When estimating the eigenvalues of a given observable, even fault-tolerant quantum computers will be subject to errors, namely algorithmic errors. These stem from approximations in the algorithms implementing the unitary passed to phase estimation to extract the eigenvalues, e.g. Trotterisation or qubitisation. These errors can be tamed by increasing the circuit complexity, which may be unfeasible in early-stage fault-tolerant devices. Rather, we propose in this work an error mitigation strategy that enables a reduction of the algorithmic errors up to any order, at the cost of evaluating the eigenvalues of a set of observables implementable with limited resources. The number of required observables is estimated and is shown to only grow polynomially with the number of terms in the Hamiltonian, and in some cases, linearly with the desired order of error mitigation. Our results show error reduction of several orders of magnitude in physically relevant cases, thus promise accurate eigenvalue estimation even in early fault-tolerant devices with limited number of qubits.
When estimating the eigenvalues of a given observable, even fault-tolerant quantum computers will be subject to errors, namely algorithmic errors. These stem from approximations in the algorithms implementing the unitary passed to phase estimation to extract the eigenvalues, e.g. Trotterisation or qubitisation. These errors can be tamed by increasing the circuit complexity, which may be unfeasible in early-stage fault-tolerant devices. Rather, we propose in this work an error mitigation strategy that enables a reduction of the algorithmic errors up to any order, at the cost of evaluating the eigenvalues of a set of observables implementable with limited resources. The number of required observables is estimated and is shown to only grow polynomially with the number of terms in the Hamiltonian, and in some cases, linearly with the desired order of error mitigation. Our results show error reduction of several orders of magnitude in physically relevant cases, thus promise accurate eigenvalue estimation even in early fault-tolerant devices with limited number of qubits.
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.
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.
Simulating real-time dynamics under a Hamiltonian is a central goal of quantum information science. While numerous Hamiltonian-simulation quantum algorithms have been proposed, the effects of physical noise have rarely been incorporated into their performance analysis, despite the non-negligible noise levels of quantum devices. We present an end-to-end complexity analysis of noisy Hamiltonian simulation combined with quantum error mitigation (QEM) to answer how many circuit runs are required to reach a given target accuracy. Because the QEM sampling overhead grows exponentially with the circuit depth while the algorithmic error decreases with it, the circuit depth becomes an optimization variable, and we derive an analytic depth-selection rule for two algorithm families. For the order-k Suzuki--Trotter formula, the optimized cost exhibits a critical error εc, below which the required number of circuit runs grows exponentially. QEM improves the noise dependence of εc from sublinear to kth-power scaling, an exponent improvement by a factor of k+1. For randomized-LCU-based simulation, optimizing the repetition number yields a square-root improvement in the simulation-time dependence of the sampling-overhead exponent over the standard parameter choice. We further quantify the noise-characterization cost required for error mitigation via gate set tomography and the recently proposed space-time noise inversion method, showing that the latter can significantly reduce this cost.
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.
Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design parameters-such as syndrome measurement frequency-to avoid diminishing returns from detection overhead. In this work, we present a comprehensive framework for fault-tolerant compilation and simulation of quantum algorithms using [[n, n-2, 2]] codes, which enable low-qubit-overhead error detection and a simple nearly fault-tolerant universal set of operations. We demonstrate and analyze our pipeline with a purely statistical interpretation and through the implementation of Grover's search algorithm. Our results are used to answer the question is quantum error detection a worthwhile avenue for early-term fault tolerance, and if so how can we get the most out of it? Simulations under the circuit-level noise model reveal that finding optimal syndrome schedules improves algorithm success probabilities by an average of 6.7x but eventual statistical limits from post-selection in noisy/resource-limited regimes constrain scalability. Furthermore, we propose a simple data-driven approach to predict fault tolerant compilation parameters, such as optimal syndrome schedules, and expected fault tolerant performance gains based on circuit and noise features. These results provide actionable guidelines for implementing QED in early-term quantum experiments and underscore its role as a pragmatic, constant-overhead error mitigation layer for shallow algorithms. To aid in further research, we release all simulation data computed for this work and provide an experimental QED compiler at https://codeqraft.xyz/qed.
This thesis presents multiple advances in the construction of noise-resistant quantum computers. In order to bridge the gap between abstract theory and practical experiments, my research encompasses a wide range of subfields within quantum computing, including error correction, silicon spin qubit physics and error mitigation. Firstly, I demonstrate that large-scale defects affecting entire regions of an error correcting code, such as cosmic rays or manufacturing defects, do not necessarily constitute a fundamental obstacle. Conversely, bespoke protocols that employ code deformation can be utilised to mitigate their impact, at a moderate resource overhead. Secondly, I investigate the feasibility of building a fault-tolerant silicon spin qubit quantum computer in the near term. I demonstrate that arranging the qubits in a 2×N array represents an effective methodology for the construction of error correcting codes, provided that the qubits can be shuttled with high fidelity. This expected characteristic of silicon spins can thus be utilised to circumvent the construction of complex dense 2D grids of qubits. Subsequently, I build upon the existing literature on the implementation of low-noise gates in silicon spin qubits, at the physical level this time. Specifically, I design a protocol for the individual control of silicon spin qubits, without the need for local magnetic fields targeting every qubit in the device. This simplification enables efficient control of silicon spins, whose single-qubit addressability was an identified bottleneck. Finally, I complement the above research with an error mitigation scheme targeted at the evaluation of eigenvalues with an early-fault-tolerant quantum computers. This scheme promises to alleviate the stringent resource costs of error correction, thereby paving the way for fault-tolerance in a nearer future.