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This code is part of Qiskit.","#","# (C) Copyright IBM 2017, 2020","#","# This code is licensed under the Apache License, Version 2.0. You may","# obtain a copy of this license in the LICENSE.txt file in the root directory","# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.","#","# Any modifications or derivative works of this code must retain this","# copyright notice, and modified files need to carry a notice indicating","# that they have been altered from the originals.","\"\"\"","Random symplectic operator functions","\"\"\"","","from __future__ import annotations","","import numpy as np","from numpy.random import default_rng","","from .clifford import Clifford","from .pauli import Pauli","from .pauli_list import PauliList","","","def random_pauli("," num_qubits: int, group_phase: bool = False, seed: int | np.random.Generator | None = None","):"," \"\"\"Return a random Pauli.",""," Args:"," num_qubits (int): the number of qubits."," group_phase (bool): Optional. If True generate random phase."," Otherwise the phase will be set so that the"," Pauli coefficient is +1 (default: False)."," seed (int or np.random.Generator): Optional. Set a fixed seed or"," generator for RNG.",""," Returns:"," Pauli: a random Pauli"," \"\"\""," if seed is None:"," rng = np.random.default_rng()"," elif isinstance(seed, np.random.Generator):"," rng = seed"," else:"," rng = default_rng(seed)"," z = rng.integers(2, size=num_qubits, dtype=bool)"," x = rng.integers(2, size=num_qubits, dtype=bool)"," phase = rng.integers(4) if group_phase else 0"," pauli = Pauli((z, x, phase))"," return pauli","","","def random_pauli_list("," num_qubits: int,"," size: int = 1,"," seed: int | np.random.Generator | None = None,"," phase: bool = True,","):"," \"\"\"Return a random PauliList.",""," Args:"," num_qubits (int): the number of qubits."," size (int): Optional. The length of the Pauli list (Default: 1)."," seed (int or np.random.Generator): Optional. Set a fixed seed or generator for RNG."," phase (bool): If True the Pauli phases are randomized, otherwise the phases are fixed to 0."," [Default: True]",""," Returns:"," PauliList: a random PauliList."," \"\"\""," if seed is None:"," rng = np.random.default_rng()"," elif isinstance(seed, np.random.Generator):"," rng = seed"," else:"," rng = default_rng(seed)",""," z = rng.integers(2, size=(size, num_qubits)).astype(bool)"," x = rng.integers(2, size=(size, num_qubits)).astype(bool)"," if phase:"," _phase = rng.integers(4, size=(size))"," return PauliList.from_symplectic(z, x, _phase)"," return PauliList.from_symplectic(z, x)","","","def random_clifford(num_qubits: int, seed: int | np.random.Generator | None = None):"," \"\"\"Return a random Clifford operator.",""," The Clifford is sampled using the method of Reference [1].",""," Args:"," num_qubits (int): the number of qubits for the Clifford"," seed (int or np.random.Generator): Optional. Set a fixed seed or"," generator for RNG.",""," Returns:"," Clifford: a random Clifford operator.",""," Reference:"," 1. S. Bravyi and D. Maslov, *Hadamard-free circuits expose the"," structure of the Clifford group*."," `arXiv:2003.09412 [quant-ph] `_"," \"\"\""," if seed is None:"," rng = np.random.default_rng()"," elif isinstance(seed, np.random.Generator):"," rng = seed"," else:"," rng = default_rng(seed)",""," had, perm = _sample_qmallows(num_qubits, rng)",""," gamma1 = np.diag(rng.integers(2, size=num_qubits, dtype=np.int8))"," gamma2 = np.diag(rng.integers(2, size=num_qubits, dtype=np.int8))"," delta1 = np.eye(num_qubits, dtype=np.int8)"," delta2 = delta1.copy()",""," _fill_tril(gamma1, rng, symmetric=True)"," _fill_tril(gamma2, rng, symmetric=True)"," _fill_tril(delta1, rng)"," _fill_tril(delta2, rng)",""," # For large num_qubits numpy.inv function called below can"," # return invalid output leading to a non-symplectic Clifford"," # being generated. This can be prevented by manually forcing"," # block