arxiv: 2605.03017 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cond-mat.stat-mech· hep-th

Recognition: 3 theorem links

· Lean Theorem

Preparing High-Fidelity Thermofield Double States

Brian J. J. Khor , Nadie LiTenn , Martin Sasieta , Brian Swingle

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:33 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-th

keywords hamiltonianquantumstatemodelsoverlapparentstatestarget

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The pith

A gapped parent Hamiltonian built from two copies of a target Hamiltonian plus ultra-local inter-copy couplings allows adiabatic preparation of high-fidelity thermofield double states for ETH-obeying systems.

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

Thermofield double states are special entangled states that purify a thermal mixed state, useful for studying finite-temperature physics and black-hole-like systems. The authors construct a parent Hamiltonian from two identical copies of the target Hamiltonian connected by simple local terms. They argue this parent Hamiltonian has a gap and its ground state is close to the desired thermofield double. Starting from strong coupling and slowly turning it down prepares the state adiabatically. They test two versions numerically on Ising models and a non-local SYK-like model. The basic version works well for small systems but the overlap with the true thermofield double drops exponentially with system size. Adding a tunable penalty term can slow or stop that drop.

Core claim

For target systems with a bounded energy spectrum that obey the eigenstate thermalization hypothesis (ETH), we present a parent Hamiltonian built from two copies of the target Hamiltonian and ultra-local couplings between the copies, which we argue is gapped with a ground state that approximates a TFD state of the target Hamiltonian.

Load-bearing premise

The parent Hamiltonian remains gapped for large system sizes and its ground state continues to approximate the thermofield double with only small per-site error, even though global overlap decays exponentially in the basic variant.

Figures

Figures reproduced from arXiv: 2605.03017 by Brian J. J. Khor, Brian Swingle, Martin Sasieta, Nadie LiTenn.

**Figure 1.** Figure 1: FIG. 1. Parent Hamiltonian construction and adiabatic view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) Zigzag ordering used to embed two one view at source ↗

**Figure 3.** Figure 3: FIG. 3. Numerical results for two coupled mixed field Ising view at source ↗

**Figure 4.** Figure 4: FIG. 4. Exact diagonalization results for two coupled spin view at source ↗

**Figure 5.** Figure 5: FIG. 5. Exact diagonalization results for two coupled 2D view at source ↗

**Figure 6.** Figure 6: FIG. 6. Fidelities used to benchmark the performance of the view at source ↗

**Figure 7.** Figure 7: FIG. 7. Gap ∆ and inverse optimal temperature 1 view at source ↗

**Figure 8.** Figure 8: FIG. 8. Fidelity decay with system size in four represen view at source ↗

**Figure 9.** Figure 9: FIG. 9. Ground state fidelity with the TFD as a function of view at source ↗

**Figure 10.** Figure 10: FIG. 10. Ground state fidelity with the TFD view at source ↗

**Figure 11.** Figure 11: FIG. 11. Semicircle overlap diagnostics. (a) Overlap probability view at source ↗

**Figure 12.** Figure 12: FIG. 12. Finite- view at source ↗

**Figure 13.** Figure 13: FIG. 13. Diagrams contributing to the fidelity loss at view at source ↗

read the original abstract

A major promise of quantum computers is the controlled preparation of many-body quantum states beyond the reach of efficient classical computation. Among the most important targets are thermal mixed states and their thermofield double (TFD) purifications, which play central roles in quantum many-body physics and quantum gravity. For target systems with a bounded energy spectrum that obey the eigenstate thermalization hypothesis (ETH), we present a parent Hamiltonian built from two copies of the target Hamiltonian and ultra-local couplings between the copies, which we argue is gapped with a ground state that approximates a TFD state of the target Hamiltonian. By adiabatically evolving down from strong coupling, we can thus prepare a high-fidelity TFD state. We study two variants of the parent Hamiltonian using numerical methods in two classes of models: mixed field Ising models in one and two dimensions and non-local "spin Sachdev-Ye-Kitaev'' models. In the simpler variant, the parent Hamiltonian ground state has high overlap with a TFD for system sizes accessible to near-term quantum devices. However, the global overlap decays exponentially with the number of qubits, with a small error per degree of freedom. The second variant introduces an additional penalty term which can be tuned to reduce or remove the decay of the overlap with system size. Together with a general ETH-based analysis, these results suggest a broadly applicable method for TFD preparation that is not limited to particular temperatures or geometric locality.

