Dissipative algorithms for preparing thermal and ground states typically require a total Hamiltonian simulation time that grows at least linearly with system size $N$, even when their mixing time grows only polylogarithmically. We achieve sublinear simulation time for free fermions by engineering local interactions with a bath whose size grows in proportion to the system size. Starting from arbitrary initial states, our thermal state preparation algorithm achieves trace-norm error $ε$ in total simulation time $\widetilde{O}(\sqrt{N}/ε)$ at fixed inverse temperature and bounded single-particle bandwidth, while ground state preparation of gapped systems requires only $\mathrm{polylog}(N/ε)$ time. We further establish lower bounds for thermal state preparation within the bath protocols considered, identifying two obstacles to faster preparation, namely fourth-order deviations from the target thermal state and fluctuations induced by a random frequency shared by all bath qubits. Our results demonstrate that the linear simulation time barrier in dissipative state preparation can be overcome and clarify the bath and filter-design constraints that govern further complexity improvements.
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The paper asks whether dissipative state preparation can use sublinear total Hamiltonian evolution time while retaining local system–bath couplings. It studies number-conserving free-fermion Hamiltonians $H=\sum_{ij}h_{ij}c_i^\dagger c_j$, using an extensive bath of $\Theta(N)$ fermionic modes, simultaneous couplings, and bath resets between rounds. Convergence guarantees hold from arbitrary initial states, not only Gaussian states.
For bounded single-particle bandwidth and fixed inverse temperature, thermal-state preparation reaches trace-norm error $\epsilon$ in $\widetilde O(\sqrt N/\epsilon)$ total evolution time. The construction uses a zero-Hamiltonian vacuum bath and a filter enforcing second-order thermal detailed balance. Transforming system and bath into matched energy modes decomposes the channel into commuting mode channels; modewise perturbation analysis and product-state error bounds control the accumulated stationary-state bias. For ground states with $\Delta=\min_k|\lambda_k|>0$, a shared random bath frequency and Gaussian filter give $\widetilde O(\Delta^{-2})$ evolution time, with polylogarithmic dependence on $N/\epsilon$. This analysis controls higher Dyson orders beyond the Lindblad approximation.
Protocol-specific lower bounds identify two thermalization obstacles: fourth-order bias gives $\Omega(\sqrt N/\epsilon)$ for the chosen zero-bath filters, while shared-frequency fluctuations give $\Omega(N/\epsilon)$ unless detailed balance holds almost surely in the sampled frequency. Fourth-order thermal preservation characterizes filters permitting an improved zero-bath upper bound, but their existence remains open. The resource measured is evolution time, not gate count; extensions to interacting systems, intermediate bath sizes, and alternative bath correlations remain unresolved.
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Dissipative algorithms for preparing thermal and ground states typically require a total Hamiltonian simulation time that grows at least linearly with system size N, even when their mixing time grows only polylogarithmically. We achieve sublinear simulation time for free fermions by engineering local interactions with a bath whose size grows in proportion to the system size. Starting from arbitrary initial states, our thermal state preparation algorithm achieves trace-norm error ε in total simulation time O(N/ε) at fixed inverse temperature and bounded single-particle bandwidth, while ground state preparation of gapped systems requires only polylog(N/ε) time. We further establish lower bounds for thermal state preparation within the bath protocols considered, identifying two obstacles to faster preparation, namely fourth-order deviations from the target thermal state and fluctuations induced by a random frequency shared by all bath qubits. Our results demonstrate that the linear simulation time barrier in dissipative state preparation can be overcome and clarify the bath and filter-design constraints that govern further complexity improvements.
The paper asks whether dissipative state preparation can use sublinear total Hamiltonian evolution time while retaining local system–bath couplings. It studies number-conserving free-fermion Hamiltonians H=∑ijhijci†cj, using an extensive bath of Θ(N) fermionic modes, simultaneous couplings, and bath resets between rounds. Convergence guarantees hold from arbitrary initial states, not only Gaussian states.
For bounded single-particle bandwidth and fixed inverse temperature, thermal-state preparation reaches trace-norm error ϵ in O(N/ϵ) total evolution time. The construction uses a zero-Hamiltonian vacuum bath and a filter enforcing second-order thermal detailed balance. Transforming system and bath into matched energy modes decomposes the channel into commuting mode channels; modewise perturbation analysis and product-state error bounds control the accumulated stationary-state bias. For ground states with Δ=mink∣λk∣>0, a shared random bath frequency and Gaussian filter give O(Δ−2) evolution time, with polylogarithmic dependence on N/ϵ. This analysis controls higher Dyson orders beyond the Lindblad approximation.
