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Quantum Gates

Quantum computing offers a new means of

computation based on quantum mechanics.

One of the cornerstones of quantum computing is

the CNOT gate. It uses one particle to

control another. SPIN UP and SPIN DOWN take

the place of ON or OFF in conventional computing

Spin is an inherent property of all

particles. This is particularly important with

the class of particles call Fermions, which

includes electrons, because of the Pauli

Exculsion.

No two particles can occupy the same state.

E.g., in the infinite well, there could be two

particles in each of the eigenstates

In actuality, spin is not just up or

down. The spin can be in any direction.

Z

By convention, we say that spin up is

the positive Z direction and spin down is the

negative Z direction.

Y

X

Therefore, to account for spin, we can no longer

just use a complex function y. Instead, we use a

spinor, which contains a spin up part y1 and a

spin down part y2.

Remember, we already had a real and imaginary

part for each waveform y, so now we have four

coupled equations to allow for spin.

A quartic potential has two coupled

potentials. The shape is controlled by a

parameter a.

The electron in A is the CONTROL and the electron

in B is the TARGET.

- Remember
- Each particle has spin UP and spin DOWN waveform.
- Each waveform has a REAL and IMAGINARY part.

Therefore, the simulation of two particles

requires The simultaneous simulation of EIGHT

waveforms.

- The simulation of two particles is far more

complicated. There are two additional terms - Coulomb interaction
- Exchange interaction.

The exchange interaction is a purely

quantum mechanical phenomenon resulting from

the exclusion principle.

The exchange interaction is key to the

gating mechanism of the CNOT gate.

CONTROL

TARGET

If the CONTROL particle is spin DOWN while the

TARGET is in a superposition of spin UP and DOWN,

only the spin DOWN part of the TARGET will feel

the exchange interaction from the CONTROL. This

will cause the TARGET to precess.

For this example, we start with both particles

DOWN, which is logical 1.

A localized By pulse turns the TARGET along the X

direction.

The gating parameter a is lowered, allowing the

particles to come closer together so there is an

exchange interaction

The exchange interaction causes the TARGET to

precess until it is in the Y direction.

A localized Bx field turns the TARGET to spin

UP, a logical 0.

(No Transcript)

If the CONTROL particle had started in the spin

UP position (logical 0), the same process would

have left the TARGET spin DOWN, i.e., unchanged

in the logical 1 position.

In other words, if the TARGET particle

is inverted only if the CONTROL particle

starts in the logical 1 position.

The simulation two particles is more complicated

than just having two separate equations. They

are coupled via the Coulomb and exchange terms.

The Coulomb is just the classical repulsion of

particles will the same charge.

The exchange is a purely quantum mechanical force.

The Hartee-Fock Approximation

Either of the integrals is enough to swamp the

FDTD calculation

e.g., the calculation of the Coulomb force on

particle 1 due to particle 2, involves

integrating over the entire problem space for

each cell!

However, if I say

Then the integral takes the form of a convolution

And by taking the 3D Fourier transforms of both

x(r) and h(r), I can calculate the integral by

multiplication

This is still the most computationally intense

part of the program, but on a supercomputer, it

is possible.

Remember that my simulation method uses two

coupled equations of real variables.

Take the time derivative of the real part

Now insert the second equation into the first

This is just that original equation that

Schrödinger decided he had to split into two part

because it was too complicated!

No one understands quantum mechanics

Richard Feynman

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