---
title: Adiabatic Controlled-Z Gates
url: https://www.emergentmind.com/topics/adiabatic-controlled-z-cz-gates
type: topic
---

# Adiabatic Controlled-Z Gates

Adiabatic controlled-\(Z\) (CZ) gates are two-qubit controlled-phase operations realized by varying Hamiltonian parameters slowly enough that the system follows instantaneous eigenstates while accumulating a conditional phase. After removing local single-qubit \(Z\) phases, the logical action is equivalent to \(\mathrm{diag}(1,1,1,-1)\). In the implementations represented here, the control parameter is platform dependent—baseband flux in superconducting circuits, detuning or exchange in semiconductor qubits, and laser detuning or electric field in neutral-atom systems—but the common principle is adiabatic conditional-phase accumulation with leakage suppression through spectral engineering or pulse shaping [2106.00725][1812.03177][1606.08198].

## 1. Logical action and adiabatic principle

The controlled-phase viewpoint is the natural formalism for adiabatic CZ gates. In superconducting-circuit notation, the ideal controlled-phase gate is written as
\[
U_{\textrm{CPHASE}}=
\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&e^{i\phi}
\end{bmatrix},
\]
and an adiabatic pulse generally produces a raw diagonal unitary
\[
U_{\textrm{raw}}=
\begin{bmatrix}
1&0&0&0\\
0&e^{i\phi_{01}}&0&0\\
0&0&e^{i\phi_{10}}&0\\
0&0&0&e^{i\phi_{11}}
\end{bmatrix}.
\]
After local \(Z\) corrections, the nonlocal phase is
\[
\phi'=\phi_{11}-\phi_{01}-\phi_{10},
\]
so the gate is a CZ when \(\phi'=\pi\) [2407.02722].

In tunable-coupler superconducting architectures, the same condition is often expressed as
\[
\phi_{\rm zz}=\int_0^{T_g}\zeta(t)\,dt,
\]
with the CZ obtained when \(\phi_{\rm zz}=\pi\) after removing single-qubit phases [2106.00725]. In quantum-dot hybrid qubits, the entangling term is written as an effective \(\omega_{zz}\) interaction and the total logical evolution remains diagonal in the adiabatic basis, again reducing the gate to a controlled phase plus local \(Z\) rotations [1812.03177]. In exchange-coupled spin qubits, the same logic appears as a pulse-area condition: the exchange-induced conditional phase must equal \(\pi\) modulo \(2\pi\), yielding a gate locally equivalent to CZ after single-qubit \(z\) corrections [2303.18015].

The defining adiabatic requirement is that transitions out of the instantaneous eigenstate manifold remain suppressed. In the floating-coupler transmon analysis this is written as
\[
\frac{\left|\langle i|\frac{dH}{dt}|k\rangle\right|}{(E_k-E_i)^2}\ll 1,
\]
and its practical consequence is that the pulse must move slowly near small gaps and may move faster where the spectrum is more benign [2604.05048]. Across platforms, the gate is therefore not defined by a specific pulse family but by adiabatic following of an eigenstate path that returns population to the computational subspace while imprinting a nonlocal phase.

## 2. Superconducting-circuit realizations

In superconducting qubits, adiabatic CZ gates are commonly implemented by flux-pulsing a tunable element so that the dressed \(\lvert 11\rangle\) branch approaches a non-computational manifold, acquires a conditional dynamical phase, and then returns to the idle point. A representative realization uses two Xmon qubits coupled by a tunable coupler. There the effective coupling \(\tilde g/2\pi\) is tunable from about \(0.4\) MHz up to \(80\) MHz, the idle-point effective \(ZZ\) interaction is below \(500\) kHz, and the adiabatic CZ is implemented with a half-period cosine flux pulse of total duration \(30\) ns. Interleaved randomized benchmarking gives \(r_{\rm CZ}=0.0046(2)\) and \(F_{\rm CZ}=99.53(8)\%\) [2010.14053].

