Qubit

   Qubit is a [1]quantum computing equivalent of a [2]bit. While bits in
   classical computers can have one of two state -- either 0 or 1 -- a qubit
   can additionally have infinitely many states "in between" 0 and 1 (so
   called [3]superposition). Physically qubits can be realized thanks to
   quantum states of particles, e.g. the polarization of a photon or the spin
   of a photon. Qubits are processed with [4]quantum gates.

   Whenever we measure a qubit, we get either 1 or 0, just like with a normal
   bit. However during quantum computations the internal state of a qubit is
   more complex. This state determines the probabilities of measuring either
   1 or 0. When the measurement is performed (which is basically any
   observation of its state), the qubit state collapses into one of those two
   states.

   Now we will be dealing with so called pure states -- these are the states
   that can be expressed by the following representation. We will get to the
   more complex (mixed) states later.

   The state of a qubit can be written as

   A * |0> + B * |1>

   where A and B are [5]complex numbers such that A^2 + B^2 = 1, |0> is a
   vector [0, 1] and |1> is a vector [1, 0]. A^2 gives the probability of
   measuring the qubit in the state 0, B^2 gives the probability of measuring
   1.

   The vectors |0> and |1> use so called bra-ket notation and represent a
   vector basis of a two dimensional state. So the qubit space is a point in
   a space with two axes, but since A and B are complex, the whole space is
   four dimensional (there are 4 variables: A real, A imaginary, B real and B
   imaginary). However, since A + B must be equal to 1 ([6]normalized), the
   point cannot be anywhere in this space. Using logic^TM we can figure out
   that the final state of a qubit really IS a point in two dimensions: a
   point on a sphere (Bloch sphere). A point of the sphere can be specified
   with two coordinates: phase ([7]yaw, 0 to 2 [8]pi, can be computed from
   many repeated measurements) and p ([9]pitch, says he probability of
   measuring 1). It holds that:

   A = sqrt(1 - p)

   B = e^(i * phase) * sqrt(p)

   The sphere has the state |0> at the top (north pole) and |1> at the bottom
   (south pole); these are the only points a normal bit can occupy. The
   equator is an area of states where the probability of measuring 0 and 1
   are equal (above the equator gives a higher probability to 0, below the
   equator to 1).

   Now a qubit may actually be in a more complex state than the pure states
   we've been dealing with until now. Pure states can be expressed with the
   state vector described above. Such a state is achieved when we start with
   a qubit of known value, e.g. if we cool down the qubit, we know it has the
   value |0>, and transforming this state with quantum gates keep the state
   pure. However there are also so called mixed states which are more complex
   and appear e.g. when the qubit may have randomly been modified by an
   external event, or if we start with a qubit of unknown state. Imagine if
   we e.g. start with a qubit that we known is either |0> or |1>. In such
   case we have to consider all those states separately. A mixed state is
   composed of multiple pure states. Mixed states can be expressed with so
   called density matrices, an alternative state representation which is able
   to encode these states.

Links:
1. quantum.md
2. bit.md
3. superposition.md
4. quantum_gate.md
5. complex_number.md
6. normalization.md
7. yaw.md
8. pi.md
9. pitch.md