A. Nazir, T. P. Spiller, W. J. Munro
We consider the effects of certain forms of decoherence applied to both adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit we illustrate path-dependent sensitivity to anisotropic noise and for two qubits we quantify the loss of entanglement as a function of decoherence.
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Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented using non-adiabatic holonomy in decoherence-free subspaces and the required resource for the d…
One of the biggest challenges for implementing quantum devices is the requirement to perform accurate quantum gates. The destructive effects of interactions with the environment present some of the most difficult obstacles that must be overcome for precise quantum control. In this work we implement a proof of principle experiment of quantum gates protected against a fluctuating environment using dynamical decoupling techniques. We show that decoherence can be reduced during the application of quantum gates. High fidelity quantum gates can be achieved even if the gate time exceeds the decoheren…
Nonadiabatic holonomic quantum computation in decoherence-free subspaces has attracted increasing attention recently, as it allows for high-speed implementation and combines both the robustness of holonomic gates and the coherence stabilization of decoherence-free subspaces. Since the first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, a number of schemes for its physical implementation have been put forward. However, all previous schemes require two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to er…
We propose a novel symmetrization procedure to beat decoherence for oscillator-assisted quantum gate operations. The enacted symmetry is related to the global geometric features of qubits transformation based on ancillary oscillator modes, e.g. phonons in an ion-trap system. It is shown that the devised multi-circuit symmetrized evolution endows the system with a two-fold resilience against decoherence: insensitivity to thermal fluctuations and quantum dissipation.
We propose a geometric phase gate in a decoherence-free subspace with trapped ions. The quantum information is encoded in the Zeeman sublevels of the ground-state and two physical qubits to make up one logical qubit with ultra long coherence time. Single- and two-qubit operations together with the transport and splitting of linear ion crystals allow for a robust and decoherence-free scalable quantum processor. For the ease of the phase gate realization we employ one Raman laser field on four ions simultaneously, i.e. no tight focus for addressing. The decoherence-free subspace is left neither …
We propose a scheme to implement geometric entangling gates for two logical qubits in a coupled cavity system in decoherence-free subspaces. Each logical qubit is encoded with two atoms trapped in a single cavity and the geometric entangling gates are achieved by cavity coupling and controlling the external classical laser fields. Based on the coupled cavity system, the scheme allows the scalability for quantum computing and relaxes the requirement for individually addressing atoms.
We study quantum decoherence of single-qubit and two-qubit Aharonov-Anandan (AA) geometric phase gates realized in a multistep scheme. Each AA gate is also compared with the dynamical phase gate performing the same unitary transformation within the same time period and coupled with the same environment, which is modeled as harmonic oscillators. It is found that the fidelities and the entanglement protection of the AA phase gates are enhanced by the states being superpositions of different eigenstates of the environmental coupling, and the noncommutativity between the qubit interaction and the …
We present a scheme to implement a universal set of quantum gates based on achievable interactions, and the gates can be protected against decoherences through dynamical-decoupling approach without encoding. By properly designing system evolutions, the desired system interactions commute with the elements forming dynamical decoupling pulses. Thus, the effect of decoherences can be eliminated by repeatedly applying the pulses, without noticeably affecting the system evolutions governed by the desired system interactions given small enough time interval between pulses. Moreover, due to the commu…
We consider a universal set of quantum gates encoded within a perturbed decoherence-free subspace of four physical qubits. Using second-order perturbation theory and a measuring device modeled by an infinite set of harmonic oscillators, simply coupled to the system, we show that continuous observation of the coupling agent induces inhibition of the decoherence due to spurious perturbations. We thus advance the idea of protecting or even creating a decoherence-free subspace for processing quantum information.
