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 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.
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.
We implement an ensemble quantum counting algorithm on three NMR spectrometers with 1H resonance frequencies of 500, 600 and 750 MHz. At higher frequencies, the results deviate markedly from naive theoretical predictions. These systematic errors can be attributed almost entirely to off-resonance effects, which can be substantially corrected for using fully-compensating composite rotation pulse sequences originally developed by Tycko. We also derive an analytic expression for generating such sequences with arbitrary rotation angles.
We describe novel composite pulse sequences which act as general rotors and thus are suitable for nuclear magnetic resonance (NMR) quantum computation. The Resonance Offset Tailoring To Enhance Nutations (ROTTEN) approach permits perfect compensation of off-resonance errors at two selected frequencies placed symmetrically around the frequency of the RF source.
We discuss the commonly encountered problem when optimizing NMR pulses using optimal control that the otherwise very precise NMR theory does not provide as excellent agreement with experiments. We hypothesize that this disagreement is due to phase transients in the pulse due to abrupt phase- and amplitude changes resulting in a large bandwidth. We apply the GROUP algorithm that gives high fidelity pulses with a low bandwidth compared to the typical GRAPE pulses. Our results obtain a better agreement between experiment and simulations supporting our hypothesis and solution to the problem.
We demonstrate experimentally the usefulness of selective pulses in NMR to perform quantum computation. Three different techniques based on selective pulse excitations have been proposed to prepare a spin system in a pseudo-pure state. We describe the design of novel ``portmanteau'' gates using the selective manipulation of level populations. A selective pulse implementation of the Deutsch-Jozsa algorithm for a two-qubit and a three-qubit quantum computer is demonstrated.
Experimental realization of quantum information processing in the field of nuclear magnetic resonance (NMR) has been well established. Implementation of conditional phase shift gate has been a significant step, which has lead to realization of important algorithms such as Grover's search algorithm and quantum Fourier transform. This gate has so far been implemented in NMR by using coupling evolution method. We demonstrate here the implementation of the conditional phase shift gate using transition selective pulses. As an application of the gate, we demonstrate Grover's search algorithm and qua…
Quantum computing using two-dimensional NMR has recently been described using scalar coupling evolution technique [J. Chem. Phys.,109,10603 (1998)]. In the present paper, we describe two-dimensional NMR quantum computing with the help of selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been impl…
A detailed description of the development of a three qubit NMR realization of the Deutsch-Jozsa algorithm [Collins et.al., Phys. Rev. A 62, 022304 (2000)] is provided. The theoretical and experimental techniques used for the reduction of the algorithm's evolution steps into a sequence of NMR pulses are discussed at length. This includes the description of general pulse sequence compilation techniques, various schemes for indirectly coupled gate realizations, experimental pulse parameterization techniques and bookkeeping methods for pulse phases.
The efficiency of dipole-dipole coupling driven coherence transfer experiments in solid-state NMR spectroscopy of powder samples is limited by dispersion of the orientation of the internuclear vectors relative to the external magnetic field. Here we introduce general design principles and resulting pulse sequences that approach full polarization transfer efficiency for all crystallite orientations in a powder in magic-angle-spinning experiments. The methods compensate for the defocusing of coherence due to orientation dependent dipolar coupling interactions and inhomogeneous radio-frequency fi…
The realisation of optically detected magnetic resonance via nitrogen vacancy centers in diamond faces challenges at high magnetic fields which include growing energy consumption of control pulses as well as decreasing sensitivities. Here we address these challenges with the design of shaped pulses in microwave control sequences that achieve orders magnitude reductions in energy consumption and concomitant increases in sensitivity when compared to standard top-hat microwave pulses. The method proposed here is general and can be applied to any quantum sensor subjected to pulsed control sequence…
We present a genetic algorithm for finding a set of pulse sequences, or rotations, for a given quantum logic gate, as implemented by NMR. We demonstrate the utility of the method by showing that shorter sequences than have been previously published can be found for both a CNOT and for the central part of Shor's algorithm (for N=15.) Artificial intelligence techniques like the genetic algorithm here presented have an enormous potential for simplifying the implementation of working quantum computers.
