arXiv:2609.26372

Lattice Lindbladian simulation by patching and merging

Kaoru Mizuta

quant-phcond-mat.otherSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations

Abstract

Simulating the dynamics of dissipative quantum many-body systems governed by local Lindbladians is a fundamental task in quantum computation. While the near-optimal gate count has been achieved for Hamiltonian simulation, comparable results for Lindbladian simulation remain elusive. In this work, we develop quantum algorithms for lattice Lindbladians by exploiting their locality. First, we consider sparsely dissipative systems, in which the dissipation is sparsely located, including boundary-driven systems. We establish a near-optimal quantum algorithm for their dynamics with gate count $O(Nt \, \mathrm{polylog}(Nt/\varepsilon))$, where $N$ is the system size, $t$ is the evolution time, and $\varepsilon$ is the allowable error. We then consider generic lattice Lindbladians with finite-range interactions and dissipation, and develop an algorithm for simulating time-evolved observables with gate count $O((Nt)^{4/3} \, \mathrm{polylog}(Nt/\varepsilon))$ per sample with the sampling complexity $Θ(\varepsilon^{-2})$. The gate count has the smallest known dependence on the system size among algorithms retaining polylogarithmic dependence on $1/\varepsilon$. Our algorithms are based on two techniques that exploit locality: patching and merging. Patching decomposes dissipative dynamics into dynamics on subsystems with exponentially small error, generalizing a key idea underlying the Haah-Hastings-Kothari-Low algorithm for near-optimal Hamiltonian simulation. Merging absorbs reversed dissipative dynamics into other parts of the evolution, substantially reducing the overhead associated with quasi-probabilistic sampling. These results demonstrate that locality can be fully exploited to achieve fast quantum simulation of dissipative many-body dynamics, opening the way to applications such as predicting nonequilibrium phenomena and preparing desirable quantum states.

Comments: 41 pages, 7 figures, 1 Table

Related papers

Near-optimal quantum simulation of lattice Lindbladian dynamics

Rahul Trivedi, Xiehang Yu, Daniel Malz

quant-pharXiv:2609.36203Sep 30, 20260 citations

Shallow Quantum Circuits for Geometrically Local Lindblad Dynamics

Zheyu Shen, Yusen Wu, Xiao Yuan +1 more

quant-pharXiv:2610.05362Oct 6, 20260 citations

Polynomial-time thermalization and Gibbs sampling from system-bath couplings

Samuel Slezak, Matteo Scandi, Álvaro M. Alhambra +2 more

quant-pharXiv:2601.16154Jan 23, 20260 citations

Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians

Jeongwan Haah, Matthew B. Hastings, Robin Kothari +1 more

arXiv:1801.03922SIAM Journal on ComputingJan 12, 2021148 citations

Optimizing digital quantum simulation of open quantum lattice models

Xie-Hang Yu, Hongchao Li, J. I. Cirac +1 more

arXiv:2509.02268Sep 2, 20250 citations

Locality and filter design for dissipative ground-state preparation

Samuel J. Elman

quant-pharXiv:2609.31266Sep 28, 20260 citations

arXiv:2609.26372

Lattice Lindbladian simulation by patching and merging

Kaoru Mizuta

quant-phcond-mat.otherSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations

arXiv HTML PDF

Abstract

O(Nt \, \mathrm{polylog}(Nt/\varepsilon))

, where

N

is the system size,

t

is the evolution time, and

\varepsilon

is the allowable error. We then consider generic lattice Lindbladians with finite-range interactions and dissipation, and develop an algorithm for simulating time-evolved observables with gate count

O((Nt)^{4/3} \, \mathrm{polylog}(Nt/\varepsilon))

per sample with the sampling complexity

Θ(\varepsilon^{-2})

. The gate count has the smallest known dependence on the system size among algorithms retaining polylogarithmic dependence on

1/\varepsilon

. Our algorithms are based on two techniques that exploit locality: patching and merging. Patching decomposes dissipative dynamics into dynamics on subsystems with exponentially small error, generalizing a key idea underlying the Haah-Hastings-Kothari-Low algorithm for near-optimal Hamiltonian simulation. Merging absorbs reversed dissipative dynamics into other parts of the evolution, substantially reducing the overhead associated with quasi-probabilistic sampling. These results demonstrate that locality can be fully exploited to achieve fast quantum simulation of dissipative many-body dynamics, opening the way to applications such as predicting nonequilibrium phenomena and preparing desirable quantum states.

