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The Journal of V N Karazin Kharkiv National University Series Physics

DISCRETE QND ENTANGLEMENT-PROTECTION PROTOCOL FOR A PHASE-SHIFTED BELL PAIR

O. M. Konovalenko, Z. А. Maizelis

Published May 28, 2025• 0 citations
Publisher

Abstract

We evaluate a discrete entanglement-protection protocol for a Bell two-qubit pair undergoing unitary evolution with a single mid-sequence phase change. The test circuit consists of state preparation, a phase shift that does not affect entanglement and emulates a useful quantum gate, and a final standard readout. We compare two variants over a total evolution time of 16 arbitrary time intervals: in the first variant, the phase operation is bracketed by two blocks of five two-qubit QND measurements without readout, while in the baseline variant these measurements are absent. Performance is quantified by the fidelity F, based on the concurrence metric; we take as the minimally acceptable lower bound, below which hardware noise can no longer be safely compensated by the available specialized tools. Simulations show that the protected scheme maintains throughout the entire observation window, whereas the unprotected scheme, under free evolution in realistic noise regimes encountered in experiments, quickly drops below this threshold even before the gate is applied. These results demonstrate that repeated projective measurements of the entangled state in the Bell basis, implemented as discrete quantum non-demolition (QND) two-qubit measurements without readout, substantially slow the loss of entanglement despite coupling to a thermal bath. The protocol requires fewer measurements, distinguishing it from high-frequency schemes that rely on the Zeno effect. The approach is directly relevant for fine-tuning superconducting and trapped-ion processors, for pure-state engineering, and for optimizing protocols that use mixed entangled input states. In particular, in these architectures the protocol naturally integrates with existing QND procedures and phase control, providing a means to curb decoherence. Practically, this implies periodic in-operation measurements every 1–2 arbitrary time steps, as well as the possibility of scaling to multi-qubit modules with improved gate fidelities and simplified calibration.

