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arXiv:2610.03673

Quantum Simulation on Riemannian Manifolds

Dylan Herman, Jacob Watkins, Guneykan Ozgul, Jiayu Shen, Brandon Augustino, Junhyung Lyle Kim, Shouvanik Chakrabarti

quant-phcs.DSSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.

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We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via a position-dependent metric in the kinetic term. We formulate QRHD in both operator and path integral formalisms and derive the corresponding quantum equations of motion, showing that quantum corrections appear in the action integral but they are suppressed at late times by the time-dependent dissipation factor. This implies that convergence near optimal points is controlled by the classical potential while quantum effects influence early-time dynamics. By analyzing the semiclassical equation, we estimate a lower bound on the convergence time and numerically demonstrate whether QRHD works as a quantum optimization algorithm in some examples. A quantum circuit implementation based on time-dependent Hamiltonian simulation is also discussed and the query complexity is estimated.

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Quantum Computational Riemannian and Sub-Riemannian Geodesics

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Nielsen et al. have introduced a Riemannian metric on the space of n-qubit unitary operators [M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Science 311 (2006), 1133]. The length of the shortest curve connecting the identity I and a desired unitary operator W defined by this metric is essentially equivalent to the quantum gate complexity of W. This metric, and thus, the Riemannian geodesic equation for this metric, has a parameter q called “penalty” that must be large enough. In this paper, we investigate the sub-Riemannian geodesic equation obtained by taking the limit of the Riemannian geodesic equation as q →∞, and show that the Riemannian geodesics for finite q can be explicitly constructed from the sub-Riemannian geodesics. We also present a numerical algorithm for finding the (sub-)Riemannian geodesics connecting I and W, which is based on the Krotov method in optimal control theory. As an example, we give an exact sub-Riemannian geodesic connecting I and the controlled-controlled-Z gate, which is obtained by guessing from the numerical results of the algorithm.

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Efficient Quantum Optimization via Dynamical Simulation

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We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f : R N → R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O ( N 2 κ 2 / h x 2 ϵ ) queries to a phase oracle to find an ϵ -approximate local optimum, where κ is the condition number of the Hessian matrix and h x is the discretization spacing. In contrast, we show that methods based on gradient descent require O ( N 3 / 2 ( 1 / ϵ ) κ log ( 3 ) / 4 ) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

arXiv:2502.04285•PRX Quantum•May 14, 2026•1 citation
We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and ℓp\ell_pℓp​ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.
We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via a position-dependent metric in the kinetic term. We formulate QRHD in both operator and path integral formalisms and derive the corresponding quantum equations of motion, showing that quantum corrections appear in the action integral but they are suppressed at late times by the time-dependent dissipation factor. This implies that convergence near optimal points is controlled by the classical potential while quantum effects influence early-time dynamics. By analyzing the semiclassical equation, we estimate a lower bound on the convergence time and numerically demonstrate whether QRHD works as a quantum optimization algorithm in some examples. A quantum circuit implementation based on time-dependent Hamiltonian simulation is also discussed and the query complexity is estimated.
Nielsen et al. have introduced a Riemannian metric on the space of n-qubit unitary operators [M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Science 311 (2006), 1133]. The length of the shortest curve connecting the identity I and a desired unitary operator W defined by this metric is essentially equivalent to the quantum gate complexity of W. This metric, and thus, the Riemannian geodesic equation for this metric, has a parameter q called “penalty” that must be large enough. In this paper, we investigate the sub-Riemannian geodesic equation obtained by taking the limit of the Riemannian geodesic equation as q →∞, and show that the Riemannian geodesics for finite q can be explicitly constructed from the sub-Riemannian geodesics. We also present a numerical algorithm for finding the (sub-)Riemannian geodesics connecting I and W, which is based on the Krotov method in optimal control theory. As an example, we give an exact sub-Riemannian geodesic connecting I and the controlled-controlled-Z gate, which is obtained by guessing from the numerical results of the algorithm.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.
We propose Quantum Constrained Hamiltonian Descent (QCHD), a continuous-time quantum framework for optimization under smooth equality constraints. The unconstrained counterpart, Quantum Hamiltonian Descent (QHD), minimizes a cost function by simulating the Schrödinger dynamics of a particle in the potential given by the cost function. Applying Dirac's formalism to the classical dynamics with constraints, we construct an intrinsic quantum Hamiltonian with a Laplace--Beltrami operator on the constraint manifold in a kinetic term. Its evolution preserves the constraints exactly without penalty terms and is expected to solve quantum-dynamically the constrained optimization problem. We analyze its relation to Quantum Hamiltonian Descent supplemented by a strong penalty term. For a squared-distance penalty, the effective tangential dynamics acquires an extrinsic-curvature potential and, in nontrivial normal-rotation sectors, a coupling to the normal-bundle connection. These embedding-dependent terms remain in the strong-penalty limit, generally yielding quantum dynamics different from QCHD. Numerical comparisons for circular, elliptical, and two simultaneous constraints show faster optimization and improved concentration near global minima for the initial states and finite penalty strengths considered.
We analyze convergence of gradient-descent methods on Riemannian manifolds. In particular, we study randomization of Riemannian gradient algorithms for minimizing smooth cost functions (of Morse-Bott type). We prove that randomized gradient descent methods, where the Riemannian gradient is replaced by a random projection of it, converge to a single local optimum almost surely despite the existence of saddle points. We consider both uniformly distributed and discrete random projections. We also discuss the time required to pass a saddle point. As a major application, we consider ground-state preparation through quantum optimization over the unitary group. In mathematical terms our randomized algorithm applied to the trace function U→tr⁡(AUρU∗)U \to \operatorname{tr}(AUρU^*)U→tr(AUρU∗) almost surely converges to its global minimum. The minimum corresponds to the smallest eigenvalue (ground state) of the selfadjoint operator AAA (Hamiltonian) if ρρρ is a rank-one projector (pure state). In this setting, one can efficiently replace the uniform random projections by implementing so-called discrete unitary 2-designs.
We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f : R N → R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O ( N 2 κ 2 / h x 2 ϵ ) queries to a phase oracle to find an ϵ -approximate local optimum, where κ is the condition number of the Hessian matrix and h x is the discretization spacing. In contrast, we show that methods based on gradient descent require O ( N 3 / 2 ( 1 / ϵ ) κ log ( 3 ) / 4 ) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.