Dissipation-Assisted Operator Evolution

• DAOE is a Heisenberg-picture method that employs artificial dissipation to selectively prune high-weight operator strings in matrix product operator simulations.
• It controls operator entanglement by damping non-local operators, thereby enabling the extended simulation of transport properties in strongly interacting systems.
• DAOE has been generalized to address finite chemical potential, weak interactions, and fermionic systems, making it a versatile tool for studying 1D lattice dynamics.

Dissipation-Assisted Operator Evolution (DAOE) is a Heisenberg-picture method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. The method modifies unitary operator evolution by inserting an artificial dissipation that reduces the weight on non-local operators, typically defined in a Pauli-string or fermionic-string basis. When the evolving observable is represented as a matrix product operator (MPO), this dissipation causes operator entanglement to turn over and decay at late times, thereby extending the accessible simulation window. Physical transport coefficients are then obtained by weakening the dissipation and extrapolating to the zero-dissipation limit (Rakovszky et al., 2020). Subsequent work generalized the approach to finite chemical potential, weakly interacting fermions, open-system nonequilibrium steady states, and quasiparticle decay problems in one dimension (Srivatsa et al., 2024, Lloyd et al., 2023, Yoo et al., 2022, Young et al., 1 Aug 2025).

1. Formal definition and operator-weight dissipation

In its original form, DAOE evolves an operator $O(t)$ under a Hamiltonian $H$ that conserves some charge $Q$, but replaces exact Heisenberg evolution by

$$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$

where $\gamma>0$ is a dissipation rate and $\mathcal D$ acts diagonally on a local operator basis such as Pauli strings. If $S$ is a Pauli string of length $\ell_S$, with $\ell_S$ defined as the number of non-identity factors, and if $\ell_*$ is a cutoff length, then

$H$0

Equivalently, strings with $H$1 are multiplied by $H$2, while strings with $H$3 are left unchanged. In the limit $H$4 or $H$5, one recovers exact unitary evolution (Rakovszky et al., 2020).

A discrete-time formulation alternates unitary evolution and dissipative pruning. Writing the Liouvillian as $H$6, one applies $H$7 and then the diagonal superoperator $H$8. This realizes DAOE as a Trotterized evolution in operator space, with the key approximation located entirely in the suppression of high-weight strings rather than in a modification of the microscopic Hamiltonian (Rakovszky et al., 2020).

The notion of “weight” is central. In spin chains it is the number of nontrivial Pauli factors in a tensor-product string, $H$9 for $Q$0. In fermionic generalizations, the relevant measure is instead a “Fermi weight,” constructed from a local basis $Q$1, $Q$2, $Q$3, $Q$4, with weights $Q$5 respectively (Lloyd et al., 2023).

The open-system literature emphasizes a closely related point: DAOE is an artificial, linear superoperator acting on operator Hilbert space, and while it is contractive and Hermitian under the Hilbert–Schmidt inner product, it does not in general preserve positivity of density matrices. In boundary-driven Lindblad problems, it therefore functions as a controlled truncation device rather than as a physical bath in the GKLS sense (Yoo et al., 2022).

2. MPO representation, vectorization, and operator-entanglement control

DAOE is designed for tensor-network simulation. One reshapes an operator $Q$6 into a vectorized state $Q$7 in a doubled Hilbert space and represents it as a matrix-product state (MPS) of bond dimension $Q$8. In this language, the dissipation superoperator $Q$9 itself becomes an MPO of bond dimension $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$0, because it only needs to count operator weight up to the cutoff and apply the appropriate multiplicative factor (Rakovszky et al., 2020).

Without dissipation, generic Hamiltonian evolution produces linear growth of operator entanglement,

$$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$1

where $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$2 is the reduced density matrix of half the chain in the vectorized state $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$3. Under DAOE with $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$4 and finite $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$5, $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$6 initially rises but then turns over and decays at late times. This is the mechanism that permits long-time evolution with fixed finite $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$7: the method suppresses the operator strings responsible for unbounded entanglement growth while preserving the low-weight sectors that dominate hydrodynamic observables (Rakovszky et al., 2020).

A practical implementation proceeds by choosing $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$8, $$\frac{d}{dt}O(t)=i[H,O(t)]-\gamma\,\mathcal D[O(t)] ,$$9, $\gamma>0$0, and a Trotter step $\gamma>0$1; initializing $\gamma>0$2 as an MPS; applying one TEBD step for $\gamma>0$3; applying the MPO for $\gamma>0$4; and monitoring truncation errors at each gate, increasing $\gamma>0$5 until a preset tolerance is met. Typical parameter regimes in the original benchmarks are $\gamma>0$6, $\gamma>0$7, and $\gamma>0$8, which allow times $\gamma>0$9 on system sizes $\mathcal D$0 with negligible finite-size or truncation effects. The computational cost scales as $\mathcal D$1 (Rakovszky et al., 2020).