Comments: 41 pages, 7 figures, 1 Table

Related papers

Near-optimal quantum simulation of lattice Lindbladian dynamics

Rahul Trivedi, Xiehang Yu, Daniel Malz

We consider the problem of simulating dynamics under geometrically local Lindbladians acting on a lattice of qudits. We present an algorithm to approximate the time evolution under these Lindbladians for evolution time $t$ and on $N$ qudits to diamond norm error $\varepsilon$ using geometrically local circuits of depth $O(t \ \polylog(Nt/\varepsilon))$. Our results thus achieve a performance comparable to that of the Haah-Hastings-Kothari-Low (HHKL) algorithm for lattice Hamiltonian simulation.We consider the problem of simulating dynamics under geometrically local Lindbladians acting on a lattice of qudits. We present an algorithm to approximate the time evolution under these Lindbladians for evolution time $t$ and on $N$ qudits to diamond norm error $\varepsilon$ using geometrically local circuits of depth $O(t \ \polylog(Nt/\varepsilon))$. Our results thus achieve a performance comparable to that of the Haah-Hastings-Kothari-Low (HHKL) algorithm for lattice Hamiltonian simulation.

quant-pharXiv:2609.36203Sep 30, 20260 citations

Shallow Quantum Circuits for Geometrically Local Lindblad Dynamics

Zheyu Shen, Yusen Wu, Xiao Yuan +1 more

Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of $n$ sites, allowing noncommuting jump operators. For time $T\ge1$ and diamond-norm error at most $ε$ , it has depth $O(T\operatorname{polylog}(nT/ε))$ and gate count $O(nT\operatorname{polylog}(nT/ε))$ . These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are $Ω(nT)$ for the expected gate count and $Ω(T)$ for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of $Ω(\log n/\log\log n)$ , showing that global accuracy can require increasing depth as the lattice grows.Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of $n$ sites, allowing noncommuting jump operators. For time $T\ge1$ and diamond-norm error at most $ε$, it has depth $O(T\operatorname{polylog}(nT/ε))$ and gate count $O(nT\operatorname{polylog}(nT/ε))$. These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are $Ω(nT)$ for the expected gate count and $Ω(T)$ for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of $Ω(\log n/\log\log n)$, showing that global accuracy can require increasing depth as the lattice grows.

quant-pharXiv:2610.05362Oct 6, 20260 citations

Polynomial-time thermalization and Gibbs sampling from system-bath couplings

Samuel Slezak, Matteo Scandi, Álvaro M. Alhambra +2 more

Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.

quant-pharXiv:2601.16154Jan 23, 20260 citations

Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians

Jeongwan Haah, Matthew B. Hastings, Robin Kothari +1 more

We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using ${\mathcal O}( nT {polylog} (nT/\epsilon))$ gates with depth ${\mathcal O}(T { polylog} (nT/\epsilon))$ . Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$ . Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires ${\widetilde{\Omega}}(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using ${\mathcal O}( nT {polylog} (nT/\epsilon))$ gates with depth ${\mathcal O}(T { polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires ${\widetilde{\Omega}}(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

