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arxiv: 2606.02817 · v2 · pith:A2INYAGEnew · submitted 2026-06-01 · 🪐 quant-ph · hep-th

Nielsen complexity with multiple cost factors

Marcos Rios Ribeiro , Diego Trancanelli This is my paper

Pith reviewed 2026-06-28 13:49 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Nielsen complexitymultiple cost factorsconjugate pointsEuler-Arnold equationsquantum geodesicsSYK modelright-invariant metric
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The pith

Introducing a hierarchy of penalties for non-localities generalizes Nielsen complexity and changes conjugate point scaling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Nielsen's geometric approach to quantum complexity, which normally uses one penalty to mark easy local operations versus hard non-local ones. It introduces several penalty levels tied to the degree of non-locality, producing a generalized right-invariant metric on the space of unitaries. From this metric the authors derive modified Euler-Arnold and Jacobi equations that govern geodesic paths and the points where those paths cease to be shortest. Explicit calculations in a single-qubit model yield approximate solutions for complexity growth that depend on the penalty values, while SYK-type models show multiple families of conjugate points whose locations shift with both the penalty hierarchy and system size. This setup supplies a finer account of how complexity accumulates when different non-local directions carry distinct costs.

Core claim

Assigning a hierarchy of penalties associated with different degrees of non-locality produces a generalized right-invariant complexity geometry whose geodesics obey modified Euler-Arnold and Jacobi equations, with the structure and scaling of conjugate points depending on the cost factors, as shown in single-qubit and SYK-type models.

What carries the argument

Generalized right-invariant complexity geometry defined by a hierarchy of penalty factors for directions of varying non-locality.

If this is right

  • Approximate analytic solutions for complexity growth exist in the single-qubit case and vary with the penalty hierarchy.
  • SYK-type models produce multiple families of conjugate points arising from distinct non-local sectors.
  • The occurrence of these conjugate points depends on both the cost hierarchy and the system size.
  • Refining the penalty structure supplies a richer description of complexity dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The multi-penalty construction could be applied to model gate costs in physical hardware where locality affects implementation expense.
  • It may link complexity geometry more directly to other quantum resource theories that already distinguish local and non-local operations.
  • Numerical integration of the modified Jacobi equation in small systems would test the predicted scaling of conjugate points with penalty values.

Load-bearing premise

A hierarchy of penalties can be assigned to directions of different non-locality while preserving right-invariance of the complexity metric.

What would settle it

A calculation for the single-qubit system in which the approximate analytic solutions for complexity growth fail to satisfy the modified Euler-Arnold equation.

Figures

Figures reproduced from arXiv: 2606.02817 by Diego Trancanelli, Marcos Rios Ribeiro.

