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arXiv:1501.02708 · Physical Review A

Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates

Lin Chen, Li Zhi Yu

Published Mar 17, 2015• 20 citations
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Abstract

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 ${d}_{A}$ 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 $n$-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.

DOI
10.1103/physreva.91.032308

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It is known that any bipartite unitary operator of Schmidt rank 3 is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank 3 can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank rrr and showed that one of the protocol uses O(r)O(r)O(r) ebits of entanglement and O(r)O(r)O(r) bits of classical communication, while these two types of costs for the other protocol scale as O(rlogr)O(rlogr)O(rlogr) but the actual values are smaller for all r<1100r<1100r<1100. Based on this we obtain upper bounds of the number of nonlocal controlled-not gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt ranks 2 and 3. We show that they are respectively 1 ebit and some value between log29−−16/9{log}_{2}9--16/9log2​9−−16/9 and log23{log}_{2}3log2​3 ebits.
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.
Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: ${log}_{2}9\ensuremath{-}16/9$ or log23{log}_{2}3log2​3 ebits. The entangling power, assisted entangling power, and disentangling power of $2\ifmmode\times\else\texttimes\fi{}{d}_{B}$ permutation unitaries of Schmidt rank four are all 2 ebits. These quantities are also derived for generalized Clifford operators. We further show that any bipartite permutation unitary of Schmidt rank greater than two has entangling power greater than 1.223 ebits. We construct the generalized controlled-not (cnot) gates whose assisted entangling power reaches the maximum. We quantitatively compare the entangling power and assisted entangling power for general bipartite unitaries and their connection to the disentangling power by proposing a probabilistic protocol for implementing bipartite unitaries.
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.