Implementation of a Toffoli gate with superconducting circuits

Abstract

The Toffoli gate is a three-quantum-bit (three-qubit) operation that inverts the state of a target qubit conditioned on the state of two control qubits. It makes universal reversible classical computation possible and, together with a Hadamard gate, forms a universal set of gates in quantum computation. It is also a key element in quantum error correction schemes. The Toffoli gate has been implemented in nuclear magnetic resonance, linear optics and ion trap systems. Experiments with superconducting qubits have also shown significant progress recently: two-qubit algorithms and two-qubit process tomography have been implemented, three-qubit entangled states have been prepared, first steps towards quantum teleportation have been taken and work on quantum computing architectures has been done. Implementation of the Toffoli gate with only single- and two-qubit gates requires six controlled-NOT gates and ten single-qubit operations, and has not been realized in any system owing to current limits on coherence. Here we implement a Toffoli gate with three superconducting transmon qubits coupled to a microwave resonator. By exploiting the third energy level of the transmon qubits, we have significantly reduced the number of elementary gates needed for the implementation of the Toffoli gate, relative to that required in theoretical proposals using only two-level systems. Using full process tomography and Monte Carlo process certification, we completely characterize the Toffoli gate acting on three independent qubits, measuring a fidelity of 68.5 ± 0.5 per cent. A similar approach to realizing characteristic features of a Toffoli-class gate has been demonstrated with two qubits and a resonator and achieved a limited characterization considering only the phase fidelity. Our results reinforce the potential of macroscopic superconducting qubits for the implementation of complex quantum operations with the possibility of quantum error correction.

Microwave-induced amplitude- and phase-tunable qubit-resonator coupling in circuit quantum electrodynamics

“…In experiments using the circuit QED architecture, not only the transition between ground |g and the first excited |e state at frequency ω ge , but also transitions between higher lying energy levels can easily be addressed [30] and complex quantum states can be realized [31]. In particular, the second excited state |f , which is separated from |e by ω ef = ω ge + α, has been used widely for quantum gates [32][33][34][35], and plays an important role in our implementation of the cavity-assisted Raman processes in a circuit QED setting.…”

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)?…”

“…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].…”

“…Indeed, a study of the minimum cost of two-qubit gates for simulating a multiqubit gate is not only of theoretical importance, but also an experimental requirement: to accomplish a quantum algorithm, even at a small size, one has to implement a relatively high level of control over the multiqubit quantum system. A lot of experiments demonstrate multiqubit controlled-NOT gates in ion traps [20], linear optics [21], superconductors [22], and atoms [23].…”

Hidden two-qubit dynamics of a four-level Josephson circuit

“…Many physical systems commonly used as qubits, such as trapped ions 4,5 and superconducting circuits 6,7 , are in reality multi-level systems whose dynamics are manually limited by researchers to two levels to preserve simplicity and fidelity. Such systems can easily be employed as multi-level qudits by accessing additional levels, allowing a significant reduction of the resources and gates required for a variety of quantum information applications [8][9][10] , yet also introducing a complex multi-level problem. The usefulness and popularity of multi-level qudits are limited by a dearth of theoretical methods for understanding and controlling their dynamics.…”