Cross-resonance gate on two coupled transmons

Task id: transmon/CRGate-v0. Model: qenvs.models.two_transmon.TwoTransmon. Environment: qenvs.envs.gate.GateControl with subspace=(0, 1, d, d + 1). Task module: qenvs.tasks.cross_resonance.

Two fixed-frequency transmons (Duffing oscillators, levels = \(d\) states each, 3 by default) with an exchange coupling \(J\). Both are driven at the frequency of the target transmon: the cross-resonance (CR) drive on the control and, on the target, a drive that can act as a cancellation tone. The agent or optimiser builds the two-qubit gate \(ZX(\pi/2) = \exp[-i (\pi/4)\, Z \otimes X]\) on the computational states \(|00\rangle, |01\rangle, |10\rangle, |11\rangle\) (control digit first). Population in any other state of the \(d^2\)-dimensional space counts as leakage.

Units

Time is in nanoseconds (ns). Angular frequencies, including the Hamiltonian, the actions and action_bound, are in rad/ns. Parameters you type are ordinary frequencies in GHz (detuning_ghz, anharmonicity_c_ghz, anharmonicity_t_ghz, coupling_ghz, max_rabi_ghz), converted with \(\omega = 2\pi f\), and coherence times in microseconds. \(\hbar = 1\).

import qenvs
from qenvs.tasks.cross_resonance import reference_actions

env = qenvs.make("transmon/CRGate-v0")  # closed, 3 + 3 levels, 64 x 5 ns = 320 ns
actions = reference_actions(env)  # calibrated flat-top CR + cancellation tone, (64, 4) rad/ns
state, rewards = env.rollout(actions)  # state.qstate = 9 x 9 propagator (common frame)
print(env.fidelity(state.qstate), env.leakage(state.qstate))  # about 0.98432, 1.3e-4

noisy = qenvs.make(
    "transmon/CRGate-v0", t1_c_us=100.0, t2_c_us=100.0, t1_t_us=100.0, t2_t_us=100.0
)  # 81 x 81 superoperator

python examples/grape_jax.py --task transmon/CRGate-v0 --steps 300 --lr 0.05 --json

Physics

Lab frame (rotating-wave approximation on the coupling), with \(a\) the control and \(b\) the target lowering operator, truncated to \(d\) levels each, \(n_c = a^\dagger a\), \(n_t = b^\dagger b\):

\[ H_{\mathrm{lab}} = \sum_{i \in \{c, t\}} \Big[\omega_i n_i + \frac{\alpha_i}{2} n_i (n_i - 1)\Big] + J\,(a^\dagger b + a b^\dagger) + \text{drives}. \]

Common frame. Both transmons are written in the frame rotating at the drive frequency \(\omega_d = \omega_t\) (the bare target frequency), \(\psi_r = e^{i\omega_d (n_c + n_t) t}\psi_{\mathrm{lab}}\). The exchange term commutes with \(n_c + n_t\), so the Hamiltonian is time independent apart from the control envelopes:

\[ H = \Delta\, n_c + \frac{\alpha_c}{2} n_c (n_c - 1) + \frac{\alpha_t}{2} n_t (n_t - 1) + J\,(a^\dagger b + a b^\dagger) + \frac{\Omega_{cI}}{2}(a + a^\dagger) + \frac{\Omega_{cQ}}{2}\, i(a^\dagger - a) + \frac{\Omega_{tI}}{2}(b + b^\dagger) + \frac{\Omega_{tQ}}{2}\, i(b^\dagger - b), \]

with \(\Delta = 2\pi(f_c - f_t)\) = 2 pi detuning_ghz, \(\alpha_i = 2\pi\) anharmonicity_*_ghz and \(J = 2\pi\) coupling_ghz. The tensor order is control \(\otimes\) target: basis index \(i_c d + i_t\), so the computational states sit at indices \((0, 1, d, d+1)\), which is TwoTransmon.computational_indices.

Controls. Four channels in the order [Omega_cI, Omega_cQ, Omega_tI, Omega_tQ], each with the operator convention of the single Transmon (index 1 = excited state, \(\Omega/2\pi\) = Rabi frequency in GHz). Both drives are at \(\omega_t\); there is no drive at the control frequency.

