
DOI: 10.22661/AAPPSBL.2018.28.1.03
The Research Status of Nuclear Magnetic Resonance
for Quantum Computing
KEREN LI,^{1, 2} TAO XIN,^{1, 2} YUANYE ZHU,^{1, 2} XIANGYU KONG,^{2} AND GUILU LONG^{1, 2, 3, *}
^{1 }STATE KEY LABORATORY OF LOWDIMENSIONAL QUANTUM PHYSICS AND DEPARTMENT OF PHYSICS, TSINGHUA UNIVERSITY
^{2 }TSINGHUA NATIONAL LABORATORY OF INFORMATION SCIENCE AND TECHNOLOGY
^{3 }THE INNOVATIVE CENTER OF QUANTUM MATTER
In the past two years, dramatic changes have taken place in the field of quantum information. Both theoretical and experimental research continues as more and more resources from national governments and military funding are provided in this direction. Various quantum computing systems have shown their potential as a practical quantum computer. Among them, superconducting and ion trap systems have led this competition. Compared with those thriving quantum systems, a nuclear magnetic resonance system has some other advantages, such as accurate control techniques and a long coherence time. In such a system, we could manipulate a 12qubit quantum system and implement a complicated 12qubit operation. In this article, we focus on the interesting works in our group, not only representing newlydeveloped techniques, but also pointing towards a thriving research area. The review mainly includes three parts: 1) how to implement a more accurate control manipulation by a new technique called MQFC; 2) how to simulate other quantum systems by quantum simulation; and 3) the introduction of a new cloudbased quantum computing system—NMRCloudQ.
* Correspondence and requests for materials should be addressed to G.L.L.: gllong@tsinghua.edu.cn
INTRODUCTION
The quantum computer is considered to be an innovative technique, because it may efficiently solve some problems which are usually hard in a classical case [53]. However, the building of universal quantum computers mostly depends on the development of quantum control techniques on real quantum devices. Eighteen years ago, David P. Divincenzo presented five crucial conditions for realizing the construction of a quantum computer [2], for instance, quantum systems should be scalable with welldistinguished qubits. At present, rapid progress in quantum engineering technology has been recently achieved, and some promising quantum systems satisfying the above mentioned five requirements are welldeveloped, such as, nulcear magnetic resonance(NMR) systems [3], quantum dots [810], superconducting circuit [4, 5] and trapped ions [6, 7]. As of today, no one can tell which one of these existing quantum systems will be a universal quantum processor in the future. In this article, we focus attention on the NMR platform and review recent achievements and its current status in quantum information processing.
The NMR spectrometer was early used to analyse the structure of complex molecules, so that numerous pulse control techniques were developed to precisely control the nuclear spins in a target molecule [11]. Over the past decade, NMR further became used for quantum processors to perform all kinds of quantum information and quantum computation tasks [1214]. Twenty years of developments in NMR have accumulated a wealth of experience and wellestablished control techniques in quantum computation [1517]. Compared with other quantum systems, the nuclear spins in NMR systems are more robust against the a hostile environment and can be more precisely controlled with a longer coherence time.
Benefiting from the rapid development of quantum control techniques including composite pulses [18, 19], shaped pulses, dynamical decoupling sequence [2123], the average hamiltonian [20], and gradient ascent pulse engineering [24], the NMR platform has wide applications in the demonstrations of quantum algorithms and realizations of quantum simulations. Some wellknown quantum algorithms that show the advantages of speed of a quantum computer have been realized in the NMR platform in the past decade, such as the Shor factorization algorithm [25], the Grover search algorithm [2628], the OrderingFinding algorithm [29], and algorithms for solving a linear equation [30]. For quantum simulation, it is a fact that simulating quantum dynamics is still a challenging problem, even when using the supercomputer, especially when handling a large quantum system. Richard Feynman [31] proposed an solution to this problem, positing that we can use some trusted quantum system to imitate the dynamics of another less controllable quantum system. Quantum simulation itself requires an exponentially large amount of physical resources to overcome the difficulties of the exponential amount of computer memory that is needed to deal with a quantum state. Quantum simulation has some natural applications in many physical problems. As one of veritable and controllable quantum systems, the NMR platform has obtained some famous achievements in quantum simulation in areas and topics, such as quantum phase transition [34, 35], manyboby interactions [33], molecular hydrogen [32], topological matter [36, 37] and energy physics [38].