Protocol-specific lower bounds identify two thermalization obstacles: fourth-order bias gives Ω(N/ϵ) for the chosen zero-bath filters, while shared-frequency fluctuations give Ω(N/ϵ) unless detailed balance holds almost surely in the sampled frequency. Fourth-order thermal preservation characterizes filters permitting an improved zero-bath upper bound, but their existence remains open. The resource measured is evolution time, not gate count; extensions to interacting systems, intermediate bath sizes, and alternative bath correlations remain unresolved.
Thermal state preparation is a central challenge in the simulation of quantum many-body systems. Yet, provably efficient algorithms for this task were only introduced recently [Chen et al. Nature 646, 561 (2025)]. These algorithms are based on dissipative Lindbladian evolution which exactly fixes the thermal state. Controlled and efficient digital simulation of this evolution, although possible in principle, remains out of reach for present-day quantum hardware. Subsequent work has therefore focused on analog approximations of the proposed Lindbladians via `collision models' with relatively modest requirements -- a resettable bath of ancilla qubits whose couplings to the system can be tuned in time-dependent fashion -- while still admitting rigorous fixed-point error bounds. Existing rigorous approaches, however, do not exploit the fact that these constructions generically implement not only the desired Lindblad dynamics, but also an additional unitary evolution generated by the system Hamiltonian which may aid convergence to the thermal state [Lloyd and Abanin arXiv:2506.21318 (2025)]. Here, we show that this unitary contribution does indeed tighten the fixed-point error bound and demonstrate that it is rigorously controlled by the system-bath coupling strength J, scaling as J2. This demonstrates that the effect of the spurious `Lamb shift' term generated by the system-bath interaction can be controlled by tuning J. We clarify the role, previously observed, of a randomized implementation in suppressing possible resonances of the drive with the many-body spectrum, and bound the additional variance that this randomization imposes on observables. Finally, we numerically study aspects of the protocol which are relevant for its practical realization, such as the mixing time.
Inspired by natural cooling processes, dissipation has become a promising approach for preparing low-energy states of quantum systems. However, the potential of dissipative protocols remains unclear beyond certain commuting Hamiltonians. This work provides significant analytical and numerical insights into the power of dissipation for preparing the ground state of noncommuting Hamiltonians. For quasi-free dissipative dynamics, including certain 1D spin systems with boundary dissipation, our results reveal a new connection between the mixing time in trace distance and the spectral properties of a non-Hermitian Hamiltonian, leading to an explicit and sharp bound on the mixing time that scales polynomially with system size. For more general spin systems, we develop a tensor network-based algorithm for constructing the Lindblad jump operator and for simulating the dynamics. Using this algorithm, we demonstrate numerically that dissipative ground state preparation protocols can achieve rapid mixing for certain 1D local Hamiltonians under bulk dissipation, with a mixing time that scales logarithmically with the system size. We then prove the rapid mixing result for certain weakly interacting spin and fermionic systems in arbitrary dimensions, extending recent results for high-temperature quantum Gibbs samplers to the zero-temperature regime. Together, these results show that dissipation can be a powerful tool for ground state preparation, with potential applications across condensed matter physics, quantum materials science, and beyond.
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 2×103. 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.
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.
Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Further, we adapt the techniques from separable qudits to the fermionic setting and prove rapid mixing of the strongly-interacting regime of the Fermi-Hubbard model, for which we also explicitly evaluate the guaranteed parameter regimes. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.
Cooling an arbitrary mixed state to a pure ground state requires transferring its entropy to an environment. Can a single coherent bath contact accomplish this without repeated bath resets? We develop a new cooling mechanism that transfers the input information into bath degrees of freedom that subsequently decouple, while the system and the remaining bath follow a ground-state path. The bath is discarded only at the end, and the system Hamiltonian remains on and unmodified throughout. We rigorously realize this mechanism for a class of weakly interacting, gapped fermionic systems with local interactions. Starting from any mixed state, our protocol prepares the interacting ground state in total physical time polylogarithmic in the system size and inverse global trace-norm error. It uses two initially empty bath modes per system mode and a single pulse shared across all onsite system--bath couplings. The same pulse prepares each admissible system's own ground state without knowledge of its microscopic parameters. The guarantees follow directly from a non-Markovian system--bath dynamics without an effective Lindbladian description.