In this review, we give an overview of the proposed applications in the early-FTQC (EFTQC) era. Starting from the error correction architecture for EFTQC device, we first review the recently developed space-time efficient analogue rotation (STAR) architecture \cite{akahoshiPartiallyFaultTolerantQuantum2024}, which is a partially fault-tolerant error correction architecture. Then, we review the requirements of an EFTQC algorithm. In particular, the class of ground state energy estimation (GSEE) algorithm known as the statistical phase estimation algorithm (SPE) is studied. We especially cast our attention on two SPE-type algorithms, the step-function filter-based variant by Lin and Tong (LT22) \cite{Lin:2021rwb} and Gaussian Filter \cite{Wang:2022gxu}. Based on the latter, we introduce the Gaussian Fitting algorithm, which uses an alternative post-processing procedure compared to \cite{Wang:2022gxu}. Finally, we systematically simulate the aforementioned algorithms and Variational Quantum Eigensolver (VQE) using the 1-uCJ ansatz with different shot counts. Most importantly, we perform noisy simulations based on the STAR architecture. We find that for estimating the ground state energy of the 4-qubit $H_2$ Hamiltonian in the STO-3G basis, SPE becomes more advantageous over VQE when the physical error rate is sufficiently low.
In this review, we give an overview of the proposed applications in the early-FTQC (EFTQC) era. Starting from the error correction architecture for EFTQC device, we first review the recently developed space-time efficient analogue rotation (STAR) architecture \cite{akahoshiPartiallyFaultTolerantQuantum2024}, which is a partially fault-tolerant error correction architecture. Then, we review the requirements of an EFTQC algorithm. In particular, the class of ground state energy estimation (GSEE) algorithm known as the statistical phase estimation algorithm (SPE) is studied. We especially cast our attention on two SPE-type algorithms, the step-function filter-based variant by Lin and Tong (LT22) \cite{Lin:2021rwb} and Gaussian Filter \cite{Wang:2022gxu}. Based on the latter, we introduce the Gaussian Fitting algorithm, which uses an alternative post-processing procedure compared to \cite{Wang:2022gxu}. Finally, we systematically simulate the aforementioned algorithms and Variational Quantum Eigensolver (VQE) using the 1-uCJ ansatz with different shot counts. Most importantly, we perform noisy simulations based on the STAR architecture. We find that for estimating the ground state energy of the 4-qubit H2 Hamiltonian in the STO-3G basis, SPE becomes more advantageous over VQE when the physical error rate is sufficiently low.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO2-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
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
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.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant quantum computers (EFTQCs) in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground state energy of molecular hydrogen in a minimal basis with error below 10−3 atomic units in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total T-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing approximately a 300-fold reduction in computational volume in some cases. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.
Quantum phase estimation (QPE) is a key quantum algorithm, which has been widely studied as a method to perform chemistry and solid-state calculations on future fault-tolerant quantum computers.Recently, several authors have proposed statistical alternatives to QPE that have benefits on early faulttolerant devices, including shorter circuits and better suitability for error-mitigation techniques.However, experimental investigations of the algorithm on real quantum processors are lacking.Here, we implement statistical phase estimation on Rigetti's superconducting processors.Specifically, we use a modification of the Lin and Tong [PRX Quantum 3, 010318 (2022)] algorithm with the improved Fourier approximation of Wan et al. [Phys.Rev. Lett.129, 030503 (2022)] and apply a variational-compilation technique to reduce the circuit depth.We then incorporate error-mitigation strategies including zero-noise extrapolation and readout-error mitigation with bit-flip averaging.We propose a new method to estimate energies from the statistical phase estimation data, which is found to improve the accuracy in the final energy estimates by 1-2 orders of magnitude with respect to prior theoretical bounds, reducing the cost of performing accurate phase-estimation calculations.We apply these methods to chemistry problems for active spaces up to four electrons in four orbitals, including the application of a quantum embedding method, and use them to correctly estimate energies within chemical precision.Our work demonstrates that statistical phase estimation has a natural resilience to noise, particularly after mitigating coherent errors, and can achieve far higher accuracy than suggested by previous analysis, demonstrating its potential as a valuable quantum algorithm for early fault-tolerant devices.