arxiv: 2605.12600 · v1 · submitted 2026-05-12 · 🪐 quant-ph

Recognition: unknown

Fermion lattices can be simulated by same-size qubit lattices with mathcal{O}(1) interaction overhead

Gregor Aigner , Berend Klaver , Martin Lanthaler , Wolfgang Lechner

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:37 UTC · model grok-4.3

classification 🪐 quant-ph

keywords fermion lattice simulationJordan-Wigner transformationquantum simulationHubbard modelqubit routingsurface codecircuit depth

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The pith

Two-dimensional fermion lattices can be simulated on equal-sized qubit lattices with no asymptotic interaction or space overhead.

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

The paper establishes a mapping that lets any geometrically local interaction among N fermions on a 2D lattice be realized by operations on N qubits. The mapping keeps the total number of two-qubit interactions linear in N and adds no extra qubits. It achieves this by repeatedly rotating the Jordan-Wigner string so that locality is preserved along one lattice direction at a time. A reader would care because standard higher-dimensional Jordan-Wigner mappings normally introduce long-range strings that scale badly with system size.

Core claim

All geometrically local interactions on an N-site two-dimensional fermion lattice can be simulated with no asymptotic overhead in the number of interactions and no space overhead. The method works by dynamically reorienting the Jordan-Wigner transformation to switch the lattice dimension along which locality is preserved. Circuit depth scales as O(sqrt(N)) on static qubit lattices, O(log N) on reconfigurable arrays, and O(1) in lattice-surgery surface-code architectures. The same approach extends directly to d-dimensional lattices and yields asymptotically optimal resource scaling for fermion routing and the fermionic fast Fourier transform.

What carries the argument

Dynamic reorientation of the Jordan-Wigner transformation, which switches the direction of preserved locality at each step to handle all lattice dimensions without extra qubits or interactions.

If this is right

Fermi-Hubbard models on square, Lieb, and kagome lattices can be simulated with the same interaction count as the underlying qubit lattice.
The fermionic fast Fourier transform runs on qubit lattices with resource scaling that matches optimal qubit-routing bounds.
All constructions extend immediately to any fixed dimension d without changing the asymptotic interaction count.
On reconfigurable qubit arrays the circuit depth for local fermion simulation drops to O(log N).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Material simulations that previously required extra ancilla qubits for string operators could now fit on the same hardware footprint.
The depth reductions on surface-code architectures suggest that lattice-surgery primitives may become the natural target for fermionic algorithms.
The routing results imply that any algorithm relying on fermionic swaps can inherit the known qubit-routing bounds without additional fermionic penalties.

Load-bearing premise

Dynamically reorienting the Jordan-Wigner string can be performed in hardware without introducing hidden asymptotic costs in gate count or connectivity beyond those already present for qubit swap networks.

What would settle it

An explicit gate-count measurement for a 2D Hubbard-model simulation on N sites that shows the total number of two-qubit gates growing faster than linearly with N.

Figures

Figures reproduced from arXiv: 2605.12600 by Berend Klaver, Gregor Aigner, Martin Lanthaler, Wolfgang Lechner.

**Figure 1.** Figure 1: FIG. 1. Overview of the main concepts. (a) Simulating fermionic systems, such as lattice models or quantum-chemical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Example of JW reordering in two dimensions from a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Switching between two-dimensional boustrophedon encodings. (a) A global [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Fermion routing. (a) An example routing problem. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Resource counts of Fermi-Hubbard simulations. The plots show CNOT counts (upper panels) and depths (lower panels) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. CNOT gates via lattice surgery [ [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. a) The partitions [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. 3D setup. a) 4 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Permutation-routing of fermions via FSNs and via [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Circuit implementation of a fermionic swap between [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. (a) We use two separate JW encodings when dealing with spinful models. Therefore, only modes of the same spin [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. Implementation of a second-order Trotter step corresponding to the NN Hubbard model on a grid lattice. (a) We [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. Implementation of the hopping terms corresponding to the NN square Hubbard model via FSNs. (a) The standard [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16. Implementation sequence (from left to right), of the [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗

**Figure 17.** Figure 17: FIG. 17. Description of Lieb lattice simulation. a) We tightly [PITH_FULL_IMAGE:figures/full_fig_p028_17.png] view at source ↗

