Abstract
It is well known that quantum computers are superior to classical computers in efficiently simulating quantum systems. Here we report the first experimental simulation of quantum tunneling through potential barriers, a widespread phenomenon of a unique quantum nature, via NMR techniques. Our experiment is based on a digital particle simulation algorithm and requires very few spin1/2 nuclei without the need of ancillary qubits. The occurrence of quantum tunneling through a barrier, together with the oscillation of the state in potential wells, are clearly observed through the experimental results. This experiment has clearly demonstrated the possibility to observe and study profound physical phenomena within even the reach of small quantum computers.
Introduction
Quantum simulation is one of the most important aims of quantum computation ever since Feynman studied the likelihood of simulating one quantum system by another^{1}. Recent years have witnessed fruitful results in the development of quantum computation, and it has been demonstrated that quantum computers can solve certain types of problems with a level of efficiency beyond the capability of classical computers^{2,3,4,5,6}, among which the simulation of the dynamics of quantum systems is especially attractive because of the exponential improvement in computational resources and speeds. Quantum simulation has become a subject of intense investigation and has been realized in various situations, such as system evolution with a manybody interaction Hamiltonian^{7,8,9,10}, the dynamics of entanglement^{11,12}, quantum phase transitions^{13,14}, and calculations of molecular properties^{15,16,17,18,19}.
Quantum tunneling plays an essential role in many quantum phenomena, such as the tunneling of superconducting Cooper pairs^{20} and alpha decay^{21}. Moreover, tunneling has been widely applied in modern devices and modern experimental techniques, such as the tunnel diode^{22}, the scanning tunneling microscope^{23} and so on. As a unique fundamental concept in quantum mechanics, the simulation of quantum tunneling is of great significance. Many important science problems, such as lattice quantum chromodynamics^{24}, can be dealt with similarly. However due to the large number of quantum gates and qubits required, the simulation of quantum tunneling in a quantum computer has remained untested experimentally. Recently Sornborger^{25} proposed a digital simulation algorithm for demonstrating the tunneling of a particle in a doublewell potential with no ancillary qubits, and at least halved the number of quantum gates. This makes it possible to simulate this important quantum effect in today's quantum information processors with only a few qubits. In this paper, we report the first experimental digital quantum simulation of this significant quantum phenomenon via a liquid nuclear magnetic resonance (NMR) quantum information processor. In the experiment, the continuous process of onedimensional tunneling of a particle through a potential barrier is clearly demonstrated, and the oscillation of the particle in potential wells is clearly observed. Our experiment has shown that with very few qubits, interesting quantum effects such as tunneling dynamics can be simulated with techniques which are within reach of current quantum architectures.
Results
Theoretical protocol
Consider a single particle moving in a square well potential in onedimensional space. The Schrödinger equation reads, where and are momentum and position operators, respectively. Throughout the text we set ℏ to 1. The evolution of the wave function with time can be straightforwardly given as In the digital quantum simulation^{25,27,28}, the continuous coordinate x is discretized. Suppose ψ (x, t) is continuous on the region 0 < x < L, and with a periodic boundary condition ψ (x + L, t) = ψ (x, t). x is discretized on a lattice with spacing Δl and the wave function is stored in an nqubit quantum register where , , and k〉 is the lattice basis state corresponding to the binary representation of number k. Equation (3) gives a good approximation to the wave function in the limit n → ∞.
In the lattice space of the digital quantum simulation, an important task is to construct the kinetic and potential operators. Because the potential operator is a function of the coordinate operator , it is a diagonal matrix in the coordinate representation, which can be decomposed as^{26} where σ_{3} is the Pauli matrix σ_{z}, and σ_{4} = I is the identity matrix in two dimensions.
The kinetic energy operator, which is diagonal in the momentum representation, can be constructed in the coordinate representation with the help of a quantum Fourier transformation (QFT). Similar to equation (3), we have where ϕ (p, t) is the wave function in the momentum representation, and is the eigenvalue of the momentum operator in the momentum representation, Therefore the kinetic energy operator in the coordinate representation can be written via the QFT as where is the QFT operator.