A later fixed-frequency-transmon architecture with a symmetric floating tunable coupler pushes the same principle to shorter durations. In that system the coupler is flux-pulsed while the qubits remain fixed-frequency, and the architecture is designed so that direct and coupler-mediated interactions have opposite signs, allowing exact residual \(ZZ=0\) at idle. The reported gate benchmark was performed at a nearby point with residual \(ZZ \approx 27.3\ \mathrm{kHz}\), while exact \(ZZ=0\) was accessible on the device. The key physical claim is that the dressed \(\lvert 11,0\rangle\) branch is the highest relevant two-excitation level in the straddling regime, so the conditional-phase rate grows without the usual problematic sequence of anticrossings. Experimentally, a \(24\) ns adiabatic CZ reaches \(99.919 \pm 0.010\%\) interleaved-RB fidelity and remains stable over \(8.5\) hours with average fidelity \(99.874 \pm 0.018\%\) [2604.05048].

Adiabatic CZ gates have also been proposed for superconducting architectures in which the tunable element is not a standard grounded coupler. A fluxonium–transmon–fluxonium design for integer fluxonium qubits uses a tunable transmon coupler to suppress static \(ZZ\) at idle and activate a large conditional phase during a flux sweep. In the reported simulations, coherent error is below \(10^{-4}\) for gate duration less than \(40\) ns, below \(10^{-5}\) for less than \(50\) ns, and below \(10^{-6}\) for less than \(70\) ns; specific optimized pulses \(P_1,P_2,P_3\) have durations \(39.5\) ns, \(47\) ns, and \(65.5\) ns with infidelities \(2.5\times10^{-5}\), \(5.8\times10^{-6}\), and \(1.5\times10^{-7}\), respectively, before dissipation is added [2509.04776].

A related but distinct superconducting proposal replaces flux control of qubit frequency with electrostatic tuning of a semiconductor Josephson junction that mediates the inter-qubit interaction. There the gate is adiabatic because the coupling transparency \(T_c\) is switched on and off smoothly with an error-function profile, the dressed computational energies accumulate a nonlocal phase at rate \(\Delta_{\rm CZ}\), and leakage is controlled by sufficiently long switching ramps. In the absence of decoherence, gate time below \(50\) ns with gate error below \(10^{-4}\) is reported for the studied architectures [1801.04291].

## 3. Semiconductor spin and quantum-dot variants

In semiconductor systems, adiabatic CZ gates are typically expressed either as detuning-controlled \(ZZ\) phase accumulation or as exchange-controlled conditional phase. For capacitively coupled quantum-dot hybrid qubits, the adiabatic mechanism uses simultaneous lowering of both qubits’ detunings so that their charge distributions become dipolar and interact electrostatically. The effective logical Hamiltonian contains an \(\omega_{zz}\) term,
\[
\mathcal{H}_{2q,\text{eff}}
\simeq
-\frac{\hbar\omega_{z1}}{2}\sigma_z^{(1)}\otimes I
-\frac{\hbar\omega_{z2}}{2}I\otimes\sigma_z^{(2)}
+\frac{\hbar\omega_{zz}}{2}\sigma_z^{(1)}\otimes\sigma_z^{(2)},
\]
and the gate is produced by integrating the conditional phase generated by \(\omega_{zz}(t)\) during a smooth ramp-hold-ramp sequence [1812.03177].

That work emphasizes a central limitation of adiabatic solid-state phase gates: no conventional static sweet spot exists where all relevant detuning derivatives vanish simultaneously. To address this, it introduces a dynamical sweet spot (DSS), defined by vanishing time-averaged derivatives,
\[
\int \frac{\partial \omega_j}{\partial \varepsilon^{(i)}}\,dt = 0.
\]
For detuning-only control, the best average process infidelity is \(\mathcal{I}\approx 9.2\times10^{-3}\), corresponding to about \(99.08\%\) fidelity, with \(\tau_{\rm ramp}=4.0\) ns, \(\tau_{\rm wait}=8.0\) ns, and \(\tau_{\rm total}=16.0\) ns. Adding tunnel-coupling pulse sequences gives a \(5\times\) improvement in fidelity relative to the detuning-only optimized case, with the best sequences reaching fidelities above \(99\%\) and discussion that they can approach \(\sim 99.9\%\) for DSS-inspired pulses [1812.03177].

