Update gates.rst

[nquesada]

Fixes wrong sign of two-mode squeezing gate in docs

[codecov[bot]]

Codecov Report

Merging #371 into master will increase coverage by 0.00%. The diff coverage is n/a.

@@           Coverage Diff           @@
##           master     #371   +/-   ##
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  Coverage   97.69%   97.70%           
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  Files          52       52           
  Lines        6478     6485    +7     
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+ Hits         6329     6336    +7     
  Misses        149      149           

Impacted Files	Coverage Δ
strawberryfields/ops.py	99.00% <ø> (ø)
strawberryfields/api/connection.py	100.00% <0.00%> (ø)
strawberryfields/api/job.py	94.11% <0.00%> (+0.93%)	:arrow_up:

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[co9olguy]

So the one on the conventions page is the correct convention employed by SF? https://strawberryfields.readthedocs.io/en/stable/code/api/strawberryfields.ops.S2gate.html

[nquesada]

The correct convention is S^2(z) = \exp( z a_1^\dagger a_2^\dagger - H.c.) with z = r e^{i \theta}. To verify this is the case one can calculate <1,1|S^2(z)|0,0> = tanh(r) e^{i \theta} /cosh(r)

Here is a simple code that checks that:

eng = sf.LocalEngine(backend="fock",backend_options={"cutoff_dim": 6})
width = 2
circuit = sf.Program(width)
z = 0.5*np.exp(1j*np.pi/6)
r = np.abs(z)
theta = np.angle(z)
with circuit.context as q:
    ops.S2gate(r,theta)|(q[0],q[1])
x = eng.run(circuit)
state = x.state
state.ket()[1,1], np.tanh(r)*np.exp(1j*theta)/np.cosh(r)

giving

((0.3549095267938162+0.20490711083237245j),
 (0.35490952679381627+0.20490711083237248j))

as expected.

The one in https://strawberryfields.readthedocs.io/en/stable/code/api/strawberryfields.ops.S2gate.html is incorrect. A likely cause of confusion is that we define the S gate as S(z) = \exp( -z a^\dagger/2 -H.c.) and thus one would hope that S^(2)(z) is defined as \exp(-z a1^\dagger a^\dagger - H.c.). All of this in turn I think goes back to how Eqs. 31 and 33 of here https://arxiv.org/pdf/1110.3234.pdf are incosistent.

[co9olguy]

Thanks. Can you update the error also in sf.ops.S2gate docstring?

[nquesada]

No problem.