Dissipation (Lindblad, enabled per transmon by t1_c_us, t2_c_us, t1_t_us, t2_t_us): \(\sqrt{1/T_1}\,a\) and \(\sqrt{2/T_\phi}\,n_c\) with \(1/T_\phi = 1/T_2 - 1/(2T_1)\), and the same for the target, exactly as in the single transmon. The jump operators act on the bare (uncoupled) modes.

Frames and the target

Qubit frames. The gate is defined in the frames rotating at each bare qubit frequency, \(\psi_q = e^{i(\omega_c n_c + \omega_t n_t) t}\psi_{\mathrm{lab}}\). Since \(\psi_r = e^{i\omega_t (n_c + n_t) t}\psi_{\mathrm{lab}}\),

\[ \psi_q = e^{+i\Delta n_c t}\,\psi_r ,\qquad U_q(T) = e^{+i\Delta n_c T}\, U_r(T), \]

(TwoTransmon.frame_unitary(T), qenvs.models.two_transmon.to_qubit_frame). Sign test: with \(J = 0\) and no drive, \(U_r = e^{-iH_0 T}\) and \(H_0 = \Delta n_c\) on the qubit states, so \(U_q = 1\) there; the opposite sign would leave \(e^{2i\Delta T}\) on \(|10\rangle, |11\rangle\) (check frame_change_identity and tests/test_cross_resonance.py::test_frame_change_sign).

Target. \(ZX(\pi/2) = \exp[-i(\pi/4) Z \otimes X] = (1 - i\, Z\otimes X)/\sqrt{2}\), control \(\otimes\) target, \(Z = \mathrm{diag}(1, -1)\) (qenvs.tasks.cross_resonance.zx_gate). It rotates the target by \(+\pi/2\) about \(x\) when the control is in \(|0\rangle\) and by \(-\pi/2\) when it is in \(|1\rangle\).

What the environment stores. The environment simulates in the common frame. The frame change is diagonal and maps the computational subspace to itself, so with \(D = \mathrm{diag}(1, 1, e^{i\Delta T}, e^{i\Delta T})\) on that subspace, \(F(U_q, ZX) = F(U_r, D^\dagger ZX)\) for both the unitary and the channel fidelity. env.target holds \(D^\dagger ZX(\pi/2)\) for \(T\) = horizon * dt (cr_target), so the final fidelity and the return refer to the qubit frames at the end of the episode (check env_return_equals_gain compares with to_qubit_frame directly). Intermediate values of info["fidelity"] compare the partial map with that end-of-episode target; they are a shaping signal, not the qubit-frame fidelity at time \(t\). With the defaults \(\Delta T = -64\pi\), so \(D = 1\) and \(F_0 = F(\mathbb{1}, ZX(\pi/2)) = 3/5\); in general \(F_0 = (2 + \cos\Delta T)/5\).

Qiskit. Qiskit's RZXGate(theta) documentation defines \(R_{ZX}(\theta)\,q_0, q_1 = \exp(-i\tfrac{\theta}{2} X \otimes Z)\) and prints its matrix in Qiskit's little-endian basis \(|q_1 q_0\rangle\), with \(Z\) on \(q_0\). Reordering that printed matrix to \(|q_0 q_1\rangle\) gives exactly zx_gate(theta) with the control on \(q_0\) (checked for \(\theta = \pi/2\) and 0.7 against the matrix transcribed from the documentation in test_zx_gate_and_qiskit_rzx_convention; Qiskit itself is not imported or run). So \(ZX(\pi/2)\) here corresponds to RZXGate(pi/2) applied with q0 = control and q1 = target, up to that basis reordering. We have not compared with Qiskit's ECRGate or with the CR pulse schedules of any backend.

