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arxiv: 2607.06293 · v1 · pith:44ZNDSNM · submitted 2026-07-07 · quant-ph · cond-mat.str-el

Calculating strongly correlated ground states from the non-Markovian dissipative dynamics of Gaussian fermions

Giuliano Chiriac\`o This is my paper

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classification quant-ph cond-mat.str-el
keywords computationaldissipativedynamicsfermionsnon-markoviansystemtrajectoriescost
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The pith

Fermi-Hubbard ground state maps to non-Markovian dissipative dynamics

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a formal mapping between the ground state of the Fermi-Hubbard Hamiltonian and the non-equilibrium steady state of a purely dissipative open quantum system of Majorana fermions. Using the framework of third quantization, every term in the Hamiltonian—hopping, on-site interaction, and chemical potential—is converted into a Lindblad jump operator acting on Majorana modes. The central discovery is that all these jump operators, including those arising from two-body density-density interactions, preserve the Gaussianity of the fermionic state along each individual quantum trajectory. This means each trajectory can be simulated using only a two-point correlation matrix, with computational cost polynomial in system size. However, any system with a nonzero particle number requires negative dissipative rates, making the dynamics intrinsically and eternally non-Markovian. The trajectory unravelling of such non-Markovian dynamics involves both positive and negative stochastic weights, producing a sign problem: fluctuations of the classical sampling weight grow exponentially in time, requiring a number of trajectories that scales exponentially with system size. The total computational cost therefore remains exponential, but the bottleneck has shifted from state representation to statistical sampling.

Core claim

The key structural result is that density-density interaction terms in the Fermi-Hubbard model map to quadratic jump operators on Majorana fermions (specifically products of two Majorana operators), and that these operators—like the linear jump operators from hopping and chemical potential terms—preserve Gaussianity along each quantum trajectory. This preservation means the full many-body state along a trajectory is captured by a covariance matrix of size polynomial in L. The exponential complexity of the correlated ground-state problem does not vanish; instead it reappears as the sign problem of the non-Markovian unravelling, where negative dissipation rates force signed stochastic weights.

What carries the argument

Third quantization mapping (Hamiltonian to Lindbladian on Majorana fermions); Gaussianity-preserving jump operators; quantum trajectory unravelling with a classical sign bit tracking negative-dissipation jumps; covariance matrix update rules for Gaussian states under Majorana jumps.

If this is right

  • If alternative non-Markovian unravellings could reduce weight fluctuations, the polynomial per-trajectory cost might become exploitable, potentially yielding a sub-exponential algorithm for correlated ground states.
  • The mapping provides a concrete bridge between sign-problem theory in quantum Monte Carlo and the theory of non-Markovian quantum trajectory unravellings, suggesting that progress on either front could benefit the other.
  • The Gaussianity-preserving property of quadratic Majorana jump operators could be useful for simulating dissipative Majorana systems in their own right, independent of the ground-state mapping.
  • The framework could be extended to other interacting fermion models beyond Fermi-Hubbard, as long as the interaction terms can be decomposed into density-density form.

Load-bearing premise

The exponential scaling of the number of trajectories relies on the assumption that the relaxation rate of the system does not decrease with system size. Near phase transitions or critical points, spectral gaps can close, which would change the scaling—potentially making it worse, or, if gapless regimes admit special structure, possibly better.

What would settle it

If one could exhibit a non-Markovian trajectory unravelling for this class of master equations in which the variance of the stochastic weight does not grow exponentially with system size, the claimed exponential trajectory count would fail.

Figures

Figures reproduced from arXiv: 2607.06293 by Giuliano Chiriac\`o.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of the model considered and of the mapping. To the left we have the standard depiction of a Fermi-Hubbard [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plot of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

