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mindquantum.algorithm.library.qudit_symmetric_decoding

mindquantum.algorithm.library.qudit_symmetric_decoding(qubit: np.ndarray, n_qubits: int = 1)

Qudit symmetric decoding, decodes a qubit symmetric state or matrix into a qudit state or matrix.

The input qubit state/matrix must preserve the symmetry required by the qudit-qubit mapping. For example, in a qutrit(d=3) to two-qubit mapping:

$$\begin{array}{r}\begin{array}{rl}|00\cdots 00⟩& \to |0⟩\\ \frac{|0\cdots 01⟩+|0\cdots 010⟩+|10\cdots 0⟩}{\sqrt{d-1}}& \to |1⟩\\ \frac{|0\cdots 011⟩+|0\cdots 0101⟩+|110\cdots 0⟩}{\sqrt{d-1}}& \to |2⟩\\ ⋮& ⋮\\ |11\cdots 11⟩& \to |d-1⟩\end{array}\end{array}$$

The symmetry requires that states in the same symmetric subspace must have equal amplitudes. For example, states ¦01⟩ and ¦10⟩ belong to the same symmetric subspace and must have equal amplitudes.

Parameters

• qubit (np.ndarray) – the qubit symmetric state or matrix that needs to be decoded, where the qubit state or matrix must preserve symmetry.

• n_qubits (int) – the number of qubits in the qubit symmetric state or matrix. Default: 1.

Returns

np.ndarray, the qudit state or matrix obtained after the qudit symmetric decoding.

Raises

ValueError – If the input qubit state/matrix does not preserve the required symmetry.

Examples

>>> import numpy as np
>>> from mindquantum.algorithm.library.qudit_mapping import qudit_symmetric_decoding
>>> # A symmetric qubit state where amplitudes in |01⟩ and |10⟩ are equal
>>> qubit = np.array([1., 2., 2., 3.])
>>> qubit /= np.linalg.norm(qubit)
>>> print(qubit)
[0.23570226 0.47140452 0.47140452 0.70710678]
>>> print(qudit_symmetric_decoding(qubit))
[0.23570226+0.j 0.66666667+0.j 0.70710678+0.j]