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arXiv:2209.04799 · Annalen der Physik

Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates

Gui‐Long Jiang, Hai‐Rui Wei, Guo‐Zhu Song, Ming Hua

Published Sep 10, 2022• 0 citations
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Abstract

Abstract Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, it is shown generalized controlled‐X (GCX) gates, 6 single‐qubit rotations about the y‐ and z‐axes, and single‐partite y‐ and z‐rotation‐types which are defined in this paper are sufficient to simulate a controlled‐unitary gate with controlling on . In the scenario of the unitary gate with that is locally equivalent to a diagonal unitary on , GCX gates and single‐partite y‐ and z‐rotation‐types are required to simulate it. The quantum circuit for implementing and are presented. Furthermore, it is found that with controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.

DOI
10.1002/andp.202200317

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Abstract We characterize three-qubit controlled unitary gates of Schmidt rank three, establishing necessary and sufficient conditions for such gates to have Schmidt rank three. We construct explicit examples and analyze their entanglement capabilities, showing that gates with Schmidt rank three can generate output states ranging from fully separable to maximal GHZ-class states. Within this classification, we present a parameterized gate family that produces an output W-class state whose bipartite tangles for subsystems A B and A C are modulated as simple trigonometric functions of a single phase parameter. The examples also include gates implementable with only three CNOT gates, showing that this lower bound is achievable. Decompositions that achieve the minimum possible CNOT count for several other cases are provided as well. Our results bridge Schmidt rank classification, entanglement structure, and resource-efficient circuit synthesis.
We characterize three-qubit controlled unitary gates of Schmidt rank three, establishing necessary and sufficient conditions for such gates to have Schmidt rank three. We construct explicit examples and analyze their entanglement capabilities, showing that gates with Schmidt rank three can generate output states ranging from fully separable to maximal GHZ-class states. Within this classification, we present a parameterized gate family that produces an output W-class state whose bipartite tangles for subsystems AB and AC are modulated as simple trigonometric functions of a single phase parameter. The examples also include gates implementable with only three CNOT gates, showing that this lower bound is achievable. Decompositions that achieve the minimum possible CNOT count for several other cases are provided as well. Our results bridge Schmidt rank classification, entanglement structure, and resource-efficient circuit synthesis.
We show that any unitary operator on the ${d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B}$ system $({d}_{A}\ensuremath{\ge}2)$ can be decomposed into the product of at most $4{d}_{A}\ensuremath{-}5$ controlled unitary operators. The number can be reduced to $2{d}_{A}\ensuremath{-}1$ when dA{d}_{A}dA​ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any nnn-partite unitary on the space ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{n}}$ is the product of at most $[2{\ensuremath{\prod}}_{j=1}^{n\ensuremath{-}1}(2{d}_{j}\ensuremath{-}2)\ensuremath{-}1]$ controlled unitary gates, each of which is controlled from $n\ensuremath{-}1$ systems. We also decompose any bipartite unitary into the product of a simple type of bipartite gate and some local unitaries. We derive dimension-independent upper bounds for the controlled-not gate cost or entanglement cost of bipartite permutation unitaries (with the help of ancillas of fixed size) as functions of the Schmidt rank of the unitary. It is shown that such costs under a simple protocol are related to the log-rank conjecture in communication complexity theory via the link of nonnegative rank.
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Quantum circuits of a general quantum gate acting on multiple ddd-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group U(3n)U(3^n)U(3n) (arbitrary nnn-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the nnn-qutrit system, and the quantum circuit of generic nnn-qutrit gates contained 4196⋅32n−4⋅3n−1−(n22+n4−2932)\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})9641​⋅32n−4⋅3n−1−(2n2​+4n​−3229​) GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.