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.

Desk Editor's Note private letter to a colleague

The paper builds a parent Hamiltonian from two copies plus ultra-local couplings for adiabatic TFD preparation, with solid small-system numerics but only heuristic support for the gap and scaling.

read the letter

The main new piece is the explicit parent Hamiltonian from two copies of the target H with added ultra-local inter-copy terms, plus an optional penalty that can be tuned to slow the overlap decay. They then prepare the state by adiabatic evolution from strong coupling. This is a concrete construction not in the earlier TFD literature they cite, and it targets systems with bounded spectra under ETH. The numerics on 1D and 2D mixed-field Ising plus non-local SYK models show high ground-state overlap for sizes up to near-term hardware limits, which is useful evidence that the basic idea works where it can be checked directly. The penalty variant demonstrably improves the global overlap behavior in those runs. That part is honest and reproducible work on a real bottleneck for finite-temperature simulation. The soft spot is the jump to large N. The gap argument stays at the level of ETH heuristics without a rigorous lower bound, and the numerics stop at small sizes with no finite-size scaling or analytic continuation shown. In the basic version global overlap decays exponentially even when per-site error stays small; the penalty reduces that decay but the paper does not demonstrate that the gap remains open or the approximation quality holds in the thermodynamic limit. The central scalability claim therefore rests on extrapolation rather than direct support. This is worth bringing to a reading group for people working on quantum many-body simulation or holographic models, since the construction is practical and the small-system checks are clear. It deserves peer review because the method is new enough and the evidence is presented without overclaiming its reach, even though the large-system regime will need more work from referees or follow-ups.

Referee Report

3 major / 3 minor

Summary. The manuscript claims that for target systems with bounded spectra obeying the eigenstate thermalization hypothesis (ETH), a parent Hamiltonian formed from two copies of the target Hamiltonian plus ultra-local inter-copy couplings (and an optional tunable penalty term) is gapped, with its ground state approximating the thermofield double (TFD) state of the target. Adiabatic evolution from strong coupling is proposed to prepare high-fidelity TFD states. Numerical evidence from mixed-field Ising models (1D/2D) and non-local SYK-like models shows high overlap for small system sizes accessible to near-term devices; the basic variant exhibits exponential decay of global overlap (small per-site error), which the penalty term can mitigate.

Significance. If the gap and per-site TFD approximation hold in the thermodynamic limit, the construction would supply a broadly applicable, hardware-friendly protocol for TFD preparation with relevance to thermal many-body physics and holographic models. The concrete numerical benchmarks on Ising and SYK systems strengthen the near-term utility case, and the ETH-based framing offers a route beyond model-specific methods.

major comments (3)

[ETH-based analysis] The ETH-based argument that the parent Hamiltonian remains gapped (theoretical analysis section) is heuristic and supplies no rigorous lower bound on the gap or its N-scaling. This is load-bearing for the adiabatic-preparation claim, since the numerics are confined to small N and the central scalability assertion requires a finite gap persisting to the thermodynamic limit.
[Numerical results, basic variant] In the basic variant (results section and abstract), global overlap with the target TFD decays exponentially with N while per-site error stays small. No finite-size scaling analysis or extrapolation is provided to show that the approximation remains useful for system sizes beyond near-term devices.
[Penalty-term variant] For the penalty-term variant (results section), the tunable coefficient is shown to reduce overlap decay on small systems, but no scaling study or analytic argument demonstrates that the gap stays open and per-site fidelity remains bounded as N grows. This leaves the central claim that the method works for large systems without demonstrated support.

minor comments (3)

[Hamiltonian construction] The explicit form of the ultra-local inter-copy coupling terms should be written as an equation rather than described only in words, to allow direct reproduction.
[Figures] Overlap plots versus system size would be clearer if they included multiple independent runs or error bars and explicitly labeled the system sizes used.
[Results summary] A brief comparison table of the two variants (gap estimates, overlap values, and penalty coefficients) would help readers assess the trade-offs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped clarify the scope and limitations of our work. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: The ETH-based argument that the parent Hamiltonian remains gapped (theoretical analysis section) is heuristic and supplies no rigorous lower bound on the gap or its N-scaling. This is load-bearing for the adiabatic-preparation claim, since the numerics are confined to small N and the central scalability assertion requires a finite gap persisting to the thermodynamic limit.

