The common pulse optimization technologies based on optimized algorithm search are the strongly modulating pulse (SMP) proposed by Cory et al.^[54,55] and gradient ascent pulse engineering (GRAPE) proposed by Glaser et al.^[56] The GRAPE technique originated from NMR has extensive applications in QIP. The basic idea of the GRAPE method is explained in the following. For an NMR system with n-qubit, the total Hamiltonian contains the internal term and the radio frequency (RF) term , 2 where ω _i and J _ij are the Larmor frequency and the scalar coupling constant, respectively. B _k and ϕ _k are the amplitude and phase of the controlling field on the k-th nucleus. The key of the GRAPE technique is to find the optimal parameters of the RF field making the time propagator (density operator ) very close to the desired target evolution (desired target operator ). Assuming that the total time of the RF field is T, the total evolution time is averaged into M discrete segments and the time of each segment is . The time propagator of the m-th segment can thus be expressed as 3 where and are the amplitude of the m-th segments of the RF field in x and y axes. So the total evolution is . The fitness function to be optimized is the distance with the target, which can be expressed as for the given target evolution or for the given target state . The key of the GRAPE algorithm is to consider fidelity and as the the optimization problem of the multivariate function. In the first-order approximation, the gradients of and as a function of parameters and are 4 5

The fitness functions can be increased in the gradient iteration as follows: 6 The GRAPE procedure starts from an initial guess of input ( , ) and then evaluates both the fitness functions and corresponding gradients, and then continues iterations until the fitness functions reach the target values.

The basic principle of the GRAPE technique introduced above is a process of creating a randomly initial guess, evaluating a fitness function f and a gradient g, determining a search direction, producing an iteration, and finishing an optimization. However all these procedures are performed in a classical computer. It inevitably contains the simulation of time propagations of the controlled quantum system, which will be impracticable for a lager system with the classical computer. To improve the performance of GRAPE, MQFC as a quantum version of GRAPE technique was proposed by constructing a hybrid quantum-classical machine.^[57–59] The critical point is to efficiently calculate the fitness function and the gradient on a reliable quantum simulator which can be set as the target system itself. The generation of the control parameters and the determination of the search direction in each iteration are still implemented on a classical computer. In such a way, it greatly saves the running time for optimization and improves the control fidelity.

In the following, we present the mathematical description of optimizing a control sequence with M slices to transfer the initial state ρ ₀ to the target state . Any state has the unique decomposition in Pauli basis , where r is no more than for a system with n-qubit, P _i is an element of the Pauli group , and c _i is the decomposed coefficient. Then the definition of fitness function f is 7 So the amount of measurements f directly depends on the form of the target state f. It is actually the number of r. There is no improvement on efficiency if the number of Pauli terms , which is equivalent to quantum state reconstruction. However, the number of r is sometimes small, or at least polynomial for some experiments. Then the gradient function of the m-th slide of the control sequence with totally M slices can be calculated to the first order as shown in Eq. (5). We consider the homonuclear case with the following equation: 8 We can further obtain the form of the gradient 9

It is clear that the measurement of gradient can be obtained by applying a local pulse between m-th and (m+1)-th slices and the subsequently measuring the distance between the finial state and . In other words, 4n experiments are necessary for measuring if r=1. The efficiency of MQFC may be doubted when the target state is complicated. Indeed MQFC only makes sense when we limit our considerations to special sparse states. There is of course no doubt that MQFC has advantages over traditional methods to control a lager system, as it is still difficult to precisely manipulate the dynamics of a lager system in the simplified case with current techniques.

Dynamical decoupling technique (DD) is also derived from NMR spectrometer,^[100] which is originally created to obtain a high-precision element spectrum. With the rise of quantum information science and technology, DD technique can also be used to preserve quantum information for a longer time by periodic sequences of instantaneous control pulses.^[101] Therefore, it plays a very important role in the field of quantum control.

Assuming that a single-qubit quantum system with the initial state is only affected by the dephasing noise, the corresponding Hamiltonian model can be expressed as , the propagator after evolution time τ is 10 It shows that the coherence of the state can be easily destroyed by the dephasing noise. The common approach of DD is to frequently apply π pulses to protect the coherence of the initial state. For instance, n instantaneous π pulses around the x-axis are alternately applied in the total free evolution line. will evolute to 11 where . It can be seen that if the noise frequency is sufficiently low and the pulses are equidistantly distributed, then . Hence the quantum state approximately remains unchanged. Dynamical decoupling is to approximately average the unwanted system-environment coupling to zero. The basic principle of dynamical decoupling^[102] is shown fig. 2.

Fig. 2.

