Experimental Demonstration of Quantum Tunneling in IBM Quantum Computer

Home , IBM Quantum Experience, Qiskit

Narendra N. Hegade1*, Bikash K. Behera2† & Prasanta K. Panigrahi2‡§

1Department of Physics, National Institute of Technology Silchar, Silchar, 788010, India

2Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata,

Mohanpur, 741246, West Bengal, India

Quantum computers are the promising candidates for simulation of large quantum systems,

which is a daunting task to perform in a classical computer. Here we report the experimen-

tal realization of quantum tunneling of a single particle through different types of potential

barriers by simulating it in the IBM quantum computer, which here acts as a universal quan-

tum simulator. We consider two-qubit and three-qubit systems for visualizing the tunneling

process illustrating its unique quantum nature. We clearly observe the tunneling and oscil-

lations of the particles in a step potential and in double- and multi-well potentials through

our experimental results. The proposed quantum circuits and simulational techniques used

here can be extended for observing the tunneling phenomena for multi-particle systems in

different shaped potentials. arXiv:1712.07326v4 [quant-ph] 25 Feb 2019 Keywords:Quantum simulation, Quantum tunneling, IBM Quantum Experience

*E-mail: [email protected] †E-mail: [email protected] ‡Corresponding author. §E-mail: [email protected]

1 1 Introduction

Quantum simulation is one of the problems that a quantum computer could perform more efﬁ-

ciently than a classical computer as it provides a signiﬁcant improvement in computational re-

sources 1–9. It has been applied to a wide range of areas of physics like quantum many-body theory

10–13, quantum entanglement 14, 15, quantum phase transitions 16, 17, molecular physics 18–22 etc. Al- gorithms have been used in simulating many quantum ﬁeld theoretic problems 23–34 where Hamil- tonian of the system splits into kinetic and potential energy operators which are then simulated using Trotter’s formula 35, 36. Experimental realizations of quantum simulation have already been made in systems like NMR 10–12, 15, 20, 37–39, ion-trap 40–44, atomic 16, 45 and photonic 46, 47 quantum

computers. The current status of this ﬁeld can be found out from these recent papers 48–57.

Quantum tunneling acts as one of the exciting and unique fundamental phenomena in quantum mechanics. It has been observed in superconducting Cooper pairs 58 and in modern technolo- gies such as narrow p-n junctions 59 and scanning tunneling microscope 60. Important science puz-

zles like lattice quantum chromodynamics can be solved using this tunneling simulation approach

61. A number of digital simulation on quantum tunneling has been performed on classical com-

puters 62, 63 and photonic systems 64. This type of simulation has remained untested in a quantum

computer due to the requirement of large number of ancillary qubits and quantum gates. Recently,

an algorithm proposed by Sornborger 65 illustrates the simulation with no ancillary qubits and a

small number of quantum gates which motivates the possibility of simulating in today’s quantum

computer consisting of a few number of qubits. Feng et al. 66 have demonstrated the tunneling

2 effect for two-qubit and three-qubit systems using NMR quantum computer. Ostrowski 67 has also explicated this process on a rectangular potential by digital simulation. Here, in the present work, we illustrate the simulation of quantum tunneling using IBM’s 5-qubit quantum computer

“IBM Q 5 Tenerife” and 14-qubit quantum computer “IBM Q 14 Melbourne”. Using two-qubit and three-qubit systems, CNOT gates, a set of Hadamard and controlled phase gates, we were able to simulate the tunneling process of a single particle in a step potential, double-well potential and multi-well potential. We utilize the IBM quantum experience’s QISKit, to simulate the tunneling

Hamiltonian, using which a number of research works have been performed 68–94.

2 Results

Theoretical Protocol. A digital quantum simulator is a controllable quantum system which can be used to efﬁciently simulate the dynamics of any other quantum system with local interactions.

The digital quantum simulation 95–102 consists of three main steps: initial state preparation, time evolution, and measurement. In this work, we are particularly interested in the time evolution of the system, so our main idea is to decompose the evolution of the system in terms of single and two-qubit gates. The Schro¨dinger’s equation, for a single particle moving in an one-dimensional space, is expressed as

∂ i |ψ(x, t)i = Hˆ |ψ(x, t)i (1) ∂t

3 where Hˆ = Kˆ + Vˆ , Kˆ and Vˆ are kinetic and potential energy operators respectively. Here we set the value of ~ to 1 throughout the manuscript. The time evolution of the wave function of the system can be given as

|ψ(x, t + ∆t)i = e−iHˆ ∆t|ψ(x, t)i

= e−i(Kˆ +Vˆ )∆t|ψ(x, t)i (2)

Using ﬁrst-order Suzuki-Trotter’s formula 103–105, the exponential operator can be decom-

posed up to order O(∆t) as follows.