inversion of the matrix."," block_inverse_threshold = 50",""," # Compute stabilizer table"," zero = np.zeros((num_qubits, num_qubits), dtype=np.int8)"," prod1 = np.matmul(gamma1, delta1) % 2"," prod2 = np.matmul(gamma2, delta2) % 2"," inv1 = _inverse_tril(delta1, block_inverse_threshold).transpose()"," inv2 = _inverse_tril(delta2, block_inverse_threshold).transpose()"," table1 = np.block([[delta1, zero], [prod1, inv1]])"," table2 = np.block([[delta2, zero], [prod2, inv2]])",""," # Apply qubit permutation"," table = table2[np.concatenate([perm, num_qubits + perm])]",""," # Apply layer of Hadamards"," inds = had * np.arange(1, num_qubits + 1)"," inds = inds[inds > 0] - 1"," lhs_inds = np.concatenate([inds, inds + num_qubits])"," rhs_inds = np.concatenate([inds + num_qubits, inds])"," table[lhs_inds, :] = table[rhs_inds, :]",""," # Apply table"," tableau = np.zeros((2 * num_qubits, 2 * num_qubits + 1), dtype=bool)"," tableau[:, :-1] = np.mod(np.matmul(table1, table), 2)",""," # Generate random phases"," tableau[:, -1] = rng.integers(2, size=2 * num_qubits)"," return Clifford(tableau, validate=False)","","","def _sample_qmallows(n, rng=None):"," \"\"\"Sample from the quantum Mallows distribution\"\"\"",""," if rng is None:"," rng = np.random.default_rng()",""," # Hadmard layer"," had = np.zeros(n, dtype=bool)",""," # Permutation layer"," perm = np.zeros(n, dtype=int)",""," inds = list(range(n))"," for i in range(n):"," m = n - i"," eps = 4 ** (-m)"," r = rng.uniform(0, 1)"," index = -int(np.ceil(np.log2(r + (1 - r) * eps)))"," had[i] = index < m"," if index < m:"," k = index"," else:"," k = 2 * m - index - 1"," perm[i] = inds[k]"," del inds[k]"," return had, perm","","","def _fill_tril(mat, rng, symmetric=False):"," \"\"\"Add symmetric random ints to off diagonals\"\"\""," dim = mat.shape[0]"," # Optimized for low dimensions"," if dim == 1:"," return",""," if dim <= 4:"," mat[1, 0] = rng.integers(2, dtype=np.int8)"," if symmetric:"," mat[0, 1] = mat[1, 0]"," if dim > 2:"," mat[2, 0] = rng.integers(2, dtype=np.int8)"," mat[2, 1] = rng.integers(2, dtype=np.int8)"," if symmetric:"," mat[0, 2] = mat[2, 0]"," mat[1, 2] = mat[2, 1]"," if dim > 3:"," mat[3, 0] = rng.integers(2, dtype=np.int8)"," mat[3, 1] = rng.integers(2, dtype=np.int8)"," mat[3, 2] = rng.integers(2, dtype=np.int8)"," if symmetric:"," mat[0, 3] = mat[3, 0]"," mat[1, 3] = mat[3, 1]"," mat[2, 3] = mat[3, 2]"," return",""," # Use numpy indices for larger dimensions"," rows, cols = np.tril_indices(dim, -1)"," vals = rng.integers(2, size=rows.size, dtype=np.int8)"," mat[(rows, cols)] = vals"," if symmetric:"," mat[(cols, rows)] = vals","","","def _inverse_tril(mat, block_inverse_threshold):"," \"\"\"Invert a lower-triangular matrix with unit diagonal.\"\"\""," # Optimized inversion function for low dimensions"," dim = mat.shape[0]",""," if dim <= 2:"," return mat",""," if dim <= 5:"," inv = mat.copy()"," inv[2, 0] = mat[2, 0] ^ (mat[1, 0] & mat[2, 1])"," if dim > 3:"," inv[3, 1] = mat[3, 1] ^ (mat[2, 1] & mat[3, 2])"," inv[3, 0] = mat[3, 0] ^ (mat[3, 2] & mat[2, 0]) ^ (mat[1, 0] & inv[3, 1])"," if dim > 4:"," inv[4, 2] = (mat[4, 2] ^ (mat[3, 2] & mat[4, 3])) & 1"," inv[4, 1] = mat[4, 1] ^ (mat[4, 3] & mat[3, 1]) ^ (mat[2, 1] & inv[4, 2])"," inv[4, 0] = ("," mat[4, 0]"," ^ (mat[1, 0] & inv[4, 1])"," ^ (mat[2, 0] & inv[4, 2])"," ^ (mat[3, 0] & mat[4, 3])"," )"," return inv % 2",""," # For higher dimensions we use Numpy's inverse function"," # however this function tends to fail and result in a non-symplectic"," # final matrix if n is too large."," if dim <= block_inverse_threshold:"," return np.linalg.inv(mat).astype(np.int8) % 2",""," # For very large matrices we divide the matrix into 4 blocks of"," # roughly equal size and use the analytic formula for the inverse"," # of a block lower-triangular matrix:"," # inv([[A, 0],[C, D]]) = [[inv(A), 0], [inv(D).C.inv(A), inv(D)]]"," # call the inverse function recursively to 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