Defaults and why

kwarg	default	unit	reason
detuning_ghz	-0.1	GHz	\(f_c - f_t\). Not in the straddling regime: straddling means \(\omega_c + \alpha_c < \omega_t < \omega_c\), i.e. \(0 < \Delta < -\alpha_c\) (+0.34 GHz at the defaults), where the first-order ZX coefficient is larger ($
anharmonicity_c_ghz, anharmonicity_t_ghz	-0.34	GHz	same value as the single-transmon tasks (see transmon for its justification)
coupling_ghz	0.003	GHz	$J/
levels	3		per transmon; $
horizon, dt	64, 5.0	steps, ns	320 ns gate. A coarse grid keeps an episode at 64 matrix exponentials of \(9 \times 9\) (closed). 5 ns is half a period of \(\Delta\); see the sampling note below
action_bound	\(2\pi \times 0.06\) = 0.37699	rad/ns	per-channel box, 60 MHz Rabi frequency, above the 47 MHz the reference pulse needs
max_rabi_ghz	(unset)	GHz	alternative: action_bound = 2 pi max_rabi_ghz; giving both raises
t1_c_us ... t2_t_us	None	us	closed by default; any of them switches to Lindblad dynamics
method	Expm()		exact for piecewise-constant controls

Derived values at the defaults: static ZZ \(\zeta/2\pi\) = 115.86 kHz (exact diagonalisation, second order 115.91 kHz) and a first-order ZX rate \(\nu_{ZX}/\Omega\) = 0.0232 (rate in GHz per GHz of CR Rabi frequency).

Sampling note. With piecewise-constant controls the control transmon, detuned by \(\Delta\), does not follow its drive adiabatically: every step of the envelope kicks it. In exploratory runs during development (not part of the test-suite; \(J/2\pi\) = 2.5 MHz, 400 ns gate, flat-top pulses calibrated as below) a 10 ns grid, one full period of \(\Delta\), made these kicks add up coherently and left a control rotation of about 0.4 rad (residual \(X\otimes X\) and \(Y\otimes I\) generators of about 0.19 each; best flat-top infidelity 0.064), while a 5 ns grid (half a period) made consecutive kicks largely cancel (best flat-top infidelity 0.012). If you change dt, expect the reference pulse to change quality for this reason.

ZZ and ZX formulas

Static ZZ. \(\zeta = E_{11} - E_{10} - E_{01} + E_{00}\) of the dressed states (exact diagonalisation of the undriven \(H\), each dressed state assigned to the bare state with the largest overlap; TwoTransmon.static_zz). With \(H_{ZZ} = (\zeta/4) Z\otimes Z\) this is the shift of \(|11\rangle\). Second order in \(J\) (our derivation: only \(|01\rangle \leftrightarrow |10\rangle\), matrix element \(J\), and \(|11\rangle \leftrightarrow |20\rangle, |02\rangle\), matrix element \(\sqrt{2}J\), contribute):

\[ \zeta = 2J^2\left[\frac{1}{\Delta - \alpha_t} - \frac{1}{\Delta + \alpha_c}\right] = \frac{2J^2(\alpha_c + \alpha_t)}{(\Delta + \alpha_c)(\Delta - \alpha_t)}, \qquad \Delta = \omega_c - \omega_t , \]

static_zz_second_order (in GHz). This is \(2\xi\) with \(\xi\) of Magesan and Gambetta (2020), Eq. (4.5), whose Hamiltonian term is \(\xi\, Z\otimes Z/2\). The sign convention: \(\zeta > 0\) means the \(|11\rangle\) energy lies above the sum of the single excitations; for the defaults \(\zeta > 0\).

Measured (check static_zz, \(\Delta/2\pi = -100\) MHz, \(\alpha_c/2\pi = -340\) MHz, \(\alpha_t/2\pi = -300\) MHz, float64):

\(J/2\pi\)	exact \(\zeta/2\pi\)	second order	relative difference
1 MHz	14.5446 kHz	14.5455 kHz	-6.0e-5
3 MHz	130.838 kHz	130.909 kHz	-5.4e-4
5 MHz	363.090 kHz	363.636 kHz	-1.5e-3

The remainder scales as \((J/\Delta)^2\) (test_static_zz_second_order_scaling).