ˆ ˆ ˆ ˆ e−i(K+V )∆t ≈ e−iK∆te−iV ∆t (3)

We discretize the continuous coordinate space x on a lattice (with spacing ∆l) within the

boundary region (0 < x < L) with a periodic boundary condition ψ(x+L, t) = ψ(x, t). The wave

function then can be mapped to a n-qubit register as,

2n−1 X |ψ(x, t)i → ψ(xk, t)|ki (4) k=0

1 Here |ki represents the particle location corresponding to binary number k, and xk = (k + 2 )∆l,

L ∆l = 2n . The mapping given in expression (4) can be a good approximation for large value

n. As an example, for n=3 qubits, the binary representation of a wavefunction on a 1D lattice is

represented in Fig. 1. Here the wave function mapped to the 3-qubit register can be written as,

(5)

P7 2 1 L where, k=0 |ψ(xk, t)| = 1, and xk = (k + 2 ) 8 . The quantum states |0i, |1i,...,|7i are mapped to 3-qubit quantum states as |000i, |001i,...,|111i respectively.

Figure 1: Binary representation of a wavefunction on a 1D lattice for 3-qubit case. The 3- qubit quantum states |000i, |001i,...,|111i represent the binary form of the states |0i, |1i,...,|7i respectively.

Implementing potential energy operator: The potential energy operator plays a key role for introducing and studying various types of potential structures. Here we implement step-well, double-well and multi-well potentials with the help of single qubit rotation operators with no ancillary qubits. For step potential, we apply a single-qubit Z-rotation gate on the highest order qubit

65.

n−1 e−iV ∆t = e−ivσz ∆t = e−ivσz∆t ⊗ I ⊗ I..., (6)

5 where v is a parameter, σz is the Pauli z-matrix, and n − j indicates the application of the operator

(e−ivσz∆t) on the jth qubit. The double-well potential can be implemented by applying the Z- rotation gate on the second highest order qubit.

n−2 e−iV ∆t = e−ivσz ∆t = I ⊗ e−ivσz∆t ⊗ I... (7)

Similarly, by applying Z-rotation operator on the next highest order qubits we can imple-

ment multi-well potentials. It can be observed that just one single qubit operation can reduce the

complexity in the quantum circuit by replacing a large number of gates and ancillary qubits.

e−iV ∆t = I ⊗ I ⊗ e−ivσz∆t... (8)

Implementing kinetic energy operator: Here the kinetic energy operator Kˆ can be ex-

ˆ pˆ2 pressed in terms of momentum operator as, K = 2m . For ﬁnding the quantum circuit for kinetic energy operator, let’s discretize the wave function of momentum as

2n−1 X |χ(p, t)i = χ(pl, t) |li (9) l=0

where χ(p, t) is the wave function in the momentum operator representation. The eigen

values of momentum pl are given by

6   2π n−1  2n l 0 6 l 6 2 pl = (10)   2π n−1 n−1 n  2n (2 − l) 2 < l < 2

In the momentum representation, the diagonal operator Pˆ is written as

2n−1 2n−1 X 2π X 2π Pˆ = l |li hl| + (2n−1 − l) |li hl| (11) 2n 2n l=0 l=2n−1+1

The kinetic energy operator is diagonal in momentum representation. It can be written in the coordinate representation by using quantum Fourier transform as follows,

ˆ2 −i(Kˆ +Vˆ )∆t −i P ∆t −1 −iVˆ ∆t e ≈ (QF T )e 2m (QF T )e

≈ (QF T )D(QF T −1)P (12) where

2n−1 1 X 2πilk QF T = √ e 2n |lihk| n 2 l,k=0 ˆ2 −i P ∆t D = e 2m

ˆ P = e−iV ∆t (13)

The equivalent quantum circuit for the Fourier transformation operator QF T can be realized using a series of Hadamard and controlled-phase gates 106. Thus after a small time interval ∆t, the time evolution of the system can be implemented by using Eqs. (2), (3), (4) and (12).