Recently, the renewal of interest in artificial intelligence and machine learning infuses new motivation into quantum computation [39]. Quantum machine learning algorithms may be faster than the classical counterparts for some special problems. The interdisciplinary research in artificial intelligence, machine learning and quantum computation will be another promising direction in natural science. For instance, a separabilityentanglement classifier via machinie learning was proposed to determine whether a given quantum state is entangled [40]. NMR has achieved some experimental developments in quantum machine learning. The experimental implementation of a quantum algorithm for solving a linear equation has been realized in NMR. It is the so called quantum HHL algorithm [41]. Z. Li et al performed the experimental implementation of the support vector machine (SVM) in a quantum version in a liquid NMR processor [42]. SVM is a popular tool used for classification via supervised learning. The successful demonstrations of quantum machine learning algorithms show that quantum computing will bring a new life to classical machine learning.
Although many breakthroughs of both theoretical protocol and hardware design in the process of a building quantum computer have been obtained, it is still uncertain when universal quantum computer with thousands of qubits will be a part of our daily life. So far, the quantum computer in laboratories that have been developed usually have about ten qubits and need professional maintenance from scientists. It is unlikely that in the near future the public will be able to directly experience the full power of quantum computing. The appearance of cloudbased quantum computing will make some wellcontrolled and small quantum computers in laboratories approachable to public users by presenting them on the cloud. In this new area of cloudbased quantum computing, the IBM company first launched their influential cloud services on a 5qubit superconducting quantum processor called IBM Quantum Experience in 2016. During its oneyearsevice, registered users was over 40,000 and over 270,000 of experiments were submitted and performed on a webbased interface provided by IBM. Now, another platform has followed providing quantum cloud service. Three research teamsfrom Prof. G. L. Long at Tsinghua University, the AliUniversity of Science and Technology of China(USTC) joint program and Quantum BenYuan at USTClaunched their cloud services on October 11, 2017. As one of the allround platforms with mature techniques in realizing quantum computation and improving quantum control level, the NMR was recently proposed for a new cloud quantum computing service  NMRCloudQ [43]. NMRCloudQ provides a fourqubit cloudbased quantum computer and provides a physical spectrometer in the lab for amateurs or professionals to demonstrate the quantum algorithms and explore quantum phenomena. In this article, we mainly review the recent advancements of NMR in quantum control techniques, quantum algorithms and quantum simulations, especially NMRCloudQ.
The paper is organized as follows: In Sec.II, we briefly review the experiment where we boost an the control on a 12qubit system. In Sec.III, we introduce an experiment on how to simulate the basic problems in quantum mechanics using the NMR quantum simulator. In Sec.IV, we import a new platform for nonexperimentalists to implement the QIP (quantum information processing) task—NMRCloud and explain it briefly. Finally, Sec. V summarizes the entire article and gives some prospectives for the future.
MQFC
In theory, a quantum circuit can be split into many singlebit gates and twobit gates, and these gates can be easily implemented in experiments. However, the practical situation is that the number of logic gates usually increases exponentially with the size of quantum system and the errors of all logic gates will be accumulated. Some optimization algorithms with high performance are necessary to overcome the above difficulties. In the last decade, experimental techniques for solving most control problems were rapidly developed or smallscale quantum systems. However, the current quantum control techniques are not available for a lager quantum system because classical computers can not effectively optimize the control problem of a lager quantum system. This is due to the fact that the entire optimization process requires a classic computer to simulate the evolution of a quantum system, and hence the time cost will increase exponentially with the number of qubits in the system.
Fig. 1: Schematic of the MQFC optimization process: Starting from the initial guess, the pulse generator generates the available pulses to act on the NMR quantum system. The fitness function ƒ and each slice gradient g are measured in the system and the results are fed back to us if ƒ is greater than a threshold; otherwise we use g to iterate out a new control field and generate a new pulse,to enter the next cycle.
We presented the idea that the control field could be optimized with our quantum processor instead ofa classical computer. This is the measurementbased quantum feedback control called MQFC [4446]. Using this scheme, we optimized a pulse that can realize the preparation of a 12coherent state. It is worth noting that this pulse is completely designed and optimized using a 12bit nuclear magnetic resonance quantum processor. It is different from the conventional process where we optimize the desired pulse either by the gradient ascent pulse engineering (GRAPE) optimization algorithm or by the SSGRAPE (subsystem optimization GRAPE) optimization algorithm.