We introduce a mapping between the ground state of interacting fermionic Hamiltonians and the non-equilibrium steady state of a purely dissipative open quantum system. Within the framework of third quantization, we map the Fermi-Hubbard Hamiltonian onto Lindblad jump operators acting on Majorana fermions. Remarkably, both hopping and interaction terms map onto jump operators that preserve the Gaussianity of Majorana fermions along individual quantum trajectories. As a result, the dynamics can be unravelled and each trajectory can be simulated efficiently using only two-point correlation functions, with a computational cost that scales polynomially with system size. We further show that finite particle number requires negative dissipative rates, leading to an intrinsically non-Markovian dynamics. The corresponding trajectory unravelling involves both positive and negative stochastic weights and exhibits a sign problem and large fluctuations, so that convergence requires an exponentially large number of trajectories. The overall computational cost remains exponential in system size despite the efficient Gaussian representation of individual trajectories, but is crucially dependent on the computational complexity of the non-Markovian unravelling, motivating further studies on the efficiency of such unravellings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces a mapping between the ground state of interacting fermionic Hamiltonians (specifically the Fermi-Hubbard model) and the non-equilibrium steady state of a purely dissipative open quantum system of Majorana fermions. Using the third quantization formalism, the author maps each Hamiltonian term (hopping, interaction, chemical potential) to a Lindblad jump operator. The key finding is that all resulting jump operators preserve the Gaussianity of Majorana fermion states along individual quantum trajectories, enabling polynomial-cost simulation of each trajectory via two-point correlation functions. However, finite particle number requires negative dissipative rates, producing an intrinsically non-Markovian dynamics whose trajectory unravelling exhibits a sign problem. The author argues that the number of trajectories required for convergence scales exponentially with system size, so the total computational cost remains exponential. The paper presents numerical demonstrations for small systems (L=1,2,4) in both Markovian and non-Markovian regimes.

Significance. The paper provides a parameter-free mathematical identity derived from the third quantization framework, with no fitted constants introduced in the mapping itself. The Gaussianity preservation is proven explicitly through the update rules for the covariance matrix (Table II, Appendix B), and the sign problem analysis is mathematically clean (Eqs. 23-24, Appendix C). The framework offers a conceptually novel perspective on how the exponential complexity of ground-state computation is transferred from state representation to statistical sampling. The mapping rules (Table I, Appendix A) are carefully derived and could serve as a useful reference. The identification of spatial correlations in the Hamiltonian mapping to temporal correlations in the OQS is an interesting observation. The paper is honest about the limitations of its approach, noting that the complexity conclusion depends on the efficiency of non-Markovian unravellings.

major comments (3)
  1. Section III, paragraph following Eq. (11): The paper states 'all the jump operators considered here (see Tab. (I)) are also Hermitian L_ν = L†_ν.' However, the two-body jump operator L_ν = w_k w_l is anti-Hermitian for k≠l, since (w_k w_l)† = w_l w_k = -w_k w_l (using {w_k, w_l} = 2δ_{k,l}). This means L†_ν L_ν = (-w_k w_l)(w_k w_l) = -w_k w_l w_k w_l. For k≠l, w_k w_l w_k w_l = w_k (-w_k w_l) w_l = -w_k^2 w_l^2 = -1, so L†_ν L_ν = -(-1) = 1, which is consistent with the trace conservation argument. However, the Hermiticity claim is incorrect for this operator. While this does not affect the mathematical conclusions (trace conservation, Gaussianity preservation), it does affect the physical interpretation: the interaction term does not induce pure dephasing as claimed. The author should correct the Hermiticity statement and revise the physical interpretation of the interaction jump term.
  2. Section V, paragraph beginning 'If typical expectation values relax...': The central negative result — N_tr ~ exp(L) — depends on the ratio γ_A/γ_R scaling linearly with L. The numerator γ_A ~ Lμ is robustly established. However, the denominator γ_R is assumed to be O(μ, J, U) independent of L, justified only by the phrase 'as usually occurs.' This assumption is not tested for J≠0 or at critical points. In many-body systems near phase transitions, spectral gaps can close polynomially or exponentially with L, which would make γ_R decrease and the scaling of N_tr potentially worse than exp(L). The paper acknowledges this is an assumption but does not discuss the risk it poses to the scaling claim. The author should either (i) provide a more careful justification for why γ_R should remain O(1) in L for the Fermi-Hubbard model away from criticality, or (ii) explicitly qualify that the exp(L)
  3. Section V and Figure 2: The numerical demonstrations are limited to L≤4 with J=0 (no hopping), which means the hopping term's contribution to the dynamics is never exercised. Since the hopping term maps to a distinct jump operator (w_k + w_l)/√2 with its own Gaussianity update rules (Table II), the absence of any J≠0 numerical test leaves a gap in the empirical validation. The author should at minimum acknowledge this limitation explicitly, and ideally provide one numerical example with J≠0, even for small L, to confirm that the Gaussianity preservation and trajectory protocol work correctly when all three Hamiltonian terms are active.
minor comments (5)
  1. The phrase 'as usually occurs' in Section V is too informal for a scaling argument that underpins the paper's central claim. A more precise statement of the physical conditions under which γ_R is expected to be O(1) in L would strengthen the argument.
  2. Figure 2: The axis labels use ⟨n̂_1⟩ without the spin index, while the text refers to ⟨n̂_{1,↑}⟩. Consistency would help readability.
  3. Section V, Eq. (25)-(26): The two definitions of fluctuations differ in structure (Eq. 25 uses ⟨n̂⟩² while Eq. 26 uses (1-⟨n̂⟩)²). While the motivation is clear (different target values), a unified definition or clearer explanation of why the definitions differ would improve clarity.
  4. The abstract states 'is crucially dependent on the computational complexity of the non-Markovian unravelling, motivating further studies.' This is somewhat vague; specifying what aspects of the unravelling are most critical would help guide future work.
  5. Reference [79] is dated 2026; if this is a forward reference, please verify availability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies a mathematical error regarding the Hermiticity of the two-body jump operator, raises an important assumption about the relaxation rate scaling, and notes the absence of numerical tests with nonzero hopping. We address each point below.

