G. Dridi, M. Mejatty, S. J. Glaser, D. Sugny
We present a general procedure to implement a NOT gate by composite pulses robust against both offset uncertainties and control field variations. We define different degrees of robustness in this two-parameter space, namely along one, two or all directions. We show that the phases of the composite pulse satisfy a nonlinear system, and can be computed analytically or numerically.
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Composite pulses provide a simple means for constructing quantum logic gates which are robust to small errors in the control fields used to implement them. Here I describe how antisymmetric composite NOT gates can be nested to produce gates with arbitrary tolerance of errors.
We show that some composite pulses widely employed in NMR experiments are regarded as non-adiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind the traditional technique of the composite pulses. To examine the robustness of such composite pulses against fluctuations, we present a practical noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a specific type of fluctuations.
We present three classes of symmetric broadband composite pulse sequences. The composite phases are given by analytic formulas (rational fractions of $π$) valid for any number of constituent pulses. The transition probability is expressed by simple analytic formulas and the order of pulse area error compensation grows linearly with the number of pulses. Therefore, any desired compensation order can be produced by an appropriate composite sequence; in this sense, they are arbitrarily accurate. These composite pulses perform equally well or better than previously published ones. Moreover, the cu…
We propose a simple formalism to design unitary gates robust against given systematic errors. This formalism generalizes our previous observation [Y. Kondo and M. Bando, J. Phys. Soc. Jpn. 80, 054002 (2011)] that vanishing dynamical phase in some composite gates is essential to suppress amplitude errors. By employing our formalism, we naturally derive a new composite unitary gate which can be seen as a concatenation of two known composite unitary operations. The obtained unitary gate has high fidelity over a wider range of the error strengths compared to existing composite gates.
We show that all geometric quantum gates (GQG's in short), which are quantum gates only with geometric phases, are robust against control field strength errors. As examples of this observation, we show (1) how robust composite rf-pulses in NMR are geometrically constructed and (2) a composite rf-pulse based on Trotter-Suzuki Formulas is a GQG.
We design composite controlled-phase gates, which compensate errors in the phase of a single gate. The errors can be of various nature, such as relative, absolute or both. We present composite sequences which are robust to relative errors up to the 6th order, with the number of the constituent gates growing just linearly with the desired accuracy, and we describe a method to achieve even higher accuracy. We show that the absolute error can be canceled entirely with only two gates. We describe an ion-trap implementation of our composite gates, in which simultaneous cancellation of the error in …
In NMR experiments and quantum computation, many pulse (quantum gate) sequences called the composite pulses, were developed to suppress one of two dominant errors; a pulse length error and an off-resonance error. We describe, in this paper, a general prescription to design a single-qubit concatenated composite pulse (CCCP) that is robust against two types of errors simultaneously. To this end, we introduce a new property, which is satisfied by some composite pulses and is sufficient to obtain a CCCP. Then we introduce a general method to design CCCPs with shorter execution time and less number…
We introduce universal broadband composite pulse sequences for robust high-fidelity population inversion in two-state quantum systems, which compensate deviations in any experimental parameter (e.g. pulse amplitude, pulse duration, detuning from resonance, Stark shifts, unwanted frequency chirp, etc.) and are applicable with any pulse shape. We demonstrate the efficiency and universality of these composite pulses by experimental data on rephasing of atomic coherences in a $\text{Pr}^{3+}\text{:}\text{Y}_2\text{SiO}_5$ crystal.
Composite pulses --- sequences of pulses with well defined relative phases --- are an efficient, robust and flexible technique for coherent control of quantum systems. Composite sequences can compensate a variety of experimental errors in the driving field (e.g. in the pulse amplitude, duration, detuning, chirp, etc.) or in the quantum system and its environment (e.g. inhomogeneous broadening, stray electric or magnetic fields, unwanted couplings, etc.). The control parameters are the relative phases between the constituent pulses in the composite sequence, an accurate control over which is re…
We propose a method to construct composite two-qubit gates with narrowband profiles with respect to the spin-spin coupling. The composite sequences are selective to the variations in the amplitude and duration of the spin-spin coupling, and can be used for highly-selective qubit addressing with greatly reduced cross talk, quantum logic spectroscopy, and quantum sensing.