7 2n−1 2n−1 X −1 X ψ(xk, t + ∆t)|ki = (QF T )D(QF T )P ψ(xk, t)|ki (14) k=0 k=0

The explicit construction of quantum circuit for QF T , D, QF T −1 and P are detailed in

Methods section.

Experimental Procedures and Results. The quantum circuit for one-time step evolution using two qubits is shown in Fig. 11. The step potential is implemented by the quantum operation

1 P = e−ivσz ∆t acting on the highest order qubit. Similarly, the double-well potential is implemented

0 by the quantum operation P = e−ivσz ∆t acting on the lowest order qubit. In our experiment, we set the parameter v=0 for free particle, and v = 50 for step potential, double-well potential and v = 10 for multi-well potential. We set the time interval ∆t = 0.1. Mass of the particle is taken to be 0.5.

In Fig. 2, the time evolution of a free particle with potential v = 0 is depicted. Initially, the particle was confined at |00i state. It can be seen that after a number of time steps, the particle probability distribution spreads over other basis states. Both the theoretical and experimental results are illustrated in Fig. 2 (a) and (b) cases respectively. To compare the result obtained from the quantum processor “ibmqx4” and the simulated result from “IBM Q QASM simulator” quantum state tomography is performed. In both the cases, the number of shots taken were 8192. The density matrix elements for the initial state |00i and the final state after six time steps are depicted in Fig. 3 and 4 respectively. The experimental fidelity for the initial state is 96.79%, and for the

8 Figure 2: Free particle (v=0) probability distribution. Case (a): Simulated result obtained from

“IBM Q QASM simulator” for free particle evolution using two-qubit system. Initially the particle

was conﬁned at |00i state, the probability distribution of the particle slowly spreads out after six time steps. Case (b): Experimentally obtained result from “ibmqx4” for probability distribution of free particle. Similar observation has been obtained as case (a). The horizontal axis represents the number of time steps and the vertical axis represents the corresponding probabilities.

ﬁnal state after six time step is 93.83%.

In Fig. 5, we illustrate the tunneling phenomena of a single particle in a ﬁnite length step potential of v=50. The potential barriers are situated at |10i and |11i states. At time, t=0, the

particle is located at |00i state, as time evolves we can observe the probability distribution of

the particle spreads to the state |01i. Notably, it can be observed that after six time steps, the

particle has some non-zero probability to be found at classically forbidden region |10i and |11i.

Which clearly conﬁrms the tunneling of particle through the potential barrier. We performed this

9 1 0.3

0.8 0.2 (a) 1 (b) 0.5 0.1 0.6 0 0 0 -1 0.4 -0.5 -0.1 |00> |00> |01> <11| 0.2 |01> <11| |10> <10| |10> <10| -0.2 |11> <01| |11> <01| <00| <00| 0 -0.3

0.04 0.8

|00> |00> -0.02 |01> <11| 0.2 |01> <11| |10> <10| |10> <10| |11> <01| |11> <01| <00| <00| -0.04 0

Figure 3: (a) and (b) are the real and imaginary parts of the density matrix elements for the initial state |00i of the free particle obtained from the “IBM Q QASM simulator”; (c) and (d) are the experimentally reconstructed density matrix elements obtained from “ibmqx4” chip. experiment on “IBM Q QASM simulator”, “ibmqx4” and “IBM Q 14 Melbourne”. Both the theoretical and experimental results are explicated in Fig. 5. In all the cases, the number of shots taken to perform the experiment is 8192.

The tunneling of particle in a double-well potential (v=50) is shown in Fig. 6. The wells are located at |01i and |11i states and the barriers are at |00i and |10i states. At time, t=0, the particle is conﬁned in the potential well situated at |01i state, as the time evolves we can clearly observe the tunneling of particle from the potential well located at |01i state to the well at |11i state. Both from the theoretical and experimental data (Fig. 6 (a) and (b)), it is observed that the particle oscillates between the two wells which signiﬁes tunneling through the potential barriers

10 0.5 0.4

0.4 (a) 1 (b) 0.5 0.2 0 0.3 0 0 -1 0.2 -0.5 -0.2 |00> 0.1 |00> |01> <11| |01> <11| |10> <10| |10> <10| |11> <01| |11> <01| <00| 0 <00| -0.4

