Geometric quantum complexity of bosonic oscillator systems

Quantum complexity, measuring the difficulty of implementing a quantum operation, is crucial in various fields, including black hole physics where it's linked to the growth of volume behind black hole horizons (Complexity=Volume conjecture). This paper focuses on operator complexity, specifically the complexity of time evolution operators, aiming to offer a general recipe for determining complexity without relying on reference or target states. The conventional gate complexity approach suffers from ambiguities due to the choice of gate sets and sensitivity to accuracy levels. Nielsen's geometric approach, relating complexity to the minimal geodesic length in a space of unitary operators, offers a more concrete and potentially unique definition, but its application to systems with infinite-dimensional Hilbert spaces presents challenges. This paper addresses these challenges by working primarily at the Lie algebra level, simplifying calculations and allowing for extensions to interacting systems and non-quadratic Hamiltonians.

The paper reviews existing approaches to quantum complexity, contrasting the gate-based approach (counting elementary gates to build a unitary operation) with the geometric approach (minimal geodesic length on the unitary group manifold). Gate complexity's drawbacks include the dependence on gate sets and accuracy levels. The geometric approach, introduced by Nielsen et al., is presented as a superior alternative, providing a continuous description applicable to continuous variable systems. However, applications to infinite-dimensional Hilbert spaces present difficulties due to the lack of guaranteed geodesic existence. Existing attempts to quantify the complexity of individual states often rely on Gaussian states and are dependent on the choice of reference state. This paper emphasizes the significance of operator complexity (complexity of time evolution operators), which is less explored but essential for understanding quantum processes.

The paper proposes a recipe for calculating complexity bounds (upper bounds) using a geometric approach: 1. Identify a set of generators forming a closed commutator algebra, defining a Lie group. For systems not possessing a closed algebra (e.g., anharmonic oscillators), higher-order generators are penalized to obtain a closed algebra; this results in upper bounds on complexity. 2. Solve the Euler-Arnold equations (using a right-invariant metric, incorporating a penalty factor matrix G_{IJ} reflecting the relative difficulty of different operations) to determine geodesics on the Lie group. 3. Compute the path-ordered exponential to determine the path in the group. 4. Impose boundary conditions (U(0) = I, U(1) = U_target) to find the initial velocity of the geodesic that reaches the target unitary operator. 5. Calculate the length of the shortest geodesic (complexity bound). This approach is applied to several systems. The harmonic oscillator is analyzed using two groups: the harmonic oscillator group (generated by position, momentum, Hamiltonian, and identity) and the symplectic group sp(2,R). The authors highlight subtleties related to non-compact groups (non-existence of finite-dimensional unitary matrix representations, non-surjectivity of the exponential map, geodesic incompleteness). The method is then extended to the inverted harmonic oscillator, coupled harmonic oscillators (using the sp(4,R) algebra), and an anharmonic oscillator with a cubic term. For the anharmonic oscillator, higher-order terms are suppressed through large penalty factors, providing complexity bounds. The paper details the solutions to the Euler-Arnold equations for each system, addressing periodicity and geodesic completeness issues.

The paper presents complexity bounds (upper bounds due to the use of truncated Lie groups and potential geodesic incompleteness) for various oscillator systems. For the harmonic oscillator, with m = ω⁻¹, the complexity bound shows a piecewise linear oscillating behavior with a period of 4π in ωt, doubling the period found in previous studies due to the inclusion of ground state energy. The addition of linear or quadratic terms to the harmonic oscillator Hamiltonian can significantly alter the complexity bound, potentially leading to divergences. The complexity of the inverted harmonic oscillator shows linear growth in time, lacking periodicity. For coupled harmonic oscillators, the complexity bound depends on the coupling constant, frequencies, and penalty factors, exhibiting oscillatory behavior. The study of an anharmonic oscillator with a cubic term necessitates approximations due to the lack of a finite-dimensional closed algebra. By assigning large penalties to higher-order generators, complexity bounds are calculated, exhibiting divergence points. The methodology used throughout incorporates a right-invariant metric and accounts for the non-exponential nature of some Lie groups and the possible geodesic incompleteness. These results are presented as upper bounds on the true complexity, due to the truncation of the infinite-dimensional unitary group.

The findings provide a more rigorous and generalized approach to calculating quantum complexity bounds, especially for non-quadratic Hamiltonians and interacting systems. The use of Lie algebra techniques simplifies calculations while addressing the subtleties of non-compact Lie groups. The paper demonstrates the applicability of the method to various physical systems, highlighting the impact of seemingly minor modifications to the Hamiltonian on the resulting complexity. The divergence points encountered for the anharmonic oscillator highlight the importance of carefully considering geodesic completeness and the limitations of truncating infinite-dimensional groups. The oscillatory behavior for the harmonic oscillator, the linear growth for the inverted harmonic oscillator, and the behavior of coupled oscillators are significant contributions to our understanding of complexity in these systems.

The paper provides a refined geometric approach to calculating upper bounds on quantum complexity, addressing limitations of previous methods. By focusing on Lie algebras and incorporating penalty factors, it offers a more computationally tractable and generalizable framework. The results for various oscillator systems, including anharmonic ones, offer valuable insights into the behavior of quantum complexity under different conditions, though the calculated values remain upper bounds due to the model's limitations. Future research should investigate the full infinite-dimensional case and explore applications to more complex quantum systems.

The paper's methodology involves truncating the infinite-dimensional Hilbert space to a finite-dimensional group for computational tractability, inevitably leading to complexity bounds rather than exact values. The choice of penalty factors (G_{IJ}) introduces some degree of arbitrariness, although the choices made are physically motivated. Geodesic incompleteness in some of the studied non-compact Lie groups further complicates the calculation, leading to potential divergences in complexity bounds. These factors highlight the inherent difficulties in precisely defining and calculating quantum complexity for systems with infinite degrees of freedom.