Full-quantum variational dynamics simulation for time-dependent Hamiltonians with global spectral discretization

Abstract

The most widely used approach for simulating the dynamics of time-dependent Hamiltonians via quantum computation depends on the quantum-classical hybrid variational quantum time evolution algorithm, in which ordinary differential equations of the variational coefficients for determining time evolution are solved via classical simulations with a time discretization method. We here present a full-quantum approach, in which ordinary differential equations of the variational coefficients are transformed into static linear equations via the Chebyshev spectral discretization method and then solved via the quantum singular value transformation algorithm. Our full quantum algorithm avoids classical feedback, achieves exponential convergence for smooth Hamiltonians, and yields a quantum circuit depth that is independent of the number of time steps. We demonstrate two implementation strategies, with a global formulation designed for fault-tolerant architectures and a sequential formulation tailored to near-term devices, and validate the approach through numerical simulations of proton-hydrogen charge-transfer dynamics, a prototypical time-dependent quantum chemistry problem. This work establishes a systematic pathway from quantum-classical hybrid variational quantum algorithms to full-quantum solvers for general time-dependent Hamiltonians, particularly those whose dynamics admit compact variational descriptions, opening a route toward full quantum computational advantages in time-dependent simulations.

Figures (5)

• Figure 1: Block structure of the coefficient matrices in the two linear system formulations. (a) The global matrix $\mathbf{L}$ exhibits a block lower-triangular structure, illustrated here for $N_\tau = 3$ subintervals. Diagonal blocks $\mathbf{L}_1 + \mathbf{L}_2(\mathbf{A}_h)$ encode the collocation equations and initial conditions within each subinterval, while the sub-diagonal blocks $-\mathbf{L}_3$ enforce continuity between adjacent subintervals. Each block has dimension $N_\alpha(n+1)$. (b) The sequential matrix $\mathbf{L}_h$ for a single subinterval has a $2 \times 2$ block structure labeled by an index qubit. The upper-left block $\mathbf{L}_1 + \mathbf{L}_2(\mathbf{A}_h)$ implements the spectral collocation equations with dimension $N_\alpha(n+1)$. The lower-left block $-\mathbf{L}_3$ extracts the endpoint values from the Chebyshev coefficients. The lower-right block $\mathbf{I}$ preserves the extracted state with dimension $N_\alpha(n+1)$.
• Figure 2: The realization of the quantum singular value transformation algorithm with quantum circuits. (a) The realization of operator $U_{\boldsymbol{\phi}}$, suppose the order $d$ is an even number. (b) The realization of each projector-controlled phase-shift operator $\Pi_{\phi}$.
• Figure 3: Representative time-dependent coefficients $\{g_\gamma(t)\}$ of the qubit Hamiltonian for the $\mathrm{H^+}+\mathrm{H}(1s)$ collision at $E=10$ keV and $b=1.6$ a.u. Seven distinct coefficients are shown, while the remaining terms exhibit identical temporal behavior and are omitted for clarity.
• Figure 4: Spectral convergence of the global formulation for the $\mathrm{H^+}+\mathrm{H}$ charge-transfer dynamics. (a) Time-dependent charge-transfer probability $P_n(t)$ for Chebyshev degrees $n=1,2,3,4,7$, compared with the exact solution. Low-order expansions ($n=1,2$) fail to capture the asymptotic transfer probability, while $n\geq 3$ accurately reproduces both the transient dynamics and the long-time limit. (b) Relative error $\delta P_n(T)$ of the asymptotic charge-transfer probability as a function of the Chebyshev degree $n$. The error decreases exponentially up to $n=4$, beyond which it saturates at $\sim 10^{-4}$.
• Figure 5: State fidelity for the $\mathrm{H^+}+\mathrm{H}$ collision with Chebyshev degree $n=4$. (a) Global formulation with equal segmentation. The fidelity of the global scheme exhibits deviations at the $10^{-5}$ level and reflects a norm drift level due to the absence of explicit intermediate normalization. (b) Sequential formulation with adaptive segmentation. The sequential scheme suppresses norm drift by normalizing at each step and maintains fidelity deviations at the level of $10^{-8}$.