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.
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.
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.
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
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
As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of during which limited error correction is available. We propose a framework for designing early fault-tolerant algorithms by trading between error-correction overhead and residual logical noise, and apply it to quantum phase estimation (QPE). We develop a quantum-Fourier-transform- (QFT) based QPE technique that is robust to global depolarizing noise and outperforms the previous state of the art at low and moderate noise rates. We further introduce the explicitly unbiased maximum-likelihood estimator (EUMLE), a data-processing technique that mitigates errors in QFT-based QPE schemes. EUMLE provides consistent asymptotically normal error-mitigated estimates, addressing the open problem of extending error mitigation beyond expectation-value estimation. Applying this scheme to the ground-state problem of the two-dimensional Hubbard model and various molecular Hamiltonians, we find that we can roughly halve the number of physical qubits with a ∼ 10 times wall-clock time overhead, but that further reduction causes a steep run-time increase. This work provides an end-to-end analysis of early fault-tolerance cost reductions and space-time trade-offs, and identifies which areas can be improved in the future.
A quantum computation can be described by the evolution of a collection of quantum bits, or qubits, according to the laws of quantum mechanics, from an initial state encoding the input of a problem, to its solution represented by the final state. A machine capable of performing this task, a quantum computer, must be robust against errors caused by the inherent imperfections in the fabrication and control of such complex systems. This thesis examines the theory of fault‑tolerant quantum computation, tailored to the physical constraints of today’s leading platforms, most notably superconducting circuits and neutral‑atom arrays.A fault-tolerant quantum computation requires to correct errors in real time, at the cost of a dedicated architectural layer whose integration may introduce additional noise. A possible simplification uses a quantum memory model where information is encoded in a pair of configurations of a multi-qubit system, stabilized by the uniform application of a simple local rule fitting within the theory of cellular automata. We propose a short-term solution, the shearing-rule which protects a one-dimensional repetition code against errors with good performance but lacks a threshold. This shortcoming is addressed by adding small local classical memories, used to define the signal-rules decoders, where error correction relies on exchanging binary signals. Combined with biased-noise qubits, such as cat qubits, these local decoders define a fully local one-dimensional quantum memory.The final part of the manuscript addresses the advantage of quantum machines over their classical counterparts. We emphasize that any such demonstration must account for the presence of noise to be meaningful — for example, while exact sampling from the output distribution of circuits drawn from certain families is believed to be hard for classical computers, the task may become tractable if one allows for an approximate solution that reflects the presence of noise. In this context, we propose a constant-depth quantum circuit that solves a sampling problem in a fault-tolerant manner. Because our approach is particularly well-suited to the atom rearrangement capabilities of neutral-atom array platforms, we hope it will contribute to a future experimental demonstration of a fault-tolerant quantum advantage.
Recent years have seen quantum computers move from theory to practice, with the first prototypes of \emph{fault-tolerant} quantum computers capable of correcting their own errors. This marks the beginning of an exciting era of \emph{early fault tolerance}, with the first practical applications on the horizon. Among the most promising is quantum simulation, where a quantum computer is used to study another quantum system, such as molecules or materials. However, a central challenge is how to reliably extract accurate information from these early imperfect devices under realistic constraints, including noise, limited qubit numbers, circuit depth, and control. This thesis addresses this challenge by developing and analyzing algorithms for quantum phase estimation in these regimes, a key primitive for determining the energy spectra. It also considers the problem of learning the underlying dynamics of a quantum system, demonstrating the advantage of active control over passive observation. Together, these results contribute to ongoing efforts to overcome the limitations of early quantum hardware through improved algorithms and classical post-processing.