Quantum correlations are investigated theoretically in a two-spin system with the dipole-dipole interactions in the NMR multiple-pulse spin-locking experiments. We consider two schemes of the multiple-pulse spin-locking. The first scheme consists of $π/2$-pulses only and the delays between the pulses can differ. The second scheme contains $φ$-pulses ($0<φ<π$) and has equal delays between them. We calculate entanglement for both schemes for an initial separable state. We show that entanglement is absent for the first scheme at equal delays between $π/2$-pulses at arbitraty temperatures. Entangl…
We derive the Hamiltonian in the rotating frame for NMR quantum computing with homonucleus molecules as its computational resource. The Hamiltonian thus obtained is different from conventional Hamiltonians that appear in literature. It is shown that control pulses designed for heteronucleus spins can be translated to pulses for homonucleus spins by simply replacing hard pulses by soft pulses with properly chosen pulse width. To demonstrate the validity of our Hamiltonian, we conducted several experiments employing cytosine as a homonucleus molecule. All the experimental results demonstrate tha…
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.
While Nuclear Magnetic Resonance (NMR) techniques are unlikely to lead to a large scale quantum computer they are well suited to investigating basic phenomena and developing new techniques. Indeed it is likely that many existing NMR techniques will find uses in quantum information processing. Here I describe how the composite rotation (composite pulse) method can be used to develop quantum logic gates which are robust against systematic errors.
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…
The nitrogen vacancy (NV) color center in diamond is an enormously important platform for the development of quantum sensors, including for single spin and single molecule NMR. Detection of weak single-spin signals is greatly enhanced by repeated sequences of microwave pulses; in these dynamical decoupling (DD) techniques, the key control parameters swept in the experiment are the time intervals, $τ$, between pulses. Here we show that, in fact, the pulse duration offers a powerful additional control parameter. While previously, a non-negligible pulse-width has been considered simply a source o…
Development of Neural Network-Based Optimal Control Pulse Generator for Quantum Logic Gates Using the GRAPE Algorithm in NMR Quantum Computer
Sharif University of Technology（谢里夫理工大学）；Centre for Quantum Engineering and photonics Technology, Sharif University of Technology（谢里夫理工大学量子工程与光子技术中心）；Koc University（科奇大学）
We show how to divide a coupled multi-spin system into a small subset of ``active'' spins that evolve under chemical shift or scalar coupling operators, and a larger subset of ``spectator'' spins which are returned to their initial states, as if their motion had been temporarily frozen. This allows us to implement basic one-qubit and two-qubit operations from which general operations on $N$-qubits can be constructed, suitable for quantum computation. The principles are illustrated by experiments on the three coupled protons of 2,3-dibromopropanoic acid, but the method is applicable to any spin…
Experimental implementation of a quantum algorithm requires unitary operator decomposition. Here we treat the unitary operator decomposition as an optimization problem and use Genetic Algorithm, a global optimization method inspired by nature's evolutionary process for operator decomposition. As an application, we apply this to NMR Quantum Information Processing and find a probabilistic way of doing universal quantum computation using global hard pulses. We also demonstrate efficient creation of singlet state (as a special case of Bell state) directly from thermal equilibrium using an optimum …
A scheme for decoupling and selectively recoupling large networks of dipolar-coupled spins is proposed. The scheme relies on a combination of broadband, decoupling pulse sequences applied to all the nuclear spins with a band-selective pulse sequence for single spin rotations or recoupling. The evolution-time overhead required for selective coupling is independent of the number of spins, subject to time-scale constraints, for which we discuss the feasibility. This scheme may improve the scalability of solid-state-NMR quantum computing architectures.
Noise-resistant off-resonance π-rotation realized in an NMR system
School of Physics and Technology, Wuhan University（武汉大学物理科学与技术学院）；School of Physics, Hangzhou Normal University（杭州师范大学物理学院）；Hefei National Laboratory（合肥国家实验室）
We demonstrate how NMR can in principle be used to implement all the elements required to build quantum computers, and briefly discuss the potential applications of insights from quantum logic to the development of novel pulse sequences with applications in more conventional NMR experiments.
