# Wigner tomography

Wigner tomography is the process of extracting the Wigner function $W(\alpha)$ (a quasi-probabilistic distribution) of the quantum state of a cavity. The Wigner function is related to the density matrix of a state, therefore the tomography procedure will allow for the construction of the density matrix and a full reconstruction of the quantum state. Through a standard procedure one can encode the state of a qubit in a superposition of coherent states in a cavity. Therefore, using Wigner tomography it'll be possible to extract the full density matrix of the qubit.

The Wigner function is defined as follows $W(\alpha) = 2/\pi \langle P\rangle_\alpha$, where $P$ is the photon parity operator and $\alpha$ is the same parameter in the coherent states and represents a complex vector in the IQ plane.

Using a qubit coupled to the cavity it's straightforward to extract the photon parity of the cavity form a repeated measurement of the qubit through an additional readout resonator. The parity is related to the qubit state as such: $\langle P\rangle \propto P_e - P_g$m where $P_e$ and $P_g$ are the probabilities of finding the qubit in the excited and ground state respectively, which can be extracted with repeated measurement.

Notice: The example describes the tomography process assuming the cavity was encoded prior.

## Config#

The configuration consists of 4 quantum elements:

• cavity_I and cavity_Q define single input elements and are the I and Q components of the cavity that we'll perform the tomography on.
• qubit is the qubit that's coupled to the cavity
• rr is the readout resonator that's coupled to the qubit and used to read its state

Each element has its own IF and LO frequencies, and connection ports. Next, for each element we define the relevant operation and pulse:

• For cavity_I and cavity_Q we define the displace_pulse, which will be the real and imaginary parts of the displace pulse. These were separated due to a needed 2d parameter sweep over the amplitudes of the pulses for the tomography.
• For the qubit we define the x_pi/2_pulse which is simply a $\pi/2$ rotation around the x axis
• For the rr we define the readout_pulse - the pulse used for measuring the resonator.

The waveforms used for the displace_pulse and x_pi/2_pulse are Gaussians with different parameters. Generally to displace a cavity one needs to apply a pulse such that it integrates to the desired $\alpha$.

## Program#

We first calculate the revival time of the qubit coupled to the cavity. Then, we decide of the $\alpha$ range we want to sample for constructing the Wigner function, and the spacing. Once we defined the required parameters, we proceed to the QUA program.

We first define the QUA fixed variable for the amplitude scaling required to shift the cavity by the desired $\alpha$ We than create 2 QUA for_ loops to iterate over the points of the IQ grid. The inner-most for_ loop is for repeated measurement of the same point in the IQ plane.

Then, in each cycle we perform the tomography procedure:

• We align both cavity components in order to be played simultaneously. We displace the I and Q components by the real and imaginary parts of $\alpha$, respectively, this is done using realtime amplitude modulation, by multiplying the pulse with the function amp(x), where x is the scaling parameter.
• Next, we align the cavity with the qubit to ensure the pulses meant for the qubit start after reaching the desired coherent state for the cavity. On the qubit we apply a x_pi/2 operation to bring it to the equator, wait for the revival time, and eventually apply a second x_pi/2 operation to project the qubit to the excited or ground state.
• Finally, we measure the state using the readout resonator and demodulate the reflected signals to get the qubits state on the IQ plane which can then determine its state.
amp_displace = list(-alpha / np.sqrt(2 * np.pi) / 4)
amp_dis = declare(fixed, value=amp_displace)
with for_(r, 0, r < points, r + 1):
with for_(i, 0, i < points, i + 1):
with for_(n, 0, n < shots, n + 1):
align("cavity_I", "cavity_Q")
play("displace_I" * amp(amp_dis[r]), "cavity_I")
play("displace_Q" * amp(amp_dis[i]), "cavity_Q")
align("cavity_I", "cavity_Q", "qubit")
play("x_pi/2", "qubit")
wait(revival_time, "qubit")
play("x_pi/2", "qubit")
align("qubit", "rr")
measure(
"rr",
"raw",
demod.full("integW_cos", I1, "out1"),
demod.full("integW_sin", Q1, "out1"),
demod.full("integW_cos", I2, "out2"),
demod.full("integW_sin", Q2, "out2"),
)
assign(I, I1 + Q2)
assign(Q, -Q1 + I2)
save(I, "I")
save(Q, "Q")
wait(
10, "cavity_I", "cavity_Q", "qubit", "rr"
) # wait and let all elements relax

## Post processing#

Having the I,Q results of repeated measurement of the qubit for different $\alpha$ we can extract the parity of the cavity at each point by counting the excited and ground state measurements. We can display the results using a heatmap which represents the IQ plane, with the axes being the real and imaginary parts of $\alpha$.