Always-On CZ Gate in Quantum Systems

• Always-on CZ gate is a two-qubit entangling operation that leverages permanent ZZ interactions to impart a controlled π phase shift on the |11⟩ state.
• It employs pulse shaping and precise timing protocols to achieve high on/off contrast and minimal crosstalk across superconducting, semiconductor, and spin qubit platforms.
• Robust control techniques and architectural strategies, such as opposite-anharmonicity pairing and AEON sweet spots, yield gate fidelities exceeding 99.9% for scalable quantum computing.

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 (Zhao et al., 2020, Long et al., 2021).
• 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 (Güngördü et al., 2019, Kanaar et al., 2021, Hai et al., 17 Mar 2025, Shim et al., 2016).
• Exchange-only logical qubits: From perturbative Ising-type terms in encoded three-spin systems (Shim et al., 2016).

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 (Zhao et al., 2020). 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 (Long et al., 2021).

• 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 (Güngördü et al., 2019):

• 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 (Kanaar et al., 2021).

• 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 (Shim et al., 2016):

• “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 (Hai et al., 17 Mar 2025):

• 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 (Zhao et al., 2020).
• 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 (Hai et al., 17 Mar 2025, Kanaar et al., 2021).
• 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.