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arXiv:2603.11527

Error-Mitigated Hamiltonian Simulation: Complexity Analysis and Optimization for Near-Term and Early-Fault-Tolerant Quantum Computers

Keisuke Murota, Synge Todo, Suguru Endo

quant-phSubmitted Mar 12, 2026• Announced Mar 13, 2026• 0 citations• Revised Aug 25, 2026 (v3)
v1 · Mar 12, 2026v2 · Aug 6, 2026v3 · Aug 25, 2026
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Abstract

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 $ε_c$, below which the required number of circuit runs grows exponentially. QEM improves the noise dependence of $ε_c$ from sublinear to $k$th-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.

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13 pages, 2 figure

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Quantum computers can efficiently simulate many-body systems. As a widely used Hamiltonian simulation tool, the Trotter-Suzuki scheme splits the evolution into the number of Trotter steps NNN and approximates the evolution of each step by a product of exponentials of each individual term of the total Hamiltonian. The algorithmic error due to the approximation can be reduced by increasing NNN, which however requires a longer circuit and hence inevitably introduces more physical errors. In this work, we first study such a trade-off and numerically find the optimal number of Trotter steps Nopt{N}_{\mathrm{opt}}Nopt​ given a physical error model in a near-term quantum hardware. Practically, physical errors can be suppressed using recently proposed error mitigation methods. We then extend physical error mitigation methods to suppress the algorithmic error in Hamiltonian simulation. By exploiting the simulation results with different numbers of Trotter steps $N\ensuremath{\le}{N}_{\mathrm{opt}}$, we can infer the exact simulation result within a higher accuracy and hence mitigate algorithmic errors. We numerically test our scheme with a five-qubit system and show significant improvements in the simulation accuracy by applying both physical and algorithmic error mitigations.
Quantum computation is expected to have significant practical applications. The main problem in quantum computing is susceptibility to noise. Quantum error correction was developed to allow large-scale applications of quantum computing by achieving fault tolerance. However, as large-scale fault-tolerant quantum computation is not available, the question of applications of near-term hardware remains important. Current noisy intermediate-scale (NISQ) quantum devices can be used for small applications or as testbeds for larger applications; noise suppression techniques are of significant importance to their applicability. This thesis addresses noise suppression via quantum error mitigation (QEM) and quantum error correction (QEC) for near-term applications. The work aims to improve the understanding of noise-suppression techniques and their performance for exploring the physical systems of interest. The results presented in this thesis are divided into QEM and QEC chapters. The majority of the work is focused on QEM. The main interest of this thesis is the application of QEM to Hamiltonian simulations. Two simulation methods: the Trotter method and a quantum signal processing-based method, are explored. Both simulation methods, enhanced by zero-noise extrapolation, are employed to simulate the transverse-field Ising model. The obtained numerical results are promising, as the QEM allows a good recovery of noiseless values in low-noise shallow circuits, even with fixed sampling resources. A further result obtained in the QEM domain is a definition of a mitigation technique based on optimal measurement (in a metrological sense). The presentation includes analytical calculations of implementation costs and error propagation, as well as a numerical application to Hamiltonian simulation, yielding promising results: noise suppression by several orders of magnitude. The QEC part of the thesis explores its application in quantum metrology. The focus is on parameter sensing in many-body Hamiltonians. The results are purely analytical and extend a known lemma to mmm-local homogeneous Pauli Hamiltonian parameter sensing. This allows discussion of the scaling achieved for sensing the Hamiltonian parameters with respect to the code properties. The results in this thesis establish applications of QEM and QEC in selected near-term applications, quantify their performance, and discuss caveats and potential future applications. This thesis presents a theoretical and numerical study with implications for future experimental implementations. The relative success of the QEM and QEC shown in this thesis relies on tailoring the techniques to a chosen problem, highlighting the importance of technique adaptation in near-term applications.