We present robust radio frequency (rf) pulses that tolerate a factor of six inhomogeneity in the B1 field, significantly enhancing the potential of toroid cavity resonators for NMR spectroscopic applications. Both point-to-point (PP) and unitary rotation (UR) pulses were optimized for excitation, inversion, and refocusing using the gradient ascent pulse engineering (GRAPE) algorithm based on optimal control theory. In addition, the optimized parameterization (OP) algorithm applied to the adiabatic BIR-4 UR pulse scheme enabled ultra-short (50 microsec) pulses with acceptable performance compar…
We present a framework that uses a continuous frequency space to describe and design solid-state NMR experiments. The approach is similar to the well established Floquet treatment for NMR, but is not restricted to periodic Hamiltonians and allows the design of experiments in a reverse fashion. The framework is based on perturbation theory on a continuous Fourier space, which leads to effective, i.e., time-independent, Hamiltonians. It allows the back calculation of the pulse scheme from the desired effective Hamiltonian as a function of spin-system parameters. We show as an example how to back…
We study a 2-qubit nuclear spin system for realizing an arbitrary geometric quantum phase gate by means of non-adiabatic operation. A single magnetic pulse with multi harmonic frequencies is applied to manipulate the quantum states of 2-qubit instantly. Using resonant transition approximation, the time dependent Hamiltonian of two nuclear spins can be solved analytically. The time evolution of the wave function is obtained without adiabatic approximation. The parameters of magnetic pulse, such as the frequency, amplitude, phase of each harmonic part as well as the time duration of the pulse, a…
We develop a new class of genetic algorithm that computationally determines efficient pulse sequences to implement a quantum gate U in a three-qubit system. The method is shown to be quite general, and the same algorithm can be used to derive efficient sequences for a variety of target matrices. We demonstrate this by implementing the inversion-on-equality gate efficiently when the spin-spin coupling constants $J_{12}=J_{23}=J$ and $J_{13}=0$. We also propose new pulse sequences to implement the Parity gate and Fanout gate, which are about 50% more efficient than the previous best efforts. Mor…
We consider the problem of time-optimal control of quadrupole nucleus with the spin I=1 by NMR. In contrast to the conventional methods based on selective pulses, the control is implemented using nonselective pulses separated by free evolution intervals. Using the Cartan decomposition, the system of equations is obtained for finding parameters of a control field. Partial time-optimal solutions for the important single-qutrit gates (selective rotations and quantum Fourier transform) are found. The strong dependence of minimum gate implementation times on global phase of the gate is observed. Th…
A dipolar recoupling technique is introduced as a new approach to quantum gate operation in solid-state NMR under magic angle spinning. The default-off property of inter-qubit interaction provides a simple way to controlled operation without requiring elaborate qubit-decoupling pulses.
The error-robust and short composite operations named ConCatenated Composite Pulses (CCCPs), developed as high-precision unitary operations in quantum information processing (QIP), are derived from composite pulses widely employed in nuclear magnetic resonance (NMR). CCCPs simultaneously compensate for two types of systematic errors, which was not possible with the known composite pulses in NMR. Our experiments demonstrate that CCCPs are powerful and versatile tools not only in QIP but also in NMR.
We report experimental implementation of discrete Fourier transformation(DFT) on a nuclear magnetic resonance(NMR) quantum computer. Experimental results agree with theoretical results. Using the pulse sequences we introduced, DFT can be realized on any L-bit quantum number in principle.
The exact solution for the free induction decay in a one-dimensional system in the multi-pulse experiment is obtained at both high and low temperatures in the approximation of nearest neighbor interactions. The experimental investigation is performed on a quasi-one-dimensional system of $^{19}$F nuclear spins in a single crystal of fluorapatite. The theoretical results are in a good agreement with the obtained experimental data.
We present highly robust, optimal control-based shaped pulses designed to replace all 90° and 180° hard pulses in a given pulse sequence for improved performance. Special attention was devoted to ensuring that the pulses can be simply substituted in a one-to-one fashion for the original hard pulses without any additional modification of the existing sequence. The set of four pulses for each nucleus therefore consists of 90° and 180° point-to-point (PP) and universal rotation (UR) pulses of identical duration. These 1 ms pulses provide uniform performance over resonance offsets of 20 kHz (1H) a…
A generalized quantum search algorithm, where phase inversions for the marked state and the prepared state are replaced by $π/2$ phase rotations, is realized in a 2-qubit NMR heteronuclear system. The quantum algorithm searches a marked state with a smaller step compared to standard Grover algorithm. Phase matching requirement in quantum searching is demonstrated by comparing it with another generalized algorithm where the two phase rotations are $π/2$ and $3π/2$ respectively. Pulse sequences which include non 90 degree pulses are given.
