Black, White, and Blackandwhite

Science has a way of advancing at parallel times in different spaces (i.e. the same technologies that have historically been invented by two separate people in different locales at the same time).

Quantum experiments in the past have used beams of light to explore the working priniciples, and more recently two or more groups have demonstrated teleportation of quantum states of massive particles.

So this isnt the first time in my life that I've become excited about the prospect of Quantum Computing given the black-box nature of Shrodinger's notion that 

*observing a system affects its outcome.

Since the most useful information derived from computing is the kind whose end state you can observe, I think of Quantum Computing as a complex rube-goldberg pendulum inside a box. You only get one result when you measure the system, but it's all the unknown paths that are of great value.

As far as we know, the most fundamental of physical laws are Quantum Laws. These five laws are inviolable and can be used to define quantum physics as it applies to Quantum Computing.

Quantum is a system like everything else.

To each physical system there corresponds a Hilbert space (1) of dimensionality equal to the system's maximum number of reliably distinguishable states (2).

A quantum state is a configuration of the system.

Each direction (ray) in the Hilbert space corresponds to a possible state of the system (3), with two states being reliably distinguishable if and only if their directions are orthogonal (inner product is zero).

A quantum state changes; it naturally wants to evolve, but it can always be undone.

Evolution of a closed system is a unitary (4) transformation on its Hilbert space.

Scaling - how parts make a whole.

The Hilbert space of a composite system is the tensor product of the Hilbert space of the parts (5).

Quantum measurements are probabilistic.

Each possible measurement (6) on a system corresponds to a resolution of its Hilbert space into orthogonal subspaces where . On state the result occurs with probability and the state after the measurement is . 

These five principles are the foundation for the whole quantum world.

 

Quantum Parallelism

The whole notion of parallel universes -- Sliding Doors, It's a Wonderful Life, Back to the Future, Groundhog Day -- may be somewhat improbable. But in no way is this implying that I'm dismissing quantum phenomena... quite the contrary. We have evidence of their physical existence, but Quantum Parallelism is different from Quantum Mechancics, which is linear. 

Although Quantum computers are built from wires and logic gates, just as classical computers are, the wonderful potential of such devices stems from the ability to manipulate superpositions of states.

Quantum algorithms solve problems which are not known to be solvable classically!

  • So, a superposition of inputs will give a superpostion of outputs!
  • Parallelism allows a computer to work on many computations simultaneously, which is what makes famous quantum algorithms, such as Shor's algorithm for factoring, or Grover's algorithm for searching. 
  • Simple quantum algorithm: Deutsch's algorithm

 

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Posted in Math, Quantum Computing

The Five Key Laws of Quantum Physics

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