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	Fig. 2. (color online) Schematic representation of dynamical decoupling pulse sequences. During the evolution from t=0 to t=T, π pulses are continually applied to the protected state with different time points. Time point of is labeled by the symbol δ i.

The first DD sequence is the Hahn spin echo in the NMR system.^[103] Based on the Hahn echo, periodic dynamical decoupling (PDD)^[104] and CPMG^[105,106] were further discovered 12 where is free evolution of time τ. Actually, the PDD sequence involves the repetitive application of the Hahn echo. Besides, CPMG has a very good inhibitory effect on the noise of high frequency cutoff.

In 2007, Uhrig showed that the manipulation of the relative pulse locations for fixed n and T leads to the modification of the filter function and then proposed the so-called Uhrig DD (UDD) sequence,^[107,108] which achieves n-order decoupling. Different from PDD and CPMG, UDD sequence is non-equidistant in the time domain. In the sequence of time T, the time of the m-th π pulse can be written as 13 It is concluded that UDD is more effective on noise of sharp high frequency cutoff, which has been demonstrated in solid-state NMR and ion trap systems.^[109–111]

The various DD sequences mentioned above are just used to fight against dephasing noise. It is also useful to design a DD sequence that suppresses both decoherence and operational errors. To realize this purpose, Khodjasteh and Lidar proposed concatenated dynamical decoupling (CDD)^[112] consisted of π pulses around X, Y, and Z axes. The iteration rule between different orders is 14 CDD can use the pulses to simultaneously achieve n-order decoupling for the two kinds of common noises, but it will be a challenge in experiments when n becomes bigger. In 2010, Lidar et al. proposed quadratic UDD (QDD)^[113–115] based on UDD, which guarantees simultaneous elimination of both transverse dephasing and longitudinal relaxation to n-order with only pulses.

Noise from the environment is a double-edged sword. In some special cases, noise is unwanted that needs to be suppressed. But it also can become a crucial factor that promotes the evolution of the system to certain states yielding new phenomena.^[116,117] In order to evaluate the performance of various DD sequences such as CDD, UDD, QDD under various noises or to study the dynamical evolution of the open quantum systems under some special noises, we usually artificially construct these noises in experiments which naturally exist in a controlled system. Here we introduce some methods of artificially injecting miscellaneous noises in NMR system,^[47,118,119] including longitudinal relaxation noise, transverse relaxation noise, and hybrid noise.

2.4.2. Transverse relaxation noise

Transverse relaxation noise usually puts a dephasing effect in a system, which mainly comes from the inhomogeneous and non-static magnetic field in NMR system in the classical case. Similar to the method for longitudinal relaxation noise, 18 the PSD and noise spectrum can be written as 19 figure 3 shows the functional form required for in order to achieve common noise PSDs.

Fig. 3.

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	Fig. 3. (color online) The functions for the given noise PSDs. The different noise models are presented, including , 1/f, white, and Ohmic noises.

Under the influence of transverse relaxation noise, the initial state with the Hamiltonian in the rotating frame after time τ will become 20 where is integral of and denotes the Larmor frequency. Hence if we attempt to simulate the evolution of a quantum system in the dephasing environment, we just rotate the angle along the z axis at the desired point in time.

Linear equations play an important role in all fields of sciences and engineering involving quantitative analysis. For example, in global positioning system (GPS), we have to receive the signals from N satellites, then construct (N-1)-dimensional linear equations through geometric relationships and mathematical changes, and obtain the position by solving these equations. The experimental implementation of HHL algorithm was recently realized in NMR quantum processor,^[72,128] which will be introduced in the following.

The algorithm of linear equations of N unknowns, even to obtain an approximate solution, in general requires time in the scale of at least N for a classical computer. One quantum algorithm purposed by Harrow et al. indicates that the HHL algorithm can solve an N-dimensional linear equation in time which is as much as exponentially faster than the classical algorithms. Details of the HHL algorithm are introduced as follows.

Given a Hermitian ( ) matrix with a spectral decomposition , where λ _i is the eigenvalue of and is the corresponding eigenstate and a unit vector . Then solving the linear equations is to find a vector that satisfies . In quantum language, can be represented by a quantum state . We can also expand in the eigenbasis of as . If is not Hermitian, we can further define 24 to make sure is Hermitian, such that we can solve Eq. (24) to obtain the in quantum computing processor. We assume that is Hermitian in the following. This algorithm needs an additional register c for the phase estimation and an ancillary qubit for the projective measurement, besieds the work qubits denoted by a register b which encodes a vector and creates a desired vector . There are two main steps to solve this in quantum principle. First, use the well-known techniques of phase estimation with the controlled unitary , where and , with l being the qubit number of register c. After this step, we can obtain the state of the registers c and b in the eigenbasis of , . Second, an ancillary qubit is added to use the as a control register to obtain the state 25 with an appropriate constant C. To realize the state transformation, we prepare the state with starting from , and then use as the control qubit to rotate the ancillary qubit. Finally, we perform the inverse phase estimation process and then obtain 26 When a projective measurement is applied on the ancilla qubit yielding with certain probabilities, the state in register b will collapse to the desired state , which is actually the answer for .

