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arxiv: 2606.17158 · v1 · pith:637T75ICnew · submitted 2026-06-15 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.str-el

Fermionic Hamiltonian engineering with local control

\"Ozg\"un Kum , Matthias Zipper , Ludwig Mathey , Martin Kliesch This is my paper

Pith reviewed 2026-06-27 03:34 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.str-el
keywords fermionic Hamiltonian engineeringlocal controlquantum simulationlinear programmingcomplex tunnellingHarper-Hofstadter modelFermi-Hubbard modelquantum simulators
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The pith

Conjugating system evolution with local fermionic unitaries realizes arbitrary complex tunnelling coefficients limited only by connectivity.

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

The paper presents a method to engineer a wide range of target Hamiltonians in fermionic quantum simulators by interleaving natural evolution under the hardware Hamiltonian with sequences of local unitaries. These sequences are computed efficiently using a linear program. The approach allows arbitrary complex tunnelling terms as long as the underlying lattice is connected. It demonstrates this on large lattices for the Harper-Hofstadter model and an interacting Fermi-Hubbard chain. This expands the class of simulatable models while avoiding energy absorption issues common in other engineering techniques.

Core claim

By conjugating free evolution under the system Hamiltonian with sequences of experimentally feasible local fermionic unitaries obtained via a linear program, the method realizes effective time evolution under a broad class of target Hamiltonians, including those with arbitrary complex tunnelling coefficients constrained only by the connectivity of the system Hamiltonian.

What carries the argument

The linear program that determines the sequences of local fermionic unitaries and free-evolution times to achieve the desired conjugation of the system Hamiltonian into the target Hamiltonian.

If this is right

  • Arbitrary complex tunnelling coefficients become accessible on any connected fermionic lattice.
  • The dynamics of the non-interacting Harper-Hofstadter model on a 1088-mode lattice can be engineered.
  • An interacting Fermi-Hubbard chain with complex tunnelling coefficients can be simulated.
  • The method provides intrinsic robustness to finite-pulse-time errors.
  • The approach avoids continuous energy absorption inherent to Floquet engineering.

Where Pith is reading between the lines

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

  • This could allow simulation of topological phases or other models requiring complex hoppings on hardware with limited native interactions.
  • The linear programming approach might be adapted to optimize for minimal sequence length or robustness to noise.
  • Similar techniques could apply to bosonic or spin systems with appropriate local controls.

Load-bearing premise

The local fermionic unitaries are experimentally feasible on the target hardware and the linear program returns sequences that exactly reproduce the desired target dynamics in the ideal-pulse limit.

What would settle it

An experiment where the measured dynamics under the engineered sequence deviate significantly from the predicted target Hamiltonian evolution, such as incorrect tunnelling phases or amplitudes.

Figures

Figures reproduced from arXiv: 2606.17158 by Ludwig Mathey, Martin Kliesch, Matthias Zipper, \"Ozg\"un Kum.

Figure 1
Figure 1. Figure 1: Schematic overview of the fermionic Hamiltonian engineering protocol. Given a description of a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fermionic Hamiltonian engineering (FHE) maps [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical simulation of the Harper–Hofstadter dynamics on a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical simulation of the edge currents for the Harper–Hofstadter model on a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The three nearest-neighbour tunnelling operators on a triangular lattice plaquette, with each arrow indicating [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A 4 × 4 square lattice with a bipartition into A and B sublattices. Hamiltonian KS is mapped to an engineered quadratic Hamiltonian KT = X {j,k}∈E βjkc † j ck + H.c. (26) This implies that the effective interaction term takes the form Ieff = X b λb V † b HintVb = ∥λ∥ℓ1Hint. (27) Applying this set of pulses to the full system Hamilto￾nian (24) therefore yields the effective Hamiltonian Heff = KT + ∥λ∥ℓ1 IS.… view at source ↗
Figure 7
Figure 7. Figure 7: Numerical simulation of engineered Fermi–Hubbard dynamics on a one-dimensional chain with [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Schematic illustrations of fermionic Hamiltonian engineering (FHE). (a) Conjugation by a local unitary [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Phase transition in the probability of obtaining a feasible matrix as a function of the ratio between the number of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a)Average quantum runtime for the simulation of a target Hamiltonian starting from a quadratic system Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
read the original abstract

Quantum simulators enable the exploration of complex quantum phenomena in condensed-matter systems by reproducing their dynamics on controllable quantum devices. However, experimental constraints often restrict the class of Hamiltonians that can be realized natively. Hamiltonian engineering addresses this limitation by expanding the set of accessible target Hamiltonians from a fixed system Hamiltonian defined by the hardware. We introduce a new framework for fermionic Hamiltonian engineering based on conjugating free evolution under the system Hamiltonian with sequences of experimentally feasible local fermionic unitaries. The required sequences and free-evolution times are obtained efficiently via a linear program. By interleaving system evolution with these local unitaries, our method realizes effective time evolution under a broad class of target Hamiltonians, with intrinsic robustness to finite-pulse-time errors. In particular, we demonstrate that arbitrary complex tunnelling coefficients can be realized, constrained only by the connectivity of the underlying system Hamiltonian. We illustrate this capability by engineering the dynamics of the non-interacting Harper-Hofstadter model on a 1088-mode lattice and an interacting Fermi-Hubbard chain with complex tunnelling coefficients. By construction, our approach avoids the continuous energy absorption inherent to Floquet engineering.

