Qubit Gates¶
Included within Lightworks is a set of standard photonic circuits for implementing the gates required for qubit-based paradigms of quantum computing. There is a number of the common single and two qubit gates, these gates are intended to act on dual-rail encoded qubits. More information on the exact requirements of each gate can be found below and in the class docstrings for each gate.
Single Qubit Gates¶
The following single qubit gates are currently implemented:
Gate |
Matrix |
|---|---|
I |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}\end{split}\]
|
H |
\[\begin{split}\frac{1}{\sqrt{2}}\begin{bmatrix}
1 & 1 \\
1 & -1 \\
\end{bmatrix}\end{split}\]
|
X |
\[\begin{split}\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}\end{split}\]
|
Y |
\[\begin{split}\begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix}\end{split}\]
|
Z |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}\end{split}\]
|
Sx |
\[\begin{split}\frac{1}{2} \begin{bmatrix}
1+i & 1-i \\
1-i & 1+i \\
\end{bmatrix}\end{split}\]
|
S |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & i \\
\end{bmatrix}\end{split}\]
|
Sadj |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & -i \\
\end{bmatrix}\end{split}\]
|
T |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & \exp(i\pi/4) \\
\end{bmatrix}\end{split}\]
|
Tadj |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & \exp(-i\pi/4) \\
\end{bmatrix}\end{split}\]
|
P |
\[\begin{split}\begin{bmatrix}
1 & 0 \\
0 & \exp(i\theta) \\
\end{bmatrix}\end{split}\]
|
Rx |
\[\begin{split}\begin{bmatrix}
\cos(\theta/2) & -i\sin(\theta/2) \\
-i\sin(\theta/2) & \cos(\theta/2) \\
\end{bmatrix}\end{split}\]
|
Ry |
\[\begin{split}\begin{bmatrix}
\cos(\theta/2) & -\sin(\theta/2) \\
\sin(\theta/2) & \cos(\theta/2) \\
\end{bmatrix}\end{split}\]
|
Rz |
\[\begin{split}\begin{bmatrix}
\exp(-i\theta/2) & 0 \\
0 & \exp(i\theta/2) \\
\end{bmatrix}\end{split}\]
|
To create a new gate, it needs to be called from the qubit sub-module of Lightworks. For example, a Hadamard gate can be created with:
H = lw.qubit.H()
These gates are functionally PhotonicCircuits, and so can also be added and combined with other PhotonicCircuits in the same way. In the following, we create a 4 mode circuit to define two qubits (a & b) and then apply X, Y & Z gates to one qubit and H, S & T to the other.
circuit = lw.Circuit(4)
# X, Y, Z to qubit a
circuit.add(lw.qubit.X(), 0)
circuit.add(lw.qubit.Y(), 0)
circuit.add(lw.qubit.Z(), 0)
# H, S, T to qubit b
circuit.add(lw.qubit.H(), 2)
circuit.add(lw.qubit.S(), 2)
circuit.add(lw.qubit.T(), 2)
# View create circuit
circuit.display(mode_labels = ["a0", "a1", "b0", "b1"])
Two Qubit Gates¶
The majority of the two qubit gates included require post-selection and/or heralding to function correctly, as well as some additional modes. The exact layout of the modes and requirements can be found in the docstrings for the chosen gate, but are also summarized in the table below. In this table, the qubit modes are also specified, where c0 and c1 are the 0 & 1 states of the control qubit respectively and t0 & t1 are the 0 & 1 states of the target qubit. The qubit gates utilise heralds within the circuit, so these do not need to be accounted for as part of the simulation objects. In some cases, some additional post-selection is required however, this is noted below.
Gate |
Qubit Modes |
Success Probability |
Post-selection/Heralding |
|---|---|---|---|
CZ |
c_0 : 0
c_1 : 1
t_0 : 2
t_1 : 3
|
1/9 |
Requires heralding and need to post-select on only measuring one photon across each of the qubit modes. |
CNOT |
c_0 : 0
c_1 : 1
t_0 : 2
t_1 : 3
|
1/9 |
Requires heralding and need to post-select on only measuring one photon across each of the qubit modes. |
CZ_Heralded |
c_0 : 0
c_1 : 1
t_0 : 2
t_1 : 3
|
1/16 |
Requires heralding but not post-selection. |
CNOT_Heralded |
c_0 : 0
c_1 : 1
t_0 : 2
t_1 : 3
|
1/16 |
Requires heralding but not post-selection. |
SWAP |
User selectable
|
1 |
N/A |
The two qubit gates can then be created in the same way as the single qubit gates. We can directly use these gates with all of the simulation objects provided within the emulator. As an example, below the heralded CNOT gate is tested with the sampler. The input \(\ket{1,0}\) (which translates to \(\ket{0,1,1,0}\) in mode language) is chosen.
# Define cnot and input
cnot = lw.qubit.CNOT_Heralded()
# c0 c1 t0 t1
input_state = lw.State([0, 1, 1, 0])
# Then sample 10,000 times
sampler = emulator.Sampler(cnot, input_state, 10000, seed = 8)
results = Backend("slos").run(sampler)
# View measured counts
print(results)
# {State(|0,1,0,1>): 615}
As expected, with the correct heralding we only measure the output state \(\ket{0,1,0,1}\), which corresponds to the qubit state \(\ket{1,1}\), demonstrating that the CNOT works as expected. Despite inputting to the system 10,000 times we only measure 615 outputs that meet the heralding conditions, this is because the heralded CNOT only has a success probability of 1/16 (= 0.0625, 615/10000 = 0.0615).
Warning
Care needs to be taken when cascading two qubit gates to ensure that any post-selection and heralding criteria can still be maintained and information on this is not lost.
Three Qubit Gates¶
There is also a number of three qubit gates included within Lightworks, these are summarized in the table below:
Gate |
Qubit Modes |
Success Probability |
Post-selection/Heralding |
|---|---|---|---|
CCZ |
ca_0 : 0
ca_1 : 1
cb_0 : 2
cb_1 : 3
t_0 : 4
t_1 : 5
|
1/72 |
Requires heralding and need to post-select on only measuring one photon across each of the qubit modes. |
CCNOT |
ca_0 : 0
ca_1 : 1
cb_0 : 2
cb_1 : 3
t_0 : 4
t_1 : 5
|
1/72 |
Requires heralding and need to post-select on only measuring one photon across each of the qubit modes. |