Quantum Circuits for General Multiqubit Gates

Abstract

We consider a generic elementary gate sequence which is needed to implement a general quantum gate acting on n qubits-a unitary transformation with 4(n) degrees of freedom. For synthesizing the gate sequence, a method based on the so-called cosine-sine matrix decomposition is presented. The result is optimal in the number of elementary one-qubit gates, 4(n), and scales more favorably than the previously reported decompositions requiring 4(n)-2(n+1) controlled NOT gates.

Constructive quantum Shannon decomposition from Cartan involutions

“…In 2004 Shende, Markov and Bullock proved the highest known lower bound on asymptotic CNOT cost, ⌈ 1 4 (4 n − 3n − 1)⌉ [30], and Vartiainen, Möttönen, and Salomaa simplified the best existing circuit using Gray codes to achieve for the first time a leading order CNOT cost of O(4 n ) (in fact, for large n, the cost was approximately 8.7 × 4 n ), a multiplicative factor away from the highest known lower bound [31]. Later that year, the same authors, along with Bergholm, presented a decomposition based on the cosine-sine matrix decomposition (CSD) which produced asymptotic behavior scaling as 4 n − 2 n+1 [32]. Vatan and Williams published a three CNOT universal two qubit gate along with a proof that fewer CNOTs could never achieve universality [33], and proposed a 40 CNOT universal three qubit gate which was, at the time, the best known [34].…”

“…e.g. [38,32,34,42]). The task of computing the Cartan factors for a specific unitary matrix is greatly facilitated by the existence of Cartan involutions.…”

Towards an optimal swap gate

“…The reason for this is due to the high cost associated with experimentally realizing a CNOT gate (Möttönen et al (2004)). Constructing quantum circuits that limit the use of CNOT gates have become a important aspect of research.…”

“…The minimization of quantum gate counts has been a concern of quantum computing and researchers have focused on universal n-qubit gates that contain fewest uses of CNOT gates (Möttönen et al (2004), Shende et al (2004Shende et al ( , 2006 and Sedlák and Plesch (2008)). The reason for this is due to the high cost associated with experimentally realizing a CNOT gate (Möttönen et al (2004)).…”

An Improved Arbitrated Quantum Scheme with Bell States

“…To realize arbitrary quantum gates, they are decomposed to a set of physically implementable gates by quantum technologies, which is called quantum logic synthesis [19], [20], [21], [22]. This set of gates typically consists of CNOT and single-qubit gates, called "basic gate" library [23] or CNOT and single-qubit rotation gates, called "elementary gate" library [24].…”