of robust quantum algorithms for the task of quantum phase estimation.
Current quantum computing research is divided into two eras: near-term intermediate scaling quantum (NISQ) and fault-tolerant quantum computing (FTQC). There is a significant gap in these both in terms of hardware and algorithms. An important question is “how to transition from NISQ to FTQC”? Some research looks at an intermediate third era of quantum computing, sometimes referred to as “early” fault-tolerant quantum computing (EFTQC). The idea is to establish a trade-off between hardware and algorithmic parameters such that we have limited size circuits, albeit large enough to allow some amount of error-correction at the cost of more number of samples and increased run-time. This also requires adjustment to algorithms for various tasks. One way to do this is to allow the algorithm to tolerate some level of noise in the input. These algorithms are called “robust” quantum algorithms. In this chapter, we will understand what EFTQC means and how we can distinguish between the three eras of quantum computing. Furthermore, we will look at the ideas behind algorithms suitable for EFTQC devices. Lastly, we will look at two examples of robust quantum algorithms for the task of quantum phase estimation.
In recent years, research in quantum computing has largely focused on two approaches: near-term intermediate-scale quantum (NISQ) computing and future fault-tolerant quantum computing (FTQC). A growing body of research into early fault-tolerant quantum computing (EFTQC) is exploring how to utilize quantum computers during the transition between these two eras. However, without agreed-upon characterizations of this transition, it is unclear how best to utilize EFTQC architectures. We argue for the perspective that this transition period will be characterized by a law of diminishing returns in quantum error correction (QEC), where the ability of the architecture to maintain quality operations at scale determines the point of diminishing returns. Two challenges emerge from this picture: how to model this phenomenon of diminishing return of QEC as the performance of devices is continually improving and how to design algorithms to make the most use of these devices. To address these challenges, we present models for the performance of EFTQC architectures, capturing the diminishing returns of QEC. We then use these models to elucidate the regimes in which algorithms suited to such architectures are advantageous. As a concrete example, we show that for the canonical task of phase estimation, in a regime of moderate scalability and using just over one million physical qubits, the “reach” of the quantum computer can be extended (compared to the standard approach) from 90-qubit instances to over 130-qubit instances using a simple early fault-tolerant quantum algorithm, which reduces the number of operations per circuit by a factor of 100 and increases the number of circuit repetitions by a factor of 10 000. This clarifies the role that such algorithms might play in the era of limited-scalability quantum computing. Published by the American Physical Society 2024
Progress in fault-tolerant quantum computation (FTQC) has driven the pursuit of practical applications with (EFTQC). These devices, limited in their qubit counts and fault-tolerance capabilities, require algorithms that can accommodate some degrees of error, which are known as EFTQC algorithms. To predict the onset of early quantum advantage, a comprehensive methodology is needed to develop and analyze EFTQC algorithms, drawing insights from both the methodologies of noisy intermediate-scale quantum and traditional FTQC. To address this need, we propose such a methodology for modeling algorithm performance on EFTQC devices under varying degrees of error. As a case study, we apply our methodology to analyze the performance of randomized Fourier estimation (RFE) [Kshirsagar, Katabarwa, and Johnson, ], an EFTQC algorithm for phase estimation. We investigate the runtime performance and the fault-tolerant overhead of RFE in comparison to the traditional quantum phase estimation algorithm. Our analysis reveals that RFE achieves significant savings in physical qubit counts while having a much higher runtime upper bound. We anticipate even greater physical qubit savings when considering more realistic assumptions about the performance of EFTQC devices. By providing insights into the performance trade-offs and resource requirements of EFTQC algorithms, our work contributes to the development of practical and efficient quantum computing solutions on the path to quantum advantage. Published by the American Physical Society 2024
Quantum computing is entering a transitional regime between noisy intermediate-scale quantum (NISQ) processing and early fault-tolerant quantum computation (FTQC), in which increasingly capable hardware is beginning to support repeated syndrome measurements, partial error correction, and logical-qubit operations, while residual physical and logical errors remain non-negligible. In this regime, error suppression, error mitigation, and quantum error correction are increasingly better viewed as complementary layers of a unified error-reduction strategy rather than as separate approaches, with each acting at a different stage of the quantum computation to improve simulation reliability. Thus, in this review, we provide a practical and forward-looking overview of the principal hardware error sources and the corresponding error suppression and mitigation methods for reducing their impact across the current NISQ-FTQC transition. We discuss hardware-aware circuit design, coherent-error suppression, readout mitigation, noise extrapolation, classical inference, and software-supported workflows, with particular emphasis on their implementation on actual quantum processors. We further examine how error mitigation techniques can be adapted to encoded and logical-qubit settings so that they can operate alongside quantum error correction to suppress residual logical errors and improve the accuracy of computation in the early fault-tolerant regime.
Abstract The hope of the quantum computing field is that quantum architectures are able to scale up and realize fault-tolerant quantum computing. Due to engineering challenges, such “cheap” error cor- rection may be decades away. In the meantime, we anticipate an era of “costly” error correction, or early fault-tolerant quantum computing. Costly error correction might warrant settling for error-prone quantum computations. This motivates the development of quantum algorithms which are robust to some degree of error as well as methods to analyze their performance in the presence of error. We introduce a randomized algorithm for the task of phase estimation and give an analysis of its performance under two simple noise models. In both cases the analysis leads to a noise threshold, below which arbitrarily high accuracy can be achieved by increasing the number of samples used in the algorithm. As an application of this general analysis, we compute the maximum ratio of the largest circuit depth and the dephasing scale such that performance guarantees hold. We calculate that the randomized algorithm can succeed with arbitrarily high probability as long as the required circuit depth is less than 0.916 times the dephasing scale.
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.