Numerous quantum error-mitigation protocols have been proposed, motivated by the critical need to suppress noise effects on intermediate-scale quantum devices. Yet, their general potential and limitations remain elusive. In particular, to understand the ultimate feasibility of quantum error mitigation, it is crucial to characterize the fundamental sampling cost-how many times an arbitrary mitigation protocol must run a noisy quantum device. Here, we establish universal lower bounds on the sampling cost for quantum error mitigation to achieve the desired accuracy with high probability. Our bounds apply to general mitigation protocols, including the ones involving nonlinear postprocessing and those yet to be discovered. The results imply that the sampling cost required for a wide class of protocols to mitigate errors must grow exponentially with the circuit depth for various noise models, revealing the fundamental obstacles in the scalability of useful noisy near-term quantum devices.
Quantum error mitigation (QEM) is critical for extracting reliable computations from noisy quantum processors, proving itself essential not only in the near term but also as a valuable supplement to fully fault-tolerant systems in the future. Despite the necessity, practical QEM deployment still confronts challenges, including the excessive cost of sampling quantum circuits and reliance on unrealistic assumptions such as gate-independent noise. In this paper, we propose Classical Noise Inversion (CNI), which shifts the QEM from sampling diverse quantum circuits to repeated single-circuit measurement, a drastic cost reduction given that the former incurs far heavier time overhead than the latter on realistic quantum hardware. We target general noise models and establish critical conditions for their classical tractability, under which CNI remains effective. For noise models that violate the condition, we propose partial CNI, which mitigates the classically tractable factor of noise via CNI and mitigates the other part via probabilistic error cancellation tailored to gate-dependent noise. To minimize the sampling overhead, we introduce noise compression, which groups noise components with equivalent effects on measurement outcomes, thereby attaining theoretically optimal error-mitigation overhead. The proposed protocols are particularly efficient for error-propagatable circuits, including adaptive universal quantum computing represented by Clifford+T fault-tolerant circuits, and generalized measurement represented by classical shadows. To demonstrate the practical merits of CNI, we integrate it with thrifty classical shadow, and use analysis and numerical simulations to show its advantages in both efficiency and accuracy over existing approaches.
Quantum error mitigation (QEM) infers noiseless expectation values from noisy variants of a target quantum circuit. Unlike quantum error correction, QEM requires no additional hardware resources and is therefore routinely employed in experiments on contemporary quantum processors. QEM strategies based on agnostic noise amplification (ANA) are intrinsically resilient to temporal noise drift during the execution of the experiment, but their sampling cost (runtime overhead) remains a major practical bottleneck. In this work, we introduce the virtual noise scaling framework and combine it with layered mitigation to further enhance performance. While virtual noise scaling consistently reduces sampling overhead, we identify a specific noise threshold for the layered mitigation approach. When the noise level is above this threshold, layered mitigation decreases the sampling overhead; conversely, when below it, the overhead increases. Notably, this threshold is circuit-independent and depends solely on the number of layers. For strong noise, the combination of virtual noise scaling and layered mitigation yields several orders of magnitude reduction in sampling overhead compared with conventional zero-noise extrapolation post-processing. As a result, mitigation tasks that once seemed unrealistic are now challenging but achievable. The proposed approach is compatible with dynamic circuits and can be seamlessly integrated with error detection and quantum error correction schemes. In addition, it is also applicable to ANA-based mitigation of mid-circuit measurements and preparation errors. Our findings extend to any constant-step amplification factors and therefore also apply to probabilistic error amplification (PEA) QEM. We validate our post-processing approach by applying it to previously reported experimental data, where we observe a substantial improvement in mitigation efficiency and accuracy.
Quantum error mitigation (QEM) has emerged as a powerful tool for the extraction of useful quantum information from quantum devices. Here, we introduce the Subspace Noise Tailoring (SNT) algorithm, which efficiently combines the cheap cost of Symmetry Verification (SV) and low bias of Probabilistic Error Cancellation (PEC) QEM techniques. We study the performance of our method by simulating the Trotterized time evolution of the spin-1/2 Fermi-Hubbard model (FHM) using a variety of local fermion-to-qubit encodings, which define a computational subspace through a set of stabilizers, the measurement of which can be used to post-select noisy quantum data. We study different combinations of QEM and encodings and uncover a rich state diagram of optimal combinations, depending on the hardware performance, system size and available shot budget. We then demonstrate how SNT extends the reach of current noisy quantum computers in terms of the number of fermionic lattice sites and the number of Trotter steps, and quantify the required hardware performance beyond which a noisy device may compete with current state-of-the-art classical computational methods.