Quantum simulations with nuclear magnetic resonance system*

Chudan Qiu(邱楚丹), Xinfang Nie(聂新芳),†, and Dawei Lu(鲁大为),2,‡

1Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology,Shenzhen 518055,China

2Guangdong Provincial Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology,Shenzhen 518055,China

Keywords: nuclear magnetic resonance,quantum simulation,quantum phase transition,quantum gravity

1. Introduction

In order to predict the evolution of a quantum system,numerical analysis is a traditional but demanding approach.Especially due to the exponential size of Hilbert space as a function of quantum subsystems, the limited number of subsystems for numerical analysis is a major obstacle to addressing quantum problems with desirable accuracy on a classical computer. Conversely,in many cases,the quantum simulation may offer a faster solution with a probably more illustrative perspective. More importantly,the qubits in quantum simulators are intrinsically able to store exponentially large amounts of information compared to classical bits. Furthermore,in recent years, quantum effects play an increasingly substantial role in modern technology, which leads to the new needs for quantum simulation. Meanwhile, the physical platforms, as well as the technologies required,have matured enough to enable the practical implementations,including superconducting circuits,[1,2]trapped ions,[3,4]neutral atoms,[5–7]photons,[8,9]spin-based systems such as impurity spins in solids[10,11]and nuclear magnetic resonance(NMR).

Among various candidates,nuclear spins manipulated using NMR are usually considered to be competent in smallscale quantum simulations.[12]Twenty years ago, many leading experiments blazed a trail in implementing complex multiqubit gate operations using NMR.[13–15]In the following decades,more sophisticated techniques have been well developed in terms of endeavouring to simulate the dynamics of quantum systems.[16,17]More recently,the state-of-the-art 12-qubit coherent control has been demonstrated.[18,19]

These experimental advances in controlling spins promise the applications in disparate areas, including many-body physics and quantum gravity. Some problems in the former are formidably hard to tackle with classical computers due to the complexity arisen from the interactions between many quantum individuals. And the quantum gravity which aims at unifying quantum mechanics and Einstein gravity is a fundamental area but with many ideas remaining unveiled experimentally.

In this review,we present some representative experimental demonstrations in many-body systems and quantum gravity realized with NMR.First,we start with an introduction of NMR fundamentals in Section 2. Next,centered around a key quantity called out-of-time correlator (OTOC), the following Section 3 expounds on three topics where the OTOC serves as an excellent probe: (1) OTOC can differentiate the integrable and non-integrable systems and quantify the butterfly velocity in quantum chaos (Subsection 3.1); (2) OTOC can observe different signatures of many-body localization(MBL)and Anderson localization (AL), as an alternative metric instead of the entanglement entropy(Subsection 3.2);(3)OTOC can detect the dynamical quantum phase transition (DQPT)and mark the critical point in equilibrium dynamical phase transition (EQPT) between the paramagnetic and ferromagnetic phases,surpassing the autocorrelation and the two-body correlation function as regards to the precision and robustness(Subsection 3.3). These results shed light on the underlying connections among quantum chaos,thermalization,and quantum gravity. Accordingly,based on Ryu–Takayanagi(RT)entropy, the profound relation between OTOC and anti-de Sitter/conformal field theory(Ads/CFT)duality directs us to Subsection 4.1. The experiment that RT entropy has been verified on perfect tensor state is elaborated. While the perfect tensors are viewed as the building blocks of the tensor network,this verification makes the first step toward studying quantum gravity experimentally. In the other Subsection 4.2, the successful simulations of geometry properties and local dynamics of quantum spacetime, with respect to quantum tetrahedrons and spinfoam vertex amplitude, respectively, open up an avenue studying loop quantum gravity(LQG).Finally,the challenges and prospects of quantum simulation with NMR are discussed in Section 5.

2. Basics of NMR

In the remainder of this article,we will restrict ourselves to the more comprehensible liquid-state NMR. In the liquid solution, where the rapid tumbling averages out the dipole–dipole interactions, the Hamiltonian with spin–spin interactions under the weak coupling approximation reads as(¯h=1)

where ωi=γiB0, with gyromagnetic ratio γiof the i-th nuclear spin and static magnetic field B0aligned along z-axis,is the Larmor angular frequency, and Jijis the scalar coupling strength between the i-th and j-th spins.

