Random Symplectic

Python library for generating uniformly random d-nary symplectic matrices for prime $d$ . These matrices form a representation of the $n$ -qudit Clifford group and can be decomposed into elementary 1- and 2-qudit gates (Hostens, 2005). Uniform sampling from the Clifford group is a critical subroutine to randomized benchmarking (Magesan, 2012).

The algorithm implements a mapping from the integers to $\text{Sp}(2n, \mathbb{Z}_d)$ , reducing sampling from the symplectic group to sampling an integer, and is an extension of the qubit-based method presented by Koenig and Smolin (2014). A manuscript detailing the algorithm is forthcoming.

Installation

To install the package for Python 3.8+, do

pip install random-symplectic

Getting Started

The package implements the DnaryArray class, a subclass of numpy's ndarray for arrays over the integers mod $d$ . Users can specify $d$ using the classmethod DnaryArray.set_d, which will return a subclass of DnaryArray specialized to the given modulus.

>>> from randomsymplectic import DnaryArray
>>> D3 = DnaryArray.set_d(3)
>>> D3([1, 2, 3, 4])
DnaryArray(d=3)([1, 2, 0, 1])

In particular, a specialized subclass exposes a plethora of useful methods for doing symplectic algebra over the group of d-nary arrays, as well as generating symplectic matrices.

Random Symplectic Matrices

DnaryArray has two classmethods for generating uniformly random symplecitc matrices: DnaryArray.random_symplectic and DnaryArray.from_index. The former operates by generating random d-nary vectors of lengths building up to the size of the matrix to be generated. The latter deterministically generates each of those vectors from the provided index.

>>> from randomsymplectic import DnaryArray
>>> D3 = DnaryArray.set_d(3)
>>> D3.from_index(0, n=1)
DnaryArray(d=3)([[1, 0],
                 [0, 1]])
>>> D3.from_index(12152, 3)
DnaryArray(d=3)([[1, 0, 1, 2, 1, 1],
                 [0, 1, 0, 2, 1, 2],
                 [2, 0, 0, 1, 1, 0],
                 [0, 0, 2, 0, 1, 0],
                 [0, 0, 0, 0, 1, 0],
                 [2, 0, 2, 0, 2, 0]])

References

E. Hostens, J. Dehaene, and B. De Moor, Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic, Phys. Rev. A 71, 042315 (2005), arXiv:quant-ph/0408190.
E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012), arXiv:1109.6887v2.
R. Koenig and J. A. Smolin, How to efficiently select an arbitrary Clifford group element, Journal of Mathematical Physics 55, 122202 (2014), arXiv:1406.2170.