ZX rate at weak drive. For a constant CR drive \(\Omega\) on the control I channel, to first order in \(\Omega\) and \(J\), with \(H_{\mathrm{eff}} = (\nu_{ZX}/2)\, Z\otimes X + (\nu_{IX}/2)\, I\otimes X + \dots\):

\[ \nu_{ZX} = -\frac{J\Omega}{\Delta}\,\frac{\alpha_c}{\alpha_c + \Delta},\qquad \nu_{IX} = -\frac{J\Omega}{\Delta + \alpha_c} \]

(zx_rate_first_order). These are Eq. (4.26) and the IX entry of Appendix C of Magesan and Gambetta (2020); their drive \(\Omega\cos(\omega_d t)(b^\dagger + b)\) becomes \((\Omega/2)(b^\dagger + b)\) in the rotating frame, which is our convention, and their Pauli coefficients multiply \(P/2\), as ours do. We also derived them independently (dress the control by the drive to first order, then project the exchange term onto each control state).

Measured (check zx_rate_weak_drive): constant drive \(\Omega/2\pi\) = 5 MHz for 2 us, \(J/2\pi\) = 2 MHz, other parameters as above. The target rotation vector for the control in \(|c\rangle\) is read from the unitary part of the \(\{|c0\rangle, |c1\rangle\}\) block of the qubit-frame map; \(\nu_{ZX} = (w_0 - w_1)/2\), \(\nu_{IX} = (w_0 + w_1)/2\) from the \(x\) components.

	simulation	first order	relative difference
\(\nu_{ZX}/2\pi\)	77.114 kHz	77.273 kHz	-2.0e-3
\(\nu_{IX}/2\pi\)	22.719 kHz	22.727 kHz	-3.8e-4

At the strong drive of the reference pulse the first-order rate is no longer accurate: the calibrated flat-top needs 47.2 MHz where the first-order formula predicts 43.4 MHz for the same pulse area. Magesan and Gambetta give third-order corrections (their Eq. (4.25)); we did not implement or check them.

Reference pulse

qenvs.tasks.cross_resonance.reference_actions(env) (also qenvs run ... --policy reference):

• CR drive on Omega_cI: a flat-top with \(\sin^2\) edges of 30 ns, starting at \(t = 0\), averaged over each control step (flat_top), amplitude \(sA\) with \(A \ge 0\) and \(s\) the sign of the first-order \(\nu_{ZX}\) (cr_amplitude_sign): \(s = +1\) at the default \(\Delta = -0.1\) GHz and \(s = -1\) at \(\Delta = +0.1\) GHz (the straddling side, where \(\nu_{ZX} < 0\)).
• Cancellation tone on Omega_tI: \(c\,s A f(t)\) with the first-order ratio \(c = J/(\Delta + \alpha_c)\) for levels >= 3 (only the control anharmonicity enters, also when \(\alpha_t \ne \alpha_c\)), which cancels \(\nu_{IX}\) above. For levels = 2 the control has no \(|2\rangle\) level and dressing it by the drive gives \(H_{\mathrm{eff}} = -(J\Omega/2\Delta)\,Z\otimes X\) with no IX term, so \(c = 0\) (cancellation_ratio); the three-level value would add an IX term there (the two-level reference pulse with it reaches F = 0.9644 instead of 0.9948, test_reference_two_level_model).
• \(A\) is calibrated by bisection so that the conditional rotation is \(\pi/2\) (\((\theta_0 - \theta_1)/2 = \pi/2\) from the target Bloch vector reached from \(|00\rangle\) and \(|10\rangle\), conditional_angles), in the closed model, for each of 16 flat-top widths, then for 16 widths around the best one; the width with the highest fidelity wins. The width matters because the control's Stark shift (a \(Z\otimes I\) phase growing with \(\int A^2 dt\)) is not removed by virtual-Z frame updates here.