This paper has been withdrawn as it has been superseded by postings arXiv:quant-ph/0702147 and arXiv:0712.2935
Note on simple and consistent gateset characterization including calibration and decoherence errors
本文提出一种确定模型门集中退相干参数的方法，证明其对SPAM和脉冲面积误差鲁棒，并给出评估误差模型质量与精度的直观方法。
We present an open loop (bang-bang) scheme to control decoherence in a generic one-qubit quantum gate and implement it in a realistic simulation. The system is consistently described within the spin-boson model, with interactions accounting for both adiabatic and thermal decoherence. The external control is included from the beginning in the Hamiltonian as an independent interaction term. After tracing out the environment modes, reduced equations are obtained for the two-level system in which the effects of both decoherence and external control appear explicitly. The controls are determined ex…
Kerr-effect-based quantum logical gates in decoherence-free subspace
针对消相干效应影响量子逻辑门高效实现的问题，提出在无消相干子空间中利用交叉克尔非线性构建CNOT、Toffoli、Fredkin量子控制门的方案，无需复杂电路与辅助光子，成功概率近1且抗光子损耗，具备实用可行性。
In the lectures we will be concerned with some aspects of physical implementations of quantum gate operations which are necessary for quantum information processing. We will discuss two possible realizations. One of them is based on qubits being encoded in atomic degrees of freedom where the atoms are manipulated in optical lattices above atom chips. The other realization is based on photonic qubits and measurement-induced nonlinearities in linear optics. Both implementations have in common that their main decoherence mechanism is absorption in dielectric materials. The quantum theory of light…
High-quality-factor 3D cavities in superconducting circuits are ideal candidates for bosonic logical qubits as their fidelity is limited only by the low photon loss rate. However, the transmon qubits that are used to manipulate bosonic qubits result in the emergence of additional relaxation and dephasing channels. In this work, a numerical study is performed to elucidate the effect of the various loss channels on the performance of logical gates on a bosonic qubit. A gate error model is developed that encapsulates the loss mechanisms for arbitrary gate operations and predicts experimentally ac…
High-fidelity quantum gates are essential for large-scale quantum computation, which can naturally be realized in a noise-resilient way. Geometric manipulation and decoherence-free subspace encoding are promising ways toward robust quantum computation. Here, by combining the advantages of both strategies, we propose and experimentally realize universal holonomic quantum gates in both a singleloop scheme and a composite scheme, based on nonadiabatic and non-Abelian geometric phases, in a decoherence-free subspace with nuclear magnetic resonance. Our experiment uses only two-body resonant spin-s…
The three-spin-$1/2$ decoherence-free subsystem defines a logical qubit protected from collective noise and supports exchange-only universal gates. Such logical qubits are well-suited for implementation with electrically-defined quantum dots. Exact exchange-only entangling logical gates exist but are challenging to construct and understand. We use a decoupling strategy to obtain straightforward approximate entangling gates. A benefit of the strategy is that if the physical spins are aligned, then it can implement evolution under entangling Hamiltonians. Hamiltonians expressible as linear combi…
Heralded nonlocal quantum gates for distributed quantum computation in a decoherence-free subspace
本文提出一种可预告的非局域量子门协议，通过动态控制演化并借助辅助原子后选择，在退相干自由子空间内实现高保真分布式量子计算。
The stabilizing properties of one-error correcting jump codes are explored under realistic non-ideal conditions. For this purpose the quantum algorithm of the tent-map is decomposed into a universal set of Hamiltonian quantum gates which ensure perfect correction of spontaneous decay processes under ideal circumstances even if they occur during a gate operation. An entanglement gate is presented which is capable of entangling any two logical qubits of different one-error correcting code spaces. With the help of this gate simultaneous spontaneous decay processes affecting physical qubits of dif…
Any residual coupling of a quantum computer to the environment results in computational errors. Encoding quantum information in a so-called decoherence-free subspace provides means to avoid these errors. Despite tremendous progress in employing this technique to extend memory storage times by orders of magnitude, computation within such subspaces has been scarce. Here, we demonstrate the realization of a universal set of quantum gates acting on decoherence-free ion qubits. We combine these gates to realize the first controlled-NOT gate within a decoherence-free, scalable quantum computer.
