Block-ZXZsynthesis of an arbitrary quantum circuit

Abstract

Given an arbitrary 2 w × 2 w unitary matrix U , a powerful matrix decomposition can be applied, leading to four different syntheses of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w = 1) to gates with arbitrary value of w. The synthesized circuit consists of controlled one-qubit gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled NOT gates in the case that U is a permutation matrix.

Constructive quantum scaling of unitary matrices

“…Moreover, our transformation is exact and can be found constructively. In contrast to [9], our transformation consists only of negator gates. The main difference is that transformation needs one-qubit ancilla.…”

“…Matrix in the middle is a block-negator matrix (which is also a negator matrix), while left and right matrices are block diagonal matrices. In [9] an algorithm of finding such decomposition was presented. Group XU(2 n ) is isomorphic to U(2 n −1) and can be generated by single-qubit negator and controlled-√ NOT gates [10].…”

“…Since after performing operations Φ and N the state is of the form |− ⊗ U |ψ , the partial trace is simply removing the ancilla system giving a pure state U |ψ . We describe an efficient algorithm that for given U returns explicit and exact form of N with decomposition into a sequence of single-qubit negator and controlled-√ NOT gates only in contrast to results of de Vos and de Baerdemacker [9,10]. In Section 2 we recall basic facts.…”

A Unified Approach to Quantum Computation and Classical Reversible Computation

“…-Thanks to the groups bXU(n) and bZU(n), we have two decompositions: the primal bZbXbZ decomposition [6] and the dual bXbZbX decomposition [7]…”

Reuse Method for Quantum Circuit Synthesis

“…One can cite as example the QR method, via Givens rotations [16]. Other decomposition methods have also been proposed, for example the recent block-ZXZ decomposition [4], or the quantum Shannon decomposition [13] that relies on the use of the sine cosine decomposition of a unitary operator U ∈ U(2 n ):…”