Measured at the defaults (float64, CPU, dynamiqs 0.3.6, jax 0.10.2):

pulse	\(F\) (closed)	leakage	\(F\) (\(T_1 = T_2 = 100\) us, both)
reference: \(A/2\pi\) = 47.2 MHz, pulse ends at 280 ns, conditional angles \((+1.5715, -1.5701)\)	0.98432	1.3e-4	0.98057
GRAPE, run_grape(steps=300, lr=0.05, seed=0) from a random start: best	0.99929
same run, last iterate (Adam oscillates)	0.99004
GRAPE, 150 steps (as in test_grape_improves_on_the_reference): best, last	0.99805, 0.98894

Other settings (closed, float64, same procedure): \(\Delta/2\pi = +0.1\) GHz, \(A/2\pi\) = \(-46.3\) MHz, pulse ends at 180 ns, conditional angles \((+1.5750, -1.5666)\): \(F\) = 0.99626, leakage 5.7e-4; levels = 2 (either sign of \(\Delta\), no cancellation tone): \(F\) = 0.99476 (the three-level default is 0.98432, so removing \(|2\rangle\) helps by 0.010). For a noisy env the reference is the same as for the closed one (calibration is on the closed model).

GRAPE passes the reference after 90 Adam steps (300 steps take 14 s on CPU, x64; GRAPE was run closed only). Writing the reference's qubit-frame map (unitary part, polar decomposition of the \(4\times4\) block) as \(ZX(\pi/2)\, e^{-iG}\), the Pauli coefficients of \(G\) above 0.01 are \(Z\otimes Y\) -0.100, \(I\otimes Z\) -0.081, \(I\otimes Y\) 0.038, \(Z\otimes Z\) 0.030, \(X\otimes I\) 0.017, \(Y\otimes X\) -0.015: mostly phase errors of the target. They come from two sources of opposite sign. (i) The static dressing of the bare frames: exact diagonalisation of the undriven \(H\) gives a target frequency shift of +89.9 kHz for the control in \(|0\rangle\) but +205.8 kHz for the control in \(|1\rangle\) (second order, \(J^2/\Delta\) = 90 kHz is only the first of these; the second adds the \(|11\rangle \leftrightarrow |20\rangle, |02\rangle\) terms), and a control shift of -89.9 kHz (target in \(|0\rangle\)) or +26.0 kHz (target in \(|1\rangle\)). Over 320 ns without any drive (qubit frames, same decomposition, \(e^{-iG_s}\)) this alone gives \(I\otimes Z\) -0.149, \(Z\otimes Z\) +0.058 (the 115.9 kHz of the static ZZ) and \(Z\otimes I\) +0.032. (ii) The Stark shifts of the drive, which for the reference partly cancel these: the measured \(I\otimes Z\) (-0.081) and \(Z\otimes Z\) (+0.030) are about 0.068 and -0.028 away from the static values (the decompositions do not simply add, since \(G\) is defined relative to \(ZX(\pi/2)\), so this is an estimate). The naive estimate from \(J^2/\Delta\) (90 kHz over 320 ns, coefficient 0.09) is only the static target shift for the control in \(|0\rangle\) and matches the measured \(I\otimes Z\) (-0.081) by coincidence of the two contributions, not because it explains it. We did not separate the drive-induced part further. This task counts all of them because it has no virtual-Z gates.

Validation

Checks live in qenvs.validation.cross_resonance.CHECKS and run in tests/test_cross_resonance.py (float64 via the test conftest, and once more in a float32 subprocess). Measured on CPU, dynamiqs 0.3.6, jax 0.10.2, qutip 5.3.1.

