The Geometric Speed Limit of Accessible Quantum State Preparation

We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians.We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor.The conjecture implies a geometric lower bound for the quantum speed limit (QSL).We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory.Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution.When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound.However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length.We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians.We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor.The conjecture implies a geometric lower bound for the quantum speed limit (QSL).We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory.Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution.When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound.However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length.We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.

arXiv:1804.05399Physical Review XFeb 20, 201995 citations

Geometric speed limit of state preparation and curved control spaces

Maximilian Goll, Robert H. Jonsson

The preparation of quantum many-body systems faces the difficulty that in a realistic scenario only few control parameters of the system may be accessible. In this context, an interesting connection between the energy fluctuations during state preparation and its geometric length as measured by the Fubini-Study metric was discussed by Bukov et al. in 2019 \cite{bukov_geometric_2019}. An inspiring conjecture lower bounding the energy fluctuations by the minimal geometric length of all accessible state preparation protocols was put forward together with supporting examples and numerical evidence. However, we here show that the conjecture does not hold but can be violated if the accessible parameter space has extrinsic curvature, when embedded into the space of all dynamically accessible states. We illustrate this by a number of generic qubit, qutrit and harmonic oscillator systems.The preparation of quantum many-body systems faces the difficulty that in a realistic scenario only few control parameters of the system may be accessible. In this context, an interesting connection between the energy fluctuations during state preparation and its geometric length as measured by the Fubini-Study metric was discussed by Bukov et al. in 2019 \cite{bukov_geometric_2019}. An inspiring conjecture lower bounding the energy fluctuations by the minimal geometric length of all accessible state preparation protocols was put forward together with supporting examples and numerical evidence. However, we here show that the conjecture does not hold but can be violated if the accessible parameter space has extrinsic curvature, when embedded into the space of all dynamically accessible states. We illustrate this by a number of generic qubit, qutrit and harmonic oscillator systems.

arXiv:2504.15175QuantumJul 15, 20260 citations

Approaching the quantum speed limit in quantum gates with geometric control

François Impens, David Guéry-Odelin

We present a geometric optimization method for implementing quantum gates by optimally controlling the Hamiltonian parameters, with the goal of approaching the Mandelstam-Tamm quantum speed limit (MT-QSL). Achieving this bound requires satisfying precise geometric conditions that govern the evolution of quantum states. We extend this geometric framework to quantum unitary operators in arbitrary dimensions and analyze the conditions necessary for saturation of the bound. Additionally, we show that the quantum brachistochrone, when generalized to operators, does not, in general, saturate the MT-QSL bound. Finally, we propose a systematic optimal control strategy based on geometric principles to approach the quantum speed limit for unitary driving. We illustrate this optimization method on a set of four well-known two-qubit quantum gates. Our procedure significantly reduces the deviation from the optimal quantum speed limit while preserving high quantum fidelity.We present a geometric optimization method for implementing quantum gates by optimally controlling the Hamiltonian parameters, with the goal of approaching the Mandelstam-Tamm quantum speed limit (MT-QSL). Achieving this bound requires satisfying precise geometric conditions that govern the evolution of quantum states. We extend this geometric framework to quantum unitary operators in arbitrary dimensions and analyze the conditions necessary for saturation of the bound. Additionally, we show that the quantum brachistochrone, when generalized to operators, does not, in general, saturate the MT-QSL bound. Finally, we propose a systematic optimal control strategy based on geometric principles to approach the quantum speed limit for unitary driving. We illustrate this optimization method on a set of four well-known two-qubit quantum gates. Our procedure significantly reduces the deviation from the optimal quantum speed limit while preserving high quantum fidelity.

arXiv:2501.05330Physical Review ASep 12, 20250 citations

Speeding up quantum adiabatic processes with a dynamical quantum geometric tensor

Jin-Fu Chen

For adiabatic controls of quantum systems, the nonadiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For the system initiated from the $n\mathrm{th}$ eigenstate, the transition probability in the optimal protocol is bounded by ${P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}$ with the operation time $\ensuremath{\tau}$ and the quantum adiabatic length ${\mathcal{L}}_{n}$ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.For adiabatic controls of quantum systems, the nonadiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For the system initiated from the $n\mathrm{th}$ eigenstate, the transition probability in the optimal protocol is bounded by ${P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}$ with the operation time $\ensuremath{\tau}$ and the quantum adiabatic length ${\mathcal{L}}_{n}$ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.

