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arXiv:2607.00462

Closed-loop control for two-qubit gates with trapped ions

Eduardo J. Páez, Seyed Shakib Vedaie, Barry C. Sanders

quant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 0 citations
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Abstract

State-of-the-art two-qubit gates with trapped ions employ open-loop control that rely on simplified models to precompute control sequences. Our aim is to introduce closed-loop control for two-qubit gates to correct disturbances as they occur during the gate implementation. We introduce a spectator ion into the ion chain used for quantum logic processing, where it couples with the other ions through collective motional modes. The spectator ion's position is continuously monitored by driving dipole transitions and detecting the resultant fluorescence. We show that incorporating a spectator ion is feasible for linear Paul trap implementations and is expected to reduce the two-qubit gate Bell-state preparation infidelity by an order of magnitude with the deleterious effects of position monitoring being negligible compared to the thermal effects that exist in the system, even in the absence of the spectator ions. Mathematically, we describe driven ion-trap dynamics, including the spectator ion, by a stochastic quantum master equation involving the amplitude-modulation multimode-motional coupling gate, motional drift, thermal effects, recoil from photon scattering, spontaneous decay, and light shift. Our on-the-fly control method employs reinforcement learning with the reward function based on the actual geometric phase of the spectator ion. A key advantage of our approach is that we introduce a control method that involves `learning' and correcting disturbances happening in the trap on-the-fly, thus achieving high-fidelity gates. Our approach will lead to a significantly higher two-qubit gate fidelity at a reduced calibration overhead owing to the small parameter drift in the control system.