Check	Reference	Tol	Error x64	Error float32
frame_change_identity	\(J = 0\), idle 37 ns: \(e^{+i\Delta n_c T}U_r = 1\) on the qubit states (the opposite sign is 1.90 away); env fidelity with the pulled-back target equals \(F(U_q, ZX)\)	1e-5	7.0e-15	3.0e-6
j0_decoupling	\(J = 0\): 16 random 4-channel steps give \(U_c \otimes U_t\) from two independent Transmon maps (\(J/2\pi\) = 3 MHz: 4e-2 away); DRAG X on the target gives \(1\otimes V\) and $F(1\otimes X) = (2\,\mathrm{Tr}\,mm^\dagger + 4	\mathrm{Tr}\,m	^2)/20$	1e-5
static_zz	relative difference exact vs second order at \(J/2\pi\) = 3 MHz	5e-3	5.4e-4	6.1e-4
zx_rate_weak_drive	max relative difference of \(\nu_{ZX}\), \(\nu_{IX}\) vs first order	1e-2	2.0e-3	2.0e-3
subspace_fidelity_closed_forms	subspace \((0, 1, 3, 4)\): \(F(1, ZX) = 3/5\), embedded \(ZX\) perfect, \(F(CZ, 1) = 2/5\), $	11\rangle \leftrightarrow	20\rangle$: \(F = 3/5\), \(L = 1/4\); unitary and superoperator; the default first-4-levels subspace gives 0.689 instead	1e-9 (x64), 10 eps (f32)
qutip_cross_check	QuTiP 5 from an independent construction: open mesolve from a random \(\rho_0\) (control \(T_1/T_2\) = 30/25 ns, target 40/20 ns) and closed propagator, 16 random 4-channel steps of 0.5 ns	1e-5	1.0e-9	1.7e-6
superop_cptp	open random pulse: $	\mathrm{tr}\,S(\rho) - 1	$, Hermiticity, Choi negativity	1e-5
solver_convergence	Expm() vs Tsit5(rtol=atol=1e-10), closed and open, 16 x 0.5 ns	1e-4	5.2e-9	2.4e-6
env_return_equals_gain	\(\sum r = F_T - F_0\) and \(F_T = F(e^{i\Delta n_c T}U, ZX)\), horizon = 12, dt = 4.3 (nontrivial frame), closed and open	1e-5	2.8e-17	6.0e-8

Discrimination: each check suite was run against in-process mutants, and every mutant fails at least one check: frame-change sign flipped; CR operator doubled; control and target channels swapped; sign of \(\Delta\) flipped; \(\alpha_c \leftrightarrow \alpha_t\); \(J\) doubled; counter-rotating coupling \(a^\dagger b^\dagger + ab\) instead of exchange; control and target \(T_1/T_2\) swapped; dephasing \(\sqrt{1/T_\phi}\,n\); computational indices \((0,1,2,3)\); the ZZ formula without its factor 2 or with \(\alpha_c \leftrightarrow \alpha_t\); the ZX formula with \(\alpha_c/(\alpha_c - \Delta)\); the IX formula with \(\Delta\) instead of \(\Delta + \alpha_c\); cr_target with \(D\) instead of \(D^\dagger\); subspace ignored in qenvs.gates; the superoperator frame change conjugated. Two further mutants are caught by dedicated tests rather than by CHECKS: GateControl.leakage dropping subspace in the open case (the reference pulse then reads 0.124 instead of 1.4e-4, test_open_env_leakage_uses_the_computational_subspace) and the operand order of to_qubit_frame reversed, unitary or superoperator (test_to_qubit_frame_operand_order).

Float32 and the matrix exponential. jax.scipy.linalg.expm (jax 0.10, used by dynamiqs.method.Expm) chooses the number of squarings as \(\lfloor\log_2(\|A\|_1/\theta)\rfloor\) (\(\theta\) = 3.93 in single precision), so the scaled matrix can have a norm up to \(2\theta\), beyond the range of its degree-7 Pade approximant. We measured single-precision errors of up to about \(10^{-3}\) per step for \(\|H\,dt\|_1 \approx 26\) (in float64 the same effect is about \(10^{-9}\)). With 3 levels per transmon \(\|H\|_1\) is dominated by the \(|22\rangle\) energy \(|2\Delta + \alpha_c + \alpha_t|\), so the largest errors sit in the phases of doubly excited states. The checks that compare whole \(9\times 9\) maps therefore use 0.5 ns steps (\(\|H\,dt\|_1 < 3.93\), no scaling). On the default task the calibrated reference pulse reaches \(F\) = 0.984331 in float32 and 0.984316 in float64 (difference 1.5e-5). Use x64 (jax.config.update("jax_enable_x64", True), which the CLI does) for this task.

