arXiv subjects

Zhiyan Ding

Publications and source records attributed to Zhiyan Ding.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space

We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution $U^\varepsilon$, which is the value function of a mean field control problem, converges uniformly as $\varepsilon \to 0$ to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions with multiscale dependence, we prove that the rate of convergence is $O(\sqrt{\varepsilon})$. When the Hamiltonian depends only on the fast variable and the momentum, we establish the sharp convergence rate $O(\varepsilon)$. To the best of our knowledge, this is the first quantitative convergence result extending the optimal rate for first-order Hamilton-Jacobi equations in finite dimensions to the Wasserstein space. Finally, we show that our analysis extends to dynamic optimal transport problems, where the terminal condition imposes a constraint on the final distribution.

math.AP↗

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.

quant-ph↗

arXiv2026

Overcoming the Lamb Shift in System-Bath Interaction Models via KMS Detailed Balance: High-Accuracy Thermalization with Time-Bounded Interactions

We investigate quantum thermal state preparation algorithms based on system-bath interactions and uncover a surprising phenomenon in the weak-coupling regime. We rigorously prove that, if the system-bath interaction is engineered so that the transition part of the approximate Lindbladian generator satisfies the Kubo--Martin--Schwinger (KMS) detailed balance condition, then the unique fixed point of the dynamics can be made arbitrarily close to the Gibbs state in the weak-coupling limit, regardless of the structure of the Lamb shift term. Importantly, this remains true even when the approximate Lindbladian differs substantially from the ideal Davies generator and the Lamb shift term does not commute with the thermal state. Our result shows that the role of the KMS detailed balance condition extends well beyond standard Lindbladian dynamics, serving as a general principle for a broader class of dissipative systems. Furthermore, by combining this with a general perturbation framework, we bound the mixing time of the dynamics and establish an end-to-end complexity of $\mathcal{O}(\varepsilon^{-1})$ for Gibbs state preparation. These guarantees apply to any Hamiltonian whose associated KMS-detailed-balance Lindbladian is known to be fast mixing.

quant-ph↗

arXiv2026

Provably Efficient Long-Time Exponential Decompositions of Non-Markovian Gaussian Baths

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval $[0,T]$ by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time $T$, inverse temperature $\beta$, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of $T$, achieving time-uniform complexity. The $T$-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.

quant-ph↗

arXiv2026

Uniform-in-time propagation of chaos for consensus-based minimax algorithm

We study the large-population convergence of a consensus-based algorithm for the saddle point problem proposed by ArXiv: 2212.12334, establishing the uniform-in-time propagation of chaos using a coupling method. Our work shows that the $L^2$-deviation has order $O(N_1^{-1} + N_2^{-1})$ uniformly in time, where $N_1$ and $N_2$ denote the numbers of particles corresponding to the two competing players. It demonstrates the convergence of the particles to some location near a saddle point of the given objective function, which confirms the computational feasibility of the algorithm. The main idea behind the proofs is the exponential decay and the concentration of the variances of the particle system.

math.PR↗

arXiv2025

Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

We establish new theoretical results demonstrating the efficiency and robustness of system-bath interaction algorithms for quantum thermal and ground state preparation. We rigorously show that, even when the coupling strength is chosen independently of the desired accuracy, the induced quantum channel admits the target thermal and ground states as approximate fixed points with arbitrarily high precision. This contrasts with prior analyses, which typically rely on the leading order Lindbladian approximation and require the coupling strength to decrease polynomially with the target error tolerance. Our proof introduces new techniques for controlling all orders of the Dyson expansion and for analyzing the associated multidimensional operator Fourier transforms. Building on these new estimation, we demonstrate an improved end-to-end complexity analysis of thermal and ground state preparation for the system-bath interaction algorithms. These bounds substantially improve upon prior results, and numerical simulations further confirm the robustness of the system-bath interaction framework across both weak and strong coupling regimes.

quant-ph↗

arXiv2025

Quantum Replica Exchange

The presence of energy barriers in the state space of a physical system can lead to exponentially slow convergence for sampling algorithms like Markov chain Monte Carlo (MCMC). In the classical setting, replica exchange (or parallel tempering) is a powerful heuristic to accelerate mixing in these scenarios. In the quantum realm, preparing Gibbs states of Hamiltonians faces a similar challenge, where bottlenecks can dramatically increase the mixing time of quantum dynamical semigroups. In this work, we introduce a quantum analogue of the replica exchange method. We define a Lindbladian on a joint system of two replicas and prove that it can accelerate mixing for a class of Hamiltonians with local energy barriers. We provide a rigorous lower bound on the spectral gap of the combined system's Lindbladian, which leads to an exponential improvement in spectral gap with respect to the barrier height. We showcase the applicability of our method with several examples, including the defected 1D Ising model at arbitrary constant temperature, and defected non-commuting local Hamiltonians at high temperature.

