Unifying the Clifford hierarchy via symmetric matrices over rings
Unifying the Clifford hierarchy via symmetric matrices over rings
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DOI：
10.1103/physreva.100.022304
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发表时间：
2019-02
影响因子：
2.9
通讯作者：
Narayanan Rengaswamy;Robert Calderbank;H. Pfister
中科院分区：
文献类型：
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作者：
Narayanan Rengaswamy;Robert Calderbank;H. Pfister
克利福德层次结构是通用量子计算（UQC）的基础概念。介绍的是表明可以通过量子传送来实现UQC，从而可以访问某些标准资源。虽然层次结构的完整结构仍不清楚，但Cui等人。 （Arxiv：1608.06596）最近描述了层次结构中对角线单位的结构。他们考虑了对角门，其在计算基础上的作用是由$ 2^k $的统一根描述的，该统一的根源提高到国家的多项式函数，并确定了层次结构中此类单位的水平。对于量子系统，我们考虑$ k $ -th级的对角门门，这些门可以通过戒指$ \ mathbb {z} _ {2^k} $ of Integers mod mod mod $ 2^k $描述。这些涉及$ \ Mathbb {z} _ {2^k} $上的对称矩阵，可用于有效地描述层次结构中的所有$ 2 $ - 局部和某些较高的位置对角门。我们还对其对Pauli矩阵的作用进行了明确的代数描述，该矩阵为对角门的对角门建立了自然递归。这涉及$ \ mathbb {z} _ {2^k} $上的符号矩阵，因此我们的观点将这些门与Clifford Gates的二进制符号框架统一。我们通过简单的示例来增强描述某些标准门。 In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits.我们的结果表明，可以通过将其他二元符号矩阵概括为整数环，可以理解一些非对角门。
The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (arXiv:1608.06596) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal gates whose action on a computational basis qudit state is described by a $2^k$-th root of unity raised to a polynomial function of the state, and they established the level of such unitaries in the hierarchy. For qubit systems, we consider $k$-th level diagonal gates that can be described just by quadratic forms of the state over the ring $\mathbb{Z}_{2^k}$ of integers mod $2^k$. These involve symmetric matrices over $\mathbb{Z}_{2^k}$ that can be used to efficiently describe all $2$-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal gates from lower levels. This involves symplectic matrices over $\mathbb{Z}_{2^k}$ and hence our perspective unifies these gates with the binary symplectic framework for Clifford gates. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates might be understood by generalizing other binary symplectic matrices to integer rings.