Operator spreading in single-particle quantum maps: the open-problem bundle of operator hydrodynamics, randomization, spectral-form-factor timescales, basis dependence, emergent classicality, and extensions to scars

A 2019 study by Moudgalya, Devakul, von Keyserlingk, and Sondhi (Phys. Rev. B 99, 094312; arXiv:1808.04889) initiated the quantitative study of Heisenberg-picture operator spreading in single-particle quantum maps — finite-$N$ unitaries obtained by quantizing chaotic classical maps on the torus — introducing the operator coefficient entropy $S_Q(t)$, the scrambling time $t_{\mathrm{scr}}$ at which it saturates to its Haar value, and heuristic connections to the dip and plateau times of the spectral form factor $K(t)=|\mathrm{Tr}\,U^t|^2$. The paper's own text (Sections IV, VI, and the Conclusion) formulates an open-problem bundle: (1) whether a coarse-grained 'operator hydrodynamics' exists for single-particle quantum chaotic systems; (2) a precise connection between map evolution and information-theoretic randomization (unitary designs, complexity growth); (3) rigorous definitions of dip and plateau times for a single finite unitary and conditions under which $t_{\mathrm{dip}}\sim t_{\mathrm{scr}}$ and $t_{\mathrm{plat}}\sim N\,t_{\mathrm{scr}}$; (4) the role and range of validity of the Egorov condition, including which operator bases map to smooth classical observables; (5) whether finite-$N$ emergent-classical permutational systems are integrable in any physically useful sense; (6) extension of the coefficient-entropy diagnostic to many-body scarred systems. No resolution of strands (1)-(3) is known in the literature; adjacent but distinct results partially inform strands (4)-(6) (symmetry-resolved coefficient entropy; Krylov complexity for driven maps and for scars; operator-space fragmentation in perturbed Floquet-Clifford circuits). The problem bundle remains open.

Background

Quantum chaos is traditionally quantified through spectral statistics and wave-packet dynamics. The source paper (Moudgalya, Devakul, von Keyserlingk, and Sondhi, Phys. Rev. B 99, 094312 (2019)) added a third diagnostic in the simplest chaotic setting: single-particle quantum maps. A quantum map is the quantization of a classical area-preserving map of the two-dimensional torus — paradigmatically the cat map, a linear hyperbolic map with Lyapunov exponent $\lambda$ — iterated in discrete time. Its quantization $U$ acts on an $N$-dimensional Hilbert space with $N=1/(2\pi\hbar)$, so the semiclassical limit is $N\to\infty$ at fixed discrete time. The analysis is in the Heisenberg picture: an operator $Q$ evolves as $Q^{(t)}=U^{-t}Q\,U^{t}$, and its 'size' is measured in the generalized Pauli basis $\{X^m Z^n\}$, where $X$ is the cyclic shift and $Z$ the clock (phase) operator on $\mathbb{C}^N$; these $N^2$ unitary operators span all operators on the Hilbert space. The operator coefficient entropy is $S_Q(t)=-\sum_{m,n}|Q^{(t)}_{m,n}|^2\log|Q^{(t)}_{m,n}|^2$, the Shannon entropy of the squared moduli of the evolved operator's coefficients in this basis. For the unperturbed cat map the evolution exactly permutes the basis elements up to phases, so nothing spreads; for a perturbed cat map with perturbation strength $\kappa$, $S_Q(t)$ grows and saturates near the value for a Haar-random operator at a scrambling time $t_{\mathrm{scr}}$ (Eq. (40) of the source), which diverges as $\kappa\to 0$. Semiclassical reasoning of the Egorov type — the standard statement that quantization and classical time evolution approximately commute for smooth classical observables — is trustworthy only up to the Ehrenfest time $t_E\sim(1/\lambda)\log N$ (Eq. (32)), beyond which wave packets cover the whole torus; in the operator-spreading context this condition additionally requires the chosen operator basis elements to correspond to smooth classical phase-space functions, which is why the $\{X^m Z^n\}$ basis is singled out. In many-body systems the analogous questions acquired an answer through an emergent 'operator hydrodynamics': Khemani, Vishwanath, and Huse showed that under local unitary evolution with conservation laws, operator weight is transported by dissipative hydrodynamic equations, later formalized in the memory-matrix treatment of McCulloch and von Keyserlingk (a Kubo-type linear-response formalism that obtains transport coefficients from matrix inverses of correlation functions of slow, conserved densities); both rely essentially on locality and conserved densities, structures absent for single-particle maps. A second thread concerns randomization of a deterministic evolution. The spectral form factor (SFF) is $K(t)=|\mathrm{Tr}\,U^t|^2$; for the circular unitary ensemble (CUE, the Haar ensemble on $N\times N$ unitaries) its ensemble average has a characteristic dip-ramp-plateau shape. Famously, for random-matrix ensembles $K(t)$ is not self-averaging (Prange, 1997), so for a single deterministic unitary — no ensemble, no averaging — the very definitions of dip time $t_{\mathrm{dip}}$ and plateau time $t_{\mathrm{plat}}$ require care, since standard randomness notions are asymptotic. Using the heuristic assumption that evolution by $U^{t_{\mathrm{scr}}}$ acts on operators like one Haar-random step, the source estimated $t_{\mathrm{dip}}\sim t_{\mathrm{scr}}$ (Eq. (50)) and $t_{\mathrm{plat}}\sim N\,t_{\mathrm{scr}}$ (Eq. (51)). Related semiclassical estimates of out-of-time-ordered-correlator growth and decay times in cat maps, connecting them to Ruelle-Pollicott resonances (the decay rates of classical correlation functions), appear in García-Mata et al. (2018). Randomization is quantified information-theoretically by unitary designs — ensembles or sequences whose moments match those of the Haar distribution up to a given order — and by the growth of state and operator complexity. Third, the source exhibited quantized maps with emergent classicality at finite $N$: permutational evolution with commensurate phases, exact recurrences at times that grow with $N$, and spectra consisting of roots of unity, and asked whether such systems are 'quantum integrable' in any physically useful sense, noting that the usual definitions of quantum integrability presuppose either the thermodynamic limit or a continuous family of finite-dimensional systems. Finally, the authors proposed extending the operator-evolution analysis to interacting many-body systems exhibiting quantum many-body scars — special nonthermal eigenstates embedded in otherwise chaotic spectra that cause weak ergodicity breaking and revivals. Together these strands constitute the open directions the source itself poses for the operator-spreading program for quantum maps: determining the deterministic content, validity range, and robustness of the central diagnostics $S_Q(t)$, $t_{\mathrm{scr}}$, and $K(t)$, and their extension to interacting systems.

