Simulate quantum states, both pure and mixed, in a Bloch-sphere with various quantum noise
channels and gates.
The effects of the various components are shown in real time on the right hand side in the
Bloch-spheres.
Two Bloch-spheres are shown here: one at the top and one at the bottom. The top sphere
represents the Bloch-sphere without any influences from settings in the tabs at the bottom
left. These changes will be made visible in the bottom sphere. The top sphere can be rotated
by the user, the bottom sphere will move with it.
Computing the vector in the Bloch-sphere that represents a density matrix is done with the
following formula:
ρ = 1/2 (I + a⋅ σ)
Here a is the Bloch vector of the system and I is the identity. σ represents the three
Hermitian, traceless Pauli matrices, also known as the matrices used as the X-, Y- and
Z-gate. The Bloch vector can be retrieved by computing the trace of the multication of these
Pauli-matrices with the density matrix. The resulting numbers will be the coordinates of the
vector.
In the top left panel up to 4 states can constructed and added to the Bloch-spheres. The
states can be constructed with two methods:
The first one is specifying the polor coordinates the state vector will have in the bloch
sphere. As the length of the vector can not be specified, this will always be 1, so these
states will always be pure states. From the coordinates the wave function can be determined
using the following formula: |Ψ> = cos(θ/2)|0> + sin(θ/2) eiφ
The density matrix can then be constructed by computing |Ψ><Ψ|
The second way is to directly fill in values in the density matrix. If the user chooses
this method, the wave function and polar coordinates will not be updated. This is
because not all density matrices can be represented this way. If the density matrix that
is filled in is a mixed state, it is impossible to find the wave function.
In the noise tab various parameters can be set to visualize how noise changes the shape of
the bloch sphere, and thus how it changes the pure states, which are on the surface of the
sphere. The provided matrices will act on the density matrix according to the following
formula: E1ρ
E1†+E2ρE2†
There are 4 predetermined noise types available: Depolarizing, Dephasing (X or Z) and
Amplitude Damping. The corresponding noise matrices and formula for the chosen type of noise
will be shown upon selection. For these 4 noise types the amount of noise can be set with
the slider at the bottom of the tab. r = 1 means no noise (matrix E1 will be the
identiy and E2 will be zero) and r = 0 means maximum noise.
It is also possible for the user to set its own noise matrices. In this case everything is
read and computed directly from the matrices the user provides, so the slider with amount of
noise 'r' will no longer be of influence. The provided noise matrices have to satisfy the
condition: ∑ Ei Ei† = I, with I the identity
matrix. If this condition is not met, the transformed Bloch-sphere will not be computed.
In this tab the user can create a mixed state. The components of the mixed state are the
existing states the user created in the top panel. The sliders next to each state show the
probability that the state will be created. The active box has to be ticked to include a
state with a probability. The sum of all probabilities always adds up to 1, so when
including more than two states it becomes difficult for the user to control the sliders to
construct the right distribution. For this the 'Lock probability' checkbox was constructed.
When ticked, the probability for that state will not change anymore and the other active
states will make sure the total probability adds up to 1.
The generated mixed state is shown as a white arrow in the bloch sphere. Mixed state Bloch
vectors have a lenght smaller than 1, so they will end in the Bloch sphere and not on the
surface. The transformations in the other tabs will also affect the mixed state.
In the last tab unitary tranformations can be applied to the states. All the states will be
multiplied by the unitary in the following way: U ρ U†
There are 5 predetermined unitaries or gates available in the dropdown menu. The first three
are the Pauli-gates, X, Y, Z. On top of that the Hadamard gate and a phase shift gate can be
chosen. For the latter one the user has to specify the phase shift in radians. By choosing
'user defined unitary' the user can decide and fill in any valid unitary.
The other option where user input is needed is the Evolving State. In this case, the unitary
is determined by a Hamiltonian and time. The unitary which is applied to the density matrix
is time-dependent and can be written as follows: U(t) = eitH, where H is the
hamiltionian. A slider at the bottom of the tab is provided to see how the states evolve
with different time settings to construct an new Unitary.