read the original abstract

Local interactions among electrons underlie many complex properties of correlated materials. While the Jordan-Wigner transformation can preserve this locality along one spatial dimension, interactions along the remaining dimensions typically incur substantial overhead. We show how to simulate all geometrically local interactions on an $N$-site two-dimensional fermion lattice with no asymptotic overhead in the number of interactions and no space overhead. The primary overhead of our method is circuit depth, which on a qubit lattice matches that of fermionic swap networks, scaling as $\mathcal{O}(\sqrt{N})$, but reduces to $\mathcal{O}(\log N)$ on reconfigurable qubit arrays and to $\mathcal{O}(1)$ in lattice-surgery-based surface-code architectures. This is enabled by dynamically reorienting the Jordan-Wigner transformation to switch the lattice dimension along which locality is preserved. Furthermore, we study fermion routing, as required for the simulation of non-local interactions. When using qubit lattices, we reach resource scaling that asymptotically matches that of qubit routing, whilst on fully connected qubit devices, a depth scaling arbitrarily close to $\mathcal{O}(\log N)$ is reached. This allows the fermionic fast Fourier transform to be implemented on qubit lattices with asymptotically optimal resource scaling under these locality constraints. Notably, all of our constructions naturally extend to $d$-dimensional lattices. Beyond scaling improvements, we show explicit examples of our method, including Fermi-Hubbard-model simulations of the square-, Lieb- and kagome lattice and the fermionic fast Fourier transform.

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 dynamic Jordan-Wigner reorientation gives a plausible route to constant interaction overhead for 2D fermions on fixed qubit lattices, but the reorientation swaps need explicit counting to confirm the claim holds.

read the letter

The main takeaway is a mapping that keeps 2D fermion lattices on same-size qubit arrays while holding total two-qubit interactions to O(1) overhead per term, achieved by flipping the Jordan-Wigner string direction on the fly instead of using fixed strings or heavy swap networks. This is new relative to the usual 1D-preserving mappings and the swap-network literature cited in the abstract. The paper also shows how the same idea scales to d dimensions, handles routing for non-local terms, and reaches depth bounds that match qubit routing on lattices while getting close to log N on fully connected devices. The explicit Hubbard-model examples on square, Lieb, and kagome lattices plus the fermionic FFT construction are concrete and immediately usable for checking resource counts. Those parts are the strongest contribution. The soft spot is the reorientation step itself. On a static 2D grid, switching string direction requires remapping the fermion-to-qubit correspondence, which is done with SWAPs. Each such sequence costs interactions linear in the current string length, and string length is O(sqrt(N)). If more than a constant number of reorientations happen per Trotter step, the total interaction count picks up an extra factor even while depth stays O(sqrt(N)). The abstract claims the primary overhead is depth and that interaction count stays asymptotically flat, but without a breakdown of amortized swap costs or explicit gate tallies, it is not obvious the O(1) interaction claim survives. The stress-test note identifies exactly this gap. This paper is for people who simulate condensed-matter or chemistry models on hardware with limited connectivity and want tighter resource bounds than standard mappings. A reader who needs to implement or benchmark the construction will get value from the examples and scaling statements. It deserves a serious referee because the claim is specific, the applications are relevant, and the potential improvement in reachable system size is worth checking even if the overhead accounting needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that all geometrically local interactions on an N-site two-dimensional fermion lattice can be simulated on a same-size qubit lattice with no asymptotic overhead in the number of interactions and no space overhead. This is achieved by dynamically reorienting the Jordan-Wigner transformation to preserve locality along one dimension at a time. The primary overhead is circuit depth, scaling as O(sqrt(N)) on qubit lattices, O(log N) on reconfigurable arrays, and O(1) in lattice-surgery surface-code architectures. The work also addresses fermion routing for non-local interactions, achieving scaling that matches qubit routing on lattices and near-O(log N) depth on fully connected devices, enabling an asymptotically optimal fermionic FFT under locality constraints. Explicit examples are given for Fermi-Hubbard simulations on square, Lieb, and kagome lattices, with natural extension to d-dimensional cases.

Significance. If the central claims on interaction overhead and depth scaling hold with rigorous bounds, the result would be a meaningful advance for quantum simulation of fermionic systems. It removes the typical space and interaction overheads of higher-dimensional Jordan-Wigner mappings while matching or approaching the resource costs of pure qubit routing. The explicit lattice examples and the O(1) overhead claim (when verified) would directly benefit near-term simulations of correlated materials. The extension to reconfigurable hardware and surface-code architectures further strengthens its practical relevance.

major comments (2)

[Abstract / §3] Abstract and the dynamic reorientation construction (likely §3): the claim of no asymptotic overhead in total interaction count requires an explicit bound showing that the number of reorientations per Trotter step is O(1) and that each reorientation's SWAP sequence costs only O(1) interactions on average. Without this, the O(sqrt(N)) string length in 2D could accumulate an extra factor when reorienting for arbitrary local terms, as each remapping involves at least linear SWAPs in the string length.
[§4] Fermion routing and FFT section (likely §4): the statement that routing resources 'asymptotically match' those of qubit routing is load-bearing for the non-local interaction claim, yet no explicit equation or table compares the total two-qubit gate count (including JW-string updates) against standard qubit swap-network costs. A concrete scaling derivation or numerical count for the fermionic FFT would be needed to confirm the 'arbitrarily close to O(log N)' depth on fully connected devices does not hide interaction overhead.

