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 spin-1/2 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.
Quantum simulation is one of the most important aims of quantum computation ever since Feynman studied the likelihood of simulating one quantum system by another1. 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 computers2,3,4,5,6, 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 many-body interaction Hamiltonian7,8,9,10, the dynamics of entanglement11,12, quantum phase transitions13,14, and calculations of molecular properties15,16,17,18,19.
Quantum tunneling plays an essential role in many quantum phenomena, such as the tunneling of superconducting Cooper pairs20 and alpha decay21. Moreover, tunneling has been widely applied in modern devices and modern experimental techniques, such as the tunnel diode22, the scanning tunneling microscope23 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 chromodynamics24, 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 Sornborger25 proposed a digital simulation algorithm for demonstrating the tunneling of a particle in a double-well 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 one-dimensional 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.
Consider a single particle moving in a square well potential in one-dimensional 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 simulation25,27,28, 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 n-qubit 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 as26 where σ3 is the Pauli matrix σz, and σ4 = 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 Δt25,27,29. Equation (2) is then approximated as From equation (8) we have where F is the usual bit-swapped Fourier transformation operator, which can be readily realized in quantum circuits via a series of Hadamard gates and controlled-phase gates30. Consequently D in equation (11) is a bit-swapped 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 two-qubit and the three-qubit implementations, respectively. The explicit construction of F, D, and Q is detailed in Methods.
Experimental procedures and results
As a small-scale demonstration, a two-qubit simulation and a three-qubit simulation were investigated. In the experiment, we studied the time evolution of a particle moving in double-well potentials using the digital algorithm via two-qubit and three-qubit NMR quantum information processors. The molecular structures and parameters of the quantum information processors are given in Fig. 3.
In the two-qubit 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 V0I σz, which can be implemented with only a single-qubit gate25. This potential represents a double-well potential of amplitude 2V0, with two peaks V0 at |00〉 and |10〉, and two troughs −V0 at |01〉 and |11〉. In our experiments we set the time interval Δt = 0.1 and the amplitude of the potential 2V0 = 20. We simulated the situation in which the particle is initially trapped inside one of the two wells by preparing the pseudo-pure state |01〉 from thermal equilibrium31 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 steps32,33. The whole evolution process of a single particle in the double-well 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 three-qubit 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 V0I σz I, which corresponds to a double-well potential of amplitude 2V0. It has a higher resolution than the double-well potential in the two-qubit 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 2V0 = 200. The initial position of the particle is set in one of the two wells, by preparing the pseudo-pure 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).
Our experiments have simulated the fundamental quantum phenomenon of tunneling. Our two-qubit experiments reflected a remarkable difference between the two situations with and without the double-well potential. In our three-qubit experiments, a higher resolution can provide a more sophisticated structure of the potential wells, making it possible to observe the in-well-oscillation of the particle.
In the two-qubit 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 three-qubit 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 N-site-lattice simulation can be efficiently implemented in experiments with only log2N qubits. The result has revealed the amazing power hidden behind the qubits and the promising future of quantum simulations.
In summary, we accomplished a small-scale demonstration of the quantum tunneling process on two- and three-qubit 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.
Experiments were carried out at room temperature using a Bruker Avance III 400 MHz spectrometer. We used chloroform dissolved in d6 acetone as the two-qubit NMR quantum processor, and diethyl-fluoromalonate dissolved in d6 acetone as the three-qubit NMR quantum information processor. The natural Hamiltonians of the two-qubit system (denoted as ) and the three-qubit system (denoted as ) are as follows, where νi is the chemical shift of the ith nucleus, and Ji,j is the J-coupling constant between the ith and the jth nuclei. The molecular structures of chloroform and diethyl-fluoromalonate, and their parameters are described in Fig. 3. The two quantum information processors are initially prepared in the pseudo-pure states |01〉 and |110〉 using the spatial average technique31, with qubit orders 1H, 13C and 1H, 13C, 19F, respectively.
Gate construction and implementation in the two-qubit simulation
Here we give the explicit construction of F, Q, and D, which are illustrated in Fig. 1. F is the bit-swapped Fourier transformation operator, which can be implemented in quantum circuits via a series of Hadamard gates and controlled-phase gates30, , where H1 and H2 are Hadamard gates on the first and second qubits, respectively, and [1, 1, 1, i] is the two-qubit controlled-phase gate. is a single-qubit gate on the second qubit. D realizes a bit-swapped version of up to an overall phase factor (we have taken m = 1/2), as D = ΦπZ1Z2, where Φπ, Z1, and Z2 can be expressed as follows,
The single-qubit gates in Fig. 1 are realized using radio-frequency pulses, and the two-qubit gates in Fig. 1 are realized by combining refocusing pulses and J-coupling evolution. The pulse sequences for implementing F, D and Q are exhibited in Fig. 8.
Gate construction and implementation in the three-qubit simulation
We used the optimal control method for pulse designing in our experiments, namely the GRAPE algorithm34, 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.
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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EPL (Europhysics Letters) (2018)