Orbital rotation versus a fermionic swap network: exact checks and 6×6 gate costs

Lees de volledige Nederlandse versie · 2D cuprate project overview

In our 6×6 compilation test, orbital rotations require approximately 14.2% more CZ gates than the existing fermionic swap network. Circuit depth remains on the same scale. On 2×3 and 3×3 lattices, both routes converge towards the same exact dynamics. This is an offline study of accuracy and circuit resources, conducted on 29 October 2026: no new hardware measurements were performed and no QPU time was used.

The same model, two circuit constructions

Both routes simulate the full fermionic 2D Fermi-Hubbard model, including nearest-neighbour hopping, diagonal hopping in both directions and the onsite interaction. The parameters are t = 1, U = 8, t′ = −0.25 and chemical potential μ = 0. We use open boundaries and the same striped Néel initial state within each comparison.

The 2×3 lattice contains four electrons, the 3×3 lattice eight and the 6×6 lattice 32, equally divided between the two spins. These sizes therefore do not constitute a sweep at fixed doping fraction. Time T is expressed in ℏ/t.

The new route diagonalizes the hopping matrix. An orbital rotation transforms kinetic evolution into a basis requiring only occupation-dependent phases, followed by the inverse rotation. This kinetic step is exact up to numerical rounding. The interaction is still approximated through a symmetric Strang splitting: half an onsite step, a full kinetic step and another half onsite step. The existing swap network uses a different symmetric ordering of pair terms. Their states need not coincide at finite time steps.

The approach was inspired by Haiqu’s public Hubbard circuit code. Our implementation retains our own geometry, initial state and fermionic conventions. This does not reproduce Haiqu’s complete hardware, mitigation or KQD workflow.

Exact checks on 2×3 and 3×3

We compare complete simulated states with evolution under an independently constructed Hamiltonian in the sector with fixed spin-resolved particle numbers. The tests cover T = 0.1 and 0.4 with two, four and eight time steps. The reference is also checked using four shorter exponentiations. These checks apply to the selected initial states; they do not establish an accuracy bound for every possible state.

The table shows eight steps at T = 0.4. D is the mean probability of double occupation per site. State error is the L2 distance after aligning the global phase, not a hardware error percentage.

The orbital route has a slightly smaller state error here, but a slightly larger doublon error on 3×3. There is no uniform improvement across observables. Across all time-step comparisons, the observed convergence order ranges from 2.001 to 2.103: approximately four times less state error when the step size is halved, as expected for a second-order method. The separate U = 0 control confirms exact kinetic evolution for the orbital route to numerical precision.

6×6: the cost of the compiled circuit

For 6×6 we compare circuit resources only: 72 logical qubits, two time steps and final Z-basis measurement. Both routes use the same stored Fez target from 21 October 2026, initial physical layout, optimization level 3 and SABRE seeds 9 and 17. Both seeds yield the counts and depths below. This is not an exhaustive compiler optimization search.

Before decomposition into native gates, orbital rotations appear cheaper: 9,044 versus 10,368 logical two-qubit gates, a reduction of 12.77%. Conversion to CZ gates reverses that advantage, producing 14.16% and 14.21% more CZ gates, respectively. CZ counts are identical with and without the Fez connectivity constraint, so routing adds no CZ gates for this layout. Two-qubit depth increases only from 564 to 570. Counting logical gates alone does not predict hardware resources here.

The counts include state preparation and Z measurements. The compiler can remove terminal phases that do not affect these measurements. These are therefore not resource counts for an arbitrary application of the complete unitary operator. Depth means gate layers, not seconds. Likewise, the target register width of 156 does not mean 156 active qubits: only 72 are active.

Verification and interpretation

All eight implementation tests pass, including a full operator check on 2×2 and a negative control for incorrect fermionic reordering. A separate verifier checks 466 numerical comparisons, recounts the eight retained QPY circuits and verifies file hashes. This audit uses saved reference states: it does not reconstruct the Hamiltonian or independently repeat the U = 0 propagation.

For this compilation, orbital rotations do not save CZ gates. The small systems support the correctness and second-order convergence of both routes. There is no state or MPS accuracy check for 6×6; equal step counts do not guarantee equal error there. These results establish no hardware advantage, speed advantage or performance comparison with Google or Haiqu. A comparison at matched 6×6 accuracy first needs an appropriate classical reference.

Download the data appendix containing the numerical tables, an audit summary and hashes of the source artifacts.