Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

Abstract: 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, which 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.

• The paper introduces a Riemannian metric tensor that minimizes adiabatic error in quantum dynamics for any ground state degeneracy.
• It demonstrates the metric's role in characterizing quantum phase transitions through fidelity susceptibility and optimal adiabatic scheduling with universal power-law scaling.
• The paper establishes connections to Grassmannian geometry, Uhlmann parallel transport, and quantum Fisher information to enhance strategies in quantum control and computation.

Intrinsic Geometry of Quantum Adiabatic Evolution and Quantum Phase Transitions

Introduction and Motivation

This paper develops a unified geometric formulation for quantum adiabatic evolution (QAE) and quantum phase transitions (QPT) by exposing a deep connection between both phenomena via a Riemannian metric tensor structure. The authors systematically derive this metric by minimizing adiabatic deviations and demonstrate its scope by generalizing to systems with any ground state degeneracy. The resulting geometric framework is shown to underpin not only the characterization of QPTs via fidelity but also optimal control strategies in adiabatic and holonomic quantum computing.

Geometric Structure of Adiabatic Evolution

The foundational technical advance is the identification of a Riemannian metric tensor on the space of Hamiltonian control parameters, constructed by minimizing the leading order contribution to the adiabatic error in quantum dynamics. For an $n$-body quantum system with a path-dependent Hamiltonian $H(\mathbf{x})$ parametrized by $\mathbf{x}$, the key construction is:

• The adiabatic error, defined via the operator norm difference between the true and adiabatic propagators, admits a decomposition into two terms, with the dominant term for large evolution time $T$ containing all geometric information.
• By employing the Frobenius (Hilbert-Schmidt) norm, a Riemannian metric emerges unambiguously for both nondegenerate and degenerate ground state subspaces.

Explicitly, for a ground state projector $P_0(\mathbf{x})$ with degeneracy $g_0$, the metric reads: $$\mathbf{g}_{ij} = \frac{1}{2g_0} \mathrm{Tr} \left[ \partial_i P_0 \partial_j P_0 \right]$$ This endows the control parameter manifold with a natural metric geometry, and the geodesics of this manifold define optimal adiabatic passage paths, i.e., those which minimize the geometric component of the adiabatic error.

Connections to Quantum Phase Transitions

The same metric is shown to characterize the fidelity susceptibility approach to QPTs: the infinitesimal drop in ground state fidelity at a quantum critical point is governed by the Riemannian structure derived above. For systems with degenerate ground spaces, this extends previously known results for nondegenerate phases. Importantly, the divergence of components of $\mathbf{g}$ signals quantum criticality, thus connecting optimal adiabatic computation with critical phenomena and universal behavior near QPTs.

A universal power-law scaling of adiabatic geodesics crossing a second-order quantum critical point is derived. Near a critical point,

$x(s) - x_c \sim (s - s_c)^\chi$

with

$\chi = \frac{2}{d \nu}$

where $d$ is spatial dimension and $\nu$ is the critical exponent of the correlation length. This ties the optimal adiabatic schedule's scaling directly to the universality class of the phase transition.

Alternative Interpretations and Natural Emergence

The geometric tensor admits multiple equivalent interpretations:

• Grassmannian Geometry: The metric arises as the natural distance on the Grassmannian manifold of ground state projectors.
• Uhlmann Parallel Transport: The metric encapsulates the differential geometry of parallel transport in purified state spaces, with the wave operator's geometric contribution ensuring minimal adiabatic error.
• Bures Metric and Quantum Fisher Information: The metric is proportional to the Bures distance of the ground state density operator; hence, it is equivalent (up to a normalization) to quantum Fisher information.

Examples and Analytical Geodesics

The formalism is illustrated analytically for several paradigmatic models:

• Adiabatic Deutsch-Jozsa Algorithm: The adiabatic path is trivial (straight line in parameter space) since the metric is constant.
• Projective Hamiltonians (Grover-type Search): The derived geodesics coincide with optimal schedules from the quantum adiabatic brachistochrone approach.
• Transverse Field Ising Model: The geodesic equations can be integrated analytically in the thermodynamic limit. The resulting schedules embody universal critical scaling near the QPT, showcasing the power of the geometric approach.

Strong analytical results are obtained for critical exponents governing the optimal schedule's passage through QPTs, supporting previously conjectured power-law scaling forms.

Comparison with Alternative Metrics and Strategies

A comparison is made to previously proposed adiabatic metrics, notably one derived from the traditional adiabatic condition leading to a stronger inverse dependence on the energy gap. The present geometric approach yields a metric with quadratic inverse gap dependence, and hence is less sensitive to critical gap closing. The practical implication is that paths derived from the geometric metric are better suited for robust control in regions near criticality.

The analysis further discusses advanced strategies for minimizing adiabatic error, clarifying that while exponential error reduction is achievable with highly regular (analytic, boundary-conditioned) Hamiltonians, the geodesic solution focuses on minimizing the dominant geometric component, with higher-order corrections and further optimization left as a refinement.

Implications and Prospects

The results have considerable implications for quantum information processing:

• The unifying geometric tensor framework provides a rigorous basis for designing optimal adiabatic quantum algorithms that are robust to criticality and applicable in both nondegenerate and degenerate scenarios, including holonomic quantum computation.
• The connection to QPTs paves the way for new characterizations of quantum matter via information geometry, with the fidelity approach being universal and not dependent on Landau order or local order parameters.
• The explicit relation to quantum Fisher information enables cross-application to quantum metrology and parameter estimation.

The theoretical foundation set here anticipates future developments in geometrically engineered quantum control, robust passage through critical points, and phase discrimination in many-body systems.

Conclusion

By deriving and unifying the intrinsic geometry of adiabatic quantum evolution and quantum phase transitions, the paper establishes Riemannian geometry as the fundamental structure organizing optimal adiabatic control, the physics of criticality, and fidelity-based quantum information metrics. Analytical results on geodesic paths and critical scaling exemplify the approach's utility for canonical quantum algorithms and strongly correlated systems. The presented framework offers a systematic route for exploiting geometry in quantum computation and phase identification, with potentially widespread influence across quantum information science (1004.0509).