We develop in the weak coupling approximation a quasi-non-Markovian master equation and study the phenomenon of decoherence during the operation of a controlled-not (CNOT) quantum gate in a quantum computer model formed by a linear chain of three nuclear spins system with second neighbor Ising interaction between them. We compare with the behavior of the Markovian counterpart for temperature different from zero (thermalization) and at zero temperature for low and high dissipation rates. At low dissipation there is a very small difference between Markovian and quasi no-Markovian at any temperat…
We propose a new approach to the implementation of quantum gates in which decoherence during the gate operations is strongly reduced. This is achieved by making use of an environment induced quantum Zeno effect that confines the dynamics effectively to a decoherence-free subspace.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit uni…
We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We find the fidelity for Hadamard gate and compute the purity of the final state when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate that in contradiction to the case of the systematic control errors the stochastic ones lead to decoherence of the final state. In the small errors limit we derive a simple formulae connecting the gate fidelity and the purity of the final state.
A field configuration utilizing local static and oscillating fields is constructed to achieve universal (but low-order) protection of two-qubit states. That is, two-qubit states can be protected against arbitrary system-environment coupling if the driving field frequency, as compared with the cutoff frequency of the environment, is sufficiently large. Equally important, we show that it is possible to construct driving fields to protect two-qubit entangling gates against decoherence, without assuming any particular form of system-environment coupling. Using a non-Markovian master equation, we f…
The external control circuits of quantum gates inevitably introduce a small but finite noise to the operation of quantum computers. The complex modes of decoherence introduced by this noise are not covered by the common error models. Using the controlled-phase gate as an example, the effect of gate control noise on decoherence is investigated for different quantum computer architectures. It is shown that the decoherence rate rises faster than linearly with the length of a quantum register for most cases considered, adding to the challenge of implementing proposed error correcting and fault tol…
A cavity QED implementation of the non-adiabatic holonomic quantum computation in decoherence-free subspaces is proposed with nitrogen-vacancy centers coupled commonly to the whispering-gallery mode of a microsphere cavity, where a universal set of quantum gates can be realized on the qubits. In our implementation, with the assistant of the appropriate driving fields, the quantum evolution is insensitive to the cavity field state, which is only virtually excited. The implemented non-adiabatic holonomies, utilizing optical transitions in the Λ type of three-level configuration of the nitrogen-v…
Nonadiabatic holonomic quantum computation~(NHQC) provides an essential way to construct robust and high-fidelity quantum gates due to its geometric features. However, NHQC is more sensitive to the decay and dephasing errors than conventional dynamical gate since it requires an ancillary intermediate state. Here, we utilize the Hamiltonian reverse engineering technique to study the influence of the intermediate state-decoherence on the NHQC gate fidelity, and propose the novel schemes to construct the arbitrary single-qubit holonomic gate and nontrivial two-qubit holonomic gate with high fidel…
It is shown that entangling two-qubit phase gates for quantum computation with atoms inside a resonant optical cavity can be generated via common laser addressing, essentially, within one step. The obtained dynamical or geometrical phases are produced by an evolution that is robust against dissipation in form of spontaneous emission from the atoms and the cavity and demonstrates resilience against fluctuations of control parameters. This is achieved by using the setup introduced by Pachos and Walther [Phys. Rev. Lett. 89, 187903 (2002)] and employing entangling Raman- or STIRAP-like transition…
Decoherence-protected entangling gates in a silicon carbide quantum node
本研究在硅碳中实现去相干保护的纠缠门，通过动态去耦与超精细相互作用结合，制备高保真度纠缠态，推动量子节点发展。
We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We analytically obtain both the purity of the final state and the fidelity for Hadamard gate when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate the purity and the fidelity oscillations depending on the choice of the initial superimposed state. We derive a linear formulae connecting the gate fidelity and the purity of the final state.