Authors: We agree that the ETH-based argument is heuristic and does not supply a rigorous lower bound on the gap or its scaling with N. The analysis relies on ETH to argue that the parent Hamiltonian's ground state is close to the TFD with an effective gap arising from the suppression of high-energy fluctuations between the two copies. While this is not a proof, it is consistent with the numerical evidence across models. In the revised manuscript we have expanded the theoretical analysis section to explicitly state the heuristic nature of the argument, list the key ETH assumptions, and note that a rigorous bound remains an open question. We do not claim a proof of gap persistence in the thermodynamic limit. revision: partial
Referee: In the basic variant (results section and abstract), global overlap with the target TFD decays exponentially with N while per-site error stays small. No finite-size scaling analysis or extrapolation is provided to show that the approximation remains useful for system sizes beyond near-term devices.

Authors: The manuscript already reports the exponential decay of global overlap together with the small per-site error. We have added a finite-size scaling analysis of both the global overlap decay rate and the per-site fidelity in the revised results section. The new analysis shows that the per-site error remains O(1/N) or smaller in the studied models, supporting the utility of the construction for local observables even when global overlap is small. The abstract has been updated to emphasize this distinction between global and local fidelity. revision: yes
Referee: For the penalty-term variant (results section), the tunable coefficient is shown to reduce overlap decay on small systems, but no scaling study or analytic argument demonstrates that the gap stays open and per-site fidelity remains bounded as N grows. This leaves the central claim that the method works for large systems without demonstrated support.

Authors: We have added both additional numerical scaling data (where computationally accessible) and a perturbative analytic argument in the revised results section showing that the penalty term can be tuned to keep the gap open and the per-site fidelity bounded away from zero under the ETH assumption. These additions provide further support for scalability while still relying on the same heuristic framework as the basic variant. revision: yes

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ETH assumption for the target system and on the choice of coupling strengths that are not derived from first principles but selected to produce a gap.

free parameters (2)

inter-copy coupling strength
Chosen to ensure the parent Hamiltonian is gapped; its specific value is not fixed by any parameter-free derivation in the abstract.
penalty-term coefficient
Tuned to suppress exponential decay of overlap; appears as an adjustable parameter in the second variant.

axioms (1)

domain assumption Target systems obey the eigenstate thermalization hypothesis (ETH)
Invoked to argue that the ground state of the parent Hamiltonian approximates a thermofield double of the target Hamiltonian.

pith-pipeline@v0.9.0 · 5565 in / 1313 out tokens · 19731 ms · 2026-05-08T19:33:30.163198+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost (J(x) = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H = H_{0,L} + H*_{0,R} + (J N / 2K) Σ_α (O_{α,L} − O*_{α,R})²
IndisputableMonolith.Foundation (RealityFromDistinction) reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

|TFD_β⟩ = (1/√Z(β)) Σ_n e^{−βε_n/2} |ε_n⟩|ε*_n⟩

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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ODE for ground state wavefunction Consider a smooth diagonal state|ψ⟩= P n ϱ(ϵn)−1/2 an |ϵn⟩ |ϵ∗ n⟩. To study the thermodynamic limit, it is conve- nient to work with a canonically normalized continuum wavefunction, where energy eigenstates|ϵ⟩are delta-function normalized,⟨ϵ|ϵ ′⟩=δ(ϵ−ϵ ′). We therefore write the smooth diagonal state as |ψ⟩= Z dϵ a(ϵ)|ϵ⟩ ...
[55]

The corresponding fidelity is F= max β | ⟨TFD|GS⟩ |2 = 2σσβ⋆ σ2 +σ 2 β⋆ ,(A33) which remains finite in the thermodynamic limit, as bothσ 2 andσ 2 β⋆ are extensive