arXiv:1801.03922SIAM Journal on ComputingJan 12, 2021148 citations

Optimizing digital quantum simulation of open quantum lattice models

Xie-Hang Yu, Hongchao Li, J. I. Cirac +1 more

Many-body systems arising in condensed matter physics and quantum optics inevitably couple to the environment and need to be modelled as open quantum systems. While near-optimal algorithms have been developed for simulating many-body quantum dynamics, algorithms for their open system counterparts remain less well investigated. We address the problem of simulating geometrically local many-body open quantum systems interacting with a stationary Gaussian environment. Under a smoothness assumption on the system-environment interaction, we develop near-optimal algorithms that, for a model with $N$ spins and evolution time $t$ , attain a simulation error $δ$ in the system-state with $\mathcal{O}(Nt(Nt/δ)^{o(1)})$ gates, $\mathcal{O}(t(Nt/δ)^{o(1)})$ parallelized circuit depth and $\tilde{\mathcal{O}}(N(Nt/δ)^{o(1)})$ ancillas. We additionally show that, if only simulating local observables is of interest, then the circuit depth of the digital algorithm can be chosen to be independent of the system size $N$ . This provides theoretical evidence for the utility of these algorithms for simulating physically relevant models, where typically local observables are of interest, on pre-fault tolerant devices. Finally, for the limiting case of Markovian dynamics with commuting jump operators, we propose two algorithms based on sampling a Wiener process and on a locally dilated Hamiltonian construction, respectively. These algorithms reduce the asymptotic gate complexity on $N$ compared to currently available algorithms in terms of the required number of geometrically local gates.Many-body systems arising in condensed matter physics and quantum optics inevitably couple to the environment and need to be modelled as open quantum systems. While near-optimal algorithms have been developed for simulating many-body quantum dynamics, algorithms for their open system counterparts remain less well investigated. We address the problem of simulating geometrically local many-body open quantum systems interacting with a stationary Gaussian environment. Under a smoothness assumption on the system-environment interaction, we develop near-optimal algorithms that, for a model with $N$ spins and evolution time $t$, attain a simulation error $δ$ in the system-state with $\mathcal{O}(Nt(Nt/δ)^{o(1)})$ gates, $\mathcal{O}(t(Nt/δ)^{o(1)})$ parallelized circuit depth and $\tilde{\mathcal{O}}(N(Nt/δ)^{o(1)})$ ancillas. We additionally show that, if only simulating local observables is of interest, then the circuit depth of the digital algorithm can be chosen to be independent of the system size $N$. This provides theoretical evidence for the utility of these algorithms for simulating physically relevant models, where typically local observables are of interest, on pre-fault tolerant devices. Finally, for the limiting case of Markovian dynamics with commuting jump operators, we propose two algorithms based on sampling a Wiener process and on a locally dilated Hamiltonian construction, respectively. These algorithms reduce the asymptotic gate complexity on $N$ compared to currently available algorithms in terms of the required number of geometrically local gates.

arXiv:2509.02268Sep 2, 20250 citations

Locality and filter design for dissipative ground-state preparation

Samuel J. Elman

Ground-state preparation is a fundamental challenge in quantum many-body physics, chemistry and materials science. Dissipative state-preparation techniques engineer a Lindbladian coupling to a bath that drives the system to its ground state, needing no initial overlap. Their cost is dominated by the Hamiltonian simulations inside the filtered jump operators, since each jump is a weighted sum of Heisenberg-evolved copies, or branches, of a coupling operator. We reduce this cost by following three simple observations to their logical end. First, on a fixed time grid, linear programming optimises the branch weights, trading the heating response against the circuit cost. Second, Lieb--Robinson bounds let each branch evolve on a patch whose radius grows with its duration. Third, if the dynamics converges quickly and truncation changes the jumps only slightly, the stationary state stays near the ground state even on patches smaller than the light cone. We validate our approach numerically. On a grid of $81$ evolution times, the optimised filter reduces the worst-case heating response, or leakage, more than $116$ -fold at equal cost. Across fifteen molecules and six chains, the cost falls by 27 to 56\% at matched design-grid leakage; at matched cost, the molecular leakage falls by a factor of $2.9$ to $2\times10^{3}$ . Density-matrix simulations show high ground-state fidelity for both filters, and on Ising and Hubbard chains the cheaper filter keeps a ground-state population of at least $0.89$ on patches of three or four sites. The same patches fail on a Heisenberg chain whose truncation error falls more slowly with patch size and whose Lindbladian gap is 12 times smaller than the Ising chain's, pointing to a role for the gap in how far patches can be truncated. At fixed radius the per-branch cost becomes independent of system size, reducing the cost of dissipative state preparation.Ground-state preparation is a fundamental challenge in quantum many-body physics, chemistry and materials science. Dissipative state-preparation techniques engineer a Lindbladian coupling to a bath that drives the system to its ground state, needing no initial overlap. Their cost is dominated by the Hamiltonian simulations inside the filtered jump operators, since each jump is a weighted sum of Heisenberg-evolved copies, or branches, of a coupling operator. We reduce this cost by following three simple observations to their logical end. First, on a fixed time grid, linear programming optimises the branch weights, trading the heating response against the circuit cost. Second, Lieb--Robinson bounds let each branch evolve on a patch whose radius grows with its duration. Third, if the dynamics converges quickly and truncation changes the jumps only slightly, the stationary state stays near the ground state even on patches smaller than the light cone. We validate our approach numerically. On a grid of $81$ evolution times, the optimised filter reduces the worst-case heating response, or leakage, more than $116$-fold at equal cost. Across fifteen molecules and six chains, the cost falls by 27 to 56\% at matched design-grid leakage; at matched cost, the molecular leakage falls by a factor of $2.9$ to $2\times10^{3}$. Density-matrix simulations show high ground-state fidelity for both filters, and on Ising and Hubbard chains the cheaper filter keeps a ground-state population of at least $0.89$ on patches of three or four sites. The same patches fail on a Heisenberg chain whose truncation error falls more slowly with patch size and whose Lindbladian gap is 12 times smaller than the Ising chain's, pointing to a role for the gap in how far patches can be truncated. At fixed radius the per-branch cost becomes independent of system size, reducing the cost of dissipative state preparation.