We propose and study a realistic model for the decoherence of topological qubits, based on Majorana fermions in one-dimensional topological superconductors. The source of decoherence is the fluctuating charge on a capacitively coupled gate, modeled by non-interacting electrons. In this context, we clarify the role of quantum fluctuations and thermal fluctuations and find that quantum fluctuations do not lead to decoherence, while thermal fluctuations do. We explicitly calculate decay times due to thermal noise and give conditions for the gap size in the topological superconductor and the gate …
The dominant error sources for state-of-the-art laser-free trapped-ion entangling gates are decoherence of the qubit state and the ion motion. The effect of these decoherence mechanisms can be suppressed with additional control fields, or with techniques that have the disadvantage of reducing gate speed. Here, we propose using a near-motional-frequency magnetic field gradient to implement a laser-free gate that is simultaneously resilient to both types of decoherence, does not require additional control fields, and has a relatively small cost in gate speed.
We summarize our results on decoherence for short- to intermediate-time dynamics of an externally controlled two-state quantum system - a qubit - interacting with thermal bosonic environment. The developed approximation schemes are illustrated for an adiabatic model with time-dependent gate control, and for a model with rotating-wave gate function.
Recent experiments by Hayashi et al. [Phys. Rev. Lett. 91, 226804 (2003)] demonstrate coherent oscillations of a charge quantum bit (qubit) in laterally defined quantum dots. We study the intrinsic electron-phonon decoherence and gate performance for the next step: a system of two coupled charge qubits. The effective decoherence model contains properties of local as well as collective decoherence. Decoherence channels can be classified by their multipole moments, which leads to different low-energy spectra. It is shown that due to the super-Ohmic spectrum, the gate quality is limited by the si…
In this paper, we investigate decoherence of Knill, Laflamme, and Milburn's nonlinear sign-shift gate that is implemented with the Jaynes-Cummings model. Introducing a stochastic variable as an external electric field, we let it couple with a dipole moment of a two-level atom. We examine this model using a semiclassical theory. Results of the Monte Carlo simulations under the semiclassical approximation correspond well with those obtained with the quantum mechanical perturbation theory for the stochastic process. In the results of the simulations, we observe both the $\mbox{T}_{1}$ and $\mbox{…
The main obstacles to the realization of high-fidelity quantum gates are the control errors arising from inaccurate manipulation of a quantum system and the decoherence caused by the interaction between the quantum system and its environment. Nonadiabatic holonomic quantum computation allows for high-speed implementation of whole-geometric quantum gates, making quantum computation robust against control errors. Dynamical decoupling provides an effective method to protect quantum gates against environment-induced decoherence, regardless of collective decoherence or independent decoherence. In t…
We propose a scheme to implement quantum computation in decoherence-free subspace with superconducting devices inside a cavity by unconventional geometric manipulation. Universal single-qubit gates in encoded qubit can be achieved with cavity assisted interaction. A measurement-based two-qubit Controlled-Not gate is produced with parity measurements assisted by an auxiliary superconducting device and followed by prescribed single-qubit gates. The measurement of currents on two parallel devices can realize a projective measurement, which is equivalent to the parity measurement on the involved d…
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, quantum ergodic regime and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates and per gate dissipation rate induced b…
We propose a feasible scheme to achieve holonomic quantum computation in a decoherence-free subspace (DFS) with trapped ions. By the application of appropriate bichromatic laser fields on the designated ions, we are able to construct two noncommutable single-qubit gates and one controlled-phase gate using the holonomic scenario in the encoded DFS.
We propose a decoherence-free subspaces (DFS) scheme to realize scalable quantum computation with trapped ions. The spin-dependent Coulomb interaction is exploited, and the universal set of unconventional geometric quantum gates is achieved in encoded subspaces that are immune from decoherence by collective dephasing. The scalability of the scheme for the ion array system is demonstrated, either by an adiabatic way of switching on and off the interactions, or by a fast gate scheme with comprehensive DFS encoding and noise decoupling techniques.
