arXiv subjects

Zikang Jia

Publications and source records attributed to Zikang Jia.

4 recordsLinked to original sources

arXiv2026

A Theory of Finite-Noise Optima and Generalization in Quantum Machine Learning

Quantum noise is expected to degrade quantum machine learning by driving circuits away from their noiseless implementations. Yet recent studies show moderate noise can reduce testing error, a behavior unexplained by weak-noise perturbative error accumulation or strong-noise trainability collapse. Here we develop a statistical learning theory connecting microscopic noise processes to macroscopic learning performance. At its heart is a noise-order purity parameter, derived from a surrogate model analysis, that predicts the noise-induced reduction in model complexity and the consequent reduction in the generalization gap. Noise simultaneously increases prediction bias. Their competition explains the intermediate-noise regime left open between these limits. It produces a finite-noise optimum whose location depends on the learning setup and can disappear in the large-sample limit. Numerical experiments validate these predictions. Noise programming can move a model towards this optimum. These results make the non-monotonic effect of noise predictable and provide a route to harness it.

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arXiv2026

Quantum Algorithms for Nonlinear Differential Equations via Pivot-Shifted Carleman Linearization

We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a Lyapunov transform and rescaling, we enlarge the class of nonlinear systems that can be efficiently simulated on quantum computers. For systems that exhibit stability in the shifted coordinates, we establish long time convergence of the truncated Carleman embedding. We prove that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. By introducing a modified nonlinearity condition, this framework entirely removes the conventional lower bound requirement on the initial condition. For more general systems that remain unstable after shifting, we provide short time convergence guarantees that are similarly free from the initial condition constraints. Numerical experiments on the logistic and the Lotka-Volterra equations demonstrate that an appropriate pivot choice improves stability and accuracy, and yields exponential error decay with truncation order. These results show that pivot shifting provides a practical and theoretically justified route for extending Carleman-based quantum algorithms to a broader class of nonlinear dynamical systems.

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arXiv2026

Programmable Signal Design for Quantum Phase Estimation via Quantum Signal Processing

Quantum phase estimation is a central primitive in quantum algorithms and sensing, where performance is governed by the sensitivity of measurement signals to the target parameter. While existing methods have developed increasingly sophisticated inference and adaptive design strategies, the signal family used for phase learning is often largely pre-specified. Here we propose a programmable signal design framework for quantum phase estimation based on quantum signal processing, which enables the measurement signal to be tailored to the current uncertainty region. We cast phase estimation as a max-min optimization problem over admissible signals and introduce a sensitivity efficiency parameter that quantifies information gain per query depth. The resulting iterative algorithm combines optimized quantum signal transformations with structured classical inference, retaining Heisenberg-limited scaling while improving sensitivity efficiency and practical resource prefactors. Numerical results show reduced estimation variance compared with standard protocols such as robust phase estimation. Our framework also extends to Hamiltonian eigenvalue estimation in higher dimensions and establishes a quantum-classical co-design paradigm through programmable signal shaping.

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arXiv2025

Enhancing the Clique Local Decoder to Correct Length-2 Space Errors in the Surface Code

The growing demand for fault-tolerant quantum computing drives the need for efficient, scalable Quantum Error Correction (QEC) strategies. Conventional decoders designed for worst-case error scenarios incur significant overhead, prompting the development of local decoders, that leverage the sparse and often trivial nature of many quantum errors, to support the conventional decoders. The previously proposed Clique decoder addresses this by handling isolated, length-1 space and time errors within the cryogenic environment with minimal hardware costs, thereby mitigating I/O bandwidth constraints between cryogenic quantum systems and room-temperature processors. Building on this foundation, we propose Clique_L2 that extends the Clique-based approach by relaxing some original constraints and incorporating additional low-cost logic to also correct length-2 error chains in space, which become non-trivial occurrences at higher physical error rates and code distances. This enhanced capability not only further reduces out-of-the-fridge data transmission but also adapts more effectively to clustered errors observed under a variety of noise models. Specifically, under data-qubit-only errors and uniformly random noise, Clique_L2 achieves up to 8.95x decoding bandwidth reduction over the original Clique (or Clique_L1) decoder, especially beneficial at higher code distances. When clustered errors and longer error chains are more likely to occur, Clique_L2 achieves up to 18.3x decoding bandwidth reduction over Clique_L1, achieving substantial benefits across a wide range of physical qubit error rates.

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