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Physical Review A

Quantum gates using a pulsed bias scheme

P. K. Gagnebin, S.R. Skinner, Elizabeth Behrman, James E. Steck, Zhongyuan Zhou, Siyuan Han

Published Oct 7, 2005• 18 citations
Publisher

Abstract

We propose a scheme for realizing quantum gates [controlled-NOT (CNOT) and Toffoli] using a two-level quantum system by means of a pulsed bias. We show how to realize a CNOT gate using a two-qubit, two-level coupled system and a Toffoli gate using a three-qubit, two-level coupled system. For the CNOT gate, we show how to reduce the $4\ifmmode\times\else\texttimes\fi{}4$ Hamiltonian of the coupled system to a $2\ifmmode\times\else\texttimes\fi{}2$ Hamiltonian in order to solve for parameter values of the system. When these values are substituted into the coupled system equations for the two-qubit system, the simulation results confirm the CNOT gate operation. We further demonstrate our proposed scheme for a CNOT gate operation using a multilevel system of two coupled superconducting quantum interference devices as a particular example. We next use the same approach to implement a Toffoli gate using a three-qubit system.

DOI
10.1103/physreva.72.042311

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•Jan 1, 2003•4 citations
We propose a scheme for realizing quantum gates [controlled-NOT (CNOT) and Toffoli] using a two-level quantum system by means of a pulsed bias. We show how to realize a CNOT gate using a two-qubit, two-level coupled system and a Toffoli gate using a three-qubit, two-level coupled system. For the CNOT gate, we show how to reduce the 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 Hamiltonian of the coupled system to a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 Hamiltonian in order to solve for parameter values of the system. When these values are substituted into the coupled system equations for the two-qubit system, the simulation results confirm the CNOT gate operation. We further demonstrate our proposed scheme for a CNOT gate operation using a multilevel system of two coupled superconducting quantum interference devices as a particular example. We next use the same approach to implement a Toffoli gate using a three-qubit system.
A theory scheme was proposed for implementing a Toffoli gate with three-level structure superconducting quantum-interference devices(SQUID),based on the cavity quantum electrodynamics(CQED).In this proposal,quantum information was encoded into the two lower levels of a SQUID while a higher level serves as the gate manipulation.Applying the coupling of the cavity field with a SQUID,which was driven by a classical microwave pulse to control the evolution of quantum state,the scheme could be extended to achieve N-qubit Toffoli gate.And,the experimental feasibility of Toffoli gate was disscussed,and the total operation time of four-qubit Toffoli gate was computed only 30 nanometers,which was much shorter than the decay time of the cavity mode or the energy relaxation time of the higher level.And the increasing rate of the gate response time became slow with the increase of the qubit number.
In two and three coupled Josephson charge qubits, we exemplify how to take pulses for realizing quantum gates from fidelity-limited pioneering stages to the decoherence limit of near time optimal high-fidelity controls. Thus, a CNOT gate can be obtained with a fidelity >1>1>1--${10}^{\ensuremath{-}9}$ for the two qubits. Even when including higher charge states, the leakage is below 11%1, although the pulses are nonadiabatic. The controls are five times faster than the pioneering experiment [Nature (London) 425, 941 (2003)] for otherwise identical parameters---i.e., a progress towards the error-correction threshold by a factor of 100. We outline schemes to generate these shaped pulses by Cauer synthesis, or more generally by few LCR circuits. The approach generalizes to larger systems: e.g., directly realizing a TOFFOLI gate in three linearly coupled charge qubits is shown to be 13 times faster than decomposing it into a circuit of nine CNOT gates of the above experimental work. In view of the next generation of fast pulse shapers, the combination of methods is designed to find wide application in quantum control of pseudospin and macroscopic quantum systems.
An alternative approach is proposed to realize an n -qubit Toffoli gate with superconducting quantum-interference devices (SQUIDs) in cavity quantum electrodynamics (QED). In the proposal, we represent two logical gates of a qubit with the two lowest levels of a SQUID while a higher-energy intermediate level of each SQUID is utilized for the gate manipulation. During the operating process, because the cavity field is always in vacuum state, the requirement on the cavity is greatly loosened and there is no transfer of quantum information between the cavity and SQUIDs.
The Toffoli gate is a three-qubit controlled-controlled-NOT gate, which plays a central role in quantum information processing and quantum computing. Recently, hybrid systems and hybrid quantum gates have attracted much attention. We propose an efficient scheme to implement a hybrid Toffoli gate with two nitrogen-vacancy center ensembles (NV ensembles) as the control qubits and a single superconducting flux qubit as the target qubit. This hybrid Toffoli gate is realized by employing two NV ensembles coupled to a superconducting qutrit. The hybrid Toffoli gate is constructed by a controlled-controlled-phase (CCPHASE) gate sandwiched between two Hadamard gates applied on the target qubit. Since the CCPHASE gate is realized using only a single-step operation and no conventional gate decomposition protocol is applied, the Toffoli-gate implementation is greatly simplified. Because the higher energy level of the flux qutrit is not populated, the decoherence of this level is greatly suppressed. In addition, our hybrid Toffoli gate can be used to generate a hybrid Greenberger–Horne–Zeilinger entangled state of a flux qubit and two NV ensembles. Numerical simulations demonstrate that high-fidelity implementation of a hybrid Toffoli gate is feasible within current experimental technology. This proposal is universal and can be applied to accomplish the same task in a wide range of physical systems.
We experimentally constructed an all-microwave scheme for the controlled-NOT (cNOT) gate between two superconducting transmon qubits in a three dimensional cavity. Our cNOT gate is based on the microwave-activated phase (MAP) gate, which requires an additional procedure to compensate the accumulated phases during the operation of the MAP gate. We applied Z-axis phase gates using microwave hyperbolic secant pulse on both qubits with adequate rotation angles systematically calibrated by separate measurements. We evaluated the gate performance of the