arXiv:2203.03164Physical Review ResearchJun 28, 20228 citations

Geodesic Paths for Quantum Many-Body Systems

Michael Tomka, Souza, Tiago, Steven Rosenberg +1 more

We propose a method to obtain optimal protocols for adiabatic ground-state preparation near the adiabatic limit, extending earlier ideas from [D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012)] to quantum non-dissipative systems. The space of controllable parameters of isolated quantum many-body systems is endowed with a Riemannian quantum metric structure, which can be exploited when such systems are driven adiabatically. Here, we use this metric structure to construct optimal protocols in order to accomplish the task of adiabatic ground-state preparation in a fixed amount of time. Such optimal protocols are shown to be geodesics on the parameter manifold, maximizing the local fidelity. Physically, such protocols minimize the average energy fluctuations along the path. Our findings are illustrated on the Landau-Zener model and the anisotropic XY spin chain. In both cases we show that geodesic protocols drastically improve the final fidelity. Moreover, this happens even if one crosses a critical point, where the adiabatic perturbation theory fails.We propose a method to obtain optimal protocols for adiabatic ground-state preparation near the adiabatic limit, extending earlier ideas from [D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012)] to quantum non-dissipative systems. The space of controllable parameters of isolated quantum many-body systems is endowed with a Riemannian quantum metric structure, which can be exploited when such systems are driven adiabatically. Here, we use this metric structure to construct optimal protocols in order to accomplish the task of adiabatic ground-state preparation in a fixed amount of time. Such optimal protocols are shown to be geodesics on the parameter manifold, maximizing the local fidelity. Physically, such protocols minimize the average energy fluctuations along the path. Our findings are illustrated on the Landau-Zener model and the anisotropic XY spin chain. In both cases we show that geodesic protocols drastically improve the final fidelity. Moreover, this happens even if one crosses a critical point, where the adiabatic perturbation theory fails.

arXiv:1606.05890Jun 19, 201613 citations

Geometric Instability and Self-Limitation in Driven Quantum Systems

A. M. Tishin

We develop a unified geometric framework for local non-adiabaticity in driven quantum systems. We show that the previously introduced AMT non adiabaticity parameter arises as a special realization of a more general geometric instability criterion governed by the normalized Fubini Study distinguishability speed. The local geometric evolution speed is identified as the physically relevant quantity controlling the onset of non-adiabatic instability. We introduce a universal dimensionless instability parameter measuring the competition between quantum-state evolution speed and spectral-gap protection. This quantity provides a local, gauge-invariant, and basis-independent criterion for arbitrary driven Hamiltonians. Near quantum critical points, the instability parameter diverges through inverse gap amplification, recovering the Kibble Zurek freeze-out condition directly from local geometric data. We prove that monotonic occupation-dependent nonlinear regulators geometrically compress the quantum metric, establishing a self-limitation theorem in which nonlinear spectral deformation confines the accessible region of projective Hilbert space under strong driving. The multimode extension yields a matrix-valued instability criterion that identifies collective instability channels invisible to scalar descriptions. The framework naturally extends to open quantum systems through the Bures metric and quantum Fisher geometry, where thermal mixing and Lindblad decay increase the instability threshold through geometric suppression of state distinguishability. The instability threshold further implies a universal geometric lower bound on coherent control time and quantum gate duration.We develop a unified geometric framework for local non-adiabaticity in driven quantum systems. We show that the previously introduced AMT non adiabaticity parameter arises as a special realization of a more general geometric instability criterion governed by the normalized Fubini Study distinguishability speed. The local geometric evolution speed is identified as the physically relevant quantity controlling the onset of non-adiabatic instability. We introduce a universal dimensionless instability parameter measuring the competition between quantum-state evolution speed and spectral-gap protection. This quantity provides a local, gauge-invariant, and basis-independent criterion for arbitrary driven Hamiltonians. Near quantum critical points, the instability parameter diverges through inverse gap amplification, recovering the Kibble Zurek freeze-out condition directly from local geometric data. We prove that monotonic occupation-dependent nonlinear regulators geometrically compress the quantum metric, establishing a self-limitation theorem in which nonlinear spectral deformation confines the accessible region of projective Hilbert space under strong driving. The multimode extension yields a matrix-valued instability criterion that identifies collective instability channels invisible to scalar descriptions. The framework naturally extends to open quantum systems through the Bures metric and quantum Fisher geometry, where thermal mixing and Lindblad decay increase the instability threshold through geometric suppression of state distinguishability. The instability threshold further implies a universal geometric lower bound on coherent control time and quantum gate duration.

quant-pharXiv:2606.00259Jun 2, 20260 citations

n\mathrm{th}

eigenstate, the transition probability in the optimal protocol is bounded by ${P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}$ with the operation time $\ensuremath{\tau}$ and the quantum adiabatic length

{\mathcal{L}}_{n}

induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.