Symplectic moments #43
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nquesada
reviewed
Dec 4, 2025
Comment on lines
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| monte_carlo_integral = sum( | ||
| prod( | ||
| cse[i-1, j-1] | ||
| for i, j in zip(seq_i, seq_j) | ||
| ) | ||
| for cse in cse_gen | ||
| ) / sample_size | ||
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Contributor
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It would preferable to precalculate the values on separate notebook and harccode the answers here. I would suggest to not do Montecarlo inside the test.
nquesada
reviewed
Dec 4, 2025
haarpy/tests/test_symplectic.py
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| def generate_random_usp(d): | ||
| """Generate a Haar-random USp(2d) matrix of size (2d x 2d).""" | ||
| # Step 1: Random quaternionic d x d matrix with i.i.d. N(0,1) components | ||
| A = np.random.normal(size=(d, d)) | ||
| B = np.random.normal(size=(d, d)) | ||
| C = np.random.normal(size=(d, d)) | ||
| D = np.random.normal(size=(d, d)) | ||
| # Prepare list to store orthonormal quaternion columns (each as 4 real component arrays) | ||
| basis = [] | ||
| # Step 2: Quaternionic Gram-Schmidt | ||
| for j in range(d): | ||
| # j-th column as quaternion vector components | ||
| v0, v1, v2, v3 = A[:, j].copy(), B[:, j].copy(), C[:, j].copy(), D[:, j].copy() | ||
| # Make orthogonal to previous basis vectors | ||
| for (u0, u1, u2, u3) in basis: | ||
| # Compute quaternion inner product: q = <u, v> = sum_i conj(u_i)*v_i (a quaternion) | ||
| p0, p1, p2, p3 = u0, -u1, -u2, -u3 # components of conj(u) | ||
| q0 = p0 @ v0 - p1 @ v1 - p2 @ v2 - p3 @ v3 # (conj(u)·v)_real | ||
| q1 = p0 @ v1 + p1 @ v0 + p2 @ v3 - p3 @ v2 # (conj(u)·v)_i | ||
| q2 = p0 @ v2 - p1 @ v3 + p2 @ v0 + p3 @ v1 # (conj(u)·v)_j | ||
| q3 = p0 @ v3 + p1 @ v2 - p2 @ v1 + p3 @ v0 # (conj(u)·v)_k | ||
| # Subtract projection: v := v - u * q | ||
| t0 = u0 * q0 - u1 * q1 - u2 * q2 - u3 * q3 | ||
| t1 = u0 * q1 + u1 * q0 + u2 * q3 - u3 * q2 | ||
| t2 = u0 * q2 - u1 * q3 + u2 * q0 + u3 * q1 | ||
| t3 = u0 * q3 + u1 * q2 - u2 * q1 + u3 * q0 | ||
| v0 -= t0; v1 -= t1; v2 -= t2; v3 -= t3 | ||
| # Normalize v (so that <v,v> = 1) | ||
| norm = np.sqrt(v0@v0 + v1@v1 + v2@v2 + v3@v3) | ||
| v0 /= norm; v1 /= norm; v2 /= norm; v3 /= norm | ||
| # Step 3: Multiply by a random unit quaternion (random phase) | ||
| q0, q1, q2, q3 = random_unit_quaternion() | ||
| t0 = v0*q0 - v1*q1 - v2*q2 - v3*q3 | ||
| t1 = v0*q1 + v1*q0 + v2*q3 - v3*q2 | ||
| t2 = v0*q2 - v1*q3 + v2*q0 + v3*q1 | ||
| t3 = v0*q3 + v1*q2 - v2*q1 + v3*q0 | ||
| v0, v1, v2, v3 = t0, t1, t2, t3 | ||
| basis.append((v0, v1, v2, v3)) | ||
| # Step 4: Assemble the complex 2d x 2d matrix from quaternion basis | ||
| d2 = 2 * d | ||
| U = np.zeros((d2, d2), dtype=np.complex128) | ||
| # Fill block entries for each quaternion column | ||
| for j, (u0, u1, u2, u3) in enumerate(basis): | ||
| U[:d, j] = u0 + 1j*u1 # top-left block | ||
| U[:d, j+d] = u2 + 1j*u3 # top-right block | ||
| U[d:, j] = -u2 + 1j*u3 # bottom-left block | ||
| U[d:, j+d] = u0 - 1j*u1 # bottom-right block | ||
| return U | ||
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Contributor
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the code to generate symplectics should not be here. Instead we should separately Montecarlo teh answer and the just hardcode them here.
nquesada
approved these changes
Dec 4, 2025
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