For neighboring spin qubits, the conventional adiabatic or static exchange gate is formulated through the Hamiltonian
\[
H = B(\hat S_{1z}+\hat S_{2z}) + \Delta B(\hat S_{2z}-\hat S_{1z}) + J(t)\left(\mathbf{\hat S}_1\cdot \mathbf{\hat S}_2-\frac{1}{4}\mathbbm{1}\right).
\]
In the static case \(J(t)=J_0\), the CZ condition reduces to
\[
J_0\tau_n=\pi+2n\pi,
\]
while the oscillating-exchange generalization becomes
\[
J_0\tau_n+\frac{J_1}{\omega}\sin(\omega \tau_n)=\pi+2n\pi.
\]
The resulting operation is generally locally equivalent to CZ rather than literally \(\mathrm{diag}(1,1,1,-1)\) in the bare basis, with only single-qubit \(z\) rotations needed to recover the canonical form. In the noisy model based on Gaussian fluctuations of \(J_0\), resonant oscillating-exchange CZ and iSWAP gates are reported as comparable to the conventional static CZ for low charge noise, roughly \(\sigma/J_0 \lesssim 1\%\), while the static CZ can remain more robust at larger noise because it uses shorter gate times in the parameter sets considered [2303.18015].

## 4. Neutral-atom and Rydberg implementations

Neutral-atom adiabatic CZ gates exploit Rydberg-mediated conditional phase shifts, but the physical realizations differ substantially in how the adiabatic path is traversed. One route uses Stark-tuned Förster resonances. In the Cs example \(\lvert 90S_{1/2},96S_{1/2}\rangle \rightarrow \lvert 90P_{1/2},95P_{1/2}\rangle\), the two-atom system undergoes a double adiabatic passage through resonance under a time-dependent electric field. The essential result is deterministic phase accumulation: after two identical passages the initial state returns with a phase shift of \(\pi\), so only the doubly excited logical branch acquires the CZ sign change [1606.08198].

Another route is adiabatic Rydberg dressing. In the \(^{133}\)Cs proposal using clock states, the entangling interaction is the difference between the two-atom dressed light shift and twice the single-atom light shift,
\[
J(t)=E_{2\text{-atom}}^{\text{LS}}(t)-2E_1^{\text{LS}}(t),
\qquad
\Phi_J=\int_0^T J(t)\,dt.
\]
When \(\Phi_J=\pi\), the operation is a CZ up to local phases. The proposal further uses a counterpropagating \(\sigma_+/\sigma_-\) geometry to cancel first-order Doppler sensitivity in the adiabatic subspace. For the representative parameters, the predicted CZ gate time is under \(10~\mu\)s, specifically around \(2.3~\mu\)s in the example shown, with error probability on the order of \(10^{-3}\) [1411.2622].

Symmetric adiabatic pulses applied to both atoms provide a third family. In the one-photon ARP scheme, the states \(\lvert 01\rangle\) and \(\lvert 10\rangle\) adiabatically cycle through singly excited Rydberg manifolds while \(\lvert 11\rangle\) evolves through the symmetric state \((\lvert r1\rangle+\lvert 1r\rangle)/\sqrt2\). The gate becomes maximally entangling when \(\phi_1=\phi_2=\pi\). With smooth ARP pulses and Cs parameters, the predicted Bell-state fidelity reaches \(\mathcal{F}=0.9994\), and the abstract summarizes this regime as \(\mathcal{F}>0.999\). A two-photon, globally optimized STIRAP-inspired sequence gives \(\mathcal{F}=0.997\) [1912.02977].

