Many Worlds

1
Many Worlds Theory

• Originated with Dr Hugh Everett, III
• The theory implies that there are many copies of
  you in many different universes
• We are not aware of other copies because there
  can be no communication between the universes
• Supported by some of the leading investigators
  in the field of quantum computation David
  Deutsch, Stephen Hawking, Murray Gell-Mann, and
  Richard Feynman.
• Against Roger Penrose

2
Many Worlds Interpretation

• "Political scientist" L David Raub reports a
  poll of 72 of the "leading cosmologists and other
  quantum field theorists" about the "Many-Worlds
  Interpretation"
• The following response breakdown is given

3
Superpositioned State

• Consider a particle in its ground state in
  universe X
• Supplying an amount of energy at the correct
  frequency for a certain period of time T will
  excite the particle
• Supplying the energy at the correct frequency
  for only half of this period will put the
  particle in a superpositioned state
• The universe X will then split in to two
  universes in universe U1 the particle is excited
  and in universe U2 the particle is still in its
  ground state

4
Constructive or Destructive Interference

• Consider another route for the universe U1 to be
  created
• Universe Y can also split in to two universes,
  one of which is identical to U1
• The probability of U1 occurring must now be
  affected as there are two paths leading to this
  universe
• The amplitude of U1 is determined by the a
  mathematical combination of the amplitudes of X
  and Y and of the amplitudes of X leading to U1
  and Y leading U1
• The universes may well interfere destructively
  or constructively
• It is important to note that the universes must
  be absolutely identical for interference to occur
  

5
Decoherence

• Measure the particle in universe X
• In universe U1 you would see that the particle
  is excited and you and the particle would enter
  universe U3
• in U2 you would see that it is in the ground
  state and you and the particle enter universe U4
  
• By this process of measurement, the state of
  billions of particles in your brain have been
  affected
• The difference between U3 and U4 would not be
  one particle, as with the differences of U1 and
  U2, but billions of particles
• U3 and U4 will never interfere again
• This process is called decoherence

6
Qubits

• For n superpositioned particles in two possible
  states we get 2n different universes with every
  possible combination of the n particles values
  being observed
• A collection of bits is called a quantum
  register
• A quantum register of length n bits can hold up
  to 2n values simultaneously with each value
  observed in an otherwise identical universe

7
Quantum Computer

• The quantum register could be used as the input
  to some circuit
• The circuit will act simultaneously on these 2n
  different inputs, perform 2n different
  calculations and output 2n superpositioned
  results
• The trick is to get all of these universes to
  interfere with each other in such a way as to
  produce an output that is of some use to us
• Consider the situation where we have the
  functions in the 2n universes outputting a
  different value with equal probability. If we
  were to perform a measurement on the output value
  the systems would decohere and the value read
  would be a random value from the 2n outputs,
  which wouldn't really tell us very much
• What's required is to arrange for the universes
  to interfere with each other in such a way so
  that the output value(s) of interest have a much
  higher probability of being observed and,
  conversely, those values which are not of
  interest having a much smaller probability of
  being observed

8
Quantum Gates

• REVERSIBILITY
• Classical Boolean AND gate not reversible
• UNIVERSAL GATES - they can be used to create any
  logic circuit
• NAND gate in classical circuits
• Universal Toffoli gate
• Fredkin gate
• Controlled NOT gate

9
UNIVERSAL TOFFOLI GATE
The Universal Toffoli Gate has three inputs. The
first two inputs are copied to the first two
output pins and the third output is the Exclusive
OR of the third input and the AND of the first
two inputs, as shown in the above figure
10
FREDKIN GATE
The Fredkin gate has three inputs. The last two
inputs are swapped if the first input is 0 and
are left untouched otherwise, as shown in the
above figure
11
THE CONTROLLED NOT GATE
The Controlled NOT gate has two inputs. The
second input is negated only if the first input
is true as shown in the above figure.
12
SQUARE ROOT OF NOT GATE
A single square root of NOT gate produces a
completely random output with equal probabilities
of the output being 0 or 1. Two such gates
linked sequentially produce an output that is the
inverse of the input, and thus behave in the same
way as the classical NOT gate.
13
Two square root of NOT gates linked sequentially
Consider X 1gt
The 0gt part of the superposition will get
transformed in to
The 1gt portion of the superposition will get
transformed in to
The total state of the system is then
Note that the possibility of the output of the
system being 1gt is cancelled. Here you can see
the interference of the quantum universes
working.
14
Observations

• The universes can only interfere if they are
  identical in every regard except for this
  superpositioned particle
• Suppose that we were to place a detector at M to
  tell us the value of the output to gate X
• One copy of ourselves would note that the value
  is 0 while another copy in a different universe
  would note that the value is 1
• Thus the universes would be different by
  billions of particles and could never again
  interfere
• So just to know the value of the output of X
  invalidates the output of gate Y, even if our
  detector could measure the output of X accurately
  without disturbing the system

15
Reversible Computations

• The laws of physics are completely reversible.
  That is, from any physical process we can always
  deduce the inputs from the outputs.
• Without reversible gates the quantum system
  would radiate heat and the quantum interference
  which is essential for the correct operation of
  the system would stop working.
• It is not enough to simply have reversible gates
  - the entire computation must be reversible.
• We cannot copy or destroy values within the
  system without it decohering. The classical
  instructions XY and X0 lose information as
  the original value of X is obliterated by the
  instruction. Thus they are not reversible, and so
  could never be used in a quantum computation
  algorithm.
• It has been shown that any deterministic
  computation can be made reversible. C. Bennet
  (1973), Logical reversibility of computation, IBM
  J. Res. Develop., 17, pp. 525-532.