DOI
10.26565/2222-5617-2025-42-06

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We evaluate a discrete entanglement-protection protocol for a Bell two-qubit pair undergoing unitary evolution with a single mid-sequence phase change. The test circuit consists of state preparation, a phase shift that does not affect entanglement and emulates a useful quantum gate, and a final standard readout. We compare two variants over a total evolution time of 16 arbitrary time intervals: in the first variant, the phase operation is bracketed by two blocks of five two-qubit QND measurements without readout, while in the baseline variant these measurements are absent. Performance is quantified by the fidelity F, based on the concurrence metric; we take as the minimally acceptable lower bound, below which hardware noise can no longer be safely compensated by the available specialized tools. Simulations show that the protected scheme maintains throughout the entire observation window, whereas the unprotected scheme, under free evolution in realistic noise regimes encountered in experiments, quickly drops below this threshold even before the gate is applied. These results demonstrate that repeated projective measurements of the entangled state in the Bell basis, implemented as discrete quantum non-demolition (QND) two-qubit measurements without readout, substantially slow the loss of entanglement despite coupling to a thermal bath. The protocol requires fewer measurements, distinguishing it from high-frequency schemes that rely on the Zeno effect. The approach is directly relevant for fine-tuning superconducting and trapped-ion processors, for pure-state engineering, and for optimizing protocols that use mixed entangled input states. In particular, in these architectures the protocol naturally integrates with existing QND procedures and phase control, providing a means to curb decoherence. Practically, this implies periodic in-operation measurements every 1–2 arbitrary time steps, as well as the possibility of scaling to multi-qubit modules with improved gate fidelities and simplified calibration.
We analyze repeated rank-1 projections onto a single Bell state as an entanglement-preserving protocol for a noisy two-qubit register. The register interacts with a Markovian dissipative environment described by a Lindblad master equation, while each qubit also experiences classical frequency fluctuations with specified noise statistics. We focus on Ornstein–Uhlenbeck noise, using random telegraph noise and Poissonian jump noise as complementary models. Analytic asymptotic expressions are derived for the noise-averaged dynamics, and performance is evaluated using concurrence and a survival-gain metric comparing measurement-assisted evolution with free decay. We identify parameter regimes where repeated Bell-basis projections provide a robust advantage, as well as regimes where the benefit depends on noise statistics, revealing the role of non-Gaussian frequency fluctuations in entanglement protection.
The calibration of high-quality two-qubit entangling gates is an essential component in engineering large-scale, fault-tolerant quantum computers. However, many standard calibration techniques are based on randomized circuits that are only quadratically sensitive to calibration errors. As a result, these approaches are inefficient, requiring many experimental shots to achieve acceptable performance. In this work, we demonstrate that robust phase estimation can enable high-precision, Heisenberg-limited estimates of coherent errors in multi-qubit gates. Equipped with an efficient estimator, the calibration problem may be reduced to a simple optimization loop that minimizes the estimated coherent error. We experimentally demonstrate our calibration protocols by improving the operation of a two-qubit controlled-Z gate on a superconducting processor, and we validate the improved performance with gate set tomography. Our methods are applicable to gates in other quantum hardware platforms such as ion traps and neutral atoms, and on other multi-qubit gates, such as CNOT or iSWAP.
Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ trapped-ion processor and certify the X\sqrt{\mathrm{X}}X​-gate rotation eigenvalue using 22 00022\,00022000 circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of 99.94(3)%99.94(3)\%99.94(3)% with 99%99\%99% confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.
While relatively easy to engineer, static transverse coupling between a qubit and a cavity mode satisfies the criteria for a quantum non-demolition (QND) measurement only if the coupling between the qubit and cavity is much less than their mutual detuning. This can put significant limits on the speed of the measurement, requiring trade-offs in the circuit design between coupling, detuning, and decoherence introduced by the cavity mode. Here, we study a circuit in which the qubit-cavity and the cavity-feedline coupling can be turned on and off, which helps to isolate the qubit. We do not rely on the rotating-wave or dispersive approximations, but solve the full transverse interaction between the qubit and the cavity mode. We show that by carefully choosing the detuning and interaction time, we can exploit a recurrence in the qubit-cavity dynamics in a way that makes it possible to perform very fast, high fidelity, QND measurements. Here, the qubit measurement is performed more like a gate operation between the qubit and the cavity, where the cavity state can be amplified, squeezed, and released in a time-sequenced fashion. In addition, we also show that the non-demolition property of the off-resonant approximation breaks down much faster than its dispersive property, suggesting that many of the dispersive measurements to date have been implemented outside the QND regime.
In a circuit QED architecture, we experimentally demonstrate a hardware-efficient and qubit-state-dependent Single-Step Phase-Engineered (SSPE) pulse scheme for actively depopulating a readout cavity. The protocol appends a reset segment with tailored amplitude and phase to a square readout pulse. Within the linear-response regime, the optimal reset amplitude scales proportionally with the readout amplitude, while the optimal reset phase remains invariant, significantly simplifying the experimental calibration procedure. For a known qubit state, time-resolved measurements demonstrate that the SSPE pulse achieves faster cavity depletion than the CLEAR protocol. Crucially, this approach enables arbitrarily fast, overshoot-free depletion of the cavity photon population, with the ultimate reset rate constrained by the available drive amplitude and the finite analog bandwidth of the measurement chain. A comprehensive evaluation of the QND character reveals no measurable increase in non-QNDness relative to the free-decay and CLEAR protocols. The remaining non-QNDness is predominantly attributable to state switching induced by qubit relaxation during the readout process. These results establish the SSPE scheme as a practical and scalable approach for achieving rapid and smooth cavity reset in superconducting quantum circuits in the case of a known qubit state.
High-fidelity and rapid qubit readout is essential for superconducting quantum processors, typically realized through the quantum non-demolition (QND) dispersive interaction within a qubit-resonator architecture. However, the achievable readout speed and fidelity are fundamentally limited by measurement-induced state transitions (MIST). For a transmon qubit, MIST is highly sensitive to the offset charge ngn_gng​ due to the charge dispersion of its higher-lying energy levels. In this work, we systematically investigate ngn_gng​-dependent MIST dynamics governed by the diabaticity and symmetry of pulse shaping within a charge-sensitive transmon architecture. We engineer fast-load and fast-clear pulses that effectively suppress resonator photon overshoots, thereby demonstrating a highly practical strategy to mitigate MIST without requiring complex waveforms or real-time feedback. Utilizing active gate-voltage control and rapid feedback, the measurement-induced transition probability is precisely mapped against ngn_gng​ and the steady-state resonator photon number, exhibiting strong agreement with numerical Floquet branch analysis. Ultimately, we evaluate the ngn_gng​-averaged total error probabilities for both readout and post-readout stages, verifying that a straightforward three-step pulse scheme consistently minimizes overall readout errors. Within the framework of large-scale superconducting quantum processors, this practical, hardware-free approach inherently offers a better trade-off between the readout signal-to-noise ratio and QND preservation.