A 2025 JHEP study benchmarked quantum chaos in the Pullen–Edmonds coupled quartic oscillator by computing an upper bound on the Nielsen geometric (circuit) complexity of the time-evolution unitary, using a generator set truncated to quartic order, a leading-order Dyson evaluation that drops path ordering, and phenomenological penalty factors identified with the coupling as q∼1/λq\sim1/\lambda, p∼λp\sim\lambda. The bound grows linearly in time precisely when the coupling is nonzero, in agreement with the Gaussian-Orthogonal-Ensemble level statistics that certify the model's chaoticity, suggesting geometric complexity as a diagnostic of quantum chaos in interacting non-Gaussian systems. The open question addressed here is whether this complexity–chaos correspondence is robust: whether it survives relaxation of the generator and Dyson truncations, whether the penalty factors can be derived from first principles or the signature shown insensitive to the penalty choice, and whether the correspondence extends to interacting quantum field theories, non-Gaussian cMERA (continuous multi-scale entanglement renormalization ansatz) circuits, and other chaotic models.

Background

Quantum circuit complexity measures the minimal number of elementary operations (gates) needed to build a target state or unitary from a reference. Nielsen and collaborators recast this optimization as a geometry problem: the available gates are generated by a Lie algebra of operators, and the complexity is the length of the shortest geodesic on the resulting group manifold with respect to a right-invariant metric whose entries GIJG_{IJ} act as penalty factors — numerical weights that make motion along 'hard' (high-order) generator directions more costly than along 'easy' (low-order) ones. The optimal paths satisfy the Euler–Arnold equations, which are the geodesic equations on the group. Random matrix theory (RMT) supplies the standard spectral diagnostic of quantum chaos: the unfolded nearest-neighbor energy-level spacings of a time-reversal-invariant chaotic Hamiltonian follow the Wigner–Dyson statistics of the Gaussian Orthogonal Ensemble (GOE), whereas integrable systems show Poissonian statistics. The Pullen–Edmonds model, H=12(pA2+pB2)+12(xA2+xB2)+λxA2xB2H=\frac{1}{2}(p_A^2+p_B^2)+\frac{1}{2}(x_A^2+x_B^2)+\lambda x_A^2x_B^2 for two coupled oscillators, is a textbook system that crosses over from quasi-periodic to chaotic classical dynamics with increasing energy and coupling, and correspondingly exhibits GOE level statistics for λ>0\lambda>0. Prior work had connected complexity growth to chaos at large NN (Balasubramanian, DeCross, Kar and Parrikar showed for chaotic Hamiltonians such as the Sachdev–Ye–Kitaev (SYK) model — a solvable, strongly interacting quantum-mechanical system that is maximally chaotic — that a geodesic of linearly growing length always exists and proposed the Eigenstate Complexity Hypothesis — a conjectured bound on the complexity of energy eigenstates that controls when linear complexity growth saturates at a value exponential in system size) and in cosmological, Gaussian (squeezed-state) systems (Bhattacharyya, Das, Haque and Underwood), while Chowdhury, Bojowald and Mielczarek developed the geometric-complexity framework for bosonic oscillator systems. The source paper (Bhattacharyya, Brahma, Chowdhury and Luo, JHEP 03 (2025) 177) applied this framework to the time-evolution unitary U(t)=e−iHtU(t)=e^{-iHt} of the Pullen–Edmonds model — the first such computation for a genuinely interacting, non-Gaussian quantum-mechanical system. Because the operator algebra generated by xx and pp does not close (commutators of quartic operators produce ever higher orders), the authors truncated it to quadratic and quartic generators; they assigned a common penalty pp to quadratic directions and a much larger common penalty qq to quartic directions, fixing them phenomenologically through q∼1/λq\sim1/\lambda, p∼λp\sim\lambda; and they evaluated the path-ordered exponential along the candidate geodesic only to leading order in the Dyson series (the standard nested-integral expansion of a path-ordered exponential, whose full resummation is intractable for centerless algebras). The result is an upper bound C≈p(v12+v22)+qλ2t2C\approx\sqrt{p(v_1^2+v_2^2)+q\lambda^2t^2} that grows linearly in time exactly when λ>0\lambda>0, while for λ=0\lambda=0 it stays bounded by periodicity — matching the GOE-versus-integrable spectral distinction and suggesting geometric complexity as a computable chaos diagnostic. The authors stress that this signature is established only within these approximations and only for this two-mode toy model: the truncated algebra may admit shorter geodesics in the discarded directions, the neglected Dyson terms could alter both the coefficient and the functional time-dependence, and the penalty identification lacks first-principles justification. An alternative perturbative route — expanding the unitary as a power series in time, which makes early-time growth universal and generator-set-independent (Haque, Jafari and Underwood) — accesses a different (early-time) regime and does not answer the question either.

Problem Statement

Determine whether the correspondence between interaction-induced linear-in-time growth of the geometric (Nielsen) complexity of the time-evolution unitary and RMT-certified quantum chaos in the Pullen–Edmonds oscillator is a robust property of the interacting system or an artifact of the approximations, and whether it generalizes beyond the two-mode toy model. Concretely, an answer should establish, for the Pullen–Edmonds model or a comparable interacting non-Gaussian quantum system: (i) controlled treatment of the generator truncation — quantify or bound how operators beyond quartic order modify the complexity, or otherwise show whether the linear-growth chaos signature survives or fails when the truncation of the non-closing operator algebra is relaxed; (ii) controlled treatment of the path-ordering/Dyson truncation — compute or bound the contributions of the omitted higher-order Dyson terms to the time-dependence and coefficient of the complexity; (iii) a principled treatment of the penalty factors — either a scheme deriving the quadratic and quartic penalties pp and qq from the microscopic Hamiltonian, or a robustness analysis showing that the discrimination between chaotic and integrable regimes is insensitive to the penalty choice within a justified class (the source's identification q∼1/λq\sim1/\lambda, p∼λp\sim\lambda being purely heuristic); and (iv) a demonstration of whether the complexity–chaos correspondence persists in at least one larger interacting system — for example an interacting scalar quantum field theory, a non-Gaussian cMERA (continuous multi-scale entanglement renormalization ansatz) circuit, or another bounded chaotic many-body model — verified against an independent chaos standard such as GOE level-spacing statistics. These components jointly serve one scientific objective: deciding whether geometric circuit complexity is a reliable quantifier of quantum chaos in interacting non-Gaussian systems. Each closes a distinct failure channel for the same correspondence claim: a negative answer to any of (i)–(iii) would show the observed linear growth of the complexity upper bound to be an artifact of that approximation, while (iv) tests whether the claim is a property of chaos rather than of one two-mode model; conversely, robustness of the signature under all of (i)–(iii) together with a confirming extension in (iv) would establish the diagnostic.

References

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