---
title: Always-On CZ Gate in Quantum Systems
url: https://www.emergentmind.com/topics/always-on-cz-gate
type: topic
---

# Always-On CZ Gate in Quantum Systems

An always-on CZ (controlled-Z) gate is a two-qubit entangling operation physically realized by harnessing the ever-present, “always-on” $ZZ$-type interaction in quantum hardware. Rather than employing tunable couplers to modulate the $ZZ$ interaction, always-on CZ gates exploit this residual coupling directly, often combining it with pulse shaping or timing protocols to achieve high on/off contrast, minimal crosstalk, and robustness to device noise. This approach figures prominently in superconducting, semiconductor spin, and exchange-only qubit platforms, with scalable protocols demonstrated for systems lacking dynamic isolation between qubits.

## 1. Physical Origin and Theoretical Framework

Always-on CZ gates are founded on the presence of a static two-qubit $ZZ$ interaction or its analog, which can be generically described by the effective Hamiltonian
\[
H_{\mathrm{eff}} \simeq J_{ZZ} \, \sigma_z^{(1)}\sigma_z^{(2)} + (\mathrm{single-qubit\,terms})\,.
\]
Here, $J_{ZZ}$ is the $ZZ$-coupling strength (units of frequency), and $\sigma_z^{(i)}$ acts on qubit $i$. This coupling arises in multiple architectures:
- **Superconducting circuits:** From virtual transitions between $|11\rangle$ and higher-excited states in coupled transmons, flux qubits, or hybrid devices [2002.07560, 2103.12305].
- **Semiconductor spins:** From the always-on Heisenberg exchange $J\,\mathbf{S}_1\cdot\mathbf{S}_2$ in double dots or linear chains, projecting to $ZZ$-type phases in appropriate bases [1911.12351, 2101.08840, 2503.12795, 1602.00320].
- **Exchange-only logical qubits:** From perturbative Ising-type terms in encoded three-spin systems [1602.00320].

Allowing the system to evolve for a precise duration $T_{\mathrm{CZ}}$ imparts the conditional $\pi$ phase on $|11\rangle$, enacting the CZ gate:
\[
U(T_{\mathrm{CZ}}) = \exp(-i\,J_{ZZ}\,T_{CZ}\;\sigma_z^{(1)}\sigma_z^{(2)}) = \mathrm{CZ} \quad (\mathrm{up\,to\,local\,phases})
\]
with $T_{\mathrm{CZ}} = \pi/(2 J_{ZZ})$. These identities are exact when single-qubit terms are compensated or tracked in software.

## 2. Realizations in Superconducting Qubits

### AB-Type Opposite-Anharmonicity Architectures

An advanced scheme uses pairs of superconducting qubits with opposite-sign anharmonicities—transmon ($\alpha_a<0$) and C-shunt flux qubit ($\alpha_b>0$)—coupled via a fixed capacitive link [2002.07560]. Key elements:
- **Hamiltonian engineering:** The $ZZ$ interaction can be controlled or nulled by tuning the detuning $\Delta = \omega_a-\omega_b$ to satisfy $\Delta = \alpha_b$ (the “zero-ZZ” point), exploiting cancellation of second-order virtual processes between $|11\rangle\leftrightarrow|02\rangle$ and $|11\rangle\leftrightarrow |20\rangle$.
- **High on/off ratio:** At the off point, residual $|J_{ZZ}|\lesssim60$ kHz, while at the on point, $J_{ZZ,\,\mathrm{on}} \sim 2g$ (e.g., $30$ MHz for $g/2\pi=15$ MHz), yielding an on/off contrast $\gtrsim500$.
- **CZ protocol:** Keep qubit $b$ at its parking frequency; pulse $a$ to the interaction frequency to enable maximal $ZZ$; hold for a time to produce a $\pi$-phase on $|11\rangle$; return to parking point.
- **Performance:** Leakage and SWAP errors $\lesssim10^{-4}$ for symmetric devices, $F_{CZ}>99.999\%$; typical fabrication variation still results in $F_{CZ}\gtrsim99.9\%$.
- **Crosstalk suppression and scaling:** Lattices of alternating A–B qubits can be tiled so all AB links operate at the zero-ZZ idle point, suppressing static crosstalk below $100$ kHz, with parallel CZ operations achievable by pulsing only the targeted A qubits.