minor comments (2)

[Introduction] Notation: the distinction between 'interaction overhead' (total gate count) and 'circuit depth' is used throughout but could be clarified with a short table in the introduction comparing the two metrics across hardware models.
[Figures 2-3] Figure clarity: the diagrams illustrating JW-string reorientation would benefit from explicit labels showing the sequence of SWAP operations and the resulting interaction graph after each reorientation step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. The comments have prompted us to strengthen the rigor of the claims on interaction overhead and routing costs. We address each major comment below and have incorporated revisions to provide the requested explicit bounds and comparisons.

read point-by-point responses

Referee: [Abstract / §3] Abstract and the dynamic reorientation construction (likely §3): the claim of no asymptotic overhead in total interaction count requires an explicit bound showing that the number of reorientations per Trotter step is O(1) and that each reorientation's SWAP sequence costs only O(1) interactions on average. Without this, the O(sqrt(N)) string length in 2D could accumulate an extra factor when reorienting for arbitrary local terms, as each remapping involves at least linear SWAPs in the string length.

Authors: We agree that an explicit bound is required to fully substantiate the no-asymptotic-overhead claim. The construction in §3 uses only a constant number of reorientations (specifically two: row-wise to column-wise and back) per Trotter step to cover all geometrically local interactions in 2D. However, the average SWAP cost per reorientation was stated implicitly rather than bounded. In the revised manuscript we have added Lemma 3.1, which proves that the expected interaction cost per reorientation remains O(1) per site because JW-string updates are performed via local parity checks that do not require traversing the full string length on average. This prevents any accumulation of the O(sqrt(N)) factor, keeping the total interaction count asymptotically identical to the pure-qubit case. The abstract has been updated to reference this lemma. revision: yes
Referee: [§4] Fermion routing and FFT section (likely §4): the statement that routing resources 'asymptotically match' those of qubit routing is load-bearing for the non-local interaction claim, yet no explicit equation or table compares the total two-qubit gate count (including JW-string updates) against standard qubit swap-network costs. A concrete scaling derivation or numerical count for the fermionic FFT would be needed to confirm the 'arbitrarily close to O(log N)' depth on fully connected devices does not hide interaction overhead.

Authors: We concur that a direct quantitative comparison is necessary. Section 4 derives asymptotic equivalence by showing that JW-string maintenance is folded into the existing swap-network layers with only constant-factor overhead. To make this explicit, the revised version adds Equation (4.5) giving the precise two-qubit gate count (including string updates) and Table 1, which tabulates total gate counts for both fermionic and standard qubit routing on lattices of size up to 16×16. The table and accompanying derivation confirm that the overhead remains O(1) per routed particle, preserving the same leading-order scaling. For the fermionic FFT on fully connected devices we now include a concrete depth count showing the interaction overhead is bounded by a small constant (≤2) while the depth approaches O(log N) arbitrarily closely, with no hidden super-logarithmic term. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on standard primitives via explicit construction

full rationale

The paper's central claim rests on an explicit construction that dynamically reorients the Jordan-Wigner transformation to preserve locality in 2D, combined with standard fermion routing and swap-network primitives. No equations, parameters, or results are shown to be fitted to a subset of the target data and then renamed as a prediction; no self-citation is invoked as a uniqueness theorem that forbids alternatives; and no ansatz is smuggled via prior work by the same authors. The scaling statements (O(sqrt(N)) depth on fixed lattices, O(log N) on reconfigurable arrays) follow directly from the geometry of the reorientation and routing steps rather than reducing to the input claim by definition. The method is therefore self-contained against external benchmarks such as conventional Jordan-Wigner plus swap networks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction relies on the standard Jordan-Wigner transformation (locality preserved along one dimension) and known fermionic swap networks; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Jordan-Wigner transformation preserves locality along one spatial dimension
Standard result invoked to justify the base mapping before dynamic reorientation.

pith-pipeline@v0.9.0 · 5576 in / 1225 out tokens · 45618 ms · 2026-05-14T20:37:11.894447+00:00 · methodology

discussion (0)

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Apply the CZ gates as defined in Eq

Move the qubitj 0 =m ′−1(0) from its current posi- tion in the orderm(j 0), to position 0 as desired by m′. Apply the CZ gates as defined in Eq. (A1) Y j′∈J CZj0,j′, J={k:m(k)< m(j 0)}. This results in a intermediate ordering where qubit j0 has reached its target position 0, but otherwise the relative order between qubitsj̸=j 0 has not changed. It follows...
[44]

Apply the CZ gates as defined in Eq

Move the qubitj 1 =m ′−1(1) from its current po- sition in the orderm(j 1), to position 1 as desired bym ′. Apply the CZ gates as defined in Eq. (A1), but taking into account that there is no need to interact withj 0 Y j′∈J CZj1,j′, J={k:m(k)< m(j 1)} \ {j0}. In line with step 1, we reach an intermediate or- dering withj 0 andj 1 at their target positions...
[45]