To prepare a 12coherent state, we optimized the pulse by the MQFC scheme as shown in Fig. 1. In fact, the MQFC algorithm is also an optimization method based on gradient calculation. We also need to measure its fitness function ƒ and the gradient g at the end of each pulse. How these physical quantities were measured can be found in the original work. In the process, the contribution that classical computers have provided to generate the next pulse via the measured gradient g. It is worth mentioning that this process will only involve some algebraic addition and subtraction operations and hence the calculation of its time consumption can be almost ignored.
The pulse profile we use throughout the MQFC optimization is determined by our molecular structure. The entire MQFC optimized pulse has a total of 5.56 milliseconds with 278 small pulses of 20 microseconds pieces, among which 168 small pulses remain at 0 in the optimization process, representing a period of free evolution. The remaining 110 small pulses are the target we need to optimize. Therefore, we only need to consider these 110 slices in the process of measuring the gradient, thus greatly simplifying our experimental time. If n is the size of system and M is the number of optimized pulse pieces, a literation requires 4nM experiments.
Fig. 2: Results for 12qubit MQFC experiment: (a) NMR spectrum for 12 coherent state: the red line is experimental result while the blue one is the simulated one; (b) Spectra of a 12coherence state after every iteration of MQFC optimization; (c) Comparison between different methods to prepare 12coherence state. solid line is the simulated consequence while dash lines are the experiments. blue ones used the SSGRAPE and red ones used MQFC optimization method; (d) The final fidelities of 12coherence state obtained via SSGRAPE and MQFC.
Figure 2 shows the experimental results. Compared with the classical optimization algorithm, there are mainly two advantages: effectiveness and accuracy. The time consumption of MQFC depends on the complexity of the target state. In most case, the target states are not always complicated. For sparse states, the time consumption increases linearly with the size of the system. MQFC achieved the preparation of 12 coherent states that could not be achieved in the past. Performance rose by nearly 10% as compared to the result obtained by using classical optimization, since MQFC overcomes some unknown errors in quantum systems. MQFC can be also transplanted into many other systems. We can claim that our results provide a new idea for actualizing quantum system control in the future, bringing us closer to the goal of demonstrating.
QUANTUM SIMULATIONS
Since the 1980s, we have seen that the real world, which follows quantum mechanics. Can not be simulated efficiently using traditional methods. One of the biggest challenges for simulation is that the number of parameters required increase exponentially, and the storage resources of classical computers presently used also increase exponentially. The storage needs may exceed the storage capacity of today's supercom puters and even the most advanced CPUs cannot deal with excessively large amounts of data. For this reason, in 1982, Feynman first imagined using quantum systems to simulate real physical phenomena [31], which is now considered to be the origin of the concept of the quantum computer. In 1996, S. Lloyd proposed a quantum computer that could be used as a universal quantum simulator to simulate any real system [47]. However, simulating a specific problem, using a specific quantum simulator, appears to be much easier, than creating a universal quantum simulator (quantum computer).
Quantum simulators have been successfully realized in several quantum platforms, such as ion traps, optical systems and superconducting systems. Our wellknown nuclear magnetic resonance (NMR) system has gone through six decades of development, and its quantum control technology can achieve precise control of 12qubit molecules. Liquid NMR quantum information processing is often considered as an excellent quantum simulation platform due to its long decoherence time and accurate radio frequency pulse control.
Fig. 3: 3qubit quantum circuit for Δt evolution, F^{1} is a threebit Fourier transform inverse transform operator; D is a kinetic operator evolution operatorw. S, S is the number of bits inversion operator and is the potential operator.
Quantum tunneling refers to a quantum behavior. For instance, electrons and other microparticles can cross a barrier energy that is higher than their total energy. It can not happen in classical world; however, quantum theory can explain this situation in a probabilistic way.
In 2013, Feng et al simulated the tunneling of particles in a doublewell in a onedimensional space using a digital quantum simulation algorithm on a NMR quantum simulator and demonstrated that the physical phenomena we know can be studied in a quantum simulator .
Considering a particle that is trapped in a onedimensional doublewell trap, the law of its motion necessarily follows the Schrödinger equation,

