Nielsen complexity with multiple cost factors

Abstract: We investigate Nielsen's geometric approach to quantum complexity in the presence of multiple cost factors, extending the standard framework where a single penalty distinguishes easy from hard directions of the group manifold. By introducing a hierarchy of penalties associated with different degrees of non-locality, we develop a generalized right-invariant complexity geometry and analyze its implications for geodesic evolution. We derive the modified Euler-Arnold and Jacobi equations and study how multiple cost factors reshape the structure and scaling of conjugate points, where geodesic optimality breaks down. The formalism is illustrated in two settings: a single-qubit system with two cost factors, where we derive approximate analytic solutions for the complexity growth and its dependence on penalty hierarchies, and SYK-type models, where we analyze both free and chaotic regimes. In these many-body systems, we show that distinct non-local sectors generate multiple families of conjugate points whose occurrence depends on both the cost hierarchy and the system size. Our results highlight how refining the penalty structure provides a richer and more realistic description of quantum complexity and its dynamical behavior.

• The paper generalizes Nielsen complexity from one penalty factor to hierarchies of costs, deriving coupled Euler–Arnold and Jacobi equations that dynamically connect non-local sectors.
• Multiple penalties shift conjugate times by factors of 1+μₚ in decoupled sectors, while sector mixing creates richer frequency structures in which greater overall costs delay obstructions but stronger anisotropy can make them appear earlier.
• Applications to single-qubit and SYK models show that complexity dynamics can freeze costly directions and produce distinct local and non-local conjugate-point families, making cost assignments important for estimating when geodesics remain minimizing.

Nielsen's geometric approach to quantum complexity defines the complexity of a unitary operation as the length of a geodesic on the group manifold $SU(2^N)$, measured with a right-invariant Finsler metric that penalizes non-local directions of the Lie algebra. In "Nielsen complexity with multiple cost factors" (2606.02817), Rios Ribeiro and Trancanelli generalize this framework from a single penalty factor $\mu$ to a hierarchy of penalties $\mu_1 < \mu_2 < \cdots$, each associated with a distinct degree of non-locality, and analyze how this refinement reshapes geodesic dynamics and the structure of conjugate points — the loci where geodesics cease to be locally minimizing and hence cease to give the true complexity.

Generalized complexity geometry

The starting point is the standard Nielsen construction: paths $V(s)$ on $SU(2^N)$ obey the Dyson equation $\dot V = -i h(s) V$ with control functions $\gamma^I(s)$ expanded in generalized Pauli matrices, and complexity is the minimal Finsler length over such paths. The authors adopt the diagonal metric tensor $G_{IJ} = \Omega_I \delta_{IJ}$ with

$\Omega_I = 1 \text{ (easy)}, \qquad \Omega_I = 1 + \mu_p \text{ ($p$-hard)},$

where the hard subspace is decomposed into $D$ sectors $\mu$0 of increasing non-locality. The Euler–Arnold equations are derived for this multi-penalty metric: local velocity components couple to all hard sectors through terms weighted by $\mu$1, while each $\mu$2-hard component evolves under couplings weighted by ratios $\mu$3. A notable structural feature is that different hard sectors are dynamically coupled through penalty-dependent coefficients, so the hierarchy of costs has genuine dynamical content rather than being a mere relabeling of directions.

For perturbations around constant (exponential) geodesics, the Jacobi equations become a coupled linear system across the non-local subspaces. For $\mu$4, the authors solve the system via Laplace transform: the influence of the most non-local sector $\mu$5 on $\mu$6 appears as an effective self-energy term $\mu$7 in an algebraic equation for the transformed Jacobi field. This self-energy structure is the formal expression of the inter-sector coupling introduced by multiple penalties.

When the commutator operator $\mu$8 is simultaneously diagonalizable within each hard sector (no mixing), the equations decouple and conjugate times associated with a $\mu$9-hard direction are simply shifted by the corresponding penalty:

$\mu_1 < \mu_2 < \cdots$0

Larger penalties therefore delay conjugate times, generalizing the known single-cost scaling $\mu_1 < \mu_2 < \cdots$1. When hard sectors mix, the analysis yields two frequencies $\mu_1 < \mu_2 < \cdots$2 and a transcendental condition for conjugate times; parametrizing the penalties as $\mu_1 < \mu_2 < \cdots$3, $\mu_1 < \mu_2 < \cdots$4, the authors find numerically that increasing the anisotropy angle $\mu_1 < \mu_2 < \cdots$5 produces earlier conjugate times, while increasing the overall scale $\mu_1 < \mu_2 < \cdots$6 delays them. In the limit $\mu_1 < \mu_2 < \cdots$7, where $\mu_1 < \mu_2 < \cdots$8 effectively merges with the local sector, the first conjugate time becomes approximately independent of $\mu_1 < \mu_2 < \cdots$9 (approaching $V(s)$0 for $V(s)$1). At $V(s)$2 the result reduces to the standard single-penalty formula $V(s)$3, providing a consistency check.

Single qubit with two cost factors

The formalism is first applied to $V(s)$4 with one easy direction ($V(s)$5) and two hard directions ($V(s)$6, $V(s)$7) carrying distinct penalties $V(s)$8. Unlike the equal-penalty case, where the Euler–Arnold system is linear in the hard components, unequal penalties render it genuinely nonlinear, with solutions expressed through Jacobi elliptic functions. Under the strong hierarchy $V(s)$9, the authors obtain approximate analytic solutions in which motion along the hardest direction freezes ($SU(2^N)$0) and the remaining velocities oscillate at the large frequency $SU(2^N)$1. Large penalties thus dynamically constrain the geodesic flow toward less costly subspaces.

