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arxiv: 2605.05315 · v1 · submitted 2026-05-06 · 🪐 quant-ph

Recognition: unknown

Two Layers, No Swaps: Biplanar SPOQC Architecture Improves Runtime of Fermi-Hubbard Simulation

Boris Bourdoncle , Peter-Jan Derks , Th\'eo Dessertaine , Johannes Frank
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Pith reviewed 2026-05-08 17:12 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Fermi-Hubbard modelquantum simulationTrotterizationquantum error correctionFloquet coderesource estimationbiplanar architecture
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The pith

Biplanar quantum architecture maps spin sectors to separate planes to eliminate fermionic swaps in Fermi-Hubbard simulations.

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

The paper demonstrates that encoding the two spin sectors of the Fermi-Hubbard model on separate planes of a biplanar spin-optical quantum computing architecture removes the need for fermionic swap gates during Trotterized time evolution. This mapping, paired with honeycomb Floquet code encoding and lattice surgery, reduces each Trotter step to a logical depth of 4t_synth + 90 timesteps instead of the 6t_synth + 354 required in single-plane layouts. All algorithmic, logical, magic-state, and synthesis errors are jointly bounded inside a single 1% diamond-norm budget. For an 8 by 8 lattice at U/t = 8 the resulting estimate is a total runtime of roughly two hours on 1.35 million physical qubits.

Core claim

Mapping the two spin components of the Fermi-Hubbard model onto two physical planes eliminates fermionic swap operations and lowers the compiled depth of each plaquette-based Trotter step to 4t_synth + 90 logical timesteps while keeping the total error within a 1% diamond-norm budget.

What carries the argument

Biplanar plane assignment of the two spin sectors, which replaces swap networks with direct transversal CNOTs between corresponding logical qubits.

Load-bearing premise

The biplanar architecture can perform transversal CNOT gates between planes at the stated logical timestep cost without adding unmodeled errors beyond the 1% diamond-norm allowance.

What would settle it

An explicit resource calculation or small-scale simulation showing that the combined Trotter plus logical-plus-synthesis error for the L=8 circuit exceeds 1% diamond norm.

Figures

Figures reproduced from arXiv: 2605.05315 by Boris Bourdoncle, Johannes Frank, Peter-Jan Derks, Th\'eo Dessertaine.

Figure 11
Figure 11. Figure 11: We now explain how to compile a full Trotter step into fault-tolerant instructions. We track the number of |T⟩ states consumed, lattice surgery cubes, transversal inter-plane CNOTs, and logical timesteps. Section IV C 1 is dedicated to the fault-tolerant synthesis of arbitrary-angle single-qubit Z-rotations shared by all sub-evolutions. Sections IV C 2, IV C 3, and IV C 4 derive, respectively, the circuit… view at source ↗
Figure 14
Figure 14. Figure 14: When no correction is needed, the patch sits idle, since all plaquette Hamiltonians are time-evolved in view at source ↗
read the original abstract

We estimate the cost of simulating the two-dimensional Fermi-Hubbard model on a biplanar spin-optical quantum computing (SPOQC) architecture. Qubits are encoded in the honeycomb Floquet code, and we use a circuit-level noise model with explicit timings for each native physical operation. We benchmark lattice surgery and magic state preparation within each plane, and transversal CNOT gates between corresponding logical qubits across planes. We compile a plaquette-based Trotterization of the time evolution operator, mapping the two spin sectors of the Fermi-Hubbard model onto two physical planes. This architectural co-design eliminates fermionic swap operations and reduces the depth of each Trotter step to $4t_{\mathrm{synth}} + 90$ logical timesteps, where $t_\mathrm{synth}$ is the logical timestep cost of arbitrary-angle rotations, compared to $6t_\mathrm{synth} + 354$ in prior single-plane compilations. All error sources - algorithmic (Trotter), logical noise, magic state infidelity, and rotation synthesis - are treated jointly within a single 1% diamond norm budget. For an $L\times L$ lattice with hopping amplitude $t$ and on-site interaction strength $U$, setting $L=8$ and $U/t=8$, we estimate a total runtime of approximately $2$ hours using $1.35\times 10^6$ physical qubits. We find that fallback-based rotation synthesis methods become a scalability bottleneck: the probability that all $L^2$ parallel rotations succeed on the first attempt vanishes exponentially with system size, causing the failure branch to dominate the expected runtime already at moderate $L$.

