Implement Mixed Magnitude Approximation

[Acciaccatura]

hi! i tried to attempt a solution for mixed magnitude approximation (https://arxiv.org/pdf/2203.10064) for #1759 - though I am unsure how to test this implementation. it appears I need a diamond_norm_distance_to_unitary for ProbabilisticChannel implemented.

I've attempted to compute the bounds from Proposition 3.22 in the paper and have received favorable results - e.g.:

def mixed_magnitude_approx(
    ...
    # Error Calculation
    theta_under = config.arccos(abs(rx_under_rotation.to_matrix().numpy(config)[0, 0]))
    theta_over = config.arccos(abs(rx_over_rotation.to_matrix().numpy(config)[0, 0]))
    e1 = prob_channel.c1.diamond_norm_distance_to_unitary(
        rsad.rz(alpha, config)
        @ rsad.rx(theta_under * 2, config)
        @ rsad.rz(beta, config),
        config,
    ) * prob_channel.probability
    e2 = prob_channel.c2.diamond_norm_distance_to_unitary(
        rsad.rz(alpha, config)
        @ rsad.rx(theta_over * 2, config)
        @ rsad.rz(beta, config),
        config,
    ) * (1 - prob_channel.probability)
    e3 = rz_prob_approx.diamond_norm_distance_to_rz(-theta / 2, config)
    print(e1 + e2 + e3)
    ...

config = rs.with_dps(50)
us = stats.unitary_group(2).rvs(10, 0)
eps = 1e-3
for u in us:
    res = rs.mixed_magnitude_approx(u, eps, 50, config)

0.00063295290417069638736463688951349364676632387644143
0.00044495825518227126465986035869209125040344705482603
0.00068678532347280860266220444494659546893651156301564
0.00063227894577241044968894582320848073120030037333944
0.00044696254374123455166776821713373312007025357696207
0.00042205905072231409651741923166395286247539076098184
0.00037267871666233144221246383761310906871129709819125
0.0004292293819301834248569711465714788766709143123124
0.00054495476265531771051471509429865353542798466889997
0.00049815457660287823583789800426226054774201469469522

any advice? thanks for taking a look!

[gemini-code-assist]

Code Review

This pull request introduces the mixed_magnitude_approx protocol for approximating unitaries, along with its corresponding exports. The review feedback highlights opportunities to pass the relative_norm_solver parameter down to internal calls (mixed_diagonal_protocol and diagonal_unitary_approx) to respect custom solvers, and suggests avoiding redundant .to_matrix() conversions by passing UnitaryChannel objects directly to from_unitaries.

[NoureldinYosri]

thanks for your contribution, I will take a look asap

[NoureldinYosri]

at a first glance looks fine, but we need tests. there is a open ticket for implementing the dimanond norm distance #1755 can you help with that?

[NoureldinYosri]

take the error budget split as an optional parameter that defaults to this split if not given

[Acciaccatura]

ok! but i think the original paper (Proposition 3.22) mentions that the diamond distance is guaranteed to be $\leq 3\epsilon$. I set eps = eps / 3 to ensure by this guarantee, that the approximation will be at most eps away from the target unitary. if we were to allow the user to change this split, I am concerned the diamond distance may no longer be guaranteed to be less than eps. wdyt?

i added eps_split and a disclaimer in the docstring on this point.