quant-pharXiv:2609.31266Sep 28, 20260 citations

O(Nt \, \mathrm{polylog}(Nt/\varepsilon))

, where

N

is the system size,

t

is the evolution time, and

\varepsilon

is the allowable error. We then consider generic lattice Lindbladians with finite-range interactions and dissipation, and develop an algorithm for simulating time-evolved observables with gate count

O((Nt)^{4/3} \, \mathrm{polylog}(Nt/\varepsilon))

per sample with the sampling complexity

Θ(\varepsilon^{-2})

. The gate count has the smallest known dependence on the system size among algorithms retaining polylogarithmic dependence on

1/\varepsilon

t

and on

N

qudits to diamond norm error

\varepsilon

using geometrically local circuits of depth $O(t \ \polylog(Nt/\varepsilon))$. Our results thus achieve a performance comparable to that of the Haah-Hastings-Kothari-Low (HHKL) algorithm for lattice Hamiltonian simulation.

n

sites, allowing noncommuting jump operators. For time

T\ge1

and diamond-norm error at most

ε

, it has depth

O(T\operatorname{polylog}(nT/ε))

and gate count

O(nT\operatorname{polylog}(nT/ε))

. These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are

Ω(nT)

for the expected gate count and

Ω(T)

for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of

Ω(\log n/\log\log n)

, showing that global accuracy can require increasing depth as the lattice grows.

n

qubits for time

T

up to error

\epsilon

using

{\mathcal O}( nT {polylog} (nT/\epsilon))

gates with depth

{\mathcal O}(T { polylog} (nT/\epsilon))

. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in

nT

and polylogarithmic in

1/\epsilon

. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires

{\widetilde{\Omega}}(nT)

gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

N

spins and evolution time

t

, attain a simulation error

δ

in the system-state with

\mathcal{O}(Nt(Nt/δ)^{o(1)})

gates,

\mathcal{O}(t(Nt/δ)^{o(1)})

parallelized circuit depth and

\tilde{\mathcal{O}}(N(Nt/δ)^{o(1)})

ancillas. We additionally show that, if only simulating local observables is of interest, then the circuit depth of the digital algorithm can be chosen to be independent of the system size

N

. This provides theoretical evidence for the utility of these algorithms for simulating physically relevant models, where typically local observables are of interest, on pre-fault tolerant devices. Finally, for the limiting case of Markovian dynamics with commuting jump operators, we propose two algorithms based on sampling a Wiener process and on a locally dilated Hamiltonian construction, respectively. These algorithms reduce the asymptotic gate complexity on

N

compared to currently available algorithms in terms of the required number of geometrically local gates.

81

evolution times, the optimised filter reduces the worst-case heating response, or leakage, more than

116

-fold at equal cost. Across fifteen molecules and six chains, the cost falls by 27 to 56\% at matched design-grid leakage; at matched cost, the molecular leakage falls by a factor of

2.9

2\times10^{3}

. Density-matrix simulations show high ground-state fidelity for both filters, and on Ising and Hubbard chains the cheaper filter keeps a ground-state population of at least

0.89

on patches of three or four sites. The same patches fail on a Heisenberg chain whose truncation error falls more slowly with patch size and whose Lindbladian gap is 12 times smaller than the Ising chain's, pointing to a role for the gap in how far patches can be truncated. At fixed radius the per-branch cost becomes independent of system size, reducing the cost of dissipative state preparation.