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 / 1 minor

Summary. The manuscript estimates the resource requirements for simulating the two-dimensional Fermi-Hubbard model on a biplanar spin-optical quantum computing (SPOQC) architecture using the honeycomb Floquet code. By mapping the two spin sectors to separate planes, fermionic swap operations are eliminated, reducing the depth of each Trotter step to 4t_synth + 90 logical timesteps (versus 6t_synth + 354 in single-plane approaches). All error sources are budgeted jointly under a 1% diamond norm. For L=8 and U/t=8, the estimated runtime is approximately 2 hours using 1.35×10^6 physical qubits, with fallback rotation synthesis identified as a scalability bottleneck.

Significance. If the hardware assumptions regarding transversal CNOTs and the Floquet code performance hold, this work demonstrates a meaningful reduction in simulation runtime through architectural co-design. The explicit treatment of multiple error sources in a unified budget and the identification of synthesis as a bottleneck are valuable contributions to the field of quantum simulation resource estimation. The use of circuit-level noise models with timings strengthens the analysis.

major comments (3)
  1. [Abstract] Abstract: The depth reduction claim to 4t_synth + 90 logical timesteps (from 6t_synth + 354) and elimination of fermionic swaps rests on the biplanar transversal CNOT timings and the plaquette-based Trotterization mapping; without the explicit compilation steps or derivation of the 90-timestep count, the central runtime improvement cannot be verified.
  2. [Error budget section] Error budget section: The joint 1% diamond norm envelope for Trotter, logical noise, magic-state infidelity, and synthesis errors is stated to cover all sources for L=8, U/t=8, but lacks an explicit breakdown or table of individual contributions and how the honeycomb Floquet code lattice surgery/magic state operations fit within the modeled noise rates.
  3. [Scalability analysis] Scalability analysis: The claim that fallback synthesis dominates expected runtime at moderate L because the success probability for L^2 parallel rotations vanishes exponentially requires the explicit probability formula and numerical verification for the given parameters.
minor comments (1)
  1. [Abstract] The notation for t_synth (logical timestep cost of arbitrary-angle rotations) should be defined more precisely in the first use, including whether it is per rotation or per parallel layer.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting areas where additional detail will improve verifiability. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the biplanar architecture, error budgeting, and scalability analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The depth reduction claim to 4t_synth + 90 logical timesteps (from 6t_synth + 354) and elimination of fermionic swaps rests on the biplanar transversal CNOT timings and the plaquette-based Trotterization mapping; without the explicit compilation steps or derivation of the 90-timestep count, the central runtime improvement cannot be verified.

    Authors: We agree that the 90-timestep derivation and compilation steps should be presented more explicitly. In the revised manuscript we will add a new subsection (or expanded appendix) that walks through the plaquette-based Trotterization of the Fermi-Hubbard Hamiltonian, shows the mapping of spin sectors to the two planes, details the sequence of transversal CNOTs and intra-plane operations, and tabulates the logical-timestep count that yields the 4t_synth + 90 figure. This will allow direct verification of the depth reduction relative to the single-plane baseline. revision: yes

  2. Referee: [Error budget section] Error budget section: The joint 1% diamond norm envelope for Trotter, logical noise, magic-state infidelity, and synthesis errors is stated to cover all sources for L=8, U/t=8, but lacks an explicit breakdown or table of individual contributions and how the honeycomb Floquet code lattice surgery/magic state operations fit within the modeled noise rates.