(1) 
where ħ is the famous Planck constant, P̂ and X̂ are the momentum and coordinate operators in quantum mechanics respectively, and is the Hamiltonian of the particles in the potential well. Therefore, we obtain the evolution of the system wave function over time as follows:

(2) 
The wave functions of the particle are continuously distributed in onedimensional space. To effectively simulate this, the continuous onedimensional space should be discrete. Assuming that the wave function ψ(x, t) obeys the periodic distribution in this onedimensional space and 0 < x < L, then we have discrete ψ(x, t) and could store it in a nqubit quantum computer:

(3) 
where corresponds to the vector of nqubit quantum register.
It is obvious that the potential operator V(X̂) is a function of the coordinate X̂, so the matrix V(X̂) under the coordinate representation is diagonal. The value of one diagonal element corresponds to the potential energy value of V(X̂). For the momentum operator, first, we start with a matrix form of its momentum representation and do the quantum Fourier transform (QFT) to obtain its matrix form. Next we divide the time into many pieces at the length of Δt. According to the circuit shown in Fig. 3, we implemented the time evolution of Hamiltonian by the Trotter formula:

(4) 
We can split the kinetic and the potential energy operator into many parts, where kinetic operators and potential energy operators act in succession in a very short time Δt shown in Fig. 3.
In Fig. 3, F^{1} is a threebit Fourier transform inverse transform operator. D is a kinetic operator evolution operator and . S is the number of bits inversion operator and is the potential operator.