We present a method to construct high-fidelity quantum phase gates, which are insensitive to errors in various experimental parameters. The phase gates consist of a pair of two sequential broadband composite pulses, with a phase difference $π+α/2$ between them, where $α$ is the desired gate phase. By using composite pulses which compensate systematic errors in the pulse area, the frequency detuning, or both the area and the detuning, we thereby construct composite phase gates which compensate errors in the same parameters. Particularly interesting are phase gates which use the recently discove…
We perform comprehensive experimental tests of various composite pulse sequences using one of open-access IBM's quantum processors, based on superconducting transmon qubits. We implement explicit pulse control of the qubit by making use of the opportunity of low-level access to the backend, provided by IBM Quantum. We obtain the excitation profiles for a huge variety of broadband, narrowband, and passband composite pulses, producing any pre-chosen target probabilities, ranging from zero to one. We also test universal composite pulses which compensate errors in any experimental parameter. In al…
Composite pulses have found widespread use in both conventional Nuclear Magnetic Resonance experiments and in experimental quantum information processing to reduce the effects of systematic errors. Here we describe several families of time symmetric and antisymmetric fully compensating composite pulses, inspired by the previous Fn, Gn and BB1 families family developed by Wimperis. We describe families of composite 180 degree pulses (NOT gates) which exhibit unprecedented tolerance of pulse strength errors without unreasonable sensitivity to off-resonance errors, and related families with more …
Composite pulse sequences, which produce arbitrary pre-defined rotations of a two-state system at an angle $θ$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $π/2$ sequences --- symmetric and asymmetric --- the phases of which are given by simple formulas in terms of rational multiples of $π$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $π/2$ composite rotatio…
We introduce a fault-tolerant construction to implement a composite quantum operation of four overlapping Toffoli gates. The same construction can produce two independent Toffoli gates. This result lowers resource overheads in designs for quantum computers by more than an order of magnitude. The procedure uses Clifford operations and 64 copies of the non-Clifford gate $T = \exp[i π(I - σ^z) /8]$. Quantum codes detect errors in the circuit. When the dominant source of error is $T$-gate failure with probability $p$, then the composite Toffoli circuit has postselected failure rate of $3072p^4$ to…
We propose various composite $π$-pulse sequences for implementing robust z-axis rotation gates widely used in quantum information processing (QIP) scenarios, and discuss their error tolerance of the pulse strength error (PSE) and off-resonance error (ORE), i.e., two typical systematic errors in NMR systems. Compared to existing composite gates, our designed composite pulses exhibit higher gate fidelity and shorter sequence time cost. Furthermore, such short pulse sequences are generalized by simple analytic formulas which can suppress these errors to any desired order simultaneously with linea…
Composite pulses are an efficient tool for robust quantum control. In this work, we derive the form of the composite pulse sequence to implement robust single-qubit gates in a three-level system, where two low-energy levels act as a qubit. The composite pulses can efficiently cancel the systematic errors up to a certain order. We find that the three-pulse sequence cannot completely eliminate the first order of systematic errors, but still availably makes the fidelity resistant to variations in a specific direction. When employing more pulses in the sequence ($N>3$), the fidelity can be insensi…
We introduce a novel control method for robust quantum information processing suited for quantum integrated photonics. We utilize off-resonant detunings as control parameters to derive a new family of composite pulses for high-fidelity population transfer within the quantum error threshold. By design, our detuning-modulated N-piece composite sequences correct for any control inaccuracies including pulse strength, duration, resonance offsets errors, Stark shifts, unwanted frequency chirp, etc. We reveal the symmetries of the pulses that allow for straightforward scaling with minimal pulse overh…
We describe the use of composite rotations to combat systematic errors in single qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of NMR quantum computing these sequences should be applicable to any implementation of quantum computation.