    Authors: We will revise the error-budget section to include a dedicated table that decomposes the total 1% diamond-norm budget into its four constituent sources for the L=8, U/t=8 instance. The table will list the allocated diamond-norm contribution of each source together with the underlying noise-rate assumptions for the honeycomb Floquet code, lattice-surgery operations, and magic-state preparation. We will also add a short paragraph explaining how these rates are derived from the circuit-level noise model and how they remain within the overall envelope. revision: yes

  3. Referee: [Scalability analysis] Scalability analysis: The claim that fallback synthesis dominates expected runtime at moderate L because the success probability for L^2 parallel rotations vanishes exponentially requires the explicit probability formula and numerical verification for the given parameters.

    Authors: We will insert the explicit success-probability formula p_success = p_single^(L^2) (where p_single is the per-rotation success probability of the fallback synthesis routine) and provide numerical verification for L = 4, 6, 8 and selected larger values. A short table or plot will show the resulting expected runtime, demonstrating the crossover where the failure branch begins to dominate. These additions will be placed in the scalability subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; depth reduction and runtime follow from explicit circuit compilation and external hardware benchmarks

full rationale

The paper's central claims derive from a plaquette-based Trotterization mapped across two planes, with the reduced depth (4t_synth + 90 logical timesteps) obtained by direct operation counting after eliminating fermionic swaps. Total runtime and qubit counts are computed by scaling with the number of Trotter steps, allocating a joint 1% diamond-norm error budget across Trotter, logical, magic-state, and synthesis errors, and multiplying by benchmarked timings for lattice surgery, magic-state preparation, and transversal CNOTs. These steps use stated external models and explicit counts rather than internal parameter fits or self-referential definitions. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked to force the result.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central runtime and depth claims rest on assumed properties of the Floquet code and a circuit-level noise model whose parameters are taken as inputs rather than derived.

free parameters (3)
  • L=8
    Chosen lattice size for the concrete runtime estimate
  • U/t=8
    Selected interaction ratio for the numerical example
  • 1% diamond norm budget
    Arbitrary but fixed total error tolerance used to combine all error sources
axioms (2)
  • domain assumption Honeycomb Floquet code supports the required lattice surgery, magic state preparation, and transversal CNOTs at the modeled noise levels
    Invoked when benchmarking operations within each plane and between planes
  • domain assumption The plaquette-based Trotterization can be mapped to the biplanar layout without additional logical overhead beyond the stated depth
    Central to the depth reduction claim

pith-pipeline@v0.9.0 · 5615 in / 1757 out tokens · 44575 ms · 2026-05-08T17:12:47.039968+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

97 extracted references · 20 canonical work pages · 3 internal anchors

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    A single transversalCNOTbetween the two planes at each site serves as the diagonalization circuit, reducing the two-qubit interaction to a single-qubitZrotation

    Interaction termH I The interaction sub-HamiltonianH I = U 4 P j Zj↑Zj↓ couples the spin-up and spin-down layers at each lattice site via a two-qubitZZterm. A single transversalCNOTbetween the two planes at each site serves as the diagonalization circuit, reducing the two-qubit interaction to a single-qubitZrotation. The sameCNOTalso serves as the inverse...

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    Pink hopping termH p h H p h is a sum of mutually commuting plaquette Hamiltonians [17]. We define theplaquette Hamiltonianfor a plaquettePas HP =−t X σ=↑,↓ X (j,k)∈∂P c† jσ ckσ +c † kσcjσ =H ↑ P +H ↓ P .(18) Since the pink plaquettes are all structurally identical bulk plaquettes, we derive the time evolution circuit for a single plaquette and apply it i...

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    The diagonalizing circuitC †F † 2,4F † 3,1,

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    two synthesized single-qubitZ-rotations executed in parallel,

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    This totals inL 2 single-qubitZ-rotations ( L2 2 per layer)

    the inverse circuitF 3,1F2,4C. This totals inL 2 single-qubitZ-rotations ( L2 2 per layer). The fault-tolerant cost of theZ-rotations is taken from Section IV C 1. The fault-tolerant cost ofC †F † 2,4F † 3,1 can be read off from Figures 15 and 16 and is summarized in Table VI. The inverse circuitF 3,1F2,4Cis structurally identical and incurs the same cost...

