`TDMprogram` requires `timebins % concurrent_modes == 0`

[fab1an-q]

Issue description

The TDMProgram will produce wrong results if the number of timebins is not an integer multiple of the number of concurrent modes. This is because the tdmprogram.unroll() function simpy multiplies the unique circuit with the number of shots, not taking into account which qumode has been measured most recently.

To illustrate the bug, let's consider this TDM program:

theta = [0] * 3
r = 3
shots = 2

prog = sf.TDMProgram(N=2)
with prog.context(theta) as (p, q):
    ops.Xgate(50) | q[1]
    ops.MeasureHomodyne(p[0]) | q[0]
eng = sf.Engine("gaussian")
res = eng.run(prog, shots=shots)
samples = res.samples

print(samples)

• Expected behavior: In average, you would expect samples to look like

  [[[ 0 50 50]]
  [[50 50 50]]]

  And, indeed, for an equivalent program with four timebins (theta = [0] * 4) we get, as expected,

  [[[ 1.159638 48.51887915 50.96767511 50.38571056]]
  
  [[50.02626644 51.60097292 49.01545834 49.81480916]]]

• Actual behavior: However, what we get for three timebins (theta = [0] * 3) is:

  [[[ -0.22303699  51.48148974  48.93924123]]
  
  [[  1.09462711 101.65679385  49.80258551]]]

  Note that the second mode of the second shot is ~100 as it has undergone the displacement gate twice. This happens because self.circuit = self.unrolled_circuit * shots multiplies the unique circuit with the number of shots where each shot starts with the same qumode index. This can be illustrated by printing the qumode index whenever a gate is applied. The above circuit with two shots yields:

  
  displace mode 1  # shot 0, timebin 0
  measure mode  0  # shot 0, timebin 0
  displace mode 0  # shot 0, timebin 1
  measure mode  1  # shot 0, timebin 1
  displace mode 1  # shot 0, timebin 2
  measure mode  0  # shot 0, timebin 2

displace mode 1 # shot 1, timebin 0 -> the displacement is applied to mode 1 although the most recently measured mode is 0 measure mode 0 # shot 1, timebin 0 displace mode 0 # shot 1, timebin 1 measure mode 1 # shot 1, timebin 1 displace mode 1 # shot 1, timebin 2 measure mode 0 # shot 1, timebin 2

The second shot starts with displacement of qumode 1. But qumode 1 is the mode that just _has_ been displaced at the end of shot 0! This issue will always occur when the number of timebins is not an integer multiple of the number of concurrent modes.

It took me pretty long to figure this out, and I am available for a chat and/or further demonstrations. I thought I could fix it myself by applying an additional qumode shift at the end of each shot, but I didn't succeed.

* *Reproduces how often:* Always