^{1,a)}and D. S. Citrin

^{2,b)}

### Abstract

This paper describes the simulation of two coupled particles using the finite-difference time-domain method. We include both unrestricted spin and spatial degrees of freedom for the two particles within the Hartree-Fock approximation. Using spin as a basis, a two-qubit controlled-NOT gate as well as several single qubit gates are simulated in . We find that the double-occupancy problem can be largely circumvented.

The authors are grateful to the National Partnership for Advance Computational Infrastructure (NPACI) and the Texas Advanced Computing Center (TACC) for making available the computer resources used in the work.

I. INTRODUCTION

II. FORMULATION

III. LOCALIZED B FIELDS

IV. SIMULATION OF ONE- AND TWO-PARTICLE GATES

A. CNOT gate

B. Hadamard gate

C. Phase gate

D. Gate

V. NOTE ON COMPUTER RESOURCES

VI. SUMMARY

### Key Topics

- Tunneling
- 10.0
- Finite difference time domain calculations
- 9.0
- Quantum computing
- 4.0
- Quantum dots
- 4.0
- Qubits
- 4.0

## Figures

Cross-sectional view of the quartic potential. The two dots, A and B, resemble two-dimensional harmonic oscillator potentials with . Each dot will contain a particle. The interaction of the particles is controlled by the parameter .

Cross-sectional view of the quartic potential. The two dots, A and B, resemble two-dimensional harmonic oscillator potentials with . Each dot will contain a particle. The interaction of the particles is controlled by the parameter .

Location of the localized magnetic sources relative to the dots. The particle in dot B is to be manipulated without disturbing the particle in dot A. and are the primarily sources. and are there to nullify the effect on dot A.

Location of the localized magnetic sources relative to the dots. The particle in dot B is to be manipulated without disturbing the particle in dot A. and are the primarily sources. and are there to nullify the effect on dot A.

Relative strengths of the magnetic fields resulting from the localized sources in Fig. 2.

Relative strengths of the magnetic fields resulting from the localized sources in Fig. 2.

The Bloch sphere is a way of visualizing the state of a single qubit (Ref. 1). The expected value of the state of the qubit is calculated by the expected value of the operator .

The Bloch sphere is a way of visualizing the state of a single qubit (Ref. 1). The expected value of the state of the qubit is calculated by the expected value of the operator .

The time lapse of the wave functions during a CNOT transformation. The contour diagrams on the plane indicate the quartic potential of Fig. 1.

The time lapse of the wave functions during a CNOT transformation. The contour diagrams on the plane indicate the quartic potential of Fig. 1.

Graph of the important parameters during the CNOT transformation. Particle I is the control particle, which does not change during the transformation. Particle II, the target particle, begins in the logical 1 state, indicated by at . The first step is the application of the field, which turns II around the axis to . At this point, the parameter , which controls the gating between the two particles, is lowered, and we begin to see some exchange energy at about . Since the control particle is , it is only the lower half of particle II that feels the exchange energy, causing the angle of particle II to precess around the axis. When it reaches at , a pulse turns it around the axis to the direction, i.e., a logical 0.

Graph of the important parameters during the CNOT transformation. Particle I is the control particle, which does not change during the transformation. Particle II, the target particle, begins in the logical 1 state, indicated by at . The first step is the application of the field, which turns II around the axis to . At this point, the parameter , which controls the gating between the two particles, is lowered, and we begin to see some exchange energy at about . Since the control particle is , it is only the lower half of particle II that feels the exchange energy, causing the angle of particle II to precess around the axis. When it reaches at , a pulse turns it around the axis to the direction, i.e., a logical 0.

Graph of the tunneling between the two particles during the CNOT transformation. The top plot is the parameter. The second plot illustrates the position of the two particles relative to center. The last two plots show the tunneling of particle I to dot B and particle II to dot A, respectively.

Graph of the tunneling between the two particles during the CNOT transformation. The top plot is the parameter. The second plot illustrates the position of the two particles relative to center. The last two plots show the tunneling of particle I to dot B and particle II to dot A, respectively.

Graph of the important parameters during a CNOT transformation, but with the control particle at , i.e., logical 0. The process parallels the one in Fig. 6, but since II is spin up, only the upper part of particle II feels the exchange and precesses in the opposite direction. The result is, particle II ends in , a logical 1.

Graph of the important parameters during a CNOT transformation, but with the control particle at , i.e., logical 0. The process parallels the one in Fig. 6, but since II is spin up, only the upper part of particle II feels the exchange and precesses in the opposite direction. The result is, particle II ends in , a logical 1.

Time lapse of the wave functions during a one-particle Hadamard transformation. Notice that particle I is unaffected.

Time lapse of the wave functions during a one-particle Hadamard transformation. Notice that particle I is unaffected.

The Hadamard transformation. Only particle II is manipulated by the fields.

The Hadamard transformation. Only particle II is manipulated by the fields.

The phase transformation. Particle II is manipulated by the field. Particle I is unaffected because it is always up or down.

The phase transformation. Particle II is manipulated by the field. Particle I is unaffected because it is always up or down.

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