read point-by-point responses

  1. Referee: Section III, paragraph following Eq. (11): The paper states 'all the jump operators considered here (see Tab. (I)) are also Hermitian L_ν = L†_ν.' However, the two-body jump operator L_ν = w_k w_l is anti-Hermitian for k≠l... The author should correct the Hermiticity statement and revise the physical interpretation of the interaction jump term.

    Authors: The referee is mathematically correct, and we will revise the manuscript accordingly. The two-body jump operator L_ν = w_k w_l is indeed anti-Hermitian for k≠l, since (w_k w_l)† = w_l w_k = -w_k w_l. Our statement that all jump operators are Hermitian is incorrect for this operator. We note that the referee's own calculation confirms that L†_ν L_ν = 1 still holds, so the trace conservation argument and Gaussianity preservation are unaffected. However, the physical interpretation must be revised: the interaction term does not induce standard pure dephasing as claimed. We will correct the Hermiticity statement in Section III and revise the physical interpretation of the interaction jump term. The phrase 'all jump operators are Hermitian' will be replaced with an accurate characterization, and the description of the interaction term as inducing 'time-correlated dephasing' will be qualified appropriately. revision: yes

  2. Referee: Section V: The central negative result — N_tr ~ exp(L) — depends on the ratio γ_A/γ_R scaling linearly with L. The denominator γ_R is assumed to be O(μ, J, U) independent of L, justified only by the phrase 'as usually occurs.' This assumption is not tested for J≠0 or at critical points... The author should either (i) provide a more careful justification... or (ii) explicitly qualify that the exp(L) scaling may be a lower bound.

    Authors: The referee raises a valid concern. We cannot provide a rigorous proof that γ_R remains O(1) in L for all parameter regimes of the Fermi-Hubbard model, and the referee is correct that near critical points spectral gaps can close, which would worsen the scaling. We will adopt option (ii): we will explicitly qualify in Section V that the assumption γ_R ~ O(μ, J, U) independent of L holds away from criticality, and that if γ_R decreases with L (as may occur near phase transitions), the scaling N_tr ~ exp(L) should be understood as a lower bound on the trajectory complexity. This qualification is consistent with the paper's existing caveats about the complexity conclusion depending on the efficiency of non-Markovian unravellings. We will add a discussion of this point in the revised manuscript. revision: yes

  3. Referee: Section V and Figure 2: The numerical demonstrations are limited to L≤4 with J=0 (no hopping), which means the hopping term's contribution to the dynamics is never exercised... The author should at minimum acknowledge this limitation explicitly, and ideally provide one numerical example with J≠0.