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 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.
In this work, average Hamiltonian theory is used to study selective excitation in a spin-1/2 system evolving under a series of small flip-angle $θ-$pulses $(θ\ll 1)$ that are applied either periodically [which corresponds to the DANTE pulse sequence] or aperiodically. First, an average Hamiltonian description of the DANTE pulse sequence is developed; such a description is determined to be valid either at or very far from the DANTE resonance frequencies, which are simply integer multiples of the inverse of the interpulse delay. For aperiodic excitation schemes where the interpulse delays are ch…
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 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…
The paper describes the design of broadband chirp excitation pulses in NMR. We first develop a three stage model for understanding chirp excitation in NMR. We then show how a chirp $π$ pulse can be used to refocus the phase of the chirp excitation pulse. The resulting magnetization still has some phase dispersion in it. We show how a combination of two chirp $π$ pulses instead of one can be used to eliminate this dispersion, leaving behind a small residual phase dispersion. The excitation pulse sequence presented here allow exciting arbitrary large bandwidths without increasing the peak rf-amp…
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 …
Searching for marked items from an unsorted database is an important scientific problem and a benchmark for computing devices as well. Using a 7-qubit liquid NMR quantum computer, we have demonstrated successfully an hybrid quantum fetching algorithm that finds marked items using only a single query. The essential idea is the operation of quantum computers in parallel. We gave the detailed pulse sequence for coherent control of the 7 qubits. The pulse sequence demonstrated here is not only useful for ensemble quantum computation, but also can be regarded as a general purpose control-gate which…
This paper presents a systematic study of applying composite method approximations with locally dense basis sets (LDBS) to efficiently calculate NMR shielding constants in small and medium-sized molecules. The pcSseg-n series of basis sets are shown to have similar accuracy to the pcS-n series when n $\geq1$ and can slightly reduce compute costs. We identify two different LDBS partition schemes that perform very effectively for density functional calculations. We select a large subset of the recent NS372 database containing 290 H, C, N, and O shielding values evaluated by reference methods on …
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…
Optimal labeling schemes lead to efficient experimental protocols for quantum information processing by nuclear magnetic resonance (NMR). A systematic approach of finding optimal labeling schemes for a given computation is described here. The scheme is described for both quadrupolar systems and spin-1/2 systems. Finally, one of the optimal labeling scheme has been used to experimentally implement a quantum full-adder in a 4-qubit system by NMR, using the technique of transition selective pulses.
Pursuit and Review of Magnetic Resonance Imaging (MRI) based Quantum Computing -- Qubit Generation, Spin Purification, Tailored RF Pulses and MRI Sequences for Quantum Computing
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.
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…
In this paper, we report an experimental realization of quantum switch using nuclear spins and magnetic resonant pulses. The nuclear spins of H and C in carbon-13 labelled chloroform are used to carry the information, then nuclear magnetic resonance pulses are applied to perform either bypass or cross function to achieve the switching. Compared with a traditional space or time domain switch, this switching architecture is much more scalable, therefore a high throughput switching device can be built simply by increasing the number of I/O ports. In addition, it can be used not only as a device t…
The propagation of excitation along a one-dimensional chain of atoms is simulated by means of NMR. The physical system used as an analog quantum computer is a nucleus of 133-Cs (spin 7/2) in a liquid crystalline matrix. The Hamiltonian of migration is simulated by using a special 7-frequency pulse, and the dynamics is monitored by following the transfer of population from one of the 8 spin energy levels to the other.
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…
The past decade has demonstrated increasing interests in using optimal control based methods within coherent quantum controllable systems. The versatility of such methods has been demonstrated with particular elegance within nuclear magnetic resonance (NMR) where natural separation between coherent and dissipative spin dynamics processes has enabled coherent quantum control over long periods of time to shape the experiment to almost ideal adoption to the spin system and external manipulations. This has led to new design principles as well as powerful new experimental methods within magnetic re…
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…
