Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1606.05890

Geodesic Paths for Quantum Many-Body Systems

Michael Tomka, Souza, Tiago, Steven Rosenberg, Anatoli Polkovnikov

Published Jun 19, 2016• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1606.05890

Related papers

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.

arXiv:2203.03164•Physical Review Research•Jun 28, 2022•8 citations

Geometric Speed Limit of Accessible Many-Body State Preparation

Marin Bukov, Dries Sels, Anatoli Polkovnikov

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:1804.05399•Physical Review X•Feb 20, 2019•95 citations

Geodesic steering for rapid dissipative entanglement distribution

Keshav Venkataraman, Pablo Martinez-Azcona, Marko Ljubotina +2 more

We describe an optimal-control strategy designed to accelerate the dissipative preparation of complex, highly entangled quantum many-body states. Analogous to optimal steering of isolated quantum systems, our scheme identifies control pulses that guide the evolution of open quantum systems along trajectories of instantaneous steady states while minimizing leakage resulting from non-adiabatic transitions. In many cases of interest, these control parameters can be derived analytically and without the need for computationally expensive numerical simulations of the full many-body master equation. We apply this technique for the preparation of bi- and multipartite entangled states in qubit systems coupled to a common photonic waveguide or a distributed squeezed reservoir. In both scenarios, our optimal-control strategy significantly outperforms the bare relaxation dynamics after a quench, reducing preparation times of near-maximally entangled states by orders of magnitude. This drastic speedup makes dissipative entanglement distribution protocols compatible with high-fidelity quantum information processing applications and illustrates more broadly the potential of leakage minimization for accelerating reservoir-engineering schemes.

quant-pharXiv:2610.16854•Oct 13, 2026•0 citations

Search for optimal driving in finite quantum systems with precursors of criticality

Felipe Matus, Jan Střeleček, Pavel Stránský +1 more

Using the adiabatic perturbation theory of driven dynamics [Phys. Rev. A 78, 052508 (2008)] we design a hierarchy of quantum state preparation protocols that systematically increase the fidelity at very long driving times. We test these and other protocols, including those based on the geometric analysis of the parameter space, in a single-qubit system and in a fully connected multiqubit system showing in its infinite-size limit several quantum phase transitions. Our protocols excel in the asymptotic driving regime, above a crossover time from the Landau-Zener regime which increases with a decreasing minimal energy gap along the driving path (with the size of the system). In the medium-time domain, the performance of all tested protocols is indecisive.

arXiv:2210.07027•Physical Review A•Jan 23, 2023•4 citations

The Geometric Speed Limit of Accessible Quantum State Preparation

Marin Bukov, Dries Sels, Anatoli Polkovnikov

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 which 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 target individual excited states in quantum chaotic systems, the geometric bound for the QSL 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 non-integrable spin chains using optimal control.

•Apr 15, 2018•2 citations

Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

A. T. Rezakhani, Damian F. Abasto, Daniel A. Lidar +1 more

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

arXiv:1004.0509•Physical Review A•Jul 20, 2010•83 citations
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.
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 nthn\mathrm{th}nth 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 Ln{\mathcal{L}}_{n}Ln​ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising 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.
We describe an optimal-control strategy designed to accelerate the dissipative preparation of complex, highly entangled quantum many-body states. Analogous to optimal steering of isolated quantum systems, our scheme identifies control pulses that guide the evolution of open quantum systems along trajectories of instantaneous steady states while minimizing leakage resulting from non-adiabatic transitions. In many cases of interest, these control parameters can be derived analytically and without the need for computationally expensive numerical simulations of the full many-body master equation. We apply this technique for the preparation of bi- and multipartite entangled states in qubit systems coupled to a common photonic waveguide or a distributed squeezed reservoir. In both scenarios, our optimal-control strategy significantly outperforms the bare relaxation dynamics after a quench, reducing preparation times of near-maximally entangled states by orders of magnitude. This drastic speedup makes dissipative entanglement distribution protocols compatible with high-fidelity quantum information processing applications and illustrates more broadly the potential of leakage minimization for accelerating reservoir-engineering schemes.
Using the adiabatic perturbation theory of driven dynamics [Phys. Rev. A 78, 052508 (2008)] we design a hierarchy of quantum state preparation protocols that systematically increase the fidelity at very long driving times. We test these and other protocols, including those based on the geometric analysis of the parameter space, in a single-qubit system and in a fully connected multiqubit system showing in its infinite-size limit several quantum phase transitions. Our protocols excel in the asymptotic driving regime, above a crossover time from the Landau-Zener regime which increases with a decreasing minimal energy gap along the driving path (with the size of the system). In the medium-time domain, the performance of all tested protocols is indecisive.
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 which 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 target individual excited states in quantum chaotic systems, the geometric bound for the QSL 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 non-integrable spin chains using optimal control.
We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.