In experiments, Du et al. implemented the HHL algorithm using NMR quantum processor.^[72] They chose iodotrifiuoroethylene dissolved in d-chloroform as a four-qubit sample, where a¹³C nucleus and three¹⁹F nuclei constitute a four-qubit quantum system. The experiment can be divided into three steps as shown in fig. 4. First, prepare the system into a pseudo-pure state (PPS), perform a rotation operation around the y axes with an angle θ to the third qubit, and obtain the initial state , with the normalized state . For the simplification, they considered a specified matrix as follows: 27 Second, they implement the quantum circuit of the algorithm shown in fig. 4 on the initial state ρ ₀. All these operations are realized using an RF pulse which is optimized by the GRAPE technique. Finally, the single-qubit state tomography is carried out to read out the target vector from the final state. It is the first experimental demonstration of the HHL algorithm when a (2×2) linear equation is considered using an NMR quantum information processor.

Fig. 4.

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	Fig. 4. (color online) Quantum circuit for implementing the HHL algorithm in NMR with r=2 and . Single-qubit operations S and H are respectively phase gate and Hadamard gate. , . is the rotation operation around y axis. . The vertical segment with × means an SWAP operation.

Recently, the artificial intelligence and machine learning become more and more popular. Classical learning machines often require huge computational resources.^[129] With the development of quantum technology, it is gradually known that a quantum machine learning algorithm may be exponentially faster than the classical counterparts with quantum parallelism for some special problems.^[130,131] The participation of quantum computing injects a new driving force for conventional machine learning. Quantum principles can optimize the classical machine learning algorithms and improve the performance.^[132,133] Therefore, one promising application of QIP is to combine the artificial intelligence and machine learning.

Many theoretical and experimental developments about machine learning in quantum information field have been achieved.^[134–137] One of these is the experimental implementation of quantum version of the support vector machine (SVM). It is believed that SVM is a popular one and learns from training data and classifies vectors in a feature into one of two sets among the supervised machine learning algorithms.^[136] Li et al. applied the SVM algorithm to the optical character recognition (OCR) problem.^[137] It shows the ability to learn standard character fonts and then recognize handwritten characters from a set with two candidates. In detail, they took the printed images of the standard fonts of ‘6’ and ‘9’ as the training sets and classified the handwritten images of characters “6” and “9”. As introduced before, the essential step of the SVM algorithm is the HHL algorithm. Thus the experiment process of SVM includes the HHL process, plus the additional two unitary operations which can be realized by the GRAPE techniques in NMR platform. The experimental results show that it can classify handwritten images of character “6” and “9” successfully.

Topological quantum computation (TQC) is one of the architectures of quantum computation, which was proposed by Alexei Kitaev.^[146] The basic elements used in topological quantum computing are a kind of two-dimensional quasi-particles which are called anyons. Their braids are formed in a (2+1)-dimensional space by crossing their world lines and the unitary operations in TQC are generated by these braids. Compared with other quantum computing models, TQC has a natural advantage that the local errors have no influence on these braids, hence this architecture of quantum computation is invulnerable against small perturbations or errors. However, this fantastic and promising computation architecture still stays at theoretical realm since the basic elements of TQC in the real physical system are still difficult to be confirmed. The development of a controlled quantum device provides us with an alternative way to topological matter. In this way, we could simulate them in another quantum system rather than built it in a real physical system. For instance, the topological order between Abelian and non-Abelian can be efficiently simulated. Among the problems on research, one is to distinguish two kinds of topological orders. It is shown that a topological order can be uniquely determined by the following three factors: the anyon types, the topological properties, and the topological protected ground states. The anyon types and the topological protected ground states are equivalent on the torus, hence identification of a topological order merely depends on the topological properties of anyons which include self-statistics, braiding, and fusion. One of the common methods is to consider their modular S and T matrices, which provide the full information of the topological properties of anyons.

