Custom Gate

 

Wanted to get some information about the custom gates that enable continuous

evolution in the Quirk Teleportation Protocol demo. Good news from Merlin

who bumped me up to Gemini pro:

Could you the explain the math behind this gate. It is from: Quirk: 5 wires, 22 ops, H•X………………messageBloch•X… (algassert.com)

MessageG.PNG

algassert.com

The gate you're asking about, labeled "message Gate [871j]", appears to be a custom gate created in Quirk, a drag-and-drop quantum circuit simulator.   This simulator allows you to create custom gates, which can be represented by a circuit diagram and a corresponding unitary matrix.  Without more information about the specific construction of "message Gate [871j]" within the Quirk environment, it's impossible to provide the exact mathematical representation.

However, I can explain the math behind the individual components shown in the circuit diagram:

Yt Gate: This likely represents a rotation around the Y-axis of the Bloch sphere by an angle of 't'. The general form for such a gate is:

text

RY(t) = [[cos(t/2), -sin(t/2)],

         [sin(t/2), cos(t/2)]]

e^(-iYt) Gate: This notation suggests an evolution under a Hamiltonian proportional to the Pauli-Y operator. The general form for such an evolution is:

text

exp(-iYt) = cos(t/2) * I - i * sin(t/2) * Y 

          = [[cos(t/2), -sin(t/2)],

             [sin(t/2), cos(t/2)]]

Where 'I' is the identity matrix and 'Y' is the Pauli-Y matrix:

text

Y = [[0, -i],

     [i, 0]]

X Gate: This is the Pauli-X gate, also known as the NOT gate. It flips the state of a qubit:

text

X = [[0, 1],

     [1, 0]]

To get the overall mathematical representation of the "message Gate [871j]", you would need to:

Determine the specific value of 't'.

Multiply the matrices of the individual gates in the correct order. Since Quirk operates from left to right, you would multiply the matrices in reverse order from right to left.

For example, if 't' was π/2, the math would look like this:

message Gate [871j] = exp(-i(π/2)Y) * X * RY(π/2)

You would then substitute the matrices for each gate and perform the matrix multiplication to obtain the final 2x2 unitary matrix representing the "message Gate [871j]". 

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