Solving the Dyson equation via a high-frequency approximation, and taking the physical Hamiltonian along the easiest direction $SU(2^N)$2 (motivated by experimental accessibility of transverse controls in qubit platforms), the boundary-value problem admits the simple analytic solution

$SU(2^N)$3

The dependence on $SU(2^N)$4 enters only through the prefactor, explaining why the intermediate penalty has a mild effect on both slope and maximum of the complexity. Periodic extension as a Fourier series reproduces the expected sawtooth behavior inherited from the periodicity of $SU(2^N)$5. The authors are explicit that this simple growth-and-decrease pattern is a consequence of the small dimensionality of $SU(2^N)$6 and should not persist on $SU(2^N)$7, where many near-degenerate geodesics compete. To emulate plateau formation, they average the complexity over Gaussian-distributed couplings $SU(2^N)$8; the oscillatory contributions cancel at late times and the averaged complexity saturates around $SU(2^N)$9.

Conjugate points in SYK-type models

The second application concerns SYK models with $\dot V = -i h(s) V$0 Majorana fermions, using two cost factors assigned by body count of the Pauli-string-like generators. For free SYK ($\dot V = -i h(s) V$1), the matrix $\dot V = -i h(s) V$2 is block-diagonal in a suitable basis, the Jacobi operator decomposes into local and non-local blocks, and its full spectrum can be computed analytically. Non-local conjugate times obey $\dot V = -i h(s) V$3, so the least non-local sector produces the earliest non-local obstructions when eigenvalues are comparable in magnitude. Numerically, for $\dot V = -i h(s) V$4 with $\dot V = -i h(s) V$5, the first $\dot V = -i h(s) V$6 conjugate times occur at $\dot V = -i h(s) V$7 for increasing $\dot V = -i h(s) V$8, and the first $\dot V = -i h(s) V$9 point appears at $\gamma^I(s)$0 for $\gamma^I(s)$1 — in quantitative agreement with the analytic shift. Crucially, the earliest zeros in the spectrum are independent of both penalties, identifying them as local-sector conjugate points.

For chaotic SYK with $\gamma^I(s)$2 (group manifold $\gamma^I(s)$3, 255 directions), the locality cutoff $\gamma^I(s)$4 gives 162 easy directions, 56 five-fermion directions with cost $\gamma^I(s)$5, and 37 six-or-more-fermion directions with cost $\gamma^I(s)$6. Exact solutions of the coupled Jacobi equations are obtained in specially constructed bases: for SYK3, the hardest sector admits simultaneous diagonalization except for a single mode governed by a frequency $\gamma^I(s)$7; for SYK4, the non-local dynamics is controlled by double-commutator eigenvalues defining $\gamma^I(s)$8. The spectral analysis reveals two qualitatively distinct families of conjugate times in both Hamiltonians: a penalty-independent family attributed to the local sector, and a penalty-dependent family shifting to later times as costs increase. The three-body case exhibits a denser and more asymmetric pattern than the four-body case, because in SYK3 the hardest sector carries internal evolution controlled by $\gamma^I(s)$9, so varying $G_{IJ} = \Omega_I \delta_{IJ}$0 changes both inter-sector coupling and intra-sector dynamics, whereas SYK4 responds symmetrically to variations of $G_{IJ} = \Omega_I \delta_{IJ}$1 and $G_{IJ} = \Omega_I \delta_{IJ}$2.

These results carry a direct implication for complexity growth diagnostics: refining the penalty structure does not merely rescale existing obstructions but generates additional families of conjugate points whose ordering depends on the cost hierarchy, so conclusions about the validity window of geodesic complexity estimates are sensitive to how finely non-locality is resolved.

Limitations and open questions

Several restrictions bound the scope of the results. The analytic treatment of conjugate points relies on exponential geodesics generated by constant velocities; general geodesics remain analytically intractable. The single-qubit solution uses a high-frequency approximation valid only under the strong hierarchy $G_{IJ} = \Omega_I \delta_{IJ}$3, and the identification of the physical Hamiltonian with the easiest direction is an assumption tailored to specific experimental platforms. In the SYK analysis, system sizes are limited to $G_{IJ} = \Omega_I \delta_{IJ}$4 by the diagonalization of $G_{IJ} = \Omega_I \delta_{IJ}$5, and the claim that local conjugate points persist to exponential times in chaotic systems is not directly tested here; whether the observed separation between local and non-local families survives at larger $G_{IJ} = \Omega_I \delta_{IJ}$6, and how the dense early-time structure of SYK3 scales, are questions the paper leaves open. The Laplace-space solution for $G_{IJ} = \Omega_I \delta_{IJ}$7 also requires inversion of a matrix whose difficulty depends on the form of the self-energy, and no closed-form treatment is given for $G_{IJ} = \Omega_I \delta_{IJ}$8.

Conclusion

This work extends Nielsen complexity geometry to metrics with a hierarchy of penalty factors, deriving the modified Euler–Arnold and Jacobi equations and showing that multiple cost factors couple distinct non-local sectors dynamically and generate multiple families of conjugate points. The decoupled case recovers the single-penalty result with shifts $G_{IJ} = \Omega_I \delta_{IJ}$9, while mixing between sectors produces richer zero-mode structures controlled by both the magnitude and the relative ordering of the penalties. Applications to a single qubit and to free and chaotic SYK models confirm these structures analytically and numerically, establishing that the internal organization of hard directions materially affects when geodesics stop minimizing complexity.