A recent experiment has shown that they could characterize topological orders and identify them via their modular and matrices using a three-qubit NMR quantum simulator (fig. 5).^[150] In their experiment, their three-qubit quantum system was loaded on the¹³C-labeled trichloroethylene (TCE) molecule which was dissolved in d-chloroform solvent. Three steps were adopted in their each experiment: preparation of the topological protected ground states, driving the system into a modular transformation, and measurement. Their experiment successfully simulated the states for Z ₂ toric code, doubled semion order, and the doubled Fibonacci order and then identified them by experimental results. In the same period, Luo et al.^[151] answered a new question “How much detail of the physics of topological orders can in principle be observed using state of the art technologies”? Based on a five-qubit NMR quantum simulator, which is represented by the 1-bromo-2,4,5-trifluorobenzene molecule, they adiabatically prepared the random ground state and measured the modular S and T matrices. They found that they could identify a Z ₂ toric code model using a Hamiltonian with minimal input from state preparation to measurements. This is the first experiment which realizes the topological order and tests their robustness against perturbations. It may open a new territory to identify the topological orders whose Hamiltonians are not exactly solvable.

Fig. 5.

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	Fig. 5. (color online) The work is to simulate the minimal honeycomb lattice on a torus. This torus only contains one plaquette, two vertices, and three edges. 1,2,3 are three edges which can be simulated by three qubits, respectively.[150]

The other aspect lies on the area of high energy physics. Some other systems have proved that some high energy problems can be mimicked on a quantum simulator.^[152] As for NMR quantum simulator, there are also some achievements about high energy physics simulation. Recently, they used an NMR quantum simulator, measuring holographic entanglement entropy, to mimic the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence (fig. 6).^[153]

Fig. 6.

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Fig. 6. (color online) Description of the RT formula. The bulk minimal surface (blue line) is anchored at the boundary of the boundary region A. (b) An example of tensor network state. The (hexagon) nodes represent tensors. The links represent the contraction of tensors. The dangling legs are physical DOFs in many-body system. The red dashed curve illustrates the virtual surface S anchored to region A, which cuts a minimal number of links.[153]

In classical spacetime, it is shown that physical variables commute with each other. However, some variables do not commute in quantum spacetime and as a result, these variables which use to be continuous may become discrete now. Quantum spacetime is such a generalization of the concept of spacetime that it is an area emerging quantum mechanics and general relativity. At present, there are two mainstream approaches by expanding either quantum mechanics or general relativity to investigate it. First, the famous string theory is constructed by considering the gravity in quantum field theory, to finally become a theory of quantum gravity. Second, as the leading competitor to string theory, loop quantum gravity (LQG) begins with relativity and tries to add quantum properties. In the past two decades, AdS/CFT correspondence is one of the most important conjectured relationship to quantum theory of gravity. It predicts that the quantum gravity theory in the bulk anti-de Sitter spacetime has an equivalent relation to the conformal field theories which are from quantum field theories on a lower-dimensional space.

Starting from the entanglement in the boundary field theory, Li et al. rebuilt the bulk geometry holographically. To identify the entanglement entropy , the Ryu–Takayanagi (RT) formula was proposed 28 where is the Newton constant. A is a -dimensional boundary region, with the bulk -dimensional minimal surface anchored to A. On the other hand, recent prosperous area called tensor network has introduced a way making the RT formula a discrete version. With the assist of the above developments, the quantum simulator will become available for the implementation of these demonstrations. For instance, a six-qubit NMR quantum simulator was recently adopted to realize such a demonstration. The spin-half¹³C in¹³C-labeled dichloro-cyclobutanone dissolved in d6-acetone was regarded as six qubits. They used control techniques include GRAPE and a pulse feedback program to boost the accuracy of the RF pulses. Experiments successfully presented the simulation of the minimal PT of rank-6. Although they just implemented the minimum case for the RT formula, their results further suggest that the RT formula or the AdS/CFT correspondence can be simulated on a larger quantum system.

Towards quantum gravity, loop quantum gravity leads another description of the quantum spacetime. In the frame of LQG, at a certain time point, the geometry can be thought as concentrating on one-dimensional structures. It is simply a one-dimensional oriented network, which is a so-called spin network. It was first proposed by Penrose and the oriented lines are linked together at their end points to form a whole net. In the spin-network, quantum spacetime can be decomposed into some kind of quantum tetrahedron. This tetrahedron corresponds to a certain state and can also be simulated in current quantum devices such as NMR quantum simulator platform. Additionally, some endeavors could be made to simulate their dynamics if more qubits could be controlled. Moreover, the NMR platform can also be used to simulate other relevant models such as a black hole which could be seen as quantum scrambling in a chaotic system,^[154,155] the Sachdev–Ye–Kitaev model,^[156] and the space–time geometry emerged from entanglement.^[157]