Problem Statement

Address the open questions of the operator-spreading program for single-particle quantum maps as posed by Moudgalya, Devakul, von Keyserlingk, and Sondhi: determine the deterministic content, validity range, and robustness of the framework's central diagnostics — the operator coefficient entropy $S_Q(t)$, the scrambling time $t_{\mathrm{scr}}$, and the spectral form factor $K(t)=|\mathrm{Tr}\,U^t|^2$ — and extend them to interacting scarred systems. Concretely, the source poses six linked but independently resolvable questions: each strand may be attacked and resolved on its own, all six are in scope, and the resolution of one neither presupposes nor implies the resolution of the others. A complete solution addresses the bundle as a whole, while a solution that fully resolves any subset of the strands constitutes a partial resolution. (1) Operator hydrodynamics: formulate, or rule out, a version of 'operator hydrodynamics' for single-particle quantum chaotic systems — determine whether coarse-grained, continuum equations govern late-time operator-weight transport in operator space, analogous to hydrodynamic descriptions of operator spreading in many-body systems. (2) Randomization: make precise the connection between operator evolution in these finite maps and information-theoretic notions of randomization such as unitary designs and state-complexity growth; in particular, quantify how evolution by $U^t$ approaches design-like behavior as $t$ increases and how this is tied to $t_{\mathrm{scr}}$. (3) Spectral-form-factor timescales: clarify the precise physical interpretation and rigorous definitions of the dip and plateau times for a single finite-$N$ unitary, given that $K(t)$ is not self-averaging and randomness notions are asymptotic; specifically, determine whether, and under what precise conditions, $t_{\mathrm{dip}}\sim t_{\mathrm{scr}}$ and $t_{\mathrm{plat}}\sim N\,t_{\mathrm{scr}}$ hold beyond the heuristic estimates presented. (4) Egorov condition and bases: understand the role and range of validity of the Egorov condition and possible weaker classical-limit conditions; classify which operator bases map to smooth classical observables, and determine how alternate sensible choices of bases in the quantum and classical limits, as could arise from weaker versions of the Egorov condition, affect the operator-spreading diagnostics. (5) Emergent classicality and integrability: investigate whether systems exhibiting emergent classicality at finite $N$ (permutational evolution with commensurate phases) should be classified as integrable in any physically useful sense. (6) Many-body scars: extend the operator-evolution analysis to many-body systems exhibiting quantum scars, determining whether the operator-coefficient entropy reveals analogous non-thermalizing structures and to what extent the single-particle findings generalize to interacting models.

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