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    Golden hopping termH g h After the pink sub-evolution, the qubit patches of each golden plaquette are separated by an MSF aisle (see Figure 10, plaquette 6-7-11-10). Every second column of patches is shifted through the aisle by growing into the MSF during the first step of the golden diagonalization circuit and shrinking from the trailing side during the...

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    Error budgeting We set the total simulation accuracy target at 1% in diamond norm, U − ˜E ⋄ ≤1%,(23) whereU(·) =U(·)U † is the ideal time-evolution channel and ˜Eis the channel implemented by the noisy fault-tolerant device. Working in diamond norm allows algorithmic approximation errors and hardware noise to be combined in a single budget: U − ˜E ⋄ ≤2ϵ a...

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    Withϵ alg = 0.50%/2.01, Eq

    Trotter step count The second-order Trotter error for the PLAQ decomposition satisfies [17, 18] ∥U(T sim)−S 2(Tsim/r)r∥ ≤W PLAQ T 3 sim r2 ,(26) whereT sim is the total simulation time andW PLAQ ≤κL 2t3 with κ= 1 24 " 3 2 U t 2 + 2 U t (2 √ 5 + 16) + 10 # .(27) which gives the following lower bound for the number of Trotter steps [18]: r≥ √κ L(T simt)3/2 ...

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    The expected T-count is given asn T = 0.57 log2(1/ϵsynth) + 8.83

    Synthesis cost per rotation We choose the mixed-fallback rotation synthesis method described in [20] and also used in [18]. The expected T-count is given asn T = 0.57 log2(1/ϵsynth) + 8.83. Hereϵ synth =ϵ rot/Nrot is the synthesis error per rotation, and Nrot = 4×L 2 ×r≈1.25×10 8, soϵ synth ≈2×10 −13. The expectedT-count then becomesn T = 33. The success ...

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    Total active cube cost, code distance and physical qubit count We have 4×L 2 Z-rotations per Trotter step, which corresponds to 434×4L 2 = 1.11×10 5 active cubes. The interaction subevolution adds 2×L 2 transversalCNOTgates. The two pink time evolutions add 205×L 2 = 1.31×10 4 active cubes. Withw msf = 2, the golden time evolution adds another 1.92×10 4 a...

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    Total number of|T⟩states consumed and magic state factory sizing We have 4L2 Z-rotations per Trotter step, each consuming 33|T⟩states, which adds 33×4L 2 = 8448|T⟩states. Each of the three hopping sub-evolutions (two pink and one golden) requires 4L 2 |T⟩states for the diagonalization circuits (Table VI), contributing an additional 3×4L2 = 768|T⟩states. T...

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    The transversalCNOTgates in the interaction sub-evolution add no logical timesteps, as they appear just twice and do not take up full surgery cubes

    Time The number of logical timesteps needed to implement one Trotter step is 4t synth + 90, wheret synth is the number of logical timesteps required for gate synthesis and 90 accounts for the diagonalization circuits of the two pink and the golden sub-evolutions. The transversalCNOTgates in the interaction sub-evolution add no logical timesteps, as they a...

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    An abort outcome when no success nor failure have been recorded within the allocatedN RU S attempts, which a probabilityp a(NRU S, ε).. We treat this outcome as a failure, see [13]. Notation Expression Description Noiseless value p0(NRU S, ε)p s(ε)1−p r(ε)NRU S 1−p r(ε) Probability to perform a successful RUS-CZwithout losses 1− 1 2NRU S pk(NRU S, ε)p s(ε...

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    An abort outcome when no success nor failure have been recorded within the allocatedN RU S attempts, which a probabilityq a(NRU S, ε). We treat this outcome as a successful measurement followed by both a phase erasure and a measurement erasureC a ⊗ 1 2([I] + [X]) , see [13]. Notation Expression Description Noiseless value q0(NRU S, ε)p s(ε)1−p r(ε)NRU S 1...

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