Quantum Gates

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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
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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
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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
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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.
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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.
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• 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.
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• 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.
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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.
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For this example, we start with both particles
DOWN, which is logical 1.
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A localized By pulse turns the TARGET along the X
direction.
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The gating parameter a is lowered, allowing the
particles to come closer together so there is an
exchange interaction
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The exchange interaction causes the TARGET to
precess until it is in the Y direction.
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A localized Bx field turns the TARGET to spin
UP, a logical 0.
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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.
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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.
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The Hartee-Fock Approximation
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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!
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However, if I say
Then the integral takes the form of a convolution
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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.
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Remember that my simulation method uses two
coupled equations of real variables.
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Take the time derivative of the real part
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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!
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No one understands quantum mechanics
Richard Feynman