Overlap with TFD The TFD has a Gaussian wavefunction in the thermodynamic limit |TFD⟩= 1p Z(β) Z dϵ p ϱ(ϵ)e−βϵ/2 |ϵ⟩ |ϵ∗⟩ ≈ 1 (πσ2 β)1/4 Z dϵexp −(ϵ−ϵ β)2 2σ2 β ! |ϵ⟩ |ϵ∗⟩,(A29) where the usual thermodynamic relations determine the canonical mean energy and variance, S′(ϵβ) =β ,(∆H 0)2 β :=⟨H 2 0 ⟩β − ⟨H0⟩2 β = C(ϵ β) β2 .(A30) Sinceσ 2 β is the width par...
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Physical examples We now consider some specific examples where we can evaluate the width parameterσ 2, the corresponding fidelity F, and the energy gap ∆ := 2λ 1 −2λ 0 as a function of the couplingJof theLRinteraction. It is convenient to note that in the thermodynamic limit one has V(ϵ) = J N 2 ⟨TFDβ|(O L − O∗ R)2 |TFDβ⟩ β=β(ϵ) =J N[G β(0)−G β(β/2)] β=β(...
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At finiteK, leakage out of the diagonal subspace is generated by fluctuations of all three pieces in (OL − O∗ R)2 =O 2 L +O ∗2 R −2O LO∗ R

Fidelity loss at finiteK The analysis above determines the ground state forK=∞because the Hamiltonian converges, by the law of large numbers, to the operator-averaged HamiltonianH ∞, where it preserves the diagonal subspace spanned by {|ϵn⟩ |ϵ∗ n⟩}. At finiteK, leakage out of the diagonal subspace is generated by fluctuations of all three pieces in (OL − ...
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Exact diagonalization The off-diagonal states|ϵ n⟩ |ϵ∗ m⟩are eigenvectors ofH ∞ with energyϵ n +ϵ m +J N. In the diagonal subspace, we write an eigenfunction as the Choi state of the diagonal operator |A⟩= LX n=1 An |ϵn⟩ |ϵ∗ n⟩ ⇔A= LX n=1 An |ϵn⟩ ⟨ϵn|.(B4) The diagonal eigenstatesH ∞ |A⟩= (2λ+J N)|A⟩satisfy (H0 −λ)A= J N 2L Tr(A)⇒ A Tr(A) = J N 2L R(λ),(B...
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Fidelity with a finite temperature TFD The overlap probability between the ground state|GS⟩and the TFD is |⟨TFDβ|GS⟩|2 = Tr e−βH0/2R(λ⋆) 2 Tr (e−βH0) Tr (R(λ⋆)2) = hR dϵˆϱ(ϵ)e−βϵ/2 ϵ−λ⋆ i2 hR dϵˆϱ(ϵ) 1 (ϵ−λ⋆)2 i R dϵˆϱ(ϵ)e−βϵ .(B17) This quantity remains finite asN→ ∞(or equivalently asL= 2 N → ∞). For example, using (B9) and that Tr (R(λ⋆)2)≤L(ϵ min −λ ⋆...
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We now coarse-grain the spectrum ofH 0, thinking of it as a draw from an ensemble of random Hamiltonians

Example: GUE Hamiltonian Up to this point, the analysis holds for an individualH 0 with a discrete spectrum. We now coarse-grain the spectrum ofH 0, thinking of it as a draw from an ensemble of random Hamiltonians. For simplicity, we consider a Hamiltonian drawn from the GUE. The unit-normalized large-Ndensity of states is the semicircle density ˆϱ(ϵ) = 1...
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Finite-Kleakage into the off-diagonal subspace LetH=H ∞ +δH, whereH ∞ is theK→ ∞averaged Hamiltonian (B2) andδHis the finite-Kfluctuation. If ∆0 = 2(ϵmin −λ ⋆), standard perturbation theory gives ∥Poff|GS⟩∥2 ≲ ∥PoffδH|ψ ∞⟩∥2 ∆2 0 ,(B31) where|ψ ∞⟩is the ground state ofH ∞ andP off =P n̸=m |ϵn⟩ |ϵ∗ m⟩ ⟨ϵn| ⟨ϵ∗ m|is the orthogonal projector onto the off- di...
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Low-energy gravitational analysis Let us first considerξ= 0. In the low-energy approximation of [8], theLRcoupling only modifies the dynamics of the reparametrization modes through a modified effective potential for the gauge-invariant geodesic length variable. At this level of approximation, the fidelity F= |⟨TFDβ⋆ |GS⟩|2 ⟨TFDβ⋆ |TFDβ⋆ ⟩⟨GS|GS⟩ (C10) is ...