quant-ph↗

arXiv2025

Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

Fine-grained spectral properties of quantum Hamiltonians, including both eigenvalues and their multiplicities, provide useful information for characterizing many-body quantum systems as well as for understanding phenomena such as topological order. Extracting such information with small additive error is $\#\textsf{BQP}$-complete in the worst case. In this work, we introduce QFAMES (Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra), a quantum algorithm that efficiently identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physically motivated assumptions, which allows us to bypass worst-case complexity barriers. QFAMES also enables the estimation of observable expectation values within targeted energy clusters, providing a powerful tool for studying quantum phase transitions and other physical properties. We validate the effectiveness of QFAMES through numerical demonstrations, including its applications to characterizing quantum phases in the transverse-field Ising model and estimating the ground-state degeneracy of a topologically ordered phase in the two-dimensional toric code model. We also generalize QFAMES to the setting of mixed initial states. Our approach offers rigorous theoretical guarantees and significant advantages over existing subspace-based quantum spectral analysis methods, particularly in terms of the sample complexity and the ability to resolve degeneracies.

quant-ph↗

arXiv2025

Simple and efficient end-to-end quantum thermal and ground state preparation

Quantum computing algorithms for many-body physics, chemistry and materials science typically require the preparation of thermal or ground states for a given Hamiltonian. We propose quantum algorithms based on system-bath interactions to prepare thermal and ground states for a range of physically relevant Hamiltonians. These algorithms require only forward evolution under a system-bath Hamiltonian in which the bath is a single reusable ancilla qubit, making them especially well-suited for early fault-tolerant quantum devices. By carefully designing the bath and interaction Hamiltonians, we prove that the fixed point of the dynamics accurately approximates the desired quantum state. Furthermore, we establish theoretical guarantees on the mixing time, and thereby provide a rigorous justification for the end-to-end efficiency of system-bath interaction models in thermal and ground state preparation for the physically relevant models we consider.

quant-ph↗

arXiv2025

Operator-Level Quantum Acceleration of Non-Logconcave Sampling

Sampling from probability distributions of the form $\sigma \propto e^{-\beta V}$, where $V$ is a continuous potential, is a fundamental task across physics, chemistry, biology, computer science, and statistics. However, when $V$ is non-convex, the resulting distribution becomes non-logconcave, and classical methods such as Langevin dynamics often exhibit poor performance. We introduce the first quantum algorithm that provably accelerates a broad class of continuous-time sampling dynamics. For Langevin dynamics, our method encodes the target Gibbs measure into the amplitudes of a quantum state, identified as the kernel of a block matrix derived from a factorization of the Witten Laplacian operator. This connection enables Gibbs sampling via singular value thresholding and yields up to a quartic quantum speedup over best-known classical Langevin-based methods in the non-logconcave setting. Building on this framework, we further develop the first quantum algorithm that accelerates replica exchange Langevin diffusion, a widely used method for sampling from complex, rugged energy landscapes.

quant-ph↗

arXiv2025

Polynomial-Time Preparation of Low-Temperature Gibbs States for 2D Toric Code

We propose a polynomial-time algorithm for preparing the Gibbs state of the two-dimensional toric code Hamiltonian at any temperature, starting from any initial condition, significantly improving upon prior estimates that suggested exponential scaling with inverse temperature. Our approach combines the Lindblad dynamics using a local Davies generator with simple global jump operators to enable efficient transitions between logical sectors. Our proof also shows that the Lindblad dynamics with a digitally implemented low-temperature local Davies generator is able to efficiently drive the quantum state towards the ground state manifold.

quant-ph↗

arXiv2025

Rapid quantum ground state preparation via dissipative dynamics

Inspired by natural cooling processes, dissipation has become a promising approach for preparing low-energy states of quantum systems. However, the potential of dissipative protocols remains unclear beyond certain commuting Hamiltonians. This work provides significant analytical and numerical insights into the power of dissipation for preparing the ground state of noncommuting Hamiltonians. For quasi-free dissipative dynamics, including certain 1D spin systems with boundary dissipation, our results reveal a new connection between the mixing time in trace distance and the spectral properties of a non-Hermitian Hamiltonian, leading to an explicit and sharp bound on the mixing time that scales polynomially with system size. For more general spin systems, we develop a tensor network-based algorithm for constructing the Lindblad jump operator and for simulating the dynamics. Using this algorithm, we demonstrate numerically that dissipative ground state preparation protocols can achieve rapid mixing for certain 1D local Hamiltonians under bulk dissipation, with a mixing time that scales logarithmically with the system size. We then prove the rapid mixing result for certain weakly interacting spin and fermionic systems in arbitrary dimensions, extending recent results for high-temperature quantum Gibbs samplers to the zero-temperature regime. Together, these results show that dissipation can be a powerful tool for ground state preparation, with potential applications across condensed matter physics, quantum materials science, and beyond.

