Krylov Complexity Saturated Beyond Heisenberg Length in Quantum Systems

Andrew Lucas and Amit Vikram of the University of Colorado, Boulder have demonstrated that Krylov complexity, a measure of how difficult a quantum state is to create, demonstrably stops growing a little past the Heisenberg length in any low-energy state, including the thermofield double state. The researchers found that orthogonal polynomials, essential to calculating this complexity on a discrete energy spectrum, crowd away from regions where the density of states is low at large Krylov index, revealing a specific mechanism for this saturation. This limit to complexity growth, not merely a theoretical possibility, has implications for ongoing work linking quantum information to quantum gravity.

Krylov Complexity and the Volume Conjecture

Recent research reveals a definitive limit to how complex quantum systems can become, challenging existing assumptions about the relationship between computational difficulty and the geometry of spacetime. This finding is a mathematically demonstrable constraint on complexity growth, with implications for ongoing investigations into quantum gravity. The team’s work centers on understanding how Krylov states, constructed iteratively from an initial quantum state, evolve in complexity. Orthogonal polynomials built on a discrete energy spectrum crowd away from regions where the density of states is low at large Krylov index. This crowding is not random; it’s a consequence of the mathematical properties of orthogonal polynomials, suggesting a specific mechanism by which complexity saturates. The researchers leveraged established bounds from the theory of orthogonal polynomials to prove this saturation, even when considering the entire, potentially infinite, Hilbert space of the system.

Previous calculations often focused on low-energy sectors, raising concerns about whether complexity would continue to grow indefinitely at higher energies. The implications extend to the volume conjecture, a proposal linking quantum computational complexity to the geometry of wormholes in 2D quantum gravity. The study finds that the saturation of Krylov complexity occurs at timescales a little past the Heisenberg length, the point at which the system reaches maximal complexity given its energy. This alignment with the expected behavior of wormhole volume in quantum gravity suggests how Krylov state complexity may be consistent with “complexity = volume” even in a non-perturbative theory of quantum gravity. The team’s results hold for any low-energy state, including the commonly studied thermofield double state, solidifying their broad applicability.

Current investigations into quantum complexity increasingly focus on Krylov state complexity as a potential bridge between quantum information theory and quantum gravity, building on the proposal initially made by Susskind in 2016. Recent work has demonstrated quantitative relationships between semiclassical Hamiltonian calculations of 2D quantum gravity and the dynamics of Krylov complexity within the double-scaled Sachdev-Ye-Kitaev (DSSYK) model, a microscopic model for 2D gravity. A key concern has been whether Krylov complexity, when calculated across the entire energy spectrum, would saturate only at infinitely high temperatures, rather than at the energy-dependent Heisenberg time scale expected for meaningful comparison with gravitational models.

Researchers at the University of Colorado, Boulder are refining calculations of quantum complexity, revealing a definitive limit to its growth. The team’s approach centers on the mathematical properties of orthogonal polynomials. These polynomials, essential for calculating Krylov complexity on a discrete energy spectrum, crowd away from regions where the density of states is low at large Krylov index. This crowding is not a numerical artifact; it’s a consequence of the underlying mathematical structure governing the complexity calculation. This saturation occurs at a scale determined by the bandwidth of the energy spectrum in DSSYK, a model system used to study 2D quantum gravity. The team’s work suggests that Krylov state complexity may indeed be consistent with the volume conjecture, even in a non-perturbative theory of quantum gravity, opening avenues for further exploration of relationships with other complexity measures like quantum ergodicity.

The physical implication of this result is that, even using a minimalist Krylov state complexity defined over the entire spectrum, the Krylov complexity provably stops growing a little past the Heisenberg length in every low-energy state. This has implications for the proposal that Krylov complexity is related to the wormhole length in 2D quantum gravity.

Recent work demonstrates a definitive limit to this complexity, a finding rooted in the mathematical properties of orthogonal polynomials and their behavior at large Krylov index. The key to this understanding lies in the behavior of orthogonal polynomials used to construct the Krylov states. Previous work had raised concerns about whether a Krylov complexity calculation encompassing the entire energy spectrum would accurately reflect the expected saturation at the Heisenberg time. This boundedness is crucial because it ensures that complexity does not continue to increase indefinitely.

A definitive limit to the growth of Krylov complexity, extending beyond theoretical possibility, fundamentally alters calculations within quantum gravity, according to new research. The team’s analysis of Lanczos coefficients implies a bound on complexity.