Equations (3), (4) and (8) give the discretized forms of the wave function, the potential operator and the kinetic energy operator, respectively. With these expressions in hand, we can efficiently implement the time evolution of the system within a small interval Δt, by using a modified Trotter formula, which is correct up to Δt^{25,27,29}. Equation (2) is then approximated as From equation (8) we have where F is the usual bitswapped Fourier transformation operator, which can be readily realized in quantum circuits via a series of Hadamard gates and controlledphase gates^{30}. Consequently D in equation (11) is a bitswapped version of . Thus equation (10) can be rewritten as where .
From equation (12) the one time step evolution quantum circuit is straightforwardly obtained. In Figs. 1 and 2 we draw the twoqubit and the threequbit implementations, respectively. The explicit construction of F, D, and Q is detailed in Methods.
Experimental procedures and results
As a smallscale demonstration, a twoqubit simulation and a threequbit simulation were investigated. In the experiment, we studied the time evolution of a particle moving in doublewell potentials using the digital algorithm via twoqubit and threequbit NMR quantum information processors. The molecular structures and parameters of the quantum information processors are given in Fig. 3.
In the twoqubit case, the four basis states 00〉, 01〉, 10〉 and 11〉 register the four lattice sites 1, 2, 3 and 4 as discretized position variables of the particle. We consider the potential V_{0}I σ_{z}, which can be implemented with only a singlequbit gate^{25}. This potential represents a doublewell potential of amplitude 2V_{0}, with two peaks V_{0} at 00〉 and 10〉, and two troughs −V_{0} at 01〉 and 11〉. In our experiments we set the time interval Δt = 0.1 and the amplitude of the potential 2V_{0} = 20. We simulated the situation in which the particle is initially trapped inside one of the two wells by preparing the pseudopure state 01〉 from thermal equilibrium^{31} as the initial state.
The tunneling process of the particle was simulated using nine experiments, where in each experiment the operations in Fig. 1 were performed. Quantum state tomography (QST) was performed on the density matrices of the final states after 1 to 9 steps^{32,33}. The whole evolution process of a single particle in the doublewell potential can be described by depicting the diagonal elements of the density matrix, which correspond to the probability distribution of the particle, at each step. They are illustrated in Fig. 4 (b). It is clearly observed that the particle tunnels through the potential barrier between the two wells while its probability to be found in the barrier remains scarce, which accords well with theoretical calculations in Fig. 4 (a).
For the sake of comparison we also experimentally simulated the evolution of a free particle with zero potential using the same experimental schemes and parameters, except the removal of the potential operator Q in the circuit. The experimental results, together with theoretical calculations, are plotted in Fig. 4 (c) and (d). It is not surprising to find that the probability is distributed more evenly on all the four sites, showing the particle is free.
Similarly, in the threequbit case, the eight basis states 000〉, 001〉, 010〉, 011〉, 100〉, 101〉, 110〉 and 111〉 register the eight lattice sites 1, 2, 3, 4, 5, 6, 7 and 8. The potential considered here is V_{0}I σ_{z} I, which corresponds to a doublewell potential of amplitude 2V_{0}. It has a higher resolution than the doublewell potential in the twoqubit simulation, with each well or peak occupying two sites: one well occupying 010〉 and 011〉, and the other well occupying 110〉 and 111〉. In our experiments we set the time interval to Δt = 0.4 and the amplitude of the potential 2V_{0} = 200. The initial position of the particle is set in one of the two wells, by preparing the pseudopure state 110〉 as the initial state.
Five experiments have been carried out, where the evolution time is nΔt, with n being in 1,2,3,4 and 5, respectively. The reconstructed diagonal elements of the density matrices, which correspond to the probability distribution of the particle at each site, are illustrated in Fig. 5 (b). Not only the particle's tunneling from one well to the other, but also its oscillations within each well, disappearing and then appearing again, are clearly observed from Fig. 5 (b), which are in good agreement with theoretical calculations in Fig. 5 (a).
Discussion
Our experiments have simulated the fundamental quantum phenomenon of tunneling. Our twoqubit experiments reflected a remarkable difference between the two situations with and without the doublewell potential. In our threequbit experiments, a higher resolution can provide a more sophisticated structure of the potential wells, making it possible to observe the inwelloscillation of the particle.