A single-temporal-modulation protocol for \(^{87}\mathrm{Rb}\) realizes a full family of
\[
\mathrm{CZ}_\theta=\mathrm{diag}(1,1,1,e^{-i\theta})
\]
gates by adiabatic following of different eigenpaths for \(\lvert 11\rangle\) and the singly excited states. For \(\theta\in[0.08\pi,\pi]\), reported fidelities are above \(99.7\%\) in less than \(1~\mu\)s even with spontaneous emission included. For the \(\theta=\pi\) benchmark, the predicted fidelity after correcting measurement errors is about \(98.4\%\) [2201.05994].

Echoing rapid adiabatic passage yields a further variant in which two identical RAP pulses generate a closed adiabatic loop. In the blockade regime, the paper states that dynamical phases from the two pulses cancel while a geometric phase of \(\pi\) remains, producing an effective two-qubit gate
\[
\mathcal{U}_{\rm CZ}=\mathrm{diag}(1,-1,-1,-1).
\]
With Cs parameters, the reported CZ fidelity is above \(0.9995\), around \(0.9997\) when \(V_0=(2\pi)\,3\) GHz, and \(0.9963\) in a conservative benchmark with \(\Omega=(2\pi)\,5\) MHz, lifetime \(130\,\mu\)s, and \(V_0=(2\pi)\,400\) MHz [2408.16274].

## 5. Pulse synthesis, superadiabatic acceleration, and analytical design

Because adiabatic CZ gates are limited by the speed–leakage tradeoff, a substantial part of the literature focuses on pulse synthesis. A direct acceleration strategy is shortcut-to-adiabaticity (STA). In coupled Xmon qubits, the relevant subspace is \(\{\lvert 11\rangle,\lvert 20\rangle\}\), and the avoided crossing is used exactly as in a conventional adiabatic gate. The STA construction adds counter-diabatic driving so that the system follows the adiabatic path in shorter time. Experimentally, the resulting pulse has duration about \(52\) ns, with measured fidelity higher than \(96.0\%\) in both quantum process tomography and randomized benchmarking; the reported process fidelity is \(96.59\%\) [1811.08096].

A related but conceptually distinct acceleration is superadiabatic control under parametric modulation. In an 8-qubit transmon chip, modulation of a tunable transmon drives the \(\lvert 11\rangle \leftrightarrow \lvert 20\rangle\) transition, and the superadiabatic construction is used to preserve adiabaticity at a speed close to the quantum limit. The SA-CZ gate uses a total evolution time of \(120\) ns, reaches \(94.2\%\) fidelity with equivalent error rate \(5.8\%\), and is reported as limited largely by decoherence. Simulation without decoherence gives \(99.94\%\), and the paper also reports robustness against \(\pm 10\%\) perturbations in pulse amplitude and duration for the superadiabatic protocol [1906.02992].

Within tunable-coupler superconducting circuits, adiabatically weighted pulse (AWP) design turns the multilevel adiabatic problem into a directly engineered waveform. The derivative of the coupler frequency is expanded as
\[
\frac{d\omega_{\rm c}}{dt}=\sum_{m=1}^{m_{\max}}\lambda_m\sin\!\left(\frac{2\pi m t}{T_g}\right),
\]
and the weighting is informed by a multilevel adiabaticity metric. The reported result is that nonadiabatic errors can reach \(\sim 10^{-5}\) for gate times above about \(24\) ns with only one AWP component, and the paper argues that, with improved coherence, the scheme could potentially achieve a two-qubit gate error rate near \(10^{-5}\) [2106.00725].

