Qubit basis states

Single-qubit computational basis states

The two orthogonal z-basis states of a qubit are defined as:

  • 0\vert 0\rangle
  • 1\vert 1\rangle

When we talk about the qubit basis states we implicitly refer to the z-basis states as the computational basis states.

The two orthogonal y-basis states are:

  • R=0+ı12\vert R\rangle =\frac{\vert 0 \rangle + \imath \vert 1 \rangle}{\sqrt{2}}
  • 10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
  • 50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,

35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"/>0+ı1

  • L=0ı12\vert L\rangle =\frac{\vert 0 \rangle - \imath \vert 1 \rangle}{\sqrt{2}}
  • 10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
  • 50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,

35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"/>0ı1

The two orthogonal x-basis states are:

  • +=0+12\vert +\rangle =\frac{\vert 0 \rangle + \vert 1 \rangle}{\sqrt{2}}
  • 10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
  • 50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,

35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"/>0+1

  • =012\vert -\rangle =\frac{\vert 0 \rangle - \vert 1 \rangle}{\sqrt{2}}
  • 10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
  • 50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,

35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"/>01

The basis states are located at opposite points on the Bloch sphere:

Bloch sphere courtesy of http://www.laborsciencenetwork.com

Multi-qubit computational basis states

A single-qubit has two computational basis states. In the z-basis these are 0\vert 0 \rangle and 1\vert 1 \rangle. A two-qubit system has 4 computational basis states denoted as 00\vert 00 \rangle, 01\vert 01 \rangle, 10\vert 10 \rangle, 11\vert 11 \rangle.

A multi-qubit system of N qubits has 2N2 ^{N} computational basis states denoted as 00...00\vert 00...00 \rangle, 0001\vert 00 \cdots 01 \rangle, 0010\vert 00 \cdots 10 \rangle ... 1111\vert 11 \cdots 11 \rangle.

Probability amplitudes

Associated with each computational basis state is a probability amplitude αi\alpha_{i}, which is a complex number.

As an example, a system of three qubits is described by the expression:

Ψ=α0000+α1001+α2010++α7111\lvert \Psi \rangle = \alpha_{0} \lvert 000 \rangle + \alpha_{1} \lvert 001 \rangle + \alpha_{2} \lvert 010 \rangle + \cdots + \alpha_{7} \lvert 111 \rangle

where αi\alpha_{i} are the probability amplitudes associated to the computational basis states.

Initialization and measurement bases

By default, all qubits are initialized in the 0|0\rangle state in the z-basis.

State initialization in a specific basis can be done explicitly with the cQASM instructions prep_z, prep_y and prep_x, which prepare qubits in the 0\vert 0 \rangle, R\vert R \rangle and +\vert + \rangle states respectively.

By default, qubits are measured with the measure or measure_all instruction in the z-basis.
Qubit measurement in a specific basis can be done explicitly with the cQASM instructions measure_x, measure_y and measure_z.

Declared states

  • When a qubit is in the 0\vert 0 \rangle state (1\vert 1 \rangle state), a measurement in the z-basis will result in 0 (1)
  • When a qubit is in the R\vert R \rangle state (L\vert L \rangle state), a measurement in the y-basis will result in 0 (1)
  • When a qubit is in the +\vert + \rangle state (\vert - \rangle state), a measurement in the x-basis will result in 0 (1)

Notes

R\vert R \rangle and L\vert L \rangle stand for Right and Left. Other notations that are often used for these states are ı\vert \imath \rangle and ı\vert - \imath \rangle.