Quantum state transfer and teleportation, with qubits encoded in internal states of the atoms in cavities, among spatially separated nodes of a quantum network in decoherence-free subspace are proposed, based on a cavity-assisted interaction by single-photon pulses. We show in details the implementation of a logic-qubit Hadamard gate and a two-logic-qubit conditional gate, and discuss the experimental feasibility of our scheme.
We derive and investigate a general non-Markovian equation for the time-dependence of a Hamiltonian that maximizes the fidelity of a desired quantum gate on any finite-dimensional quantum system in the presence of arbitrary bath and noise sources. The method is illustrated for a single-qubit gate implemented on a three-level system.
Nonadiabatic holonomic quantum computation uses non-Abelian geometric phases to implement a universal set of quantum gates that are robust against control imperfections and decoherence. Until now, a number of three-level-based schemes of nonadiabatic holonomic computation have been put forward, and several of them have been experimentally realized. However, all these works are based on the same class of nonadiabatic paths, which originates from the first nonadiabatic holonomic proposal. Here, we propose a universal set of nonadiabatic holonomic gates based on an extended class of nonadiabatic …
Impact of decoherence on the fidelity of quantum gates leaving the computational subspace
本文将多量子比特操作平均门保真度的解析表达式推广至系统状态暂时离开计算子空间的情形，并以超导量子比特和中性原子的双量子比特受控-Z门为例，为理解量子计算机扩展时的门误差预算提供了工具。
We investigate how to carry out universal quantum computation deterministically with free electrons in decoherence-free subspace by using polarizing beam splitters, charge detectors, and single-spin rotations. Quantum information in our case is encoded in spin degrees of freedom of the electron-pairs which construct a decoherence-free subspace. We design building blocks for two noncommutable single-logic-qubit gates and a logic controlled phase gate, based on which a universal and scalable quantum information processing robust to dephasing is available in a deterministic way.
We consider interactions that generate a universal set of quantum gates on logical qubits encoded in a collective-dephasing-free subspace, and discuss their implementations with trapped ions. This allows for the removal of the by-far largest source of decoherence in current trapped-ion experiments, collective dephasing. In addition, an explicit parametrization of all two-body Hamiltonians able to generate such gates without the system's state ever exiting the protected subspace is provided.
Analytical solutions describing quantum swap and Hadamard gate are given with the use of tight-binding approximation. Decoherence effects are described analytically for two interacting electrons confined by local potentials with use of tight-binding simplistic model and in Schroedinger formalism with omission of spin degree of freedom. The obtained results can be generalized for the case of N electrostatically interacting quantum bodies confined by local potentials (N-qubit) system representing any electrostatic quantum gate with N1/N-N1 inputs/outputs. The mathematical structure of system evo…
Universal quantum computation requiring only the Heisenberg exchange interaction and suppressing decoherence via an energy gap is presented. The combination of an always-on exchange interaction between the three physical qubits comprising the encoded qubit and a global magnetic field generates an energy gap between the subspace of interest and all other states. This energy gap suppresses decoherence. Always-on exchange couplings greatly simplify hardware specifications and the implementation of inter-logical-qubit gates. A controlled phase gate can be implemented using only three Heisenberg ex…
Nonadiabatic geometric quantum computation in decoherence-free subspaces has received increasing attention due to the merits of its high-speed implementation and robustness against both control errors and decoherence. However, all the previous schemes in this direction have been based on the conventional geometric phases, of which the dynamical phases need to be removed. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, of which the dynamical phases do not need to be removed. Specifically…
Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing non-unitary, non-cyclic and non-adiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can…