Baseband-flux pulse design has also been cast explicitly as a signal-processing problem. In a two-level avoided-crossing abstraction with diabatic states \(\lvert 11\rangle\) and \(\lvert 20\rangle\), the approximate leakage probability is
\[
P_e=\frac{\left|\int \frac{d\theta}{dt}e^{-i\int^t\omega(t')dt'}dt\right|^2}{4},
\]
which, after a nonlinear time transformation, connects leakage to the Fourier transform of the control trajectory. Using that framework, a Chebyshev-based anti-symmetric trajectory is compared with the standard Slepian-based trajectory. In simulation, the Chebyshev design lowers leakage by about \(6.0\%\) in certain cases, maintains similar or smaller pulse duration by about \(0.6\) ns on average, and reduces gate infidelity by \(23.3\%\) on average; the paper notes that this advantage is clearest in the moderate target-infidelity range \(10^{-3}\) to \(10^{-5}\) and can be mostly recovered under hardware constraints around \(F_s=5\ \text{GSa/s}\) and \(bw=2\ \text{GHz}\) [2407.02722].

## 6. Error channels, benchmarking, and conceptual boundaries

The dominant error mechanisms in adiabatic CZ gates are platform specific but structurally similar: leakage near avoided crossings, incoherent decay during longer pulses, flux- or detuning-dependent dephasing, and residual couplings at idle. The tunable-coupler Xmon experiment provides a direct adiabatic–diabatic comparison: the adiabatic CZ uses \(30\) ns and reaches \(99.53(8)\%\), while the diabatic CZ uses \(19\) ns and reaches \(98.72(2)\%\). Purity benchmarking attributes about \(64\%\) of the total two-qubit Clifford error to incoherence in the adiabatic case and about \(47\%\) in the diabatic case, so the faster gate reduces incoherent exposure but pays in increased leakage sensitivity and calibration complexity [2010.14053].

A recent analytical treatment makes explicit that adiabatic superconducting CZ gates are affected by time-dependent dissipation, not merely static \(T_1\) and \(T_\phi\). For a tunable-coupler gate with time-dependent coupler frequency \(\omega_c(t)\), the master equation is written with explicitly time-dependent Lindblad rates,
\[
\dot{\hat \rho}(t) = -\frac{i}{\hbar}[\hat H(t),\hat \rho(t)] + \sum_k \Gamma_k(t)\mathcal D[\hat L_k]\hat \rho(t),
\]
and the paper identifies two principal degradations: flux-dependent coupler dephasing, modeled through \(\Gamma_\phi^c(t)\propto \left|\partial \omega_c/\partial \Phi\right|\), and hybridization, which transfers coupler relaxation and dephasing into dressed logical states through factors of order \(g_{ic}^2/\Delta_{ic}^2\). A central conclusion is that static time-averaged error estimates are insufficient because they miss the time dependence of both the rates and the dressed eigenbasis [2606.20501].

Benchmarking conventions also differ across platforms. Superconducting experiments in this corpus use interleaved randomized benchmarking, purity benchmarking, quantum process tomography, and MLE-based RB analysis [2010.14053][1811.08096][2604.05048]. Neutral-atom works often report Bell-state fidelity or fidelity on uniform superposition inputs, reflecting the relative importance of leakage and phase correctness in blockade-mediated evolution [1912.02977][2408.16274]. This suggests that reported numbers are not interchangeable unless the benchmarking protocol is taken into account.

A recurring conceptual boundary concerns what should not be called an adiabatic CZ gate. Not every controlled-phase operation generated by \(ZZ\) is adiabatic. In the fixed-frequency transmon experiment with strong always-on \(ZZ\), the CZ is implemented by free evolution under
\[
H^{\rm rot}_{q}/\hbar = g_z\,\widetilde{\lvert 11\rangle}\widetilde{\langle 11\rvert},
\]
with \(g_z/2\pi=9.29~\text{MHz}\), gate time \(t_g=\pi/g_z=53.8~\text{ns}\), and reported average gate fidelity \(97.8\%\). That work explicitly realizes a free-evolution conditional-phase CZ, not a pulse-shaped or adiabatic entangling gate [2103.12305]. The distinction matters because adiabatic CZ refers specifically to protocols in which a control parameter is swept slowly enough to preserve eigenstate following, whereas free-evolution, diabatic, or microwave-activated schemes generate the same logical gate by different physical mechanisms.

Source: https://www.emergentmind.com/topics/adiabatic-controlled-z-cz-gates