Branch-resolved Pauli-block spectroscopy of residual conditional phase in two-qubit gates

Abstract: Recent progress in quantum physics and quantum technologies is driving quantum computing from the noisy intermediate-scale (NISQ) era toward fault-tolerant operation. High-precision control of two-qubit gates is among the most critical requirements in this transition and hinges on accurate two-qubit calibration. For controlled-phase and CZ-style operations, the residual conditional phase (the nonlocal ZZ-type deviation after local compensation) is weakly resolved at leading order in average infidelity and randomized benchmarking, and repeated Ramsey amplification does not reliably isolate it from ordinary target detuning, SPAM errors, and contrast loss in long sequences. We introduce branch-resolved Pauli-block spectroscopy to estimate the per-cycle residual ZZ-rotation angle theta_c with its sign, from which the controlled-phase residual follows by a fixed convention. The protocol repeats a fixed probe for N cycles, measures the closed Pauli block IX, IY, ZX, and ZY, and forms branch coherences C+ and C- conditioned on the control qubit; theta_c splits the two branch phase slopes in opposite directions, while local target phase beta_c shifts them together. An echoed-cycle variant suppresses removable local terms while preserving the nonlocal contribution. Numerical simulations with injected theta_c, detuning, damping, and SPAM confirm unbiased signed readout where scalar-sector alternatives fail and distinguish opposite-sign errors at equal infidelity. On one superconducting cloud qubit-coupler pair, a pulse-level calibration closed loop shows near-linear injection, preserved branch contrast, and tracking of the native residual conditional phase through one iteration. The approach yields a low-overhead, signed per-cycle estimate of residual conditional phase that standard fidelity benchmarks underresolve at leading order.

• The paper presents a novel branch-resolved Pauli-block spectroscopy protocol that directly estimates the residual conditional phase in two-qubit gates.
• It employs a matrix-pencil method to extract differential branch phase slopes, offering unbiased, sign-resolving measurements even in noisy environments.
• Experiments on superconducting devices demonstrated a 16× reduction in residual phase error, enabling precise and efficient gate calibration.

Branch-Resolved Pauli-Block Spectroscopy of Residual Conditional Phase in Two-Qubit Gates

Introduction and Motivation

The transition from NISQ-scale systems toward fault-tolerant quantum computation exposes the necessity for high-fidelity, repeatable two-qubit gates, particularly for superconducting qubits where coherent miscalibration, rather than stochastic errors, is a primary limitation. Controlled-phase (CZ-type) gates suffer from residual conditional phase—an effective $ZZ$ interaction remaining after compensation of local phases—which is weakly detected by conventional methods such as randomized benchmarking or average infidelity metrics due to their insensitivity to coherent phase errors and absence of sign information. Single-sector Ramsey approaches are also inadequate due to signal degradation via admixture with local detuning, SPAM, and contrast decay over long sequences.

This work introduces a protocol termed branch-resolved Pauli-block spectroscopy for direct, signed, per-cycle estimation of the residual conditional phase in two-qubit gates, addressing pitfalls of prevailing methodologies and providing a low-overhead, metrologically robust diagnostic for calibration and control.

Technical Foundation

The residual conditional phase of a generic diagonal two-qubit gate is encoded as:

$$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$

The physically relevant parameter is the deviation $\delta_{\mathrm{CP}}$ from the target controlled-phase angle, which manifests in a calibrated cycle as an effective $ZZ$ rotation parametrized by angle $\vartheta_c$. The executable cycle is generally:

$$U_{\text{nf}}(\beta_c, \vartheta_c) = \exp[-\frac{i}{2}( \beta_c IZ + (\omega_0 + \vartheta_c) ZZ )]$$

where $\beta_c$ captures ordinary target-qubit detuning. Repeating the cycle to achieve metrological amplification introduces both increased sensitivity and enhanced noise susceptibility (via decoherence and SPAM).

The $ZZ$ rotation rotates specific pairs of Pauli operators, so nonlocal phase identification relies on reconstructing a closed Pauli block, explicitly $\{IX, IY, ZX, ZY\}$. These observables permit defining branch coherences $C_\pm$ corresponding to the control-qubit eigenstates, allowing the protocol to distinguish the effect of $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$0 (opposite, differential slope for each branch) from $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$1 (common-mode shift).