For each value ofj i we apply the gates Y j′∈J CZji,j′ J={k:m(k)< m(j i)} \ {l:m ′(l)< m ′(ji)} ={k:m(k)< m(j i)} ∩ {l:m ′(l)> m ′(ji)}

Continue with this strategy for the remainingN−2 qubits. For each value ofj i we apply the gates Y j′∈J CZji,j′ J={k:m(k)< m(j i)} \ {l:m ′(l)< m ′(ji)} ={k:m(k)< m(j i)} ∩ {l:m ′(l)> m ′(ji)}. This sequence maps the orderingmto the orderingm ′, with the applied CZ gates satisfying the condition stated in Lemma 2. Note that Lemma 2 is equivalent to Lemma ...
[46]

This is possible becausem C follows anSpattern within every subgridC i ∈ C

Transform them C encoding into anm Z encoding by applyingC 2D on each subgridC i ∈ C. This is possible becausem C follows anSpattern within every subgridC i ∈ C. The result is a collection of localZ-patterns on the subgridsC i, which together form aZ-pattern over the entire grid
[47]

This sequence requiresO(N) gates and only theC 2D circuit contributes non-constant depth

Transform the resultingm Z encoding into them C′ encoding by applyingC 2D on each subgridC ′ i ∈ C ′. This sequence requiresO(N) gates and only theC 2D circuit contributes non-constant depth. Appendix C: Constant depth encoding switching via lattice surgery This section demonstrates how the proposed encod- ing switches can be implemented in constant gate ...
[48]

Introduce an auxiliary qubit initialized in the|+⟩ state, between the logical control and logical target of the desired CNOT gate
[49]

Merge and split the smooth boundary of the control qubit and the auxiliary qubit
[50]

Merge and split the rough boundary of the target qubit and the auxiliary qubit
[51]

Measure the auxiliary qubit in theZbasis
[52]

Correct the control qubit with aZgate if theXX measurement outcome is−1
[53]

Corollary 11.The circuitC 2D can be implemented us- ingO(N)gates inO(1)depth via surface codes using lattice surgery

Correct the target qubit with anXgate if the prod- uct of theZZ-andZmeasurement is−1. Corollary 11.The circuitC 2D can be implemented us- ingO(N)gates inO(1)depth via surface codes using lattice surgery. Proof.The depth of theC 2D circuit, as discussed in Defi- nition 7, is determined by CNOT ladder depths. Here we discuss how to implement a CNOT ladder w...
[54]

Initialize in them C1 boustrophedon encoding
[55]

Apply a FSN within each subgrid inG 0,0 in parallel
[56]

Apply a FSN within each subgrid inG 1,0 in parallel
[57]

Switch from boustrophedon encodingm C1 tom C2 as described in Corollary 10
[58]

Apply a FSN within each subgrid inG 0,1 in parallel
[59]

All steps except the encoding switches can be imple- mented in constant depth

Apply a FSN within each subgrid inG 1,1 in parallel. All steps except the encoding switches can be imple- mented in constant depth. Thus, the asymptotic gate- count and depth scaling is determined by the cost of switching between the boustrophedon encodings. Corollary 17.Including additional fermionic interac- tions between modes∥x i −x j∥∞ ≤δ+O(1)to the ...
[60]

Go to the compressed basis by applyingL
[61]

ApplyC ′ 2D in the compressed basis
[62]

Combing lemma 4 with lemma 23 to find theCZ gates, we find the resulting ordering ism (c,p,r) S

Return from the compressed basis by applyingL †. Combing lemma 4 with lemma 23 to find theCZ gates, we find the resulting ordering ism (c,p,r) S . This code switch is illustrated in Fig. 9b) in the compressed basis. The parallel CNOT laddersLin Eq. (22) for the lowest dimension can be generalized for dimensionias Li = 0Y i′=L−2 CNOT⟨i′⟩,⟨i′+1⟩,(F2) where ...
[63]

Initialize in one of them Cc,Cr boustrophedon en- codings
[64]

Apply a FSN within each subgrid of a correspond- ingG b
[65]

Apply a FSN within each subgrid of the secondG b that corresponds to this boustrophedon encoding
[66]

Switch to another boustrophedon encoding using Lemma 26
[67]

Thus, the asymptotic gate- count and depth scaling is determined by the cost of switching between the three-dimensional boustrophedon encodings

Continue until all geometrically local pairwise in- teractions have been implemented All steps except the encoding switches can be imple- mented in constant depth. Thus, the asymptotic gate- count and depth scaling is determined by the cost of switching between the three-dimensional boustrophedon encodings. d-Dimensional boustrophedon encodings The method...