We present a general systematic approach to design robust and high-fidelity quantum logic gates with Raman qubits using the technique of composite pulses. We use two mathematical tools -- the Morris-Shore and Majorana decompositions -- to reduce the three-state Raman system to an equivalent two-state system. They allow us to exploit the numerous composite pulses designed for two-state systems by extending them to Raman qubits. We construct the NOT, Hadamard, and rotation gates by means of the Morris-Shore transformation with the same uniform approach: sequences of pulses with the same phases f…
We discuss the implementation of arbitrary precision composite pulses developed using the methods of Brown et al. [Phys. Rev. A 70 (2004) 052318]. We give explicit results for pulse sequences designed to tackle both the simple case of pulse length errors and for the more complex case of off-resonance errors. The results are developed in the context of NMR quantum computation, but could be applied more widely.
We derive composite pulse sequences that achieve high-fidelity excitation of two-state systems in an optically dense, inhomogeneously broadened ensemble. The composite pulses are resistant to distortions due to the back-action of the medium they propagate in and are able to create high-fidelity inversion to optical depths $αz>10$. They function well with smooth pulse shapes used for coherent control of optical atomic transitions in quantum computation and communication. They are an intermediary solution between single $π$-pulse excitation schemes and adiabatic passage schemes, being far more e…
We introduce a method for detection of chiral molecules using sequences of three pulses driving a closed-loop three-state quantum system. The left- and right-handed enantiomers have identical optical properties (transition frequencies and transition dipole moments) with the only difference being the sign of one of the couplings. We identify twelve different sequences of resonant pulses for which chiral resolution with perfect contrast occurs. In all of them the first and third pulses are $π/2$-pulses and the middle pulse is a $π$-pulse. In addition, one of the three pulses must have a phase sh…
We present two methods for efficient detection of chiral molecules based on sequences of single pulses and Raman pulse pairs. The chiral molecules are modelled by a closed-loop three-state system with different signs in one of the couplings for the two enantiomers. One method uses a sequence of three interaction steps: a single pulse, a Raman pulse, and another single pulse. The other method uses a sequence of only two interaction steps: a Raman pulse, and a single pulse. The second method is simpler and faster but requires a more sophisticated Raman pulse than the first one. Both techniques a…
We introduce a method to rotate arbitrarily the excitation profile of universal broadband composite pulse sequences for robust high-fidelity population inversion. These pulses compensate deviations in any experimental parameter (e.g. pulse amplitude, pulse duration, detuning from resonance, Stark shifts, unwanted frequency chirp, etc.) and are applicable with any pulse shape. The rotation allows to achieve higher order robustness to any combination of pulse area and detuning errors at no additional cost. The latter can be particularly useful, e.g., when detuning errors are due to Stark shifts …
Composite Mølmer-Sørensen gate
本文提出一种振幅调制的复合 Mølmer-Sørensen 门，通过复合脉冲设计实现对门定时、失谐、耦合误差及交流斯塔克位移和漂移失谐的高鲁棒性，从而提升离子阱双量子比特门的保真度。
Narrowband and passband composite rotational quantum gates
本研究转换视角探索对误差高灵敏的量子门的应用潜力，借助复合脉冲技术推导三类窄带/通带基本量子门，并提出多种新的序列设计方法，性能优于现有文献中的经典序列。
Composite pulse sequences, which produce arbitrary pre-defined rotations of a qubit on the Bloch sphere, are presented. The composite sequences contain up to 17 pulses and can compensate up to eight orders of experimental errors in the pulse amplitude and the pulse duration. Composite sequences for three basic quantum gates -- X (NOT), Hadamard and arbitrary rotation -- are derived. Three classes of composite sequences are presented -- one symmetric and two asymmetric. They contain as their lowest members two well-known composite sequences -- the three-pulse symmetric SCROFULOUS pulse and the …
Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance (NMR) realises such a robust operation by employing a sequence of possibly poor quality pulses. In this article, we demonstrate that two kinds of composite pulses, one compensates for a pulse length error in a one-qubit system and the other compensates for a J-coupling error in a twoqubit system, have vanishing dynamical phase and thereby can be seen as geometric quantum gates, which implement unitar…
Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error epsilon. Independent of the initial state of the system, current techniques allow the error to be reduced to O(epsilon^3). Here, we extend the composite pulse technique to cancel errors to…
We introduce flexible high-fidelity passband (PB) composite pulse sequences constructed by concatenation of recently derived arbitrarily large and arbitrarily accurate broadband $\mathcal{B}$ and narrowband $\mathcal{N}$ composite sequences. Our PB sequences allow to produce flexible and tunable nearly rectangular two-state inversion profiles as a function of the individual pulse area because the width and the rectangularity of these profiles can be adjusted at will. Moreover, these PB sequences suppress excitation around pulse area $0$ and $2π$, and suppress deviations from complete populatio…
I describe the use of techniques based on composite rotations to combat systematic errors in quantum logic gates. Although developed and described within the context of Nuclear Magnetic Resonance (NMR) quantum computing these sequences should be applicable to other implementations of quantum computation.
