Particle-Number Conserving Dissipative Protocol

• The protocol is a framework that modifies system evolution equations to allow energy loss while strictly maintaining particle number, ensuring U(1) symmetry.
• It facilitates the preparation of topological phases, efficient numerical simulations, and benchmarked quantum device performance by restricting dynamics to fixed particle-number sectors.
• Applications span quantum state engineering, error correction, and non-equilibrium dynamics, although instabilities like reaction-diffusion effects pose practical challenges.

A particle-number conserving dissipative protocol is a framework for open quantum or classical systems whose dynamics are engineered so that, although energy and coherence may be lost due to coupling with an environment (bath), the total number of particles in the system remains strictly constant at all times. This restriction is central in quantum state engineering, non-equilibrium statistical mechanics, topological quantum computation, and numerical simulation, where conservation of a U(1) charge or analogous quantity is fundamental. The implementation of such protocols ranges from microscopic dissipative evolution in modified Schrödinger equations, GKSL master equations, and random quantum circuits, to numerically stable simulation schemes and dynamic protocols for strongly correlated phases. Their utility spans the preparation and manipulation of topological states, stabilization against decoherence, efficient benchmarking, and structure-preserving integration of kinetic models.

1. Mathematical Formalisms and Modifications for Conservation

The core mathematical strategy is the modification of the system's evolution equations to include dissipation while preserving particle number. In wavefunction dynamics, this typically involves augmenting the Schrödinger equation by a non-Hermitian dissipative operator constructed from occupation probabilities such that the norm of the state (interpreted as total probability or number) is invariant:

$$i\hbar \frac{d}{dt}|\psi(t)\rangle = [H_0 + iD] |\psi(t)\rangle,$$

where $$D = (\hbar/2) \sum_i (\partial_t \ln P_i) |\psi_i\rangle\langle\psi_i|$$ and $\sum_i P_i(t) = \text{const}$ enforces conservation (Veenendaal et al., 2010).

In open-system (Lindblad/GKSL) formalisms, jump operators $L_\alpha$ must satisfy

$[N, L_\alpha] = 0$

for the total particle number operator $N$, ensuring commutation with the generator of dynamics and thus exact conservation at the level of the master equation (Nosov et al., 2023, Lyublinskaya et al., 2023, Lyublinskaya et al., 9 Aug 2024).

Particle-number conserving random quantum circuits are constructed by restricting operations to the Hilbert space sector of fixed particle number, reducing the system's accessible states from $2^N$ to $\binom{N}{n}$ for $n$ particles, enabling both tractable simulation and accurate fidelity estimation in large devices (Kaneda et al., 16 May 2025).

2. Physical Examples and Model Implementations

Protocols enforcing particle-number conservation are realized in a variety of settings:

• Electronic and Vibrational Decay: For direct electronic decay (Fano problem), the protocol tracks population transfer between occupied and unoccupied states, conserving total probability for two-level systems. For local phonon damping, number-conserving Markovian equations model relaxation without loss of vibrational quanta (Veenendaal et al., 2010).
• Topological Superconductors and Superfluids: Lindblad dynamics with number-conserving jump operators enable dissipative preparation of p-wave superconductor-like steady states in one-dimensional wires and ladders. Bulk p-wave correlations persist, but Majorana zero modes and topological order require special geometries (e.g., two-leg ladder with parity sectors) (Iemini et al., 2015). In particle-number conserving Kitaev-type chains, parity-switch criteria and exact solutions via Bethe ansatz delineate topologically trivial and non-trivial superfluid phases (Ortiz et al., 2016). Extension to realistic Majorana physics shows that strict number conservation modifies Berry phases, potentially alters braiding statistics, and must contend with superselection constraints (Lin et al., 2018).
• Reaction-Diffusion Quantum Models: In dissipative two-band models, jump operators are chosen to transfer particles between bands while conserving total number. At intermediate scales, diffusive density propagation appears; at long scales, nonlinear effects yield Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) reaction-diffusion equations, driving the system away from engineered dark states toward finite steady densities in both bands (Nosov et al., 2023, Lyublinskaya et al., 9 Aug 2024).
• Numerical Particle Dynamics: Structure-preserving particle-in-cell schemes for continuity equations utilize compatibility of discrete energy derivatives and particle weights, allowing strict mass conservation and energy dissipation in fully discrete integration of kinetic equations such as the Landau model (Hu et al., 29 Jun 2024). Stable dissipative particle dynamics with energy and local number conservation are achieved via Metropolis-Hastings corrections and multiple timestep strategies, avoiding negative energies and other unphysical states (Stoltz, 2016).
• Benchmarking Quantum Devices: Quantum benchmarking protocols (e.g., MLXEB) utilize particle-number conservation to shrink the Hilbert space for random circuit simulation, maintaining Porter-Thomas statistics within the particle-number sector and permitting classical benchmarking on >100 qubits (Kaneda et al., 16 May 2025).