The core estimator for $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$2 is the differential branch phase slope:

$$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$3

where $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$4 are slopes of unwrapped phases of $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$5 as a function of cycle number. A matrix-pencil approach is adopted for robust parameter extraction in the presence of noise and finite sampling.

Protocol and Implementation

The protocol proceeds as:

1. State Preparation: Both qubits are initialized in $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$6, maximizing the relevant Pauli coherences without introducing $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$7-populations.
2. Cycle Evolution: A calibrated cycle, possibly including an echoed sequence with $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$8 refocusing pulses, is applied $$\phi_{\mathrm{cond}} = \phi_{00} - \phi_{01} - \phi_{10} + \phi_{11}$$9 times to amplify the relevant phase.
3. Measurement: Tomographically complete measurement of $\delta_{\mathrm{CP}}$0 is performed with only two distinct pre-rotations (target $\delta_{\mathrm{CP}}$1 or $\delta_{\mathrm{CP}}$2 basis).
4. Classical Post-processing: Virtual-$\delta_{\mathrm{CP}}$3 and echo frame corrections are applied, branch coherences are assembled, and the residual phase is extracted via the matrix-pencil estimator.

Experimental validations involve both numerical simulation (including realistic SPAM, finite coherence, and shot noise) and application to a cloud-based superconducting platform with pulse-level access.

Numerical and Experimental Results

A series of simulation batches demonstrated:

• Breakdown of Scalar Readouts: Two-moment (sector) analytics fail for nonzero $\delta_{\mathrm{CP}}$4 due to frame mixing, producing biased and discontinuous phase estimates, while branch-resolved estimators yield unbiased, sign-resolving estimates across all tested parameter regimes.
• Linearity and Resolution: The branch-resolved estimate of $\delta_{\mathrm{CP}}$5 is unbiased and near-unit-slope against injected values, with empirical resolution limits set by statistical noise. Sign recovery is robust down to injection levels of $\delta_{\mathrm{CP}}$6 rad/cycle.
• Robustness to Noise: The estimator suffers negligible bias in the presence of contrast decay and SPAM offsets and displays predictable statistical scaling with shot budget and sequence length.
• Echoed Cycle Variant: Application of $\delta_{\mathrm{CP}}$7 echoes efficiently cancels local $\delta_{\mathrm{CP}}$8 terms while preserving the $\delta_{\mathrm{CP}}$9 signal. Frame correction is essential, and the technique supports computing residuals per echoed cycle (for use with composite or echoed native gate cycles).

In direct comparison, randomized-benchmarking-derived average infidelity is symmetric with respect to $ZZ$0 and misses sign information, while the presented method resolves both magnitude and sign, enabling decisive pulse correction.

In hardware, the protocol was deployed on a cloud-accessible superconducting device (QuantumCTek, G55 coupler). By scanning native pulse actuators, it was shown that only specific detune controls (here, control-qubit detune) permit clean and linear modulation of $ZZ$1 without collapse of readout contrast. The extracted gain, sign, and branch contrast were consistent with theoretical expectations. A single feedback step, guided by the signed estimate, resulted in a point-estimate suppression of the native residual by approximately $ZZ$2 without loss of contrast.

Implications and Outlook

This protocol concretely demonstrates that targeted, low-overhead spectroscopy of coherent residuals is feasible and practical for closed-loop calibration of two-qubit gates in contemporary superconducting devices. The methodology obviates the need for full process or gate-set tomography and exceeds fidelity-oriented protocols in information content by capturing the sign and accumulation rate of the nonlocal coherent error channel, critical for recursive pulse tuning and error correction.

For practical quantum hardware scaling, this approach supports efficient, drop-in integration with calibration pipelines. It also provides theoretical ground for extending Hamiltonian-learning methods and phase estimation routines to multi-qubit and time-dependent error models where sign and branch resolution are indispensable.

Potential future directions include extension to higher connectivity or multi-frequency error syndromes, integration with leakage/defect-aware tomographic methods, and deployment at scale in automated control platforms.

Conclusion

Branch-resolved Pauli-block spectroscopy is established as a theoretically sound, operationally practical procedure for direct signed measurement and calibration of residual conditional phase errors in two-qubit gates. By leveraging branch-resolved trajectories and Pauli-block closure, the protocol disambiguates local detuning and nonlocal coherent errors, enabling robust, sign-aware pulse updates. Simulation and cloud-hardware experiments confirm its reliability across noise and circuit complexity relevant for the next generation of quantum processors (2607.10249).