Named gates and a richer model

A design note for the next phase: moving from "any perfect entangler" to a named gate, and from the paper's idealised Hamiltonian to a model closer to current hardware.

Status

The named gate is done (v2.16.0, gecko): option B below, the average gate fidelity to √iSWAP with free virtual-Z corrections (--objective sqrt_iswap). The richer model is being built as jackal (i10): the tunable-coupler device of Sung et al. without the RWA, with direct coupling, a SQUID flux curve, bounded flux, 1 ns AWG samples and a flux-line filter. Decoherence and the architecture after it are still open decisions.

Should we target a named gate?

Yes. A perfect entangler is a set of gates, so an agent's result cannot be compared with published two-qubit gate fidelities, compiled into circuits, or benchmarked on hardware without a second step that turns it into a specific gate. Every experiment below reports a fidelity to a named gate.

The natural first target is √iSWAP. The paper's control follows the parametric tunable-bus scheme of McKay et al., PR Applied 6, 064007 (2016), which modulates the bus at the qubit-qubit detuning to create a resonant XX + YY (iSWAP-type) interaction. √iSWAP needs half the rotation of iSWAP, so it should be reachable in about the same ~10 ns. CZ is possible too (through the |11⟩-|20⟩ or |11⟩-|02⟩ transitions, which live in the same N = 2 sector) and is the gate most platforms report, but it needs a different modulation frequency.

How the reward changes

Today: \(J_T = 1 - (C + 3U)/4\), with \(C\) the concurrence (1 anywhere in the perfect-entangler polyhedron) and \(U\) the unitarity.

Option A: same gate up to local operations. Replace the concurrence by a local-invariant functional that is minimal only at the target's Weyl point, for example √iSWAP at \((c_1, c_2, c_3) = (1/4, 1/4, 0)\) in units of π (Watts et al., PRA 91, 062306 (2015)). Single-qubit corrections are left free. Small change: metrics.c1c2c3 already gives the coordinates.

Option B: average gate fidelity with free Z corrections (recommended). On hardware, Z rotations are "virtual" and cost nothing (McKay et al., PRA 96, 022330 (2017)), but X/Y single-qubit gates are not. So the fair target is the average gate fidelity to √iSWAP after the best single-qubit Z phases. For a gate \(G\) restricted to the computational subspace (possibly non-unitary because of leakage) and target \(V\), with \(M = V^\dagger G\) and \(d = 4\) (Pedersen et al., Phys. Lett. A 367, 47 (2007)):

\[ F_{\text{avg}} = \frac{\operatorname{Tr}(M M^\dagger) + |\operatorname{Tr} M|^2}{d(d+1)}, \qquad F = \max_{\phi_0, \phi_1} F_{\text{avg}}\big(Z_{\phi_0}\otimes Z_{\phi_1}\, G\big). \]

The first term penalises leakage (it is \(4U\)), so the separate unitarity term goes away. The reward becomes \(-\log_{10}(1 - F)\), the same shape as now. The maximisation over two phases is a small inner problem over two (or, with corrections before and after the gate, four) phases; a few Newton or Adam steps in JAX, differentiable end to end.

Decided (2026-09-25): option B

A keeps the paper's spirit (the agent finds the entangling power; single-qubit gates are compiled away). B is what an experiment measures with interleaved randomised benchmarking. B was implemented in v2.16.0 (metrics.fidelity_free_z); A remains available through the Weyl coordinates as a diagnostic.

Do we need more states?

Not with this Hamiltonian. It conserves the excitation number, so the four computational states already give the full 4×4 gate: |000⟩ is fixed, |010⟩ and |100⟩ live in the 3-dimensional N = 1 sector, |110⟩ in the 6-dimensional N = 2 sector. A named-gate reward uses the same \(G\) we already compute; only metrics.py and env.step change.

Yes, as soon as the model stops conserving excitations (next section). Then the propagator mixes sectors, and we propagate the full space: 27 states with 3 levels each, 64 with 4. That is still cheap in JAX: one 27×27 eigh per sample instead of a 3×3 and a 6×6, roughly 10-20× slower than now, but still far faster than v1. Jackal's device model does better: without the RWA, parity is still conserved, so the space splits into an even block of 7 and an odd block of 10 states (up to 3 excitations), checked against the full 27-level space to 1e-5 in fidelity.

Limits of the current Hamiltonian

Approximation	Why it matters
Rotating-wave approximation on the couplings (drops \(a_j b\), \(a_j^\dagger b^\dagger\))	It is what makes excitation number conserved. The dropped terms shift the levels by about \(g^2/(\omega_q + \omega_c) = (0.1\,\text{GHz})^2 / 12.5\,\text{GHz} \approx 0.8\) MHz, which is larger than in-cooldown drift. A pulse tuned to J_T ~ 1e-5 in the RWA model will miss on real hardware.
Linear control \(u(t)\, b^\dagger b\)	A real coupler is tuned by flux. A SQUID transmon's frequency varies non-linearly with flux, the couplings \(g_j\) change with the coupler frequency, and the flux line filters the pulse. None of this is modelled.
Unrealistic coupler excursions	The control bound allows the coupler to move ±3.2 GHz around 7.445 GHz, and the paper's RL pulse uses up to 2.9 GHz of it, taking the coupler below both qubit frequencies (5.03 and 5.89 GHz). Real couplers have a limited tuning range and crossings cause leakage.
No direct qubit-qubit coupling	Tunable-coupler designs use a direct capacitive coupling \(g_{12}\) that interferes with the coupler path to cancel residual ZZ (Yan et al., PR Applied 10, 054062 (2018)). Without it we cannot model the idle ZZ that limits real devices.
Closed system	T1/T2 enter only at evaluation.
Two qubits	No spectator qubits or crosstalk.
3 levels per mode	Adequate here (paper appendix A), but leakage into the coupler's second excited state matters in modern couplers.

Moving towards the state of the art

State-of-the-art two-qubit gates on superconducting hardware, with the model each needs:

Architecture	Result	What it would take
Transmons + tunable coupler, CZ and iSWAP	CZ 99.76 %, iSWAP 99.87 % (Sung et al., PRX 11, 021058 (2021))	Same device family as the paper. Model of Yan et al. (2018) without RWA, with \(g_{12}\), flux-tunable coupler and bandwidth-limited flux pulses.
Double-transmon coupler (DTC)	CZ 99.90 % between highly detuned qubits, with a model-free RL pulse optimisation (Li et al., PRX 14, 041050 (2024))	Two coupler transmons; the most directly relevant: RL is already how the record was reached.
Fluxonium qubits + transmon coupler	CZ 99.922 % after RL-based optimisation (Ding et al., PRX 13, 031035 (2023))	Fluxonium Hamiltonians (many levels, low frequency); a larger change of physics.

My recommendation, in order:

1. v3 model: a realistic tunable coupler in the paper's device family. Full-space propagation without RWA, direct coupling \(g_{12}\), coupler frequency as a function of flux with a bounded range, a flux-line filter on the pulse, and a √iSWAP (then CZ) target with Option B. The circuit parameters can come from scqubits rather than being set by hand. This keeps continuity with the paper and closes the largest modelling gaps.
2. Then the double-transmon coupler. It is where RL already holds the experimental record, so a simulation study (model-based RL, robustness to recool drift, transfer across devices) would speak directly to current practice.

Decision: which architecture after v3?

DTC keeps us in transmon physics and next to an RL result from 2024; fluxonium is a bigger change with the highest reported fidelity. Is there a hardware group we could validate with? That should decide it.