0.4 0.2 (c) 1 (d) 0.5 0.1 0.3 0 0 0 -1 0.2 -0.5 -0.1 |00> |00> |01> <11| 0.1 |01> <11| |10> <10| |10> <10| |11> <01| |11> <01| -0.2 <00| <00| 0

Figure 4: (a) and (b) are the real and imaginary parts of the density matrix elements for the ﬁnal

state (i.e after six time steps) of the free particle obtained from the QASM simulator; (c) and (d)

are obtained from “ibmqx4” device.

at the other two places. To compare the result obtained from the quantum processor “IBM Q 14

Melbourne” and the ideal QASM simulator we performed quantum state tomography. To perform

the experiment, the number of shots are chosen to be 8192. The density matrix elements for the

initial state |01i and the ﬁnal state after six time steps are depicted in Fig. 7 and 8 respectively.

The experimental ﬁdelity for the initial state is 95.61%, and for the ﬁnal state after six time step

is 95.18%. Fig. 9 shows the tunneling of the particle in a multi-well potential implemented on

the “IBM Q QASM simulator”. The wells are situated at |000i , |010i , |100i and |110i positions.

Initially, at time t=0, the particle is trapped inside one of the above wells situated at |100i state, as

time evolves the particle is tunneled through the barriers situated at |001i , |011i , |101i and |111i

states. At different time steps, the tunneling of the particle through all the potential barriers is

11 Figure 5: Particle probability distribution in a step-well potential (v=50). Case (a,b): Simu-

lated results obtained from “IBM Q QASM simulator” illustrating the single particle dynamics in a

step potential. Case (c,d): Experimentally observed result from “IBM Q 14 Melbourne” showing the hopping of the particle in a step potential of finite length. Case (e,f): Experimentally observed result from “ibmqx4” depicting the hopping of the particle in a step potential of finite length. Here, the barriers are located at |10i and |11i states, From the probability distribution in all the cases, it can be observed that, as time evolves, the particle which is initially confined at |00i state hops to

the |01i state, also the particle has some non zero12 probability to be found at classically forbidden region. Figure 6: Particle probability distribution in double-well potential (v=50). Case (a) : Theo- retically calculated result of quantum tunneling in double-well potential using two-qubit system.

Initially the particle was conﬁned at |01i state. After six time step the particle is found to slowly tunnel to the |11i state. Case (b): Experimentally observed result of quantum tunneling for 6 time steps.

13 1 0.2 0.8 (a) 1 (b) 0.5 0.1 0 0.6 0 0 -1 0.4 -0.5 |00> |00> -0.1 |01> <11| 0.2 |01> <11| |10> <10| |10> <10| |11> <01| |11> <01| <00| <00| -0.2 0

0.03 0.8 0.02 (c) 1 (d) 0.5 0.6 0.01 0 0 0 0.4 -1 -0.5 -0.01 |00> |00> 0.2 |01> <11| |01> <11| -0.02 |10> <10| |10> <10| |11> <01| |11> <01| <00| <00| -0.03 0

Figure 7: The density matrix elements for the initial state |01i of a particle in double-well potential.

(a) and (b) are the real and imaginary parts of the density matrix obtained from QASM simulator,

(c) and (d) are from “IBM Q 14 Melbourne”. observed. After 10 time steps, the particle is most likely to be found at |010i state.

3 Discussion

To conclude, we have experimentally demonstrated here the quantum tunneling phenomena of a single particle in a step potential, double-well potential and multi-well potential. We have designed the equivalent quantum circuit for the Hamiltonian of the given system in the real quantum processors “ibmqx4” and “IBM Q 14 Melbourne”. We have shown the architecture of this processor with important device parameters including gate errors and readout errors. We have illustrated the tunneling process by running the quantum circuit for six time steps in two-qubit case and ten time steps in three-qubit case respectively. After comparing the theoretical and experimental results it

14 0.4 0.4 (a) 1 (b) 0.5 0.2 0.3 0 0 0.2 0 -1 -0.5 0.1 |00> |00> -0.2 |01> <11| |01> <11| |10> <10| 0 |10> <10| |11> <01| |11> <01| <00| <00| -0.1 -0.4

0.1 0.3

(c) 1 (d) 0.5 0.05 0.2 0 0 0 -1 0.1 -0.5 |00> |00> -0.05 |01> <11| |01> <11| |10> <10| |10> <10| |11> <01| 0 |11> <01| <00| <00| -0.1

Figure 8: (a) and (b) are the real and imaginary parts of the density matrix elements for the ﬁnal state (i.e after six time steps) of the particle in double-well potential obtained from the QASM simulator; (c) and (d) are obtained from “IBM Q 14 Melbourne”. is concluded that the tunneling process has been properly carried out with IBM’s quantum processors. The basic circuit mechanism used here for demonstrating quantum tunneling can be extended to observe quantum tunneling of multi-particle systems in different shaped potential wells.

