Quantum Growing Mode Dynamics

• Quantum growing modes are mechanisms where quantum states or observables exhibit exponential amplification, often with no classical counterpart.
• They manifest in diverse fields such as discrete quantum gravity, optomechanics, and many-body systems through coherent interference and instability.
• These modes provide practical insights for scaling quantum complexity, preparing topologically ordered states, and constructing robust quantum codes.

A quantum growing mode is a dynamical mechanism or mathematical scenario in which a quantum system’s state, observable, or physical quantity exhibits systematic amplification or expansion—typically exponential or otherwise accelerated—in a manner determined by quantum rules, often featuring no classical counterpart. Manifestations of quantum growing modes appear across discrete quantum gravity, quantum information, condensed matter, quantum error correction, quantum complexity, and even mathematical physics. Central unifying features include the quantization of classical growth processes, quantum amplification due to unstable dynamics, or expansion of informational content, all governed by mechanisms such as quantum interference, non-commutativity, eigenmode instability, or iterative code constructions.

1. Quantum Growing Modes in Discrete Quantum Gravity

In models of discrete quantum gravity, particularly those built on causal sets, quantum growing modes emerge via the quantization of classical sequential growth processes (CSGPs) (Gudder, 2011, Gudder, 2014). Here, the state space consists of finite posets representing possible spacetime histories. The classical dynamics, characterized by Markov-type transition rules for adding new maximal elements, are elevated to a quantum framework by associating a sequence of Hilbert spaces $H_n = L^2(\Omega_n, \mathcal{A}_n, p_c)$ describing $n$-step growth histories. Quantum probability operators (“q-probability operators”) $p_n$ are defined on these spaces and generate a quantum measure $\mu_n$ via a decoherence functional: $$D_n(A, B) = (p_n \mathbf{1}_A, \mathbf{1}_B), \qquad \mu_n(A) = D_n(A, A).$$ Quantum growing modes are realized as coherent quantum superpositions of growth paths, with quantum interference encoded in non-additive, grade-2 additive $q$-measures as opposed to classical additivity: $$\mu_n(A \cup B \cup C) = \mu_n(A \cup B) + \mu_n(A \cup C) + \mu_n(B \cup C) - \mu_n(A) - \mu_n(B) - \mu_n(C).$$ Consistent assignment of q-probability operators across filtration layers defines a “quantum sequential growth process” (QSGP); their grade-2 additivity naturally leads to quadratic algebras of suitable sets for which the quantum measure is well-defined. Concrete constructions (constant pure sequences, weighted “non-pure” sequences) model quantum growing modes in spacetime, with quantum interference potentially dramatically altering classical growth spectra and histories (Gudder, 2011).

The binary structure of c-causet (“covariant causal set”) growth in Gudder’s universe-as-quantum-computer framework further reveals quantum growing modes as the coherent unfolding of an exponentially bifurcating growth process, in which classical (bit string) universe histories become quantum (qubit) superpositions with transition amplitudes and interference structurally encoded in tensor-product rank-1 operators at each growth step (Gudder, 2014).

2. Dynamical Growth and Amplification in Open Quantum Systems

Quantum growing modes are not limited to quantum gravity models. In quantum optomechanics, for instance, the term describes controlled growth of nonclassical state features—such as the size of a superposition (“Schrödinger cat” states)—by repeated application of measurement- and coupling-induced quantum processes (Clarke et al., 2018). Protocols utilizing sequences of pulsed optomechanical interactions with heralded photon counting grow the separation of coherent components in a mechanical quantum oscillator’s state, with the macroscopicity measure increasing proportionally to the number of pulses $N$ and the effective optical-mechanical coupling $\mu$: the separation grows as $N\mu$. The initial state’s nonclassicality and macroscopicity are gradually “grown” into observable, large-scale signatures.

In dynamical systems with instability, quantum growing modes can have explicit exponential temporal scaling. The accelerated expansion of the position uncertainty $\sigma_z(t)$ of a nanoparticle in an inverted harmonic potential exemplifies this: $$\sigma_z(t) \sim \sigma_z(0) \cdot \exp(\omega_z t),$$ where $\omega_z$ is the frequency corresponding to the inverted potential. This quantum amplification can increase the wavepacket to scales comparable to the physical size of the system, offering a route to macroscopic quantum superpositions and high-precision sensing (Tomassi et al., 26 Mar 2025).

3. Quantum Growing Modes in Statistical and Many-Body Systems

In many-body dynamics, growing modes underpin protocols for preparing correlated and topologically ordered states. The “quantum growing mode” protocol for fractional quantum Hall and Chern insulator systems involves deterministic, stepwise entangling and particle insertion operations. The fusion of Thouless pumping, coherent population transfer, and strong-interaction blockade mechanisms enables the sequential “growth” of a highly nonclassical, correlated many-body state (e.g., a Laughlin state). The fidelity of these grown quantum states can be quantitatively characterized and controlled, with scaling analyses revealing how finite-size effects, nonadiabaticity, and loss mechanisms compete to limit the number of growth cycles and hence the achievable system size (Letscher et al., 2015).

A further manifestation appears in the entanglement generated by trilinear Hamiltonians under driven thermal splitting (Laha et al., 2022): the mean number of split thermal quanta directly increases the degree of non-Gaussian entanglement, with the entanglement potential and logarithmic negativity scaling with the initial thermal occupancy. Here, rather than being degraded by noise, the quantum growing mode mechanism transforms thermal excitations into a resource that boosts quantum correlations.