In the twoqubit simulation, the experimental density matrices for the initial state 01〉 and the final state after 9 time steps were fully reconstructed (see Fig. 6) with experimental fidelities 99.89% and 95.48%, respectively. In the threequbit simulation, the experimental density matrices for the initial state 110〉 and the final state after 5 time steps were also fully reconstructed (see Fig. 7) and the experimental fidelities for them are 98.63% and 93.81%, respectively. The high fidelities demonstrate our good control in the experiments.
It should be emphasized here that although the real evolution of the particle takes place in a continuous space with infinite dimensions, our quantum computer, which works with only a few qubits in limited dimensions, is already capable of undertaking some basic yet fundamental simulation tasks, such as quantum tunneling. Likewise, an Nsitelattice simulation can be efficiently implemented in experiments with only log_{2}N qubits. The result has revealed the amazing power hidden behind the qubits and the promising future of quantum simulations.
In summary, we accomplished a smallscale demonstration of the quantum tunneling process on two and threequbit NMR systems based on the digital quantum simulation method. This is the first experimental digital quantum simulation of quantum tunneling via NMR quantum information processors. The experimental results and the theoretical predictions are in good agreement.
Methods
Experiments were carried out at room temperature using a Bruker Avance III 400 MHz spectrometer. We used chloroform dissolved in d6 acetone as the twoqubit NMR quantum processor, and diethylfluoromalonate dissolved in d6 acetone as the threequbit NMR quantum information processor. The natural Hamiltonians of the twoqubit system (denoted as ) and the threequbit system (denoted as ) are as follows, where ν_{i} is the chemical shift of the ith nucleus, and J_{i,j} is the Jcoupling constant between the ith and the jth nuclei. The molecular structures of chloroform and diethylfluoromalonate, and their parameters are described in Fig. 3. The two quantum information processors are initially prepared in the pseudopure states 01〉 and 110〉 using the spatial average technique^{31}, with qubit orders ^{1}H, ^{13}C and ^{1}H, ^{13}C, ^{19}F, respectively.
Gate construction and implementation in the twoqubit simulation
Here we give the explicit construction of F, Q, and D, which are illustrated in Fig. 1. F is the bitswapped Fourier transformation operator, which can be implemented in quantum circuits via a series of Hadamard gates and controlledphase gates^{30}, , where H_{1} and H_{2} are Hadamard gates on the first and second qubits, respectively, and [1, 1, 1, i] is the twoqubit controlledphase gate. is a singlequbit gate on the second qubit. D realizes a bitswapped version of up to an overall phase factor (we have taken m = 1/2), as D = Φ_{π}Z_{1}Z_{2}, where Φ_{π}, Z_{1}, and Z_{2} can be expressed as follows,
The singlequbit gates in Fig. 1 are realized using radiofrequency pulses, and the twoqubit gates in Fig. 1 are realized by combining refocusing pulses and Jcoupling evolution. The pulse sequences for implementing F, D and Q are exhibited in Fig. 8.
Gate construction and implementation in the threequbit simulation
The operations H_{i} (i = 1, 2, 3) in Fig. 2 are Hadamard gates on the ith qubit, and . The other operations in Fig. 2 are as follows:
We used the optimal control method for pulse designing in our experiments, namely the GRAPE algorithm^{34}, which is widely used for designing and optimizing pulse sequences to produce a desired unitary propagator. The evolution operators , with n = 1,2,3,4 and 5, which are illustrated in Fig. 2, are implemented using optimized pulses generated by the GRAPE algorithm in the experiments.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11175094, 91221205), the National Basic Research Program of China (2009CB929402, 2011CB921602), and the Specialized Research Fund for the Doctoral Program of Education Ministry of China.
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Author notes
 GuanRu Feng
 & Yao Lu
These authors contributed equally to this work.
Affiliations
State Key Laboratory of Lowdimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, P. R. China
 GuanRu Feng
 , Yao Lu
 , Liang Hao
 , FeiHao Zhang
 & GuiLu Long
Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, P. R. China
 GuanRu Feng
 , Yao Lu
 , Liang Hao
 , FeiHao Zhang
 & GuiLu Long
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Contributions
G.R.F., Y.L., F.H.Z. and G.L.L. designed the experimental scheme; G.R.F., Y.L. and G.L.L. performed the experiments; G.R.F., L.H. and G.L.L. analyzed data; G.R.F., Y.L. and G.L.L. wrote the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to GuiLu Long.
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