    Authors: The referee correctly notes that all numerical demonstrations use J=0, so the hopping jump operator (w_k + w_l)/√2 and its Gaussianity update rules (Table II) are not empirically tested. We acknowledge this limitation explicitly in the revised manuscript. Regarding the request for a J≠0 numerical example: the Gaussianity preservation for the hopping jump operator is proven analytically in Appendix B (Eq. B2), and the update rules are derived from the same Majorana anticommutation algebra as the other operators. The mechanism is structurally identical — the jump acts as a unitary transformation on the covariance matrix. However, we agree that an empirical demonstration would strengthen the validation. We will attempt to include a small J≠0 example (e.g., L=2) in the revised manuscript. If computational constraints prevent a clean demonstration within the revision timeframe, we will at minimum add an explicit acknowledgment of this gap and note that the analytical proof covers this case. revision: partial

Circularity Check

0 steps flagged

No significant circularity; the mapping is a parameter-free mathematical identity and the scaling argument derives from fluctuation growth, not from assuming the answer.

full rationale

The paper's core derivation chain is self-contained. (1) The Hamiltonian-to-Lindbladian mapping (Section II, Table I, Appendix A) is a mathematical identity from third quantization (Prosen [49–52], not self-cited), with no fitted parameters. (2) Gaussianity preservation (Section IV, Eq. 17, Table II) is proven explicitly from Majorana anticommutation relations — the output (updated A matrix) is derived, not defined in terms of the desired conclusion. (3) The exponential scaling claim (Section V) is derived from the growth of s²(t) ~ e^{2γ_A t} (Eq. 24, proven in Appendix C) combined with an explicit, stated assumption that γ_R ~ O(μ,J,U) independent of L. The paper does not assume exponential cost and then derive it; it derives the trajectory count from fluctuation analysis and explicitly flags the assumption as uncertain ('as usually occurs'), noting the result is 'crucially dependent on the computational complexity of the non-Markovian unravelling.' The only self-citation is Ref. [68] (same author) for standard Gaussian fermion formulas (partition function, covariance relations), but these results are also attributed to multiple independent references [62–67, 88–91] and are not load-bearing for the paper's central claims. The consistency between the derived exponential scaling and the known complexity of ground-state computation is presented as a consistency check, not as a derivation. No step reduces to its inputs by construction. The γ_R assumption is a correctness risk (untested for J≠0, large L), not a circularity issue. Score 1 reflects the minor non-load-bearing self-citation in [68].

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The mapping is parameter-free and derived from established third quantization. The main domain assumption (γ_R scaling) is the load-bearing premise for the complexity conclusion.

free parameters (3)
  • U (interaction strength)
    Input parameter of the Fermi-Hubbard model, not fitted by the method. Appears as dissipative rate γ_U = -U/4 in Table I.
  • J (hopping amplitude)
    Input parameter of the Fermi-Hubbard model. Appears as dissipative rate γ_J = J_kl in Table I.
  • μ (chemical potential)
    Input parameter determining particle number. Appears as dissipative rate γ_μ = -μ/2 in Table I. The mapping itself introduces no free parameters.
axioms (5)
  • standard math Third quantization mapping: a density matrix ρ of an OQS corresponds to a quantum state |ψ⟩ of a correlated system with twice the degrees of freedom, with H ↔ -L.
    Established framework from Prosen (Refs. 49-52), invoked in Section II and Appendix A. The paper uses it in reverse direction.
  • standard math Gaussian fermionic states are fully characterized by their covariance matrix Γ_kl = ⟨w_k w_l⟩ - δ_kl.
    Standard result from fermionic Gaussian state theory (Refs. 62-68, 88-91), invoked in Section IV, Eq. (15).
  • domain assumption The relaxation rate γ_R scales as O(μ, J, U) and does not decrease with system size L.
    Stated in Section V: 'if we assume that γ_R is of the order of single-particle excitation energies (as usually occurs)'. This assumption directly determines the exponential scaling of N_tr.
  • domain assumption The system has an even number of fermions, avoiding parity issues in the mapping.
    Stated in Section II: 'We assume that μ is chosen so that the number of fermions in the system is even, in order to avoid parity issues when mapping to an OQS.'
  • ad hoc to paper The non-Markovian unravelling protocol with classical bit s_a (from Becker et al. Ref. 61) is representative of the best achievable unravelling.
    The paper uses one specific unravelling protocol (Eqs. 11-13) and acknowledges that alternative unravellings are an open problem (Section VI). The exponential scaling conclusion depends on this choice.
invented entities (1)
  • Classical bit s_a(t) no independent evidence
    purpose: Tracks the sign of the density matrix trace along each trajectory to handle negative dissipation rates in the non-Markovian unravelling.
    This is a computational tool within the trajectory unravelling, not a physical entity. It is adapted from Ref. 61. No new physical postulate is introduced.

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