Jin-Peng Liu

Yixuan Liang, Jin-Peng Liu

We give detailed analysis and circuit design of structure-preserving quantum algorithms for second-order linear evolutionary PDEs, including parabolic equations and hyperbolic equations with mixed Dirichlet, Neumann, and periodic boundary conditions and source terms. While prior quantum algorithms usually neglect the stability problem from the PDE-to-ODE reduction, our method-of-lines approach investigates the boundary lifting via Coons interpolation and boundary-aware discretization, so that the resulting semi-discrete systems are stable and compatible with efficient quantum ODE primitives. For the parabolic problem, we use a diagonal similarity transform to ensure the semi-discrete generator must have a positive semi-definite Hermitian part, and then solve the resulting ODE system by the optimal linear combination of Hamiltonian simulation (LCHS). For the hyperbolic problem, we rewrite the semi-discrete equation as an equivalent first-order system and solve it by Hamiltonian simulation. We implement our quantum algorithms with explicit block-encoding constructions and circuit implementations, as well as demonstrating the end-to-end complexity bounds together with spatial and quadrature error estimates. We conduct classical numerical experiments on the convection-diffusion equation, inhomogeneous heat equation, and Klein-Gordon equation to validate our structure-preserving analysis and algorithmic constructions.

Junyu Liu, Minzhao Liu, Jin-Peng Liu et al.

Large machine learning models are revolutionary technologies of artificial intelligence whose bottlenecks include huge computational expenses, power, and time used both in the pre-training and fine-tuning process. In this work, we show that fault-tolerant quantum computing could possibly provide provably efficient resolutions for generic (stochastic) gradient descent algorithms, scaling as O(T^2 polylog(n)), where n is the size of the models and T is the number of iterations in the training, as long as the models are both sufficiently dissipative and sparse, with small learning rates. Based on earlier efficient quantum algorithms for dissipative differential equations, we find and prove that similar algorithms work for (stochastic) gradient descent, the primary algorithm for machine learning. In practice, we benchmark instances of large machine learning models from 7 million to 103 million parameters. We find that, in the context of sparse training, a quantum enhancement is possible at the early stage of learning after model pruning, motivating a sparse parameter download and re-upload scheme. Our work shows solidly that fault-tolerant quantum algorithms could potentially contribute to most state-of-the-art, large-scale machine-learning problems.

Andrew M. Childs, Tongyang Li, Jin-Peng Liu et al.

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $κ$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $ε$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(κ^{1/2}d)$ and $\widetilde{O}(κ^{1/2}d^{3/2}/ε)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $κ,d,ε$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/ε^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $ε$.

Yanqiao Wang, Jin-Peng Liu

We develop a systematic sign-embedding framework of operator-output quantum algorithms for matrix equations and matrix functions. Differing from the contour-integral treatment, we start with the matrix-sign embedding route: an augmented matrix $M$ whose half-plane matrix sign compresses the target operator either as a block of $\text{sign}(M)$ or, in projector form, through $(I-\text{sign}(M))/2$; we then construct a logarithmic-sinc approximation for the half-plane sign operator and combine it with structure-aware scaled multiplexing and nodewise rebalancing of shifted inverse families. For ordinary Sylvester equations, we offer an explicit block-encoding of the target matrix solution with query complexity linear in the inverse-conditioning parameters and logarithmic in the target error tolerance, under non-normal and non-diagonalizable settings given a field-of-values (FoV) gap or strip-resolvent hypotheses. These algorithms propagate the same overlap-based normalization bookkeeping to ordinary and generalized Sylvester equations, generalized Lyapunov equations, principal square roots and inverse square roots, matrix geometric means, and continuous-time algebraic Riccati equations (CARE). These results identify matrix-sign embeddings and nodewise rebalancing as reusable design principles for structured operator-output quantum linear algebra.

Yunfei Wang, Ruoxi Jiang, Yingda Fan et al.

A diffusion probabilistic model (DPM) is a generative model renowned for its ability to produce high-quality outputs in tasks such as image and audio generation. However, training DPMs on large, high-dimensional datasets such as high-resolution images or audio incurs significant computational, energy, and hardware costs. In this work, we introduce efficient quantum algorithms for implementing DPMs through various quantum ODE solvers. These algorithms highlight the potential of quantum Carleman linearization for diverse mathematical structures, leveraging state-of-the-art quantum linear system solvers (QLSS) or linear combination of Hamiltonian simulations (LCHS). Specifically, we focus on two approaches: DPM-solver-$k$ which employs exact $k$-th order derivatives to compute a polynomial approximation of $ε_θ(x_λ,λ)$; and UniPC which uses finite difference of $ε_θ(x_λ,λ)$ at different points $(x_{s_m}, λ_{s_m})$ to approximate higher-order derivatives. As such, this work represents one of the most direct and pragmatic applications of quantum algorithms to large-scale machine learning models, presumably taking substantial steps towards demonstrating the practical utility of quantum computing.