3. Impact on Topological and Many-Body Quantum Physics

Particle-number conserving dissipative protocols enable the formation of exotic quantum phases with symmetry-imposed constraints:

• Preparation of Topological Phases: Dissipative preparation of topologically ordered steady states (e.g., hosting Majorana modes) is possible, but subject to severe trade-offs. In single-wire geometry, unique steady states exhibit long-range p-wave correlations but no topological degeneracy; ladder geometries allow twofold degeneracy and edge-localized robust Majorana zero modes (Iemini et al., 2015).
• Modified Dynamics and Stability: Exact number conservation leads to algebraic closing of dissipative gaps (relaxation rates scale as $\sim L^{-2}$), implying slow approach to steady state and enhanced sensitivity to perturbations. Reaction-diffusion instabilities, especially in the presence of impact ionization, generically destabilize engineered dark states, leading to finite population of bands meant to be “empty” (Lyublinskaya et al., 9 Aug 2024). So, topological state stabilization via particle-number conserving dissipation is fundamentally limited unless special fine-tuning or geometry is introduced.
• Many-Body Majorana Modes and Superselection: In strictly number-conserving settings, nontrivial zero modes can be constructed only as coherent superpositions of states differing by a fermion, raising obstacles for quantum control due to superselection rules. True braiding operations require environmental reference frames to overcome these restrictions (Ortiz et al., 2016, Lin et al., 2018).

4. Structure-Preserving and Computational Efficiency

Particle-number conservation offers substantial computational advantages:

• Efficient Numerical Schemes: Wavefunction-based approaches tracking $N$ coefficients rather than $N^2$ elements (density matrices) scale linearly and handle large systems up to $N \sim 10^6$ (Veenendaal et al., 2010). Fully discrete numeric methods (e.g., discrete gradient integrators) enforce energy dissipation and mass/momentum conservation exactly, ensuring correct statistical behavior and eliminating numerical instabilities (Hu et al., 29 Jun 2024).
• Quantum Benchmarking in Practice: MLXEB protocols and particle-number conserving circuits enable classical simulation and benchmarking of quantum processors well beyond the Hilbert space size limit imposed by non-conserving LXEB (Kaneda et al., 16 May 2025). Restriction to sectors of fixed particle number preserves random statistics in measurement outcomes and supports scalable fidelity estimation.
• Error Correction: Algorithms leveraging known conserved quantities (e.g., local number conservation in XY-QAOA or one-hot encoding) admit structure-tailored error correction schemes, reducing the overhead compared to generic approaches. Modified bit-flip codes require fewer ancillae and can restore symmetry-broken subspace occupation with minimal resource cost (Streif et al., 2020).

5. Universality, Instabilities, and Limitations

Despite their advantages and utility, particle-number conserving dissipative protocols run into fundamental limitations:

• Universality of Reaction-Diffusion Instabilities: Even with precisely constructed Lindblad or GKSL operators ensuring exact conservation, generic dynamics induce FKPP-type reaction-diffusion equations whose propagating fronts destabilize designed dark states at long times and large distances. This effect is universal across band models and not easily suppressed by protocol modifications (Nosov et al., 2023, Lyublinskaya et al., 9 Aug 2024).
• Hilbert-Space Fragmentation and Freezing Transitions: Kinetically constrained particle-number conserving models such as the East model exhibit sharp transitions from thermalization to dynamical freezing (Hilbert-space fragmentation), with universal critical exponents matching those of dipole-conserving fracton systems. Below critical filling, the system's dynamics are strongly arrested and many sites remain frozen indefinitely, providing a minimal yet rich setting for exploring non-ergodic phases (Wang et al., 2023).
• Experimental and Theoretical Controversies: The difference between particle-number conserving and non-conserving theoretical frameworks is not always benign. Examples include modified Josephson relations, and physical effects (such as the absence of dissipative quantum phase transitions) that contradict predictions from standard (non-conserving) approaches yet match experimental findings (Koizumi, 2022). These results underscore the necessity of carefully considering gauge phases and particle-number constraints in modeling superconductivity.

6. Applications and Future Directions

Particle-number conserving dissipative protocols are being developed and implemented in a range of experimental and computational contexts:

• Quantum simulators with ultracold atoms, trapped ions, and superconducting qubits utilize such protocols to stabilize correlated and topologically nontrivial states under controlled dissipation.
• Large-scale quantum device benchmarks use particle-number conserving random circuits to scale classical simulation and fidelity estimation.
• Numerical kinetic and many-body simulations employ structure-preserving fully discrete schemes for modeling aggregation-diffusion, Landau equations, and more.
• Protocols incorporating particle-number conservation drive the design of resource-efficient error correction strategies tuned to symmetry properties.

Continued research is aimed at mitigating instabilities that arise from inherent reaction-diffusion processes, extending structure-preserving numerical methods to broader equations, and exploring the interplay between conservation laws, kinetic constraints, and nonequilibrium quantum dynamics in both theoretical and practical settings. The constant theme is that enforcing particle-number conservation while allowing for controlled dissipation yields fundamentally distinct dynamics, constraints, and opportunities for both quantum and classical many-body systems.