Later tensor-network studies reported the same qualitative gain. In Majorana-chain simulations, DAOE converts a strictly growing bond dimension into a profile that rises and then relaxes back, allowing times $\mathcal D$2 or more in 1D chains with modest $\mathcal D$3. In finite-density transport studies, the generalized method reaches $\mathcal D$4 with moderate $\mathcal D$5, precisely because the dissipator strongly biases the entanglement spectrum toward low-weight sectors (Kuo et al., 2023, Srivatsa et al., 2024).

3. Transport diagnostics and zero-dissipation extrapolation

DAOE was introduced primarily as a transport method. For a local conserved density $\mathcal D$6, one computes the infinite-temperature correlation function

$\mathcal D$7

the mean-square displacement

$\mathcal D$8

and the time-dependent diffusion constant

$\mathcal D$9

For each fixed $S$0, $S$1 approaches a late-time value $S$2. As $S$3 is reduced, the numerical data exhibit a linear convergence

$S$4

so the physical diffusion constant is extracted as $S$5 by fitting the last few points in $S$6 versus $S$7. Repeating this for several $S$8 yields an estimate of systematic uncertainty through the spread in $S$9, typically of order a few percent (Rakovszky et al., 2020).

The original benchmarks established the method on two generic 1D models at infinite temperature. For the tilted-field Ising chain,

$\ell_S$0

with conserved density $\ell_S$1, the extrapolated energy diffusion constant is

$\ell_S$2

For the two-leg XX spin ladder,

$\ell_S$3

with conserved density $\ell_S$4, the extrapolated spin diffusion constant is

$\ell_S$5

in excellent agreement with prior estimates (Rakovszky et al., 2020).

An alternative transport diagnostic appears in boundary-driven open systems. There one computes the nonequilibrium steady-state current

$\ell_S$6

with $\ell_S$7 for ballistic transport, $\ell_S$8 for superdiffusion, $\ell_S$9 for diffusion, and $\ell_S$0 for subdiffusion. This NESS formulation does not replace the MSD analysis, but it provides a complementary lens on how DAOE affects transport scaling through its action on conserved operators (Yoo et al., 2022).

4. Finite chemical potential, BBGKY-style truncation, and hydrodynamic crossover

The $\ell_S$1 generalization modifies the basis in which operator weight is defined. At finite chemical potential one introduces the inner product

$\ell_S$2

and Gram–Schmidt orthonormalizes the local Pauli basis to $\ell_S$3, with

$\ell_S$4

The same cutoff-and-damping rule is then applied in this orthonormalized basis, through

$\ell_S$5

with $\ell_S$6 the local basis change. As in the original construction, the resulting dissipator is encoded as an MPO of bond dimension $\ell_S$7 (Srivatsa et al., 2024).

The rationale changes in an important way. At $\ell_S$8, the dominant Feynman paths in $\ell_S$9 start and end on low-weight operators, and ergodicity suggests that once a string grows beyond $\ell_*$0 it is unlikely to return. At finite $\ell_*$1, long $\ell_*$2-type strings carry nonzero overlap with $\ell_*$3 and cannot simply be discarded. In the $\ell_*$4 basis, throwing away a long $\ell_*$5-string is equivalent to replacing each $\ell_*$6, so high-order strings are replaced by disconnected ensemble averages in the spirit of the BBGKY hierarchy. This preserves the long-string physics relevant for density correlations while still dramatically reducing operator entanglement (Srivatsa et al., 2024).

This finite-density generalization resolves the ballistic-to-diffusive crossover at low filling. Defining the equilibrium density

$\ell_*$7

one finds numerically

$\ell_*$8

so for $\ell_*$9, $H$00. The same scaling follows from a memory-matrix estimate with fast-mode decay rate $H$01, which gives $H$02. The interpretation is that low-density transport has a long ballistic regime with mean free path or scattering length $H$03 before crossing over to diffusion (Srivatsa et al., 2024).

The real-space correlator exhibits this crossover explicitly. For distances and times $H$04, one has the free-particle Bessel form

$H$05

whereas for $H$06, the Fourier-space correlator becomes diffusive,

$H$07

A minimal memory-matrix model recovers the same structure by separating slow operators from an $H$08 fast subspace, approximating the fast autocorrelator as white noise, and deriving a reduced Liouvillian whose poles describe a damped ballistic mode and a diffusive contribution. A single extra fit parameter $H$09 introduced through $H$10 quantitatively reproduces the DAOE$H$11 data for all $H$12 (Srivatsa et al., 2024).

5. Fermionic DAOE, exact free limits, and weak-integrability-breaking regimes

The fermionic formulation adapts DAOE to problems where Pauli weight is not the physically natural notion of operator size. In one dimension, fermions are mapped to spins via Jordan–Wigner,

$H$13

and the dissipator is built in a Fermi-string basis. The fermionic superoperator acts diagonally as

$H$14

so strings with Fermi weight above $H$15 are exponentially suppressed (Lloyd et al., 2023).

A decisive property of fDAOE is exact recovery of free dynamics. For a quadratic Hamiltonian, any single-body operator $H$16 or $H$17 evolves only within the single-body sector, and any density $H$18 remains weight-2. If $H$19, the dissipator never acts on this sector. Consequently, fDAOE reproduces ballistic spreading with no approximation error in the free limit and provides an ideal perturbative starting point when interactions $H$20 are small (Lloyd et al., 2023).