4 Methods

We investigate quantum simulation of the tunneling process using the IBM’s 5-qubit quantum processor “ibmqx4” and 14-qubit quantum processor “IBM Q 14 Melbourne”, whose layout is depicted in Fig. 10. The connectivities between the qubits in both the processors are shown. The devices are stored in a temperature order of mK. Tables 1 & 2 illustrate the experimental parameters of the devices, where frequency, coherence time (T1), relaxation time (T2), gate error and readout

15 Figure 9: Particle probability distributions for ten time steps in a multi-well potential (v =

Circuit construction for two-qubit simulation. The explicit construction of QF T −1, D and

P are illustrated in Fig. 11. QF T −1 is prepared by two Hadamard gates and one controlled-phase

−1 gate, QF T = H0C − U1(π/2)10H1, where Hi denotes the application of Hadamard gate on the q[i] qubit, C − U1(θ)ij denotes controlled-phase gate of angle θ, where q[i] is the control qubit and

iθ q[j] is the target qubit, U1(θ) = [[1, 0], [0, e ]]. For two-qubit simulation, the diagonal elements of

16 Figure 10: Chip layout of 5-qubit (ibmqx4) and 14-qubit (IBM Q 14 Melbourne) quantum

processors.

Pˆ are calculated to be 0, π/2, π and −π/2 (See Eq. (11)). Then, the kinetic evolution operator is given by

e−iKˆ ∆t = QF T −1e−iPˆ2∆tQF T

−1 = QF T Φ10Z0Z1QF T (15)

The diagonal operator D can be expressed as a product of the following operators,

D = Φ10Z0Z1 (16)

17 Figure 11: Quantum circuit for two qubit system. Case (A): The quantum circuit for one time step (∆t = 0.1) evolution of tunneling Hamiltonian using two qubits is depicted. Case (B): The inverse quantum Fourier transform QF T −1 is built using two Hadamard gates and one controlled- rotation gate U1(π/2). Case (C): The decomposition of diagonal operator D is represented, which

2 requires two single qubit Rz rotation gates and one controlled-rotation gate U1(π /10).

0 1 −iγc0σ ∆t −iγc1σ ∆t −iγc2diag(1,1,1,−1)10∆t where Z1 = e z , Z0 = e z and φ10 = e . Here, φ10 has been

2 prepared with the C − U1(π /10)10 gate. Here Zi means application of Z operation on q[i]

π2 qubit. The constant values in (16) are obtained to be γ = 8 , c0 = −1, c1 = −4 and c2 = 4

65. The time step is taken to be ∆t = 0.1. The quantum circuit for D operator can be given

2 2 2 as, D = Rz(−π /40)1Rz(−π /10)0C − U1(π /10)10, where Rz(θ)i is the rotation operator

about Z-axis by an angle θ on the q[i]th qubit (See Fig. 11). The Rz(θ) can be expressed us-

18 π π ing standard “ibmqx4” gates as Rz(θ) = HU3(θ, φ = − 2 , λ = 2 )H, where U3(θ, φ, λ) =

[[cos(θ/2), −eiλsin(θ/2)], [eiφsin(θ/2), ei(λ+φ)cos(θ/2)]]. The potential operator P is prepared by

π π the application of HU3(θ = 5, φ = − 2 , λ = 2 )H operator. It has been applied on the qubits q[0]

and q[1] in case of step potential and double well potential respectively. The controlled phase gates

used in the experiment, are directly applied using the codes available in QISKit (See supplementary

Information).

19 Figure 12: Quantum circuit for three-qubit system. Case(A): The quantum circuit for one time step (∆t = 0.1) evolution of tunneling Hamiltonian using three qubits is depicted. Case(B): The inverse quantum Fourier transform QFT−1 is built using three Hadamard gates and three controlled phase gates of π/2 and π/4 angle. Case(C): The kinetic energy operator D is decomposed into rotation and controlled-phase gates of −π2/160, −π2/40, −π2/10, π2/10 and π2/20 angles.

