PennyLane Challenge: Universality of single-qubit gates

PennyLane Challenges
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Use this topic to ask your questions about the PennyLane Challenge: Universality of single-qubit gates.

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I have implemented the error function as

def error(U, params):
    matrix = get_matrix(params)
    z = np.ravel(np.square(np.abs(U - matrix)))
    z = np.sum(qml.math.toarray(z))
    return z.item()

which is \displaystyle\sum_i\sum_j|U_{ij}-M_{ij}|^2.

With the given seed, the optimization passes the [[1, 0], [0, -1]] case rather easily, but fo test case [[0.70710678, 0.70710678], [0.70710678, -0.70710678]] with this error function the optimization does not converge fast enough.

I have also tried with taking max instead of sum but still, even though they are good enough to pass the [[1, 0], [0, -1]] case the other case still does not converge.

    matrix = get_matrix(list(params))
    z = qml.math.toarray(np.max(np.abs((np.ravel(U - matrix)))))
    return z.item()

Any hints would appreciated.

3

Hi @LdBeth ,

It’s much simpler than that. You don’t need to use np.ravel or many of the other things you’re using. You want U and matrix to be the same. What’s the simplest way that you can calculate a single number that tells you how different these two matrices are? The solution isn’t fancy but it works!

I hope this helps.

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I have to hand roll these because the most obvious to me np.linalg.norm does not work here,

Runtime Error: Failed to execute run() function: loop of ufunc does not support argument 0 of type ArrayBox which has no callable sqrt method

If norm doesn’t work, I guess I should go for trace the of differences?

I think I know it is using the autograd package to compute the derivative but unfortunately I have no clue on where I can learn what functions are supported and apparently not all numpy function are included, for example sqrt.

5

Thanks, and it turns out the fault is not cause by my error function, in fact \sum_i\sum_j|U_{ij}-M_{ij}|^2 is the correct way to go and my first error function is almost correct. (and it is certainly fine to use np.ravel.

The issue is with my get_matrix implementation, for some strange beliefs I wrote np.matmul(rz2, rx, rz1), although give a (2, 2) shaped array, which turns out is not the same as (rz2 @ rx @ rz1), after fixed this all tests are passed.

As a suggestion, maybe it would be better if additional checks to the get_matrix result is added

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I’m glad you solved it @LdBeth ! And thanks for the suggestion, I will share it with our team.

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Oh, I forget to check the private test, which failed without revealing the reason why it has failed, and I now I have really no clues. My implementation is attached below.

def get_matrix(params):
    alpha, beta, gamma, phi = params

    a, b = np.cos(beta / 2), -1j * np.sin(beta / 2)
    rx = np.matrix([[a, b], [b, a]])
    rz1 = np.matrix([[np.exp(-1j * alpha / 2), 0], [0, np.exp(1j * alpha / 2)]])
    rz2 = np.matrix([[np.exp(-1j * gamma / 2), 0], [0, np.exp(1j * gamma / 2)]])
    return np.exp(1j * phi) * (rz2 @ rx @ rz1)

def error(U, params):
    # Put your code here #
    # Return the error
    return sum(sum(qml.math.toarray(np.square(np.abs(U-matrix)))))
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Hi @LdBeth ,

For the get_matrix function you can use qml.RZ() and qml.RX() to create a unitary. Then you can use qml.matrix() to turn that unitary into a matrix.

For the error you don’t need to square it or turn it into an array.

1 Like
9

I see, thanks! So that is all because the matrices I manually computed has caused much worse autograd performance, which gives more than 0.01 difference in the optimized result and failed to converge for other tests.

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That could be it @LdBeth !