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.
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.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant (EFT) quantum computers 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 ${10}^{\ensuremath{-}3}$ 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 an approximately 300-fold reduction in aggregate computational volume in some cases. We also compare resource requirements for QPE on Trotterized time evolution and qubitized walk operators, for this same minimal exemplar. We estimate the costs of these two approaches to be comparable, with the Trotter approach requiring slightly fewer physical qubits and the qubitized approach requiring slightly fewer code cycles. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.
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.
We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth t, and the variance is O(t−2F−2) for circuit fidelity F. This improves by a factor of t2 over a naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. This yields an overhead interpolating between the F−4 scaling typical of explicitly unbiased error mitigation and a reduced F−2 scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.
Recent experimental breakthroughs have signalled the imminent arrival of the early fault-tolerant era. However, for a considerable period in the foreseeable future, relying solely on quantum error correction for full error suppression will remain extremely challenging due to its substantial hardware overhead. Additional help from quantum error mitigation (QEM) is essential for bridging this gap towards achieving quantum advantage. The application of QEM has so far been restricted to expectation value estimation, leaving its extension to sampling-based algorithms -- which is expected to play a pivotal role in the early fault-tolerant era -- an unresolved challenge. In this work, we present a framework for applying any QEM techniques to obtain the error-mitigated output distribution, showing that this incurs no greater cost than estimating a single observable. We also devised a way to sample from this distribution and constructed an explicit scheme for applying any QEM methods to quantum phase estimation, which can be generalised to other sampling algorithms. Numerical experiments were conducted to validate the efficacy of these methods. We believe our methods significantly broaden the scope of QEM, extending its applicability to most algorithms of practical interest and forming a crucial step towards realising quantum advantage.
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.