20 Circuit construction for three-qubit simulation. The explicit construction of QF T −1,

D and P are illustrated in Fig. 12. QF T −1 is prepared by three Hadamard gates and three

−1 controlled-phase gates, QF T = H0C − U1(π/2)10H1C − U1(π/4)20C − U1(π/2)21H2. For

ˆ three-qubit simulation, the diagonal elements of Pp are calculated to be 0, π/4, π/2, 3π/4, π,

−π/4, −π/2 and −3π/4 (See Eq. (11)). The decomposition of D operator is as follows, D =

2 2 2 2 2 2 Rz(−π /160)2Rz(−π /40)1Rz(−π /10)0C − U1(π /10)21C − U1(π /20)20C − U1(−π /40)10.

0 −iγc0σ ∆t The diagonal operator D can be expressed as a product of the following operators, Z0 = e z ,

1 2 −iγc1σ ∆t −iγc2σ ∆t −iγc3diag(1,1,1,−1)21∆t −iγc4diag(1,1,1,−1)20∆t Z1 = e z ,Z2 = e z , φ21 = e , φ20 = e , φ10 =

2 e−iγc5diag(1,1,1,−1)10∆t. The constant values are obtained to be γ = − π√ , c = −1.42, c = −5.66 32 2 0 1

65 and c2 = −22.63, c3 = −22.63, c4 = 11.31 and c5 = −5.66 . It is to be noted that the

time step is taken to be ∆t = 0.1. The quantum circuit for D operator can be given as, D =

2 2 2 Rz(−π /40)1Rz(−π /10)0C − U1(π /10)10, where Rz is the rotation operator about Z-axis (See

π π Fig. 12). The potential operator P prepared by HU3(θ, φ = − 2 , λ = 2 )H operator is applied on

the qubit q[2] in case of multi-well potential. Similarly, the controlled phase gates used here are

directly applied using the codes available in QISKit (See supplementary Information).

21 || † ‡ −3 −3 ⊥ −2 Qubits Freq (GHz) T1 (µs) T2 (µs) U1 err U2 err (10 ) U3 err (10 ) RE (10 )

Q0 5.25 48.23 23.49 0 0.86 1.72 7.53

Q1 5.29 53.58 10.58 0 1.55 3.09 9.15

Q2 5.35 38.68 12.52 0 1.89 3.78 3.58

Q3 5.43 48.83 13.94 0 2.23 4.47 4.00

Q4 5.17 46.94 7.37 0 1.80 3.61 8.78 || Frequency, † Coherence time, ‡ Relaxation time, ⊥ Readout Error.

Multi-Qubit Gate Gate Error Multi-Qubit Gate Gate Error

C1X0 0.03488 C3X2 0.0866

C2X0 0.03158 C3X4 0.06682

C2X1 0.05546 C4X2 0.0842

Table 1: Experimental parameters of the device “ibmqx4” (information collected and experimented on Date: 15.02.2019).

22 || † ‡ −3 −3 ⊥ −2 Qubits Freq (GHz) T1 (µs) T2 (µs) U1 err U2 err (10 ) U3 err (10 ) RE (10 )

Q0 5.10 64.45 22.04 0 1.62 3.24 3.47

Q1 5.24 56.76 81.54 0 31.89 63.78 8.83

Q2 5.03 74.67 111.63 0 2.92 5.85 11.82

Q3 4.89 61.06 53.99 0 1.78 3.56 25.23

Q4 5.03 56.47 40.94 0 1.90 3.79 3.41

Q5 5.07 25.51 57.04 0 2.08 4.15 7.22

Q6 4.92 62.04 54.84 0 1.59 3.18 3.62

Q7 4.97 45.74 84.04 0 1.26 2.52 4.38

Q8 4.74 55.26 77.63 0 1.54 3.09 7.25

Q9 4.96 37.90 56.65 0 3.01 6.02 27.92

Q10 4.94 55.71 69.20 0 2.22 4.44 4.21

Q11 5.01 59.16 76.46 0 1.43 2.86 10.68

Q12 4.76 69.46 109.99 0 2.90 5.81 4.18

Q13 4.97 25.46 39.96 0 7.74 15.48 11.45 || Frequency, † Coherence time, ‡ Relaxation time, ⊥ Readout Error.