### Always-On Capacitive Coupling in Transmons

In capacitively coupled transmon systems, a strong residual $ZZ$ interaction ($\zeta/2\pi\sim9.29$ MHz) is exploited without dynamic coupling control [2103.12305].
- **CZ by free evolution:** Letting the system evolve under $H=\zeta\,\frac{\sigma_z^{(1)}}{2}\frac{\sigma_z^{(2)}}{2}$ for time $t_g=\pi/\zeta$ realizes the ideal CZ gate. For the cited parameters, $t_g=53.8$ ns.
- **Calibration:** The exact phase accumulation is verified by process tomography or conditional Ramsey-type measurements; single-qubit $Z$ rotations are adjusted to remove systematic phase errors.
- **Achieved fidelity:** Quantum process tomography yields $F_{CZ}=97.8\%$; master-equation simulations indicate that decoherence (T$_1$, T$_2$) and coherent leakage are the main limitations.
- **Scalability:** This CZ primitive fits naturally into architectures where tunable couplers are difficult to implement; the same mechanism can underpin single- and two-qubit gates with the always-on ZZ.

## 3. Spin Qubits and Exchange-Locked Architectures

### Silicon Double Quantum Dots and Pulse-Shaping for Robustness

Always-on $J$ in silicon quantum dots induces unwanted mixing, particularly when Zeeman splittings are nearly resonant. Nonetheless, robust CZ can be implemented [1911.12351]:
- **Hamiltonian reduction:** The full (1,1) electron manifold is block-diagonalized. The effective interaction is $JZZ$ plus single-qubit corrections.
- **Smooth pulse shaping:** A single, band-limited envelope $\Omega_2(t)$ (on one qubit) is synthesized using a “generating function” formalism (Barnes et al.), solving for $\Omega_2(t)$ such that the composite evolution yields a robust CZ up to local $ZX$ terms and is first-order insensitive to both exchange noise $\delta J$ and crosstalk.
- **Pulse design:** The envelope is parametrized as a function of a phase $\Phi(\chi)$, with nontrivial constraints enforcing the elimination of first-order Magnus errors. Additional virtual $Z$ gates compensate Bloch–Siegert shifts arising beyond the RWA.
- **Fidelity:** Numerical simulations give worst-case infidelity $<10^{-4}$ for $\delta J/J\lesssim7.5\%$ and realistic charge/qubit fluctuations.
- **Crosstalk suppression:** The pulse shaping protocol also cancels leading-order $ZX$ crosstalk—critical for multi-qubit arrays.

### Chain Architectures and Two-Tone Robust Pulses

In three-spin chains (e.g., qubits 1–2–3), always-on exchange yields simultaneous $ZZ$ interactions on adjacent pairs [2101.08840]. 
- **Single-tone protocol:** Driving only one edge qubit while choosing total pulse time $t_p=\pi/J_{23}$ ensures trivial identity evolution on unwanted pairs; the 1–2 pair acquires the CZ phase via the $ZZ$ interaction.
- **Limitations:** Single-tone pulses are not robust to realistic fluctuations: $>0.2\%$ amplitude error or $>2\%$ exchange variation degrade $F<99.99\%$.
- **Two-tone shaped pulses:** Driving both qubits with analytically constructed “noise-cancelling” pulses ensures first-order cancellation of errors in both exchange and amplitude. Numerical optimization yields robust CZ pulses (duration $\sim7\,\mu$s at $J/h\approx2$ MHz) retaining $F>99.99\%$ even under $\pm3.5\%$ disorder.

### AEON (Always-On, Exchange-Only) Qubits and Sweet Spots

Exchange-only logical qubits built from three spins in linear TQDs (AEON) enable always-on two-qubit gates while maintaining charge noise protection [1602.00320]:
- **“Full sweet spot”:** Each qubit and its coupling are operated at bias points where their respective energy splittings are first-order insensitive to detuning noise, specified by analytic conditions on the two relevant detuning parameters $(\epsilon, \epsilon_M)$.
- **CZ implementation:** An exchange $J_c$ is pulsed on between two AEON qubits, projecting to an effective Ising coupling $J_{zz}\sigma_z^A\sigma_z^B$ ($J_{zz}=J_c/36$). Evolving for $t_{CZ} = \pi\hbar/(4J_{zz})$ achieves the controlled-Z up to local Z rotations.
- **Noise resilience:** At the sweet spot, only second-order charge noise enters, and fidelities $>99.9\%$ are achievable for gate durations $t_{CZ}\sim$ hundreds of ns.