Comments
19 pages, 10 figures

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State-of-the-art two-qubit gates with trapped ions employ open-loop control that rely on simplified models to precompute control sequences. Our aim is to introduce closed-loop control for two-qubit gates to correct disturbances as they occur during the gate implementation. We introduce a spectator ion into the ion chain used for quantum logic processing, where it couples with the other ions through collective motional modes. The spectator ion's position is continuously monitored by driving dipole transitions and detecting the resultant fluorescence. We show that incorporating a spectator ion is feasible for linear Paul trap implementations and is expected to reduce the two-qubit gate Bell-state preparation infidelity by an order of magnitude with the deleterious effects of position monitoring being negligible compared to the thermal effects that exist in the system, even in the absence of the spectator ions. Mathematically, we describe driven ion-trap dynamics, including the spectator ion, by a stochastic quantum master equation involving the amplitude-modulation multimode-motional coupling gate, motional drift, thermal effects, recoil from photon scattering, spontaneous decay, and light shift. Our on-the-fly control method employs reinforcement learning with the reward function based on the actual geometric phase of the spectator ion. A key advantage of our approach is that we introduce a control method that involves `learning' and correcting disturbances happening in the trap on-the-fly, thus achieving high-fidelity gates. Our approach will lead to a significantly higher two-qubit gate fidelity at a reduced calibration overhead owing to the small parameter drift in the control system.
High-quality two-qubit gate (2QG) control policies for preparing Bell states are vital for uni-versal quantum computation, yet the successful implementations have been achieved only in few-qubit systems. State-of-the-art (SotA) two-qubit gates for trapped ions employ open-loop control that rely on simplified models to precompute control sequences. Such methods employ rigid simplifications, such as the presumption of unitary evolution to obtain a closed mathematical expression for trapped-particle dynamics and may employ greedy optimisa-tion to devise control policies. Nonetheless, quantum information processing based on cold trapped particles is susceptible to significant decoherence and fluctuation. Therefore, such control approaches do not scale properly for chains of more than two trapped ions or deliver feasible control sequences. In the present work, we aim to develop and implement novel techniques for open- and closed-loop quantum control for high-quality 2QG gates in realistic scenarios. Here, a comprehensive simulated ion trap environment is experimentally validated and used to develop and test two control protocols. First, a global optimisation algorithm finds robust solutions for a chain of seven ions with improved gate infidelity by at least one order of magnitude and a gate time of 35 μs, which is the shortest reported in the literature. Second, we demonstrate the feasibility of closed-loop control by introducing a spectator ion to the chain used for quantum logic processing for two-qubit gates to correct disturbances as they occur during gate implementations. This method improves the gate infidelity by one order of magnitude in the presence of motional heating and drift. The spectator ion’s posi-tion is continuously monitored by driving the dipole transitions and detecting the resultant fluorescence. In this work a global optimisation algorithm is able to navigate the non-convexity of the space solution to devise feasible control sequences. On the other side, we introduce the continuous monitoring of a spectator ion’s position to construct feasible control policies in realistic scenarios for implementing 2QG gates. We show that our methods for linear Paul trap implementations are expected to reduce the two-qubit gate Bell-state preparation infidelity with deleterious disturbances in realistic scenarios. The methods devised here are unique in the field of quantum control for 2QG and open up a new trend of techniques aiming to overcome the scalability challenges of quantum pro-cessor units. Our on-the-fly control method employs reinforcement learning with the reward function based on the actual geometric phase of the spectator ion. A key advantage of our approach is that we introduce a control method that involves ‘learning’ and correcting dis-turbances occurring in the trap on-the-fly, thus achieving high-fidelity gates. Our approach will lead to a significantly higher two-qubit gate fidelity with a reduced calibration overhead owing to the small parameter drift in the control system.
In an ion trap quantum computer, collective motional modes are used to entangle two or more qubits in order to execute multiqubit logical gates. Any residual entanglement between the internal and motional states of the ions results in loss of fidelity, especially when there are many spectator ions in the crystal. We propose using a frequency-modulated driving force to minimize such errors. In simulation, we obtained an optimized frequency-modulated 2-qubit gate that can suppress errors to less than 0.01% and is robust against frequency drifts over ±1 kHz. Experimentally, we have obtained a 2-qubit gate fidelity of 98.3(4)%, a state-of-the-art result for 2-qubit gates with five ions.
Two-qubit gate performance is vital for scaling up ion-trap quantum computing. Optimized quantum control is needed to achieve reductions in gate duration and gate error rate. We describe two-qubit gates with addressed Raman beams within a linear trapped-ion chain by a quantum master equation (QME). The QME incorporates the single-ion two-photon effective Rabi frequency, Autler-Townes and vibrational Bloch-Siegert energy shifts, off-resonant transitions, Raman and Rayleigh scattering, laser-power fluctuations, motional heating, cross-Kerr phonon coupling, laser spillover, asymmetric addressing beams, and an imperfect initial motional ground state, with no fitting parameters, whereas state-of-the-art methods are oblivious to these effects in the gate design procedure. We employ global optimization to design pulse sequences for achieving a robust rapid two-qubit gate for a simulated chain of seven trapped 171Yb+^{171}\mathrm{Yb}^{+}171Yb+ ions by optimizing over numerically integrated QME solutions. Here, robust means resilient against slow drift of motional frequencies, and rapid means gate execution where the effective Rabi frequency is comparable to the detuning of the laser from the ion's bare electronic transition. Our robust quantum control delivers rapid high-quality two-qubit gates in long ion chains, enabling scalable quantum computing with trapped ions.
Today's most advanced ion trap quantum computers have significant overhead due to the need for dual-species operation. Looking ahead, logical qubit register sizes will be limited by the encoding rate needed to correct generic Pauli errors. We address both of these issues by establishing high-fidelity control of metastable qubits, a key component of \textit{omg} or dual-type architectures, which enables converting a significant fraction of gate errors to erasures. We first implement an erasure conversion scheme which enables detection of ∼94%\sim 94\%∼94% of spontaneous Raman scattering errors during logic gates and nearly all errors from qubit decay. Second, we perform a two-ion geometric phase gate using far-detuned (-44\,THz) stimulated Raman transitions to produce an entangled state with a raw Bell state fidelity of 97.73\% and a SPAM-corrected Bell state fidelity of 98.61\%. When subtracting erasure errors, this fidelity becomes 99.16\%. These results, along with projections based on our detailed error budget, demonstrate metastable trapped-ion qubits as a platform for low-overhead, fault-tolerant quantum computing.
This thesis describes schemes for both a fast two-qubit gate operation and the steady-state preparation of a Bell state with trapped ions. A critical figure of merit for quantum computing with trapped ions is the gate duration relative to the decoherence timescale. We propose a fast gate scheme that off ers improvements in time, fidelity and simplicity of implementation over existing fast gate proposals. Our scheme can operate on both neighbouring and non-neighbouring ions in a long ion crystal. This provides a simpler and faster mechanism than traditional gates for complex quantum computing operations on large numbers of ions. The scheme achieves fidelities well above quantum error correction thresholds around 0.0001, and operates arbitrarily fast given arbitrary laser repetition rates. The production of these ultra-fast pulses is an experimental challenge, and fast gates have not yet been implemented; we present an implementation scheme using pulse splitting to provide a higher repetition rate and the pulse timing freedoms required for the gate scheme. We also analyse the effects of errors in the pulses on the gate operation. We analyse another strategy to generate entanglement using a driven dissipative process. Typically, environmental couplings cause decoherence. However, by combining dissipative dynamics with suitably chosen Hamiltonian evolution, the system can be steered to the desired steady states. Our steady-state scheme prepares a maximally-entangled Bell state with fidelity above 0.99, much higher than for schemes implemented with trapped ions. The driven dissipation continuously pumps the system towards the antisymmetric Bell steady-state, which is dark to the system dynamics and robust to parameter variations. The dominant loss mechanism is anomalous heating of the motional modes, reducing our fidelity by less than 0.01 for current experimental rates. Our scheme jointly addresses the ions and does not use sympathetic cooling. We enhance our scheme by combining the dissipative state preparation with the detection of photons, and obtain a significant fidelity enhancement.
Abstract Trapped-ion systems are one of the leading platforms for quantum information processing, where a key challenge is to scale up system size while maintaining high-fidelity two-qubit operations. A promising approach is to build high-performance modules interconnected via strong coupling. In particular, axial motional modes offer a practical mechanism to couple the ions in a chain, enabling the preparation of Greenberger–Horne–Zeilinger states with up to 24 ions using global operations, as well as high-fidelity two-qubit gates (96.6%–98.0%) in fully connected five-ion chains. Here, we demonstrate two-qubit quantum logic gates in a 5-ion 40 Ca + chain using axial modes, achieving fidelities exceeding 99% for adjacent pairs and over 98% for arbitrary pairs by carefully tackling dominant error sources. Our results are beneficial to the development of scalable ion-trap quantum processors, quantum simulation and quantum-enhanced metrology.