Efficient Toffoli gates using qudits

Abstract

The simplest decomposition of a Toffoli gate acting on three qubits requires five 2-qubit gates. If we restrict ourselves to controlled-sign (or controlled-NOT) gates this number climbs to six. We show that the number of controlled-sign gates required to implement a Toffoli gate can be reduced to just three if one of the three quantum systems has a third state that is accessible during the computation, i.e. is actually a qutrit. Such a requirement is not unreasonable or even atypical since we often artificially enforce a qubit structure on multilevel quantums systems (eg. atoms, photonic polarization and spatial modes). We explore the implementation of these techniques in optical quantum processing and show that linear optical circuits could operate with much higher probabilities of success.

Maximal success probabilities of linear-optical quantum gates

“…3. The best solution obtained is S ≈ 0.00340; This is an improvement over combining a CNOT gate, a CS gate, and a "passive quantum filter" to produce the Toffoli gate [1], which yields a total success rate S = (2/27) 2 × 1/2 ≈ 0.00274 using four unentangled ancilla photons.…”

“…Here we focus on more complicated gates, taking as examples the two-qubit controlled sign (CS) gate (equivalently, the CNOT gate), and the three-qubit Toffoli gate. For these physically important gates, existing theoretical results are limited to upper or lower bounds on the success probability [1,7,8].…”

“…Here we focus on more complicated gates, taking as examples the two-qubit controlled sign (CS) gate (equivalently, the CNOT gate), and the three-qubit Toffoli gate. For these physically important gates, existing theoretical results are limited to upper or lower bounds on the success probability [1,7,8].A linear-optical quantum gate, or state generator (LO-QSG) [5], can be viewed formally as a device implementing a contraction transformation (for ideal detectors) that converts pure input states into desired pure output states. The goal of the optimization problem is to find a proper linear optical network (see Fig.…”

Five two-qubit gates are necessary for implementing the Toffoli gate

“…The first experimental realization of the quantum Toffoli gate was presented in an ion-trap quantum computer, in 2009 at the University of Innsbruck, Austria [20]. Then, the Toffoli gate was realized in linear optics [21] and superconducting circuits [22,31,32].Due to its significance in quantum computing, the theoretical pursuit of efficient implementation of the Toffoli gate using a sequence of single-and two-qubit gates has a quite long history [7,8,11,12,[33][34][35][36][37]. It was explicitly stated as an open problem by Nielsen and Chuang in their influential textbook on quantum computation [14]: How many general two-qubit gates (or CNOT gates) are required to implement the Toffoli gate (see [14], p. 213, Problem 4.4)?…”

“…Due to its significance in quantum computing, the theoretical pursuit of efficient implementation of the Toffoli gate using a sequence of single-and two-qubit gates has a quite long history [7,8,11,12,[33][34][35][36][37]. It was explicitly stated as an open problem by Nielsen and Chuang in their influential textbook on quantum computation [14]: How many general two-qubit gates (or CNOT gates) are required to implement the Toffoli gate (see [14], p. 213, Problem 4.4)?…”