Multi-Qubit Gate Gate Error Multi-Qubit Gate Gate Error Multi-Qubit Gate Gate Error

C1X0 0.03624 C7X8 0.02562 C5X6 0.06062

C1X2 0.03289 C9X8 0.05133 C12X2 0.06609

C2X3 0.03919 C9X10 0.06855 C5X9 0.05182

C4X3 0.03584 C11X3 0.06448 C13X1 0.1436

C4X10 0.0272 C11X10 0.04068 C6X8 0.03824

C5X4 0.05023 C11X1223 0.04357 C13X12 0.04426

Table 2: Speciﬁcations of the parameters of each qubit in the “IBM Q 14 Melbourne” quantum computer (information collected and experimented on Date: 13.02.2019) References

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5 Data Availability statement

The data collected and the codes used in the experiment are available in the Supplementary Infor- mation. Request for any materials should be addressed to P.K.P. ([email protected]).

Acknowledgements

N.N.H. acknowledges the hospitality of Indian Institute of Science Education and Research Kolkata during the project work. B.K.B. acknowledges the ﬁnancial support of IISER-K Institute fellow- ship. We acknowledge the support of IBM Quantum Experience for using the quantum processors.

35 The views expressed are those of the authors and do not reﬂect the ofﬁcial position of IBM or the

IBM quantum experience team.

Author contributions

Theoretical analysis and design of quantum circuits were done by N.N.H and B.K.B. Simulation of quantum circuit, collection and analysis of data were done by N.N.H. The project was supervised by B.K.B. P.K.P. has thoroughly reviewed the manuscript. N.N.H. and B.K.B. have completed the project under the guidance of P.K.P.

Competing interests

The authors declare no competing ﬁnancial as well as non-ﬁnancial interests. Readers are welcome to comment on the online version of the paper. Correspondence and requests for materials should be addressed to P.K.P. ([email protected]).

36 Supplementary Information: Experimental Demonstration of Quantum Tunneling in IBM

Quantum Computer

For simulating quantum tunneling, we used QISKit to take both simulation and experimental re-

sults. The python code for the same is as follows.

1 #Quantum simulation of tunneling in double −w e l l potential

# Import the QISKit SDK

3 i m p o r t numpy as np

from qiskit import QuantumCircuit , ClassicalRegister , QuantumRegister

5 from qiskit import Aer, execute

from math import pi

7 from qiskit import IBMQ

from qiskit.tools.monitor import job_monitor

9 from qiskit.tools.visualization import plot_histogram

11 # Set yourAPI Token.

# IBMQ. enable_account(’API Token’)

# Createa Quantum Register with2 qubits.

15 q = QuantumRegister(2)

# Createa Classical Register with2 bits.

17 c = ClassicalRegister(2)

# Createa Quantum Circuit

19 qc = QuantumCircuit(q, c)

37 21 # Circuit Implementation

# Initial state

23 qc . x ( q [ 1 ] )

# Time step=1

25 # Time evolution of potential energy part

qc . h ( q [ 1 ] )

27 qc . u3 ( 5 , −pi/2, pi/2, q[1])

qc . h ( q [ 1 ] )

29 #2 − q u b i t Inverse Quantum Fourier Transform

qc . h ( q [ 0 ] )

31 qc.cu1(pi/2, q[1], q[0])

qc . h ( q [ 1 ] )

33 # Time evolution of kinetic energy part

qc . h ( q [ 0 ] )

35 qc . u3 ( −( p i **2) / 4 0 , −pi/2, pi/2, q[0])

qc . h ( q [ 0 ] )

37 qc . h ( q [ 1 ] )

qc . u3 ( −( p i **2) / 1 0 , −pi/2, pi/2, q[1])

39 qc . h ( q [ 1 ] )

qc . cu1 ( ( p i **2)/10, q[1], q[0])

41 #2 − q u b i t Quantum Fourier Transform

qc . h ( q [ 1 ] )

43 qc . cu1(− pi/2, q[1], q[0])

qc . h ( q [ 0 ] )

45 # Time evolution of potential energy part

qc . h ( q [ 1 ] )

38 47 qc . u3 ( 5 , −pi/2, pi/2, q[1])

qc . h ( q [ 1 ] )

# Measurement

51 qc.measure(q, c)

# Choose backend

53 backend = IBMQ.get_backend(’ibmq_qasm_simulator’)

# backend= IBMQ. get_backend(’ibmqx4’)

55 # backend= IBMQ. get_backend(’ibmqx2’)

# backend= IBMQ. get_backend(’ibmq_16_melbourne’)

57 s h o t s = 8192# Number of shots to run the program(experiment); maximum

i s 8192 shots.

max_credits = 10# Maximum number of credits to spend on executions.