In this work, we exploit the idea of composite pulses to achieve robust population inversion in a three-level quantum system. The scheme is based on the modulation of the coupling strength, while the other physical parameters remain unchanged. The composite pulses sequence is designed by vanishing high-order error terms, and can compensate the systematic errors to any desired order. In particular, this scheme keeps a good performance under the disturbance of waveform deformations. This trait ensures that population inversion can be nearly obtained even when the pulse sequence has a short jump …
We introduce a high-fidelity technique for coherent control of three-state quantum systems, which combines two popular control tools --- stimulated Raman adiabatic passage (STIRAP) and composite pulses. By using composite sequences of pairs of partly delayed pulses with appropriate phases the nonadiabatic transitions, which prevent STIRAP from reaching unit fidelity, can be canceled to an arbitrary order by destructive interference, and therefore the technique can be made arbitrarily accurate. The composite phases are given by simple analytic formulas, and they are universal for they do not de…
We put forward a method for achieving fast and robust for magnetization reversal in a nanomagnet, by combining the inverse engineering and composite pulses. The magnetic fields, generated by microwave with time-dependent frequency, are first designed inversely within short operation time, and composite pulses are further incorporated to improve the fidelity through reducing the effect of magnetic anisotropy. The high-fidelity magnetization reversals are illustrated with numerical examples, and visualized on Bloch sphere. The influence of damping parameters, relevant to the pulse sequence, is f…
We construct a minimal robust controlled-NOT gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number (N=3) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust SWAP gate cannot be constructed with N=3, but requires N=6 elementary two-qubit gates.
We present a general method to derive detuning-modualted composite pulses (DMCPs) as N rotations of a canonical two-state quantum system to create accurate and robust pulses that are independent of the initial state of the system. This scheme has minimal pulse overhead, and achieves pulses that are stable against amplitude errors well within the $10^{-4}$ threshold that may be suitable for quantum information processing (QIP), within the lifetime of the system. This family of pulses enables to overcome inevitable fabrication errors in silicon photonics, and relax the need for a precise initial…
We develop a systematic approach to derive narrowband (NB) and passband (PB) composite sequences which can produce any pre-selected transition probability with any desired accuracy. The NB composite pulses are derived by successive cancellation of derivative terms in the propagator. The PB pulses are built by combining two half-$π$ NB pulses. Both the NB and PB pulses feature vanishing wings on both sides of the central region around the target transition probability, and the PB pulses offer also a flat (broadband) top around the target probability. The NB sequences are particularly suitable f…
A vital requirement for a quantum computer is the ability to locally address, with high fidelity, any of its qubits without affecting their neighbors. We propose an addressing method using composite sequences of laser pulses, which reduces dramatically the addressing error in a lattice of closely spaced atoms or ions, and at the same time significantly enhances the robustness of qubit manipulations. To this end, we design novel high-fidelity composite pulses for the most important single-qubit operations. In principle, this method allows one to beat the diffraction limit, for only atoms situat…
Concatenated Composite Pulses Beyond Local Residual-Error Preservation
本文证明局部残差误差保持并非级联复合脉冲的必要条件，通过允许误差生成元共同缩放，结合CORPSE和SCROFULOUS内脉冲与等旋转角外序列，实现两种系统误差的同时一阶补偿。
Amplitude noise which inflicts a random two qubit term is one of the main obstacles preventing the implementation of a high fidelity two-body gate below the fault tolerance threshold. This noise is difficult to refocus as any refocusing technique could only tackle noise with frequency below the operation rate. Since the two qubit gate speed is normally the slowest rate in the system, it constitutes the last bottleneck towards an implementation of a gate below the fault tolerant threshold. Here we propose to use composite pulses as a dynamical decoupling approach, in order to reduce two qubit g…