## 4. Robust Control and Error Suppression in Always-On Architectures

Advanced protocols exploit pulse-shaping techniques to refocus coupling or suppress crosstalk and noise in always-on systems [2503.12795]:
- **Composite identity pulses:** Each qubit is subjected to a shaped $X_{2\pi}$ "robust identity" pulse during the $ZZ$ wait. Pulses are designed (via Fourier synthesis) such that first-order error trajectories vanish for all $Z_j$, $X_jZ_k$, and $Y_jZ_k$ contributions, i.e., static $ZZ$ noise and time-dependent $XZ/YZ$ crosstalk are suppressed.
- **CZ sequence:** Both qubits execute simultaneous robust $X_{2\pi}$ pulses for time $T_{CZ}$; evolution is solely under $J Z_1Z_2/4$, yielding the desired entangling phase. For $J/2\pi=20$ MHz, $T_{CZ}=25$ ns.
- **Error budget:** Simulations yield $1-F\lesssim10^{-4}$ for $\delta J/J\le1\%$ and $T_2\sim5\,\mu$s.
- **Scaling and mitigation of chaotic growth:** The same pulse sequences may be applied globally in 1D or 2D arrays, effectively decoupling unintended pairs and halting otherwise rapid entanglement growth characteristic of untailored always-on couplings.

## 5. Comparative Summary of Implementations

| Architecture              | $ZZ$-on/off Ratio | Typical $F_{CZ}$     | Scalability / Crosstalk Suppression          |
|---------------------------|:-----------------:|:--------------------:|:---------------------------------------------|
| Supercond. AB (Opp. Anh.) | $\gtrsim500$      | $>99.9$–99.999%      | 2D AB tiling; negligible spec. crosstalk     |
| Cap. coupled transmons    | Not switchable    | $\approx 97.8$%      | Suitable for fixed lattices                  |
| Si double dots            | Always-on         | $>99.99$% (robust)   | Pulse-shaped, local crosstalk suppression    |
| 3-spin AEON/chain         | Always-on         | $>99.9$%             | Full sweet spot; two-tone robust pulses      |
| All-spin, robust pulses   | Not needed        | $>99.99$%            | Arbitrary lattice, dynamic crosstalk suppression |

Always-on CZ gates obviate the need for high-contrast tunable couplers, enabling frequent, rapid entangling operations throughout an array. Robust-control protocols and architectural biases (e.g., opposite-anharmonicity pairing, AEON sweet spots) yield fidelities sufficient for fault-tolerant thresholds under realistic hardware constraints, with error rates set chiefly by device coherence, pulse calibration, and dynamical suppression of crosstalk.

## 6. Scalability, Spectator Errors, and System Integration

Efficient scaling of always-on CZ gates relies on the suppression of spurious $ZZ$ crosstalk and the ability to parallelize operations:
- **Superconducting AB tiling** allows all AB links to be on at once in idle; when a CZ is needed, a targeted A qubit is pulsed, while others remain at the zero-ZZ point [2002.07560].
- **Spin qubit chains/arrays** use pulse timing (via identity operations) and pulse shaping to selectively accumulate CZ phases only on intended pairs, suppressing the rest [2503.12795, 2101.08840].
- **Robust global sequences** avoid chaotic entanglement or error propagation common in naive always-on settings; scalable protocols thus ensure that high-fidelity, low-crosstalk entanglement is sustainable in deep circuits and large systems.

A plausible implication is that always-on CZ gates, when combined with advanced robust-control pulse sequences, can match or even exceed the scalability and robustness of architectures built on dynamically switchable couplers, while retaining circuit simplicity and high-speed operation.

Source: https://www.emergentmind.com/topics/always-on-cz-gate