Fig.1. CHCl3 as an example of a two-qubit sample with relevant parameters. The 13C and 1H nuclear spins serve as two qubits. The chemical shift(Hz) and J-coupling strength are given by the diagonal and off-diagonal terms in the right table, respectively. Two important parameters characterizing the longitudinal and transverse coherence properties represented by T1 and T2 (see more details in the Subsection 2.4) respectively are also provided.

2.1. Initialization

To emulate a quantum system of interest, the first step should be the preparation of an initial state of the simulated system. According to the Boltzmann distribution, the system is in the thermal equilibrium state under the room temperature,which is unpolarized(see Fig.2(a)). However,it is impossible to cool down to extremely low temperature so as to obtain a state polarized into ground state in the liquid-state NMR. Instead,a pseudo-pure state(PPS)[20,21]is widely adopted

Fig.2. Pseudo-pure state in initialization and the schematic diagram of experimental setup. For a two-qubit system in the thermal equilibrium, there is no available polarized state to employ, as shown in(a). While a PPS,as illustrated in(b),with an identity part and a tiny amount of extra population in the ground state is valid to be treated as an initialized state. (c)Gate control and the measurement are implemented by the RF coil with the following electrical circuit connected to the computer. On the other hand,the external field B0 is generated by the static field coil. This subfigure is adapted from Ref.[26].

2.2. Universal gates

In the case of NMR, the unitary gates of each qubit are carried out by the external radio-frequency (RF) pulses (see Fig.2(c)),which can be written as

2.3. Measurement

Finally, with the aid of an RF coil, the transverse magnetization of the ensemble can be obtained. This detection coil is weakly coupled to the nuclear spins so that it barely contributes to the decoherence. However, there remain interactions with the heat bath and inhomogeneity of the static field,which still leads to the decoherence of the nuclear spins.Consequently, the measurement of the nuclear spins is based on the free induction decay (FID). What FID experiments yield are the expectation values of readout operators in the x–y plane, and finally, the time-domain FID signals are converted to frequency-domain NMR spectra via Fourier transform. Though such weak measurement is unable to provide as much information of a single spin as the projective measurement,the ensemble-averaged information can be distilled.For the complete description of the state, i.e., all elements of the density matrix,one should resort to the full quantum state tomography.[13,28]

2.4. Decoherence

The decoherence originates from the unwanted interactions with the environment,which leads to the loss of information carried by qubits. As a result, the coherence is a crucial facet of examining whether a quantum system would be entitled to be the physical implementation of a quantum simulator.For the decoherence stemming from the couplings between the spins and the lattice, it is conventionally characterized by the energy relaxation(or longitudinal relaxation)time T1,which is about tens of seconds in a liquid sample. Whereas phase randomization time(or transverse relaxation)T2characterizes the decoherence resulting from the spin–spin couplings. In NMR systems,a more important parameter is denoted by T*2,which is extracted from FID experiments and in turn is termed inhomogeneous dephasing time. The T*2is supposed to be long enough during which the sequences of gates can be implemented. For the complex tasks,the more advanced techniques should be applied to extend T*2so that preserve the information, such as RF selection which improves the T*2at the cost of signal loss.[29,30]

3. Many-body physics

3.1. Quantum chaos

In the classical systems,the integrable and non-integrable systems are related to the regular and chaotic motions,respectively,where the latter display the butterfly effect that the initial diminutive deviations may give rise to considerable differences in a later time. Correspondingly, in the quantum scenario,a small perturbation that spreads over a many-body system may result in large commutators with the operators which are commutative with the perturbation initially.[31]Formally,the quantum version of butterfly effect is relevant to a core concept termed quantum information scrambling, that is, the information stored in local degrees of freedom smears over global degrees of freedom. Scrambling lies at the heart of the dynamics of quantum information.[32]From the experimental perspective, it can be understood by a key observable named out-of-time correlator(OTOC),which first appeared in the context of superconductivity[33]and was defined as

3.1.1. Integrable and non-integrable systems

The distinct behaviors of integrable and non-integrable systems can be observed via OTOC.[36]Take the paradigmatic one-dimensional transverse-field Ising chain(TFIC)model as an example,whose Hamiltonian reads as

Fig.3. Distinct behaviors of (a) integrable (g =1, h =0) and (b) nonintegrable (g=1.05, h=0.5 for left column and g=1, h=1 for right column) cases of Ising spin chain model are distinguished by OTOC F(t).(c)Measurement of butterfly velocity. The OTOCs for three different operators ˆWj are shown with different colors. The inset shows the characteristic time td versus the distance d between two operators, where the butterfly velocity vB can be obtained according to td =d/vB+c with c being the intercept. The figures are adapted from Ref.[36].