We propose a fast and efficient technique to create classes of highly entangled states of trapped ions, such as arbitrary Dicke states and superpositions of them, e.g. NOON states. The ions are initialized in the phonon ground state and are addressed globally with a composite pulse that is resonant with the first motional sideband. The technique operates on comparatively short time scales, as resonant interactions allow one to use the minimum laser pulse area. The number of single pulses from the composite sequence is equal to the number of ions, thus the implementation complexity grows only l…
Robust composite two-qubit gates for silicon-based spin qubits
基于哈密顿量逆工程，提出一种高保真度fSim门方案，通过结合几何和动态方法提升鲁棒性，实现快速构造复合双量子比特门。
In this work, we propose a composite pulses scheme by modulating phases to achieve high fidelity population transfer in three-level systems. To circumvent the obstacle that not enough variables are exploited to eliminate the systematic errors in the transition probability, we put forward a cost function to find the optimal value. The cost function is independently constructed either in ensuring an accurate population of the target state, or in suppressing the population of the leakage state, or both of them. The results demonstrate that population transfer is implemented with high fidelity eve…
The fidelity of laser-driven quantum logic operations on trapped ion qubits tend to be lower than microwave-driven logic operations due to the difficulty of stabilizing the driving fields at the ion location. Through stabilization of the driving optical fields and use of composite pulse sequences, we demonstrate high fidelity single-qubit gates for the hyperfine qubit of a $^{171}\text{Yb}^+$ ion trapped in a microfabricated surface electrode ion trap. Gate error is characterized using a randomized benchmarking protocol, and an average error per randomized Clifford group gate of $3.6(3)\times1…
We present a fast phase gate scheme that is experimentally achievable and has an operation time more than two orders of magnitude faster than current experimental schemes for low numbers of pulses. The gate time improves with the number of pulses following an inverse power law. Unlike implemented schemes which excite precise motional sidebands, thus limiting the gate timescale, our scheme excites multiple motional states using discrete ultra-fast pulses. We use beam-splitters to divide pulses into smaller components to overcome limitations due to the finite laser pulse repetition rate. This pr…
Motional state analysis of a trapped ion by ultra-narrowband composite pulses
本文提出利用超窄带复合脉冲扫描蓝边带跃迁，实现对囚禁离子运动态的无先验测量，适用于Lamb-Dicke区内外，并演示了单离子实验验证。
Correlation thresholds for effective composite pulse quantum error mitigation
本文研究复合脉冲分段中错误相关性对量子门保真度的影响，证明临界相关阈值的存在，并在集成光子学中分析其性质与数值表现。
We study the performance of composite pulses in the presence of time-varying control noise on a single qubit. These protocols, originally devised only to correct for static, systematic errors, are shown to be robust to time-dependent non-Markovian noise in the control field up to frequencies as high as ~10% of the Rabi frequency. Our study combines a generalized filter-function approach with asymptotic dc-limit calculations to give a simple analytic framework for error analysis applied to a number of composite-pulse sequences relevant to nuclear magnetic resonance as well as quantum informatio…
The resonator-induced phase gate is a multi-qubit controlled-phase gate for fixed-frequency superconducting qubits. Through off-resonant driving of a bus resonator, statically coupled qubits acquire a state-dependent phase. However, photon loss leads to dephasing during the gate, and any residual entanglement between the resonator and qubits after the gate leads to decoherence. Here we consider how to shape the drive pulse to minimize these unwanted effects. First, we review how the gate's entangling and dephasing rates depend on the system parameters and validate closed-form solutions against…