Figure 1
Figure 1. Figure 1: Schematic representation of a unitary manifold and a quantum circuit as a sequence of elementary gates. A geodesic path (black) is depicted from the identity to some target unitary U. The red straight segments represent the construction of a circuit using some elementary gates gi . In this example, the final unitary is U = g3g2g1. The geodesic is parametrized by a parameter s and it approximates the circui… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Time evolution of the minimum eigenvalue of the Jacobi operator for three anisotropy angles at fixed µ = 3 and ω = 1. Conjugate times are identified by zero crossings. As θ increases, the oscillatory frequency increases and zero crossings appear earlier. Right: First conjugate time tmin as a function of θ with ω = 1. Larger values of µ delay conjugate times in the symmetric regime, while near θ → π/2… view at source ↗
Figure 3
Figure 3. Figure 3: Complexity evolution as a function of the cost factors, using the polar parametrization introduced in (2.66). We set J = 1 and define ∆θ ≡ π/2 − θ. In the right panel, ∆θ = 0.1 is kept fixed, whereas in the left panel we fix µ = 1000. This is because, in many controlled-qubit platforms, the free evolution of the qubit is naturally generated by a Hamiltonian along σ3. As a consequence, additional control al… view at source ↗
Figure 4
Figure 4. Figure 4: Average complexity for the same random sample with σ = 1. In the parametrization of (2.66) we set µ = 200 and consider two different values of ∆θ ≡ π/2 − θ. After the initial transient, the complexity oscillates around the value Cmax ≈ π √µ2 √µ2 − 1 . Remarkably, (3.14) admits a simple analytic solution. Since the physical Hamiltonian points entirely along the σ1 direction, we set c2 = c3 = 0 in the soluti… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the minimum eigenvalue of the Jacobi operator Yµ for different values of µ1 at fixed µ2 = 15. The left panel corresponds to N = 8, while the right panel corresponds to N = 6. The insets mark the first visible non-local conjugate times associated with the NL1 directions. with the coefficients c (α) mn = ⟨m|Tα|n⟩ and ∆mn = Em − En, with Hfree|n⟩ = En|n⟩. Due to the partially diagonal struct… view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of the minimum eigenvalue of the Jacobi operator Yµ for N = 8 and different values of µ2 at fixed µ1 = 5. The inset shows the first visible non-local conjugate time associated with the NL2 directions, located at J tNL2 ≈ 8.3. are visible in figure 5 (left), while the third is outside the plotted time window. We observe the same pattern for the harder directions shown in figure 6: since max{λ… view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of the smallest eigenvalue of the Jacobi operator for N = 8 and µ2 = 15, for selected values of the parameter µ1. The left panel shows the Hamiltonian with three fermions, while the right panel shows the one with four fermions. Note that the only contribution to the local part coming from (4.27) is the constant term, since the other terms are proportional to C α β˙ 2 C β˙ 2 β˙ 1 , which can … view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the smallest eigenvalue of the Jacobi operator for N = 8 and µ1 = 5, for selected values of the parameter µ2. The left panel shows the Hamiltonian with three fermions, while the right panel shows the one with four fermions. To determine the location of conjugate times, we compute the time evolution of the smallest Jacobi operator eigenvalue for different values of the cost factors, as sho… view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the first conjugate times, both local and non-local, for the four-body (top) and three-body (bottom) Hamiltonians. The left panels show the variation with µ2 at fixed µ1 = 5, while the right panels show the variation with µ1 at fixed µ2 = 15. Each curve corresponds to a single conjugate time tn and shows how its occurrence time changes as the corresponding cost factor is varied. The non-local c… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends Nielsen's geometric formulation of quantum complexity by replacing the single penalty factor with a hierarchy of cost factors that distinguish operators according to their degree of non-locality. It constructs the associated right-invariant Riemannian metric on the unitary group, derives the modified Euler-Arnold and Jacobi equations that govern geodesics and their variations, and analyzes how the hierarchy alters the location and scaling of conjugate points. Concrete illustrations are given for a single-qubit system with two cost factors and for both free and chaotic regimes of SYK-type models, where multiple families of conjugate points appear whose occurrence depends on the penalty ratios and system size.

Significance. If the derivations are correct, the work supplies a technically straightforward but conceptually richer model of complexity geometry that can accommodate realistic distinctions between local and non-local gates. The explicit treatment of conjugate-point families in the SYK setting offers a concrete handle on how penalty structure influences the breakdown of geodesic optimality, which may prove useful for connecting geometric complexity to dynamical features of chaotic many-body systems. The approach remains fully within the standard right-invariant framework, so the technical overhead is modest.

minor comments (3)
  1. [§3] The abstract states that the modified Euler-Arnold and Jacobi equations are derived, but the main text should include an explicit step-by-step reduction from the left-trivialized geodesic equation to the new form (perhaps in §3) so that readers can verify the precise manner in which the sector-dependent inner product enters the structure constants.
  2. [single-qubit section] In the single-qubit example, the approximate analytic solutions for complexity growth are presented; it would be helpful to state the regime of validity of the approximation (e.g., small penalty ratios or short times) and to compare the analytic curves against a numerical integration of the geodesic equation.
  3. [SYK section] The SYK analysis reports that distinct non-local sectors generate multiple families of conjugate points whose occurrence depends on both the cost hierarchy and system size. A brief table or plot summarizing the leading conjugate-point times as functions of the penalty ratios for N=8,12,16 would make the scaling claim easier to assess.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the multi-cost-factor extension supplies a technically straightforward yet conceptually richer model, and for the recommendation of minor revision. The report does not enumerate any specific major comments requiring point-by-point replies.

Circularity Check

0 steps flagged

No significant circularity; derivation is standard Riemannian geometry on Lie groups

full rationale

The paper defines a right-invariant metric by partitioning the Lie algebra basis into sectors of differing non-locality and rescaling the inner product on each sector. Any positive-definite inner product yields a right-invariant Riemannian metric on the group, after which the Euler-Arnold and Jacobi equations follow from the standard left-trivialized geodesic equation with only algebraic modifications to the structure constants. No fitted parameters are relabeled as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The claimed results on conjugate-point scaling are direct consequences of the chosen metric and system size, not reductions to the inputs by construction. The derivation is therefore self-contained against external benchmarks of Lie-group geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central construction rests on the standard right-invariance assumption of Nielsen geometry.

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discussion (0)

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