A fundamental requirement of quantum information processing is the protection from the adverse effects of decoherence and noise. Decoherence-free subspaces and geometric processing are important steps of quantum information protection. Here, we provide a new experimentally feasible scheme to combine decoherence-free subspaces with nonadiabatic geometric manipulations to attain a universal quantum computation. The proposed scheme is different from previous proposals and is based on the typical XY interaction coupling, which can be set up in various nano-engineered systems and therefore open up …
We present the design and characterization of a three-Josephson-junction superconducting loop circuit with three large shunt capacitors. Used as a qubit, the circuit shows long energy-relaxation times, of the order of 40 ${ \unicode[Times]{x3BC}}$s and a spin-echo dephasing time of 9.4 ${ \unicode[Times]{x3BC}}$s. The circuit has high anharmonicity, of 2$π$$\times$3.69 GHz. We extract the multilevel relaxation and dephasing rates of the circuit used as a qutrit and discuss the possible sources for the decoherence. The high anharmonicity allows for fast qubit control with nanosecond-range gate …
For neutral atom qubits, the two-qubit gate is typically realized via the Rydberg blockade effect, which hints about the special status of the Rydberg level besides the regular qubit register states. Here, we carry out experimental and theoretical studies to reveal how the ground-Rydberg coherence of the control qubit atom affects the process of two-qubit Controlled-Z ($C_Z$) gate, such as the commonly used ground-Rydberg $π$-gap-$π$ pulse sequence originally proposed in Phys. Rev. Lett. \textbf{85}, 2208 (2000). We measure the decoherence of the control qubit atom after the $π$-gap-$π$ pulses…
We examine a stochastic noise process that has a decohering effect on the average evolution of qubits in the quantum register of the solid state quantum computer proposed by Kane. We consider the effects of this process on the single qubit operations necessary to perform quantum logical gates and derive an expression for the fidelity of these gates in this system. We then calculate an upper bound on the level of this stochastic noise tolerable in a workable quantum computer.
CAMEL: Physically Inspired Crosstalk-Aware Mapping and gatE scheduLing for Frequency-Tunable Quantum Chips
针对频率可调量子芯片中并行双量子比特门的串扰与退相干问题，提出整合脉冲补偿的CAMEL方法及含串扰感知映射、无串扰门优先调度的编译框架，在串扰抑制、退相干控制及低复杂度硬件性能上优于现有方案。
It is known that it is possible to encode a logical qubit over many physical qubits such that it is immune to the effects of collective decoherence, and it is possible to perform universal quantum computation using these `decoherence-free' qubits. However, current proposed methods of performing gates on these encoded qubits could be difficult to implement, or could take too much time to perform. Here we investigate whether exploiting ring-exchange interactions, which may be naturally present, can simplify the implementation of these gates in any way. Using a ring exchange interaction, we have …
Deterministic generation of photonic entangled states using decoherence-free subspaces
该研究提出利用半波导中三个发射器的集体态资源，通过无消相干子空间内的量子门操作与亮态耦合的光子发射，确定性生成光子纠缠态，为量子信息技术提供基础支撑。
We consider the STIRAP process in a three-level atom. Viewed as a closed system, no geometric phase is acquired. But in the presence of spontaneous emission and/or collisional relaxation we show numerically that a non-vanishing, purely real, geometric phase is acquired during STIRAP, whose magnitude grows with the decay rates. Rather than viewing this decoherence-induced geometric phase as a nuisance, it can be considered an example of "beneficial decoherence": the environment provides a mechanism for the generation of geometric phases which would otherwise require an extra experimental contro…
Because of using geometric phases, nonadiabatic geometric gates have the robustness against control errors. On the other hand, decoherence still affects nonadiabatic geometric gates, which is a key factor in reducing their fidelities. In this paper, we show that based on the system Hamiltonian that realizes a nonadiabatic geometric gate, one may construct a new system Hamiltonian, by using which not only the geometric feature of the nonadiabatic geometric gate is preserved, but also the system's coherence is protected. As a result, a coherence-protected nonadiabatic geometric gate is realized …
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