AI Links Two Key Measures of Quantum Complexity

Quantum complexity—the measure of how intricate a quantum system becomes over time—has emerged as a crucial concept in physics, linking fields from quantum information to black hole theory. Two prominent ways to quantify this complexity have developed independently: Nielsen complexity, which treats quantum circuits as geometric paths, and Krylov complexity, which tracks how operators spread during evolution. Researchers have now discovered a direct mathematical connection between these two measures, revealing that they are not separate ideas but different perspectives on the same underlying structure. This finding could help unify our understanding of quantum chaos and holography, where complexity is thought to relate to the growth of black hole interiors.

The key is that Krylov complexity can be mapped exactly to the square of the length of a specific trajectory in Nielsen's geometric framework. By embedding the Krylov basis—the set of orthonormal operators generated during evolution—into the elementary gate set used in Nielsen's approach, the researchers showed that Krylov complexity equals the cost of a straight-line path connecting the identity operator to a target unitary known as a precursor. Specifically, for a Hermitian operator O(t) evolving under a Hamiltonian, the Krylov complexity CK(O(t)) relates to Nielsen complexity CN(e^{-izO(t)}) via the inequality CN ≤ z√CK, where z is a parameter controlling the size of the precursor. This bound saturates, meaning the two complexities become equal, when the straight-line path is the shortest possible geodesic, a condition that holds for small precursors in general systems and for a finite range in the SYK model.

Ology involved constructing a bridge between the two complexity measures by choosing the Krylov basis as part of the gate set in Nielsen geometry and aligning the complexity metric with the Krylov metric. The researchers focused on precursor operators, defined as UT = e^{-izO(t)}, which link time-evolved operators to unitaries. They analyzed the straight-line trajectory U(s) = e^{-iszO} with constant control Hamiltonian Hc = zO(t), calculating its cost using Nielsen's formula. To test saturation beyond small z, they examined the SYK model—a toy model for quantum gravity—where the algebraic structure simplifies the analysis. By solving the geodesic equation and investigating conjugate points (where geodesics cease to be minimal), they determined conditions under which the straight-line path is the globally minimizing geodesic.

, Detailed in the paper, show that in the SYK model with N=8 Majorana fermions, the straight-line trajectory remains a minimal geodesic for precursors up to z=π/2, as evidenced by numerical evaluation of the determinant in Eq. (25) and Fig. 2. For penalty ratios α = Geven/Godd ranging from 0 to 1000, the first conjugate point—where minimality breaks—is always bounded below by z=π/2, with saturation at α=0. This implies that for z ≤ π/2, the inequality CN = z√CK holds exactly, establishing a precise correspondence. Additionally, the paper notes that in the large-N limit of SYK, choosing a quadratic Krylov metric fmn = n²δmn leads to a growth rate of Nielsen complexity proportional to operator size, aligning with conjectures in holography.

Of this work are significant for both theoretical physics and potential applications in quantum simulation. By linking Krylov and Nielsen complexities, it provides a unified geometric interpretation of operator growth, which is central to studying quantum chaos and thermalization. In holography, where complexity is thought to describe black hole interiors, this correspondence could help reconcile different proposals, such as complexity-action duality. For quantum information science, it offers new insights into designing efficient quantum algorithms by understanding the geometric structure of complexity. The SYK model suggest that in certain chaotic systems, complexity measures may exhibit universal behavior, aiding in the development of quantum simulators for high-energy physics.

However, the study has limitations. The precise correspondence between Krylov and Nielsen complexities is proven to saturate only for small precursors in general systems and for a finite range in the SYK model; it remains unclear if this extends to all quantum systems or larger z values. The analysis relies on specific choices, such as embedding the Krylov basis into the gate set and using homogeneous penalty factors in the SYK model, which may not hold in more complex scenarios. Additionally, the SYK model, while insightful, is a simplified toy model, and applying these to realistic physical systems or higher-dimensional holography requires further investigation. The paper also notes that solving for conjugate points with arbitrary penalty factors is analytically challenging, as explored in the Supplemental Material for the Berger sphere.