Dissipative phase decision without ground-state preparation

Technical Summary: Dissipative Phase Decision without Ground-State Preparation

Problem Statement
The paper addresses the "phase-decision problem" in quantum many-body systems: determining which ground-state phase a system realizes at a specific parameter point, given a set of candidate phases. Traditional approaches rely on a "static" perspective, requiring the preparation of highly accurate approximations to the ground state (e.g., via Quantum Phase Estimation or Adiabatic State Preparation) to compare energies or order parameters. These methods are often computationally expensive, particularly near phase boundaries where energy gaps close, necessitating long circuit depths and precise state preparation. The authors propose that for many phase-decision tasks, preparing an exact ground state is unnecessary; instead, one only needs to access a low-energy manifold where phase-sensitive observables have already converged to their ground-state values.

Methodology
The authors propose a "dynamical" perspective based on short-time dissipative cooling. Rather than waiting for the system to mix to a steady state (which may be the exact ground state), the protocol involves:

1. Initial State: Preparing a representative state of a candidate phase.
2. Dissipative Evolution: Evolving the state under Lindblad dynamics tailored to the target Hamiltonian $H$. The dynamics are governed by jump operators $K_a$ constructed using a filter function $f(s)$ and coupling operators $A_a$.
3. Realistic Filters: Unlike ideal filters that require infinite simulation time to resolve infinitesimal energy gaps, the authors utilize "realistic" filters with finite energy resolution $\Delta = \Omega(1)$. These filters are implementable using only short-time Hamiltonian simulation, avoiding the deep circuits required to resolve small gaps near critical points.
4. Early-Time Observation: Monitoring phase-sensitive observables (diagnostics) during the transient evolution, well before the system reaches the steady state. The protocol succeeds if these observables enter the diagnostic region associated with the target ground-state phase at a time $t_0 \ll t_{mix}$ (mixing time).

Key Contributions and Theoretical Analysis

• Pre-Mixing Regime Analysis: The paper investigates the regime where dissipative evolution has not yet mixed to the steady state. The authors demonstrate that for specific classes of systems, this short-time evolution is sufficient to suppress high-energy components and drive the system into a low-energy manifold where phase diagnostics are already accurate.
• Free Fermions and Bosons:
  • Free Fermions: The authors prove (Theorem 1) that for free-fermionic systems with coupling operators corresponding to canonical modes, realistic filters drive the system into a low-energy subspace with energy $E_\Delta = E_0 + \sum_{|\epsilon_k| \le \Delta} |\epsilon_k|$. The population of high-energy modes decays exponentially. Crucially, local observables in the bulk converge to ground-state values with errors scaling as $O(\Delta^2 r)$, implying that a coarse filter is sufficient to resolve phase transitions.
  • Free Bosons: For free-bosonic systems, the authors show (Theorem 2) that while a sharp energy cutoff does not exist due to the infinite ladder of Fock states, the steady-state population above an energy threshold $E$ decays exponentially ($e^{-\beta E}$). This ensures the system remains in a low-energy manifold.
  • Interacting Systems: Numerical evidence suggests that interacting fermionic systems exhibit a hybrid behavior: a "spillover" effect near the Fermi surface similar to free fermions, combined with intermode exchanges that create an exponentially decaying high-energy tail similar to free bosons.
• Resource Efficiency: The total resource cost is determined by the product of the Hamiltonian simulation time (to implement jump operators) and the Lindblad evolution time. By using coarse filters and stopping before full mixing, the protocol significantly reduces resource requirements compared to faithful ground-state preparation.

Numerical Results and Demonstrations
The authors validate the strategy on three representative models:

1. Frustrated $J_1$–$J_2$ Heisenberg Chain: The protocol successfully distinguishes between the critical Tomonaga–Luttinger liquid (TLL) phase and the gapped Valence Bond Solid (VBS) phase. By fitting the Luttinger parameter $K$ from bipartite $S^z$ fluctuations during short-time evolution, the method correctly identifies the phase even with coarse filter resolutions ($\Delta = 0.5, 1.0$) and short times, outperforming quench dynamics in detecting BKT transitions.
2. Kitaev Honeycomb Model: The method is applied to a 2D topological phase transition. Using mixed-state Chern number diagnostics derived from the covariance matrix of the dissipatively evolved state, the protocol correctly identifies the transition between Abelian and non-Abelian phases. This is achieved without opening a spectral gap or preparing the exact ground state, leveraging short-time dynamics ($t=3$) on a $21 \times 21$ lattice.
3. XXZ Spin Chain: The protocol benchmarks against adiabatic evolution for the transition between the gapless XX phase and the gapped Néel phase. While adiabatic evolution fails to reproduce the correct power-law decay of correlation functions within accessible times ($T=25$) due to the closing gap, the dissipative protocol captures the correct algebraic decay behavior of the XX phase using local spin coupling operators and tensor network simulations.

Significance and Claims
The paper claims that "dissipative phase decision" is a plausible and robust target for future utility-scale studies on early fault-tolerant quantum devices. The significance lies in:

• Reduced Complexity: It bypasses the need for high-precision ground-state preparation, which is often the bottleneck for small-gap phase decisions.
• Robustness: The binary nature of the phase-decision task (identifying a phase region rather than an exact state) makes it potentially more robust to noise and finite-size effects than full state preparation.
• Feasibility: The use of short-time Hamiltonian simulation and coarse filters aligns with the capabilities of near-term quantum hardware, offering a pathway to study complex phase diagrams (e.g., frustrated 2D models) that are currently intractable for classical methods or require prohibitive quantum resources for exact solutions.

The authors emphasize that these results are proof-of-principle tests in classically accessible regimes, intended to validate the mechanism before applying it to more challenging interacting systems where classical benchmarks are unavailable.