Algorithmically, the fermionic dissipator is implemented as a finite-state automaton MPO of bond dimension $H$21, split into parity-even and parity-odd blocks. The automaton tracks both the current Fermi weight and the fermion parity, so that the Jordan–Wigner $H$22 strings required by anticommutation do not artificially count toward Fermi weight when they lie between $H$23 and $H$24. Time evolution then alternates Trotterized two-site gates with applications of $H$25, followed by standard MPS truncation (Lloyd et al., 2023).

In a weakly interacting Fermi gas, this framework reveals a ballistic-to-diffusive crossover controlled by interaction strength. Numerically one finds $H$26 at early times, saturation at $H$27, and a late-time diffusion constant $H$28. A memory-kernel derivation and a Fermi’s golden rule estimate identify $H$29 as the fermion–fermion scattering time and the lifetime of the single-particle Green’s function (Lloyd et al., 2023).

In an interacting one-dimensional Majorana chain,

$H$30

energy transport instead yields

$H$31

contrary to naive expectations based on Fermi’s Golden rule but consistent with recent predictions based on weak integrability breaking. In the weak-interaction regime, where the fermionic nature of the system is most relevant, FDAOE is found to simulate the system more efficiently than DAOE. The same work contrasts DAOE/FDAOE with density matrix truncation (DMT): DMT preserves all operators up to a support diameter $H$32 but has no small parameter like $H$33, whereas DAOE/FDAOE provide a perturbative control parameter and a clear operator-weight criterion (Kuo et al., 2023).

DAOE has also been used in a more delicate 1D finite-temperature setting where naive Fermi’s Golden Rule breaks down. Combining DAOE with particle–particle ladder resummations and a melonic memory-matrix resummation, one study predicts a logarithmic enhancement of the quasiparticle decay rate,

$H$34

for interacting 1D lattice fermions at non-zero temperature. The DAOE simulations in that work use system sizes up to $H$35, bond dimension $H$36, cutoff $H$37, dissipation period $H$38, and $H$39, with linear extrapolation in $H$40 (Young et al., 1 Aug 2025).

6. Conserved quantities, limitations, and nomenclature

A recurring conclusion across applications is that DAOE’s effect on transport is controlled by its effect on the system’s conserved quantities. In boundary-driven XXZ-family chains and in the disordered XY model, the scaling exponent $H$41 is preserved when DAOE preserves the relevant local currents and is driven toward diffusion when DAOE breaks the operators responsible for ballistic, superdiffusive, or localized behavior (Yoo et al., 2022).

The model dependence is explicit. In the chaotic staggered-anisotropy XXZ chain, DAOE preserves the total $H$42 and the two-site spin current for $H$43, and the system remains diffusive with $H$44. In clean XXZ at $H$45, ballistic transport survives for sufficiently small $H$46 when $H$47, because the total current operator is itself a sum of weight-2 Pauli strings; if $H$48, DAOE damps the current and the system becomes diffusive. At the $H$49-symmetric point $H$50, $H$51 restores generic diffusion, while $H$52 leaves a remnant superdiffusion on simulated sizes $H$53, with an expected crossover to $H$54 only for $H$55. In the disordered XY chain, DAOE acts as a dephasing bath for localized orbitals and produces a crossover to diffusive scaling at lengths $H$56 (Yoo et al., 2022).

These results clarify a frequent misconception. DAOE is not a generic “noise” that necessarily forces diffusion; rather, it selectively damps high-weight sectors, and the hydrodynamic outcome depends on whether the physically relevant conserved densities and currents remain inside the retained low-weight subspace. A plausible implication is that operator-weight truncation is best interpreted as a symmetry-sensitive coarse graining rather than as a uniform decoherence mechanism (Yoo et al., 2022).

The method also has clear limitations. It is best suited to high temperatures; at low $H$57, the thermal density matrix itself has growing operator entanglement. The parameters $H$58 and $H$59 must be chosen so that the cutoff does not project away hydrodynamic operators, although the final answer is insensitive to their precise value as long as $H$60 exceeds the support of the conserved density and $H$61 is of order the microscopic timescale. If nonlocal strings carry important physics, DAOE must be pushed to larger $H$62 or smaller $H$63, increasing cost. In finite-density memory-matrix treatments, the white-noise approximation for the fast sector is uncontrolled (Rakovszky et al., 2020, Srivatsa et al., 2024, Young et al., 1 Aug 2025).

The acronym “DAOE” also has a separate, unrelated usage on arXiv. “Energy-Dissipative Evolutionary Deep Operator Neural Networks” uses DAOE to denote a DeepONet-based operator-learning framework for gradient-flow-type PDEs with a scalar auxiliary variable construction enforcing unconditional energy dissipation at the discrete level. That nomenclature refers to a neural-operator method for dissipative PDEs, not to Heisenberg-picture operator-weight truncation in quantum lattice dynamics (Zhang et al., 2023).