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

2 major / 2 minor

Summary. The paper presents a framework for fermionic Hamiltonian engineering that interleaves free evolution under a fixed native system Hamiltonian with sequences of local fermionic unitaries. Sequences and evolution times are obtained via linear programming; the approach is claimed to realize arbitrary complex tunnelling amplitudes (subject only to the underlying connectivity) with built-in robustness to finite pulse duration. Demonstrations are given for the non-interacting Harper-Hofstadter model on a 1088-mode lattice and for an interacting Fermi-Hubbard chain with complex hoppings; the method is positioned as avoiding the continuous driving and heating issues of Floquet engineering.

Significance. If the central claim holds, the work supplies an efficient, constructive route to a wide class of fermionic target Hamiltonians on hardware whose native interactions are limited to a fixed graph. The use of a linear program for sequence synthesis, the explicit large-scale lattice demonstration, and the avoidance of Floquet heating are concrete strengths that would make the technique immediately relevant to current quantum-simulation platforms.

major comments (2)
  1. [Abstract, §1] Abstract and §1: the assertion that 'arbitrary complex tunnelling coefficients can be realized, constrained only by the connectivity' is load-bearing for the paper's main contribution, yet the manuscript only verifies LP feasibility for the Harper-Hofstadter lattice and the 1D Hubbard chain. No general proof of feasibility for every complex coefficient set allowed by connectivity, nor an exhaustive search for infeasible counter-examples, is provided; this leaves the 'arbitrary' claim unsupported beyond the two specific models.
  2. [§3] §3 (linear-programming formulation): the reduction of the engineering task to an LP is presented as always returning a solution when the target lies within the connectivity graph, but the manuscript supplies no analytic argument or numerical stress-test confirming that the constraint matrix remains full rank or that the feasible set is non-empty for arbitrary complex phases and amplitudes. This directly affects the claim that only connectivity, not the LP itself, limits realizability.
minor comments (2)
  1. [Figures 3-4] Figure captions and axis labels in the 1088-mode lattice results should explicitly state the pulse-duration error model used to demonstrate robustness.
  2. [§4] The manuscript would benefit from a short table comparing the number of LP variables and constraints for the two demonstration systems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and positive assessment of the significance of our work. We address the major comments regarding the generality of our claims in the point-by-point responses below. We will make revisions to clarify and strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the assertion that 'arbitrary complex tunnelling coefficients can be realized, constrained only by the connectivity' is load-bearing for the paper's main contribution, yet the manuscript only verifies LP feasibility for the Harper-Hofstadter lattice and the 1D Hubbard chain. No general proof of feasibility for every complex coefficient set allowed by connectivity, nor an exhaustive search for infeasible counter-examples, is provided; this leaves the 'arbitrary' claim unsupported beyond the two specific models.

    Authors: We agree that the claim of arbitrariness would benefit from a more rigorous justification. The linear programming approach is formulated in a way that is independent of the specific target Hamiltonian, relying only on the ability to apply local unitaries and the connectivity for the free evolution. To address this, we will revise the abstract and introduction to temper the language slightly if necessary, and add a section or appendix with additional numerical experiments on random target coefficients for small lattices to demonstrate feasibility beyond the two examples. We believe this will support the claim without requiring a full analytic proof at this stage. revision: yes

  2. Referee: [§3] §3 (linear-programming formulation): the reduction of the engineering task to an LP is presented as always returning a solution when the target lies within the connectivity graph, but the manuscript supplies no analytic argument or numerical stress-test confirming that the constraint matrix remains full rank or that the feasible set is non-empty for arbitrary complex phases and amplitudes. This directly affects the claim that only connectivity, not the LP itself, limits realizability.

    Authors: The referee correctly notes the absence of an analytic argument for the non-emptiness of the feasible set. The LP is set up with variables corresponding to the durations of free evolution segments and the parameters of the local unitaries, with linear constraints derived from matching the effective Hamiltonian terms. While we do not have a proof that the matrix is always full rank, the structure of the problem suggests that the local controls provide sufficient degrees of freedom to achieve any complex coefficient consistent with connectivity. In the revision, we will include numerical stress-tests, such as solving the LP for numerous random complex targets on small graphs, to empirically confirm that solutions are found whenever the target respects the connectivity. revision: yes

Circularity Check

0 steps flagged

No circularity; constructive LP method is externally solved

full rationale

The paper derives target dynamics by solving a linear program for sequences of local fermionic unitaries interleaved with free evolution under the native Hamiltonian. This LP is an external numerical solver whose feasibility is asserted for the demonstrated cases (Harper-Hofstadter lattice, 1D Hubbard chain) without defining the target coefficients in terms of the same fitted data or reducing any claim to a self-citation chain. The central statement that realizable tunnelling is constrained only by connectivity follows from the method's construction rather than from re-labeling inputs as outputs. No load-bearing step collapses by definition or by internal fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the experimental availability of local fermionic unitaries and the assumption that the linear program yields exact effective dynamics in the ideal case; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Local fermionic unitaries can be implemented experimentally on the hardware
    Invoked when stating that the sequences are experimentally feasible.
  • domain assumption The linear program finds sequences whose interleaved evolution reproduces the target Hamiltonian
    Core of the engineering framework stated in the abstract.

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discussion (0)

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    Translating FHE into the language of quadratic Hamiltonians A pulseVb is generated by a quadratic Hamiltonian Gb = 2π 3 ∑ j∈[n] bjc† jcj,(C19) so the coefficient matrixgb for the generatorGb is given by gb = 2π 3 diag(b1,...,bn),(C20) 23 whereb j∈T≡{−1,0,1}as before. These matrices generate unitaries that act on the smallern-dimensional space vb = e−igb =...