3.1.2. Butterfly velocity

3.2. Quantum phase transition

3.2.1. Many-body localization and Anderson localization phases

Another aspect in connection with the scrambling of information is the thermalization and entanglement entropy. To be more concrete, consider an initial state which is the direct product of two pure subsystems,say A and B,and undergoes an evolution governed by a chaotic Hamiltonian. After a time longer than the thermalization time,the output state turns out to be highly entangled, which suggests that each subsystem is near maximally mixed and thus, corresponds to a thermal ensemble regarding all local measurement observables.[45]Moreover, the degree of entanglement between subsystems can be quantified by the growth of local von Neumann entropy

MBL is a regime where the entanglement entropy scales logarithmically.[47]This localization results from the strong disorder that struggles against the thermalization and accordingly suppresses the growth of entanglement entropy.Whereas the non-interacting correspondence AL has a saturation as the primary signature (see the dashed lines with non-zero (zero)interaction strength,the right axis in Fig.4).

Nevertheless, measuring entanglement entropy is challenging, the pioneering work was achieved only on the small number of particles.[48]Recently, the average correlation length Lcas an alternative has been demonstrated on the NMR platform as effective as entanglement entropy[49](see experimental results (dotted lines) and simulation (solid lines), left axis in Fig.4). The Lcis referred to the contributions of all possible spin correlations with Hamming weight.[50]Armed with the coherent averaging techniques[51]to tune both the interaction strength and the degree of disorder,the average correlation length can be extracted from the intensity of multiple quantum coherence(MQC)based on the measurement of OTOCs.

Fig.4. Simulations of entanglement entropy and correlation length for many-body localization and Anderson localization. As the increment of interaction strength, the MBL emerges, featured by a slow growth in time of Lc (solid lines,left axis),which is consistent with the approximated Lc obtained by measuring OTOC(dotted lines,left axis),the logarithmic growth of entanglement entropy(dashed lines,right axis). By pronounced contrast,in the AL phase where the interaction is absent(the black lines corresponding to zero interaction strength),Lc is saturated. The figure is adapted from Ref.[49].

3.2.2. Paramagnetic and ferromagnetic phases

As an indicator,OTOC also outperforms some other kind of correlators with respect to detecting equilibrium quantum phase transition (EQPT) and dynamical quantum phase transition (DQPT),[52]because OTOC captures the scrambling of quantum information, and both EQPT and DQPT attend rapid spread of quantum information on account of appreciable quantum fluctuations.

In a recent work,[56]the advantages of OTOC have been demonstrated compared with autocorrelation and two-body correlation function with respect to detecting DQPT and locating the critical point in EQPT.

Well in accord with the theoretical simulation,the dynamics of OTOC exhibits sharp distinctions between ferromagnetic (ggc).As illustrated in the top panel of Fig.5(a),OTOC stays at certain positive value with comparatively small fluctuations in the g

Fig.5. Experimental results of DQPT and EQPT for integrable and nonintegrable systems. The comparisons of the OTOC F(t) (top panels), autocorrelation χ(t)(bottom panels), and two-body correlation function C(t)(bottom panels)to detect DQPT in the integrable system and non-integrable system represented by(a)TFIC model and(b)ANNNI model,respectively.Critical points of(c)TFIC model and(d)ANNNI model are marked as the turning points of the long-time averaged OTOC(top panels)and two-body correlation(bottom panels)as functions of the transverse field strength g.The figures are adapted from Ref.[56].

4. Quantum gravity

4.1. Ads/CFT correspondence

4.1.1. Ryu–Takayanagi entropy

As discussed above, the OTOC has underlying connections with Ads/CFT duality. On this account,Ads/CFT correspondence links the quantum gravity theory and quantum information theory[58]meanwhile inspiring the incorporation of holographic entanglement entropy into the study of quantum gravity. Specifically, the entanglement entropy of the boundary system can be related to the bulk geometry in terms of Ryu–Takayanagi formula,[59,60]which reads as

4.1.2. Tensor network

Fig.6. Diagrammatic representation of a tensor network. The coefficients of an N-body system amount to an N-order tensor. A tensor network represents the structure and the amount of entanglement of a quantum manybody state. The tensors are linked by the lines, which correspond to the indices i1,i2,...,iN. Then, the contracted (summed) common indices are represented by the lines connecting to shapes. Then,a tensor network with m unpaired legs can be treated as an m-order tensor. In this sense, the tensor network representation reduces the complexity of quantum many-body problem.