quant-ph↗

arXiv2024

Efficient quantum Gibbs samplers with Kubo--Martin--Schwinger detailed balance condition

Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo--Martin--Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), \re{akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis.} Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.

quant-ph↗

arXiv2024

The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size $\mathcal{O}(1)$, but these analyses only show that the error $ε$ decays at a slow rate $ε=\mathcal{\tilde{O}}(n^{-1/2})$ with respect to the cutoff frequency $n$ (i.e., the maximum frequency of the measurements). In this work, we prove that under certain assumptions, the ESPRIT algorithm can attain a significantly improved error scaling $ε= \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit $ε= \mathcal{O}(n^{-1})$ given by the Nyquist-Shannon sampling theorem. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest.

cs.IT↗

arXiv2024

Quantum Multiple Eigenvalue Gaussian filtered Search: an efficient and versatile quantum phase estimation method

Quantum phase estimation is one of the most powerful quantum primitives. This work proposes a new approach for the problem of multiple eigenvalue estimation: Quantum Multiple Eigenvalue Gaussian filtered Search (QMEGS). QMEGS leverages the Hadamard test circuit structure and only requires simple classical postprocessing. QMEGS is the first algorithm to simultaneously satisfy the following two properties: (1) It can achieve the Heisenberg-limited scaling without relying on any spectral gap assumption. (2) With a positive energy gap and additional assumptions on the initial state, QMEGS can estimate all dominant eigenvalues to $ε$ accuracy utilizing a significantly reduced circuit depth compared to the standard quantum phase estimation algorithm. In the most favorable scenario, the maximal runtime can be reduced to as low as $\log(1/ε)$. This implies that QMEGS serves as an efficient and versatile approach, achieving the best-known results for both gapped and gapless systems. Numerical results validate the efficiency of our proposed algorithm in various regimes.

quant-ph↗

arXiv2024

Swarm-based gradient descent meets simulated annealing

We introduce a novel method for non-convex optimization, called Swarm-based Simulated Annealing (SSA), which is at the interface between the swarm-based gradient-descent (SBGD) [J. Lu et. al., ArXiv:2211.17157; E.Tadmor and A. Zenginoglu, Acta Applicandae Math., 190, 2024] and Simulated Annealing (SA) [V. Cerny, J. optimization theory and appl., 45:41-51, 1985; S.Kirkpatrick et. al., Science, 220(4598):671-680, 1983; S. Geman and C.-R. Hwang, SIAM J. Control and Optimization, 24(5):1031-1043, 1986]. Similar to SBGD, we introduce a swarm of agents, each identified with a position, ${\mathbf x}$ and mass $m$, to explore the ambient space. Similar to SA, the agents proceed in the gradient descent direction, and are subject to Brownian motion. The annealing rate, however, is dictated by a decreasing function of their mass. As a consequence, instead of the SA protocol for time-decreasing temperature, we let the swarm decide how to `cool down' agents, depending on their accumulated mass over time. The dynamics of masses is coupled with the dynamics of positions: agents at higher ground transfer (part of) their mass to those at lower ground. Consequently, resulting SSA optimizer is dynamically divided between heavier, cooler agents viewed as `leaders' and lighter, warmer agents viewed as `explorers'. Mean-field convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based method as a multi-dimensional global optimizer.

math.OC↗

arXiv2024

Lower bound for simulation cost of open quantum systems: Lipschitz continuity approach

Simulating quantum dynamics is one of the most promising applications of quantum computers. While the upper bound of the simulation cost has been extensively studied through various quantum algorithms, much less work has focused on establishing the lower bound, particularly for the simulation of open quantum system dynamics. In this work, we present a general framework to calculate the lower bound for simulating a broad class of quantum Markov semigroups. Given a fixed accessible unitary set, we introduce the concept of convexified circuit depth to quantify the quantum simulation cost and analyze the necessary circuit depth to construct a quantum simulation scheme that achieves a specific order. Our framework can be applied to both unital and non-unital quantum dynamics, and the tightness of our lower bound technique is illustrated by showing that the upper and lower bounds coincide in several examples.

quant-ph↗

arXiv2024

Single-ancilla ground state preparation via Lindbladians

We design a quantum algorithm for ground state preparation in the early fault tolerant regime. As a Monte Carlo-style quantum algorithm, our method features a Lindbladian where the target state is stationary. The construction of this Lindbladian is algorithmic and should not be seen as a specific approximation to some weakly coupled system-bath dynamics in nature. Our algorithm can be implemented using just one ancilla qubit and efficiently simulated on a quantum computer. It can prepare the ground state even when the initial state has zero overlap with the ground state, bypassing the most significant limitation of methods like quantum phase estimation. As a variant, we also propose a discrete-time algorithm, demonstrating even better efficiency and providing a near-optimal simulation cost depending on the desired evolution time and precision. Numerical simulation using Ising and Hubbard models demonstrates the efficacy and applicability of our method.

quant-ph↗