Composite nonadiabatic geometric quantum gates with optimization on superconducting circuits
本文提出一种优化的复合非绝热几何量子门方案，通过优化路径参数提升门的鲁棒性，适用于超导量子比特实现。
The creation of composite quantum gates that implement quantum response functions $\hat{U}(θ)$ dependent on some parameter of interest $θ$ is often more of an art than a science. Through inspired design, a sequence of $L$ primitive gates also depending on $θ$ can engineer a highly nontrivial $\hat{U}(θ)$ that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new, useful examples of $\hat{U}(θ)$ requires great intuition to perceive the possibilities, and often brute-force to find optimal implementations. We present a systematic and efficient methodol…
Robust, fast and high-fidelity composite single-qubit gates for superconducting transmon qubits
针对超导 transmon 量子比特的相干误差问题，提出基于复合脉冲的控制方法，实现鲁棒快速的高保真单量子比特门，并经仿真和四种独立技术验证。
The three-qubit conditional swap gate(Fredkin gate) is a universal gate that can be used to create any logic circuit and has many direct usages. In this paper, we experimentally realized Fredkin gate with only three transition pulses in solution of alanine. It appears that no experimental realization of Fredkin gate with fewer pulses has yet been reported up to now. In addition, the simple structure of our scheme makes it easy to be implemented in experiments.
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…
We analyze analytically and numerically quantum logic gates in a one-dimensional spin chain with Heisenberg interaction. Analytic solutions for basic one-qubit gates and swap gate are obtained for a quantum computer based on logical qubits. We calculated the errors caused by imperfect pulses which implement the quantum logic gates. It is numerically demonstrated that the probability error is proportional to $ε^4$, while the phase error is proportional to $ε$, where $ε$ is the characteristic deviation from the perfect pulse duration.
Composite pulses, originally developed in Nuclear Magnetic Resonance (NMR), have found widespread use in experimental quantum information processing (QIP) to reduce the effects of systematic errors. Most pulses used so far have simply been adapted from existing NMR designs, and while techniques have been developed for designing composite pulses with arbitrary precision the results have been quite complicated and have found little application. Here I describe techniques for designing short but effective composite pulses to implement robust NOT gates, bringing together existing insights from NMR…
Accurate quantum control is a key technology for realizing quantum information processing, such as quantum communication and quantum computation. In reality, a quantum state under control suffers from undesirable effects caused by systematic errors. A composite pulse (CP) is used to eliminate the effects of systematic errors during control. One qubit control, which is the most fundamental in quantum control, is typically affected by two errors: pulse length error (PLE) and off-resonance error (ORE). In this study, we focus on ORE-robust CPs and systematically construct ORE-robust symmetric CPs…
We present a new method for determining pulse imperfections and improving the single-gate fidelity in a superconducting qubit. By applying consecutive positive and negative $π$ pulses, we amplify the qubit evolution due to microwave pulse distortion, which causes the qubit state to rotate around an axis perpendicular to the intended rotation axis. Measuring these rotations as a function of pulse period allows us to reconstruct the shape of the microwave pulse arriving at the sample. Using the extracted response to predistort the input signal, we are able to improve the pulse shapes and to reac…
Nonadiabatic holonomic quantum computation (NHQC) has attracted significant attention due to its fast evolution and the geometric nature induced resilience to local noises. However, its long operation time and complex physical implementation make it hard to surpass the dynamical scheme, and thus hindering its wide application. Here, we present to implement NHQC with the shortest path under some conditions, through the inverse Hamiltonian engineering technique, which posseses higher fidelity and stronger robustness than previous NHQC schemes. Meanwhile, the gate performance in our scheme can be…
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