Moreover,when k ≤3,the reduced density matrix equals an identity and on the other side S(k)=min{k,6 −k}, which has also been verified,[61]with the proper refinements by taking account of the decoherence, as shown in Fig.7(c). It is noteworthy that the compensations of decoherence errors were accomplished by performing 6-qubit full tomography, which is the largest full state characterization in an NMR system to date.

Fig.7. Illustration of tensor network and perfect tensor,and the theoretical and experimental results of RT entropy. (a) The disk is a two-dimensional ads with the hexagonal tiling. The solid arc marks the minimal surfacewhich is anchored to the two ends of the boundary region illustrated by the dashed line. The tensor network on the right side is the discrete version of the left ads space. The tensor network is composed of rank-6 tensors represented by a hexagonal node with the links ℓ, each of which corresponds to a maximally-entangled state. The total tensor network state is obtained by taking inner products in ℋ, corresponding to connecting legs of nodes to links. (b)A rank-6 perfect tensor with three minimal cuts by virtual surface illustrated by the red solid line. Half of six qubits are bulk qubits and the other half are at boundary. (c) The theoretical results of entanglement entropy S(k) equalling to min{k,6 −k} are shown by orange dashed line.The refined experimental results with the compensation of decoherence represented by blue squares are consistent with the theory much better than initial results shown by red circles. The maximal entropy of a k-qubit subsystem by assuming a 6-qubit identity is plotted in green dotted line as an upper-bound reference. The figures are adapted from Ref.[61].

4.2. Loop quantum gravity

In another aspect of quantum gravity, the nuclei spin states in NMR can also be employed to simulate quantum geometries of spacetime.

4.2.1. Spin network

In loop quantum gravity, the quantum states endowed with discrete geometries of quantum spacetime at the Planck scale are represented by spin networks.[63]The time evolution of spin network[64]builds up a (3+1)-dimensional quantum spacetime,then the boundary of which is the spin network.

4.2.2. Dynamics of quantum geometry

Vertex amplitudes determine the spinfoam amplitudes,which are the transition amplitudes between the initial and the final spin networks.[68]In principle,if the two-qubit maximally entanglement states can be established between arbitrary two tetrahedra,as illustrated in Fig.8,the vertex amplitudes can be obtained by evaluating the inner product between five quantum tetrahedra states. However, it is beyond the present manageable level for a 20-qubit quantum computer.Instead,the full tomography is also helpful,through which the information about quantum tetrahedra is acquired.[66]

Fig.8. Spin network and quantum tetrahedra. (a)In a(3+1)-dimensional dynamical quantum spacetime, a 3-sphere S3 encloses a portion of quantum spacetime surrounding a vertex(in black)where the world sheets meet.(b)A spin network(blue)is represented by the intersection between world sheets and S3. Each node of spin network corresponds to a quantum tetrahedron associated with an invariant tensor state|in⟩. Five tetrahedra are glued through the faces dual to the links to form a closed S3, represented by the connections of links l, each of which carries a half-integer jl. The figures are adapted from Ref.[66].

5. Outlook

Apart from the achievements discussed above, NMR simulations also provide new insights into various subjects(see Fig.9) such as quantum state tomography,[69–73]quantum algorithm,[74–76]non-Abelian topological orders,[77,78]Sachdev–Ye–Kitaev model,[79]prethermalization,[80]disordered systems,[81]probabilistic quantum cloning,[82]eigenproblem solving,[83]anti-PT-symmetry,[84]and even photosynthetic light harvesting.[85]However,on the theoretical side,the theories of decoherence and control are required;on the experimental side, the controllability and scalability of the system remain scope to improve.In particular,the spectral crowding that occurs as the number of energy levels increases exponentially with the increasing number of spins hinders liquid NMR from scalability. Though in solid-state NMR,the scalability drawback may be overcome to some extent,the manipulation and measurement of single qubit would be difficult. A promising alternative is the nitrogen-vacancy(NV)centers in diamond,into which the well-developed techniques in controlling spins in NMR have been incorporated.[86]

Fig.9. Fields on which the NMR simulations shed light.

The progress on quantum simulation tempts us to envisage that the practical simulators will be built in the near future as the prototype of full-fledged quantum computers. Especially when the point beyond which the classical computer would be inferior to quantum simulations is marked, at least in some cases, it would be a milestone for both physics and computer science.