(5) 
In this experiment, the time interval is set as Δt = 0.4. Using the GRAPE algorithm, we designed and optimized the pulse sequence to achieve the evolution operatiach time period. As shown in Fig. 4, the initial state is 110>. It means that all particles are in the right trap. After some time evolution, the experimental results are shown in Fig. 4. The particles also have a certain probability that they are in the left trap. It shows that the tunneling occurs between two potential wells, and the tunneling probability can be obtained quantitatively.
Fig. 4: Theoretical and experimental results of dualwell tunneling using
a threebit NMR sample: (a) The theoretical distribution. (b) The experimental results. From left to right, the probability distribution of particles in eight grid points changes with time, and the time value is 0, Δt, 2Δt, 3Δt, 4Δt, 5Δt.
As a fundamental and important quantum phenomenon, quantum tunneling has very significant applications in many quantum effects, such as tunneling in α decay of nuclei and tunneling in superconducting Cooper pairs. In addition, quantum tunneling is also widely used in a variety of modern experimental devices, such as scanning tunneling microscopy, and Josephson junctions. Therefore, the simulation of the basic phenomenon of quantum tunneling is undoubtedly of great significance.
NMRCLOUDQ
The emergence of cloudbased quantum computing platforms (quantum cloud computing platform), is undoubtedly significant. Such a platform can be reached by the majority of quantum information workers and even by quantum computing fans with little background in quantum information processing, and can help everyone understand the difference between quantum and classic computing.
Quantum cloud is also a cloud service, similar to the cloud service that already provided by Amazon, Google, Alibaba et al. The difference is that the physical server is a quantum system stored in a laboratory. Users from throughout the world can access the quantum server and send the quantum program they want to execute. Then quantum cloud will then distribute some time running their procedures in that real quantum system. IBM is the leader in this field. In 2016, their superconducting quantum team made the first attempt, launching a fivebit superconducting quantum cloud platform. In the following year, IBM accumulated the experience of maintaining the quantum system, while constantly accumulating user feedback data. They have now expand the qubits of the superconducting cloud platform from 5 to 16.
On October 11, 2017, we also released our own quantum computing cloud platform (http://nmrcloudq.com/en/): NMRCloud. As mentioned earlier, the NMR quantum system plays an important role in the development of quantum computing. Based on this system, we hope to keep pace with the quantum technology revolution in this rapidly changing era and strive with the goal of increased access, so that more people are able to experience quantum computers. This work is designed for both experts and novices, to experience the fun of quantum computing.
Fig. 5: NMR quantum cloud platform components connections: Based on the 4bit NMR system, we integrate NMR control software and procedures related to quantum computing. After visiting our server, an external user can experience the operation of a quantum computer, a real physical system, through basic quantum circuits.
Quantum cloud platform, as shown in Fig 5 is the NMR spectrometer. In addition, to connect the users in the cloud, we needed a server to integrate the quantum computing algorithm package and nuclear magnetic resonance spectrometer control software. This server also assumes the mission of managing users and experimental tasks.
In general, a quantum experiment always includes three parts: initialization, quantum circuits and measurement. In NMR quantum computation, initialization is achieved by preparing a pseudo pure state (PPS). In our cloud platform, we used the spatial averaging method [48] to implement the initialization. We also evaluated the quality of the PPS by quantum state tomography and a fidelity of over 98.77% was achieved. It is known that any quantum circuit can be realized by decomposing into some basic gates, such as singlequbit rotations and twoqubit gates. In our quantum cloud platform, a large number of pulse sequences optimized in advance has been stored in the machine. These sequences include single qubit operations: Hardamard gates, π/2 rotations, π rotations and two types of phase gates for every qubit; and six kinds of controlnot gates and 3 swap gates. By using the GRAPE techniques [49], we realized the optimizations of the highfidelity control pulses in a 4bit NMR system. As shown in Fig 6: a singlequbit with a fidelity of 99.10% and a twoqubit operation with a fidelity of 97.15% were achieved by the randomized benchmarking. Finally, the density matrix of the final state is presented via quantum state tomography.
In the future, a very important factor that constrains the development of our 4bit NMRCloud is the need to improve the accuracy of the control pulses by a variety of techniques. For singlebit manipulations, we can use the 1) feedback control technique, 2) RF pulse selection technique, and 3) raise the optimization threshold to improve the basic logic gate pulse, because the error mostly comes from the inhomogeneity of the magnetic field and the control pulse optimization process performed. For twobit operations, decoherence becomes the main source of error due to the long pulse length. Since the pulse length is inversely proportional to the coupling strength between the qubits, we hope to select more strongly coupled NMR samples in the future. In this case, the pulse length can be reduced to the current singlebit gate pulse length. In addition, there are three hydrogen atoms arounding the carbon nuclei in the current sample. So we are also able to expand our NMRClounld to a sevenqubit platform.
Compared to the superconducting quantum cloud platform, our system is difficult to develop into a practical largescale universal quantum computer because of its natural unscalability. However, as the first platform to develop quantum computing, the NMR system is robust to environmental noise. Therefore, the logic gates of our quantum cloud will perform better than other systems under the same scale. As opposed to the other cloud services, we directly provide the density matrix of final state that contains all the information needed.
The appearance of the quantum cloud platform deeply promotes the development of quantum computing. Our goal is to provide public users with an accurate and reliable quantum computing physics system, so that everyone can experience, learn, and even research quantum computing on it. In the current state, there may be many problems as mentioned above, but we hope that this work will not only promote NMR control technology, but will also make a significant contribution to quantum cloud computing.
CONCLUSION
The newly developed techniques and thriving achievements in NMR have been presented in this article. First, a quantum feedback control method—MQFC has been applied a in 12qubit NMR quantum system and has boost its performance at the preparation of the state. Secondly a NMR platform could be used to mimic other quantum systems, which cannot be easily accessed, and we introduced our work on how to simulate quantum tunneling problems. Moreover, NMRCloudQ, which aims to make quantum computing accessible to the general public, has been introduced.
Fig. 6: (a) The RB sequence diagram consists of two subpictures: a standard RB sequence as a reference, where we repeat m sequences which are made up with n logic gates, and finally measure the final fidelity by a recover operation; the other subgraph is the measure of the RB sequence of the target gate, in which, unlike the standard ones, a target gate is added after every n repetitions of logic gates. (b) The singlebit logic gate RB result, the controlnot gate RB result and the swap gate RB result.
As a pioneer in the realization of QIP experimentally, we are aware that although there are almost two decades development of NMR QIP, there are still some limitations for this system, such as, nonscalability, the problem of initialization and the long time required to implement quantum gates. Hopefully, crystals or solid NMR systems may provide a way to overcome these existing limitations in liquid systems. There are some advantages to crystal and solid NMR systems: (1) high polarization; (2) long coherent time, and (3) strong coupling between qubits. These properties supply us a better way to implement future quantum computers.
For more information about NMR or NMR quantum information processing, there are some references which are easy to access, such as the book by Levitt [50] which depicts the physical basics of NMR clearly and the review article by Chuang [17], which introduces the general techniques for NMR quantum computing. Books from Suter and Mahesh [51], and Wiseman and Milburn [52] also provide a fundanmental basis for understanding NMR QIP.
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Long Guilu is a professor at Tsinghua University and has been engaged in theoretical and experimental research of quantum information. His main achievements and contributions are as follows:
(1) He helped establish and develop the direct communication of quantum security;
(2) he established a quantum search phase matching theory and constructed a quantum precise search algorithm;
(3) he helped in the verification of quantum algorithms in NMR (nuclear magnetic resonance) systems; and
(4) he established the duality quantum computation framework. 

Tao Xin received his PhD from Tsinghua University. His research areas are quantum simulation and NMR (nuclear magnetic resonance) quantum information processing. 

Li Keren is a PhD candidate at Tsinghua University. His research areas are quantum information and quantum simulation. 

Kong Xiangyu is a PhD candidate at Tsinghua University. His research area is NMR (nuclear magnetic resonance) quantum information processing. 

Yuanye Zhu is a PhD candidate at Tsinghua University. His research areas are magnetic resonance quantum processors, quantum simulation and quantum algorithms. 