Assumptions and validity

• Duffing transmons with harmonic matrix elements \(\sqrt{n}\) for drive and coupling; RWA on the coupling and on the drives (frame at \(\omega_t\); valid for \(|\Delta|, |\alpha|, J, \Omega \ll \omega\)).
• Exchange coupling \(J\) is a constant (no tunable coupler, no bus resonator); \(J\) is the effective transmon-transmon coupling.
• Bare qubit frames, not dressed ones: the target and control frequencies used for the frames are the uncoupled ones. The dressed frequencies differ: at the defaults the target is shifted by +90 kHz or +206 kHz and the control by -90 kHz or +26 kHz depending on the state of the other qubit (exact diagonalisation), which shows up as static \(I\otimes Z\), \(Z\otimes I\) and \(Z\otimes Z\) phases over the gate.
• Lindblad dissipation of the bare modes (harmonic \(T_1\) ladder, white dephasing), the same during drive and idle; no \(1/f\) noise, no measurement, no state preparation errors.
• Piecewise-constant controls on the dt grid; no transfer function, no microwave crosstalk between drive lines (Magesan and Gambetta model one; we do not).

Known limitations

• Randomisation (qenvs.wrappers.ParameterRandomization): detuning_ghz is refused in 0.3 because the target stays in the nominal frame. The supported parameters (coupling_ghz and the two anharmonicities) also shift the static dressing phases that the target keeps at their nominal values; estimated (not measured) at about 0.006 rad for a 10% coupling spread, of order 1e-5 in infidelity, growing with the randomisation strength.
• No echo pulses: there is no drive at the control frequency, so the echoed CR sequence (control \(\pi\) pulses) cannot be built. Single-qubit gates on the control are not available.
• No tunable couplers, no spectator qubits, no classical crosstalk.
• RWA on the coupling; counter-rotating coupling terms and Bloch-Siegert shifts are not modelled.
• Bare, not dressed, qubit frames (above), and no virtual-Z frame updates: Stark shifts and static phases count as errors, which makes the task stricter than a calibrated device.
• The reference pulse is a simple calibrated flat-top, not an optimised gate; its quality depends on dt (sampling note) and on the kwargs, and it raises ValueError when no width reaches the conditional rotation within action_bound. It chooses the sign of the CR amplitude from the first-order \(\nu_{ZX}\), so it works for \(\Delta > 0\) as well as \(\Delta < 0\) (checked at \(\Delta/2\pi = \pm 0.1\) GHz and for levels = 2); the sign and the cancellation ratio are first-order results, so very close to a pole of the perturbative rate (\(\Delta \approx 0, -\alpha_c\)) or with levels >= 3 far from the regime where \(J \ll |\Delta|\) they can be wrong and the calibration may fail or give a poor pulse.
• info["fidelity"] before the last step uses the end-of-episode frame (see Frames).
• Float32 accuracy of the default 5 ns steps is limited by the matrix exponential (above).
• Coherence parameters must be Python floats (the presence of jump operators is decided when the model is built).

References

1. E. Magesan and J. M. Gambetta, "Effective Hamiltonian models of the cross-resonance gate", Phys. Rev. A 101, 052308 (2020), arXiv:1804.04073. Drive convention, static ZZ \(\xi\) (Eq. (4.5)), first-order ZX (Eq. (4.26)) and IX (Appendix C).
2. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, "Charge-insensitive qubit design derived from the Cooper pair box", Phys. Rev. A 76, 042319 (2007). Transmon, negative anharmonicity.
3. P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson and W. D. Oliver, "A quantum engineer's guide to superconducting qubits", Appl. Phys. Rev. 6, 021318 (2019).
4. L. H. Pedersen, N. M. Moller and K. Molmer, "Fidelity of quantum operations", Phys. Lett. A 367, 47 (2007). Closed-form average gate fidelity.
5. C. J. Wood and J. M. Gambetta, "Quantification and characterization of leakage errors", Phys. Rev. A 97, 032306 (2018). Leakage and the fidelity of leaky channels.
6. Qiskit documentation, qiskit.circuit.library.RZXGate (matrix and qubit-order convention), https://quantum.cloud.ibm.com/docs/en/api/qiskit/qiskit.circuit.library.RZXGate.