59 job_exp = execute(qc, backend=backend , shots=shots , max_credits=max_credits)

job_monitor(job_exp)

61 result = job_exp.result()

counts_exp = result.get_counts(qc)

63 p r i n t(result .get_counts(qc))

plot_histogram(counts_exp)

doublewell_tunneling.py

39 Time Step ( t = 0.1 ) Particle Location IBM Q QASM Simulator IBM Q 5 Tenerife

1t 00 00 0.871 0.68 1 01 0.053 0.158 2 10 0.016 0.069 3 11 0.06 0.093

2t 0 00 0.576 0.426 1 01 0.17 0.277 2 10 0.084 0.096 3 11 0.17 0.201

3t 0 00 0.292 0.25 1 01 0.244 0.286 2 10 0.214 0.191 3 11 0.25 0.273

4t 0 00 0.174 0.205 1 01 0.212 0.218 2 10 0.397 0.347 3 11 0.217 0.23

5t 0 00 0.281 0.252 1 01 0.093 0.124 2 10 0.526 0.464 3 11 0.101 0.16

6t 0 00 0.446 0.3 1 01 0.008 0.104 2 10 0.539 0.508 3 11 0.008 0.087

Figure 13: Free particle probability distribution: we have used the qubits q[0] and q[1] from

IBM Q 5 Tenerife (ibmqx4) for our experiment, No. of shots=8192.

40 Time Step ( t = 0.1 ) Particle Location IBM Q QASM Simulator IBM Q 14 Melbourne IBM Q 5 Tenerife

1t 00 00 0.872 0.712 0.727 1 01 0.053 0.131 0.139 2 10 0.017 0.074 0.054 3 11 0.057 0.083 0.08

2t 0 00 0.791 0.609 0.633 1 01 0.186 0.249 0.228 2 10 0.009 0.077 0.069 3 11 0.015 0.065 0.069

3t 0 00 0.546 0.429 0.381 1 01 0.42 0.369 0.442 2 10 0.018 0.106 0.089 3 11 0.017 0.096 0.088

4t 0 00 0.331 0.327 0.246 1 01 0.61 0.429 0.554 2 10 0.038 0.122 0.093 3 11 0.021 0.122 0.107

5t 0 00 0.133 0.242 0.19 1 01 0.862 0.512 0.608 2 10 0.003 0.11 0.089 3 11 0.002 0.136 0.113

6t 0 00 0.018 0.21 0.218 1 01 0.9 0.496 0.526 2 10 0.065 0.14 0.125 3 11 0.016 0.154 0.131

Figure 14: Probability distribution for particle in step potential: we have used the qubits q[0], q[1] from ibmqx4 and q[7], q[8] from IBM Q 14 Melbourne for our experiment, No. of shots=8192.

41 Time Step ( t = 0.1 ) Particle Location IBM Q QASM Simulator IBM Q 14 Melbourne

1t 00 00 0.056 0.167 1 01 0.872 0.69 2 10 0.058 0.072 3 11 0.013 0.07

2t 0 00 0.015 0.136 1 01 0.887 0.617 2 10 0.011 0.077 3 11 0.088 0.17

3t 0 00 0.032 0.183 1 01 0.798 0.491 2 10 0.032 0.118 3 11 0.138 0.208

4t 0 00 0.044 0.179 1 01 0.624 0.393 2 10 0.04 0.145 3 11 0.292 0.283

5t 0 00 0.006 0.157 1 01 0.584 0.328 2 10 0.005 0.166 3 11 0.405 0.349

6t 0 00 0.058 0.188 1 01 0.387 0.257 2 10 0.061 0.19 3 11 0.494 0.365

Figure 15: Probability distribution for particle in double-well potential: we have used the qubits q[7] and q[8] from IBM Q 14 Melbourne for our experiment, No. of shots=8192.