4. Growth in Quantum Information and Complexity

Quantum growing modes are foundational to the scaling behavior of complexity and code structure in quantum information. In circuit complexity theory, there exists a universal quantum growing mode: the minimal geodesic length (complexity) for implementing time evolution under a time-independent Hamiltonian grows linearly with time at early times, independent of gate choice or metric,

$$C(t) \approx |\omega| t - \frac{1}{24}\frac{|\Omega_2|^2}{|\omega|^2} t^3 + \ldots,$$

where $|\omega|$ captures the Hamiltonian vector’s norm in gate space and $|\Omega_2|$ encodes Lie-algebraic commutator corrections (Haque et al., 18 Jun 2024). This early-time linear regime defines a “quantum growing mode” for complexity, universal across diverse systems from qubits to lattice-regularized field theories, with deviations appearing as negative subleading corrections from gate noncommutativity.

Similarly, in quantum error correction, “growing” refers to algorithmic procedures by which larger, sparser codes are constructed from small seed blocks. Iterative conjoining of repetition code modules—structured as tensor network contractions and not limited to isometric concatenation—produces codes (e.g., quantum LDPC and subsystem codes) in which the code distance $d$ and number of logical qubits $k$ can grow while ensuring $kd^2 = O(n)$, with sparsity preserved. This generalization of concatenation exponentially increases code variety by creative arrangements and layering of atomic code blocks, thus allowing modular design of good quantum codes (Cao et al., 17 Jul 2025).

In the context of quantum machine learning, architectures engineered to exponentially grow the number of accessible Fourier components by scaling data-encoding gates (“exponential” circuits) enable vastly increased expressivity within fixed qubit resources, constituting a quantum growing mode in representational power. This exponential scaling circumvents barren plateaus and facilitates superior function approximation compared to linear architectures (Kordzanganeh et al., 2022).

5. Quantum Growing Modes in Stochastic and Graph Growth Models

Quantum growing modes can also be realized in network and random graph contexts. The model of graph growth by continuous-time quantum walks with stochastic collapse (Jnane et al., 2020) links quantum stochastics to emergent topology: a walker evolving under $U(t) = \exp(-iA_G t)$ (with $A_G$ the adjacency matrix) is measured after a characteristic time $\tau$, and a new node is appended according to the collapse outcome. For small $\tau$, repeated reattachment at a central node yields star-like graphs; for large $\tau$, the walker explores more, enabling the emergence of scale-free degree distributions. The quantum growing mode here describes how quantum probabilistic propagation and collapse fuel topological complexity and power-law scaling in the evolving system.

6. Mathematical Structures Underpinning Quantum Growing Modes

A recurring mathematical structure is the quadratic algebra and quantum measure (q-measure) theory in quantum gravity models: grade-2 additivity relaxes the stringent requirements of classical probability, enabling nontrivial interference and nonclassical collective behavior across possible histories (Gudder, 2011). In complexity, Lie-algebraic geometry and metric considerations define the linear and sublinear growth regimes; in graph models, spectral properties of adjacency matrices and measurement-induced collapse determine the amplification or localization regimes.

The universality of quantum growing modes—across Hilbert space filtrations, tensor network architectures, field theory discretizations, and nonlinear interactions—emphasizes the role of quantum structure and interference in producing new dynamical amplification phenomena, inaccessible or anomalous compared to classical theories.

7. Broader Implications and Physical Significance

Quantum growing modes underlie several pivotal phenomena and practical protocols:

• The quantum amplification of fluctuations or state complexity beyond classical rates (linear vs. exponential, in e.g., nanoparticles under inverted potentials (Tomassi et al., 26 Mar 2025) or quantum complexity (Haque et al., 18 Jun 2024)).
• Preparation and scaling up of large, entangled and/or topologically ordered states by iterative or protocol-driven quantum steps (Letscher et al., 2015, Clarke et al., 2018).
• Modularity and scalability in quantum encoding, exemplified in the generative construction of sparse quantum codes from simple atomic blocks (Cao et al., 17 Jul 2025).
• Structural differentiation from classical random processes, as in quantum random graph growth (Jnane et al., 2020).
• Rigorous mathematical demonstration of instabilities in quantum field theory on curved backgrounds, such as the emergence of exponentially growing mode solutions driven by geometric or algebraic mechanisms (distinct from superradiance) (Zheng, 7 Oct 2024).

The paper of quantum growing modes illuminates both fundamental and applied aspects of quantum dynamics, with direct consequences for the nature of spacetime, limits of quantum information processing, optimal code construction, and the preparation and detection of macroscopic quantum states.

---

References:

• (Gudder, 2011) Models for Discrete Quantum Gravity
• (Gudder, 2014) The Universe as a Quantum Computer
• (Letscher et al., 2015) Growing quantum states with topological order
• (Clarke et al., 2018) Growing macroscopic superposition states via cavity quantum optomechanics
• (Brandão et al., 2019) Models of quantum complexity growth
• (Jnane et al., 2020) Growing Random Graphs with Quantum Rules
• (Laha et al., 2022) Entanglement growth via splitting of a few thermal quanta
• (Kordzanganeh et al., 2022) An exponentially-growing family of universal quantum circuits
• (Haque et al., 18 Jun 2024) Universal Early-Time Growth in Quantum Circuit Complexity
• (Zheng, 7 Oct 2024) Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes
• (Tomassi et al., 26 Mar 2025) Accelerated State Expansion of a Nanoparticle in a Dark Inverted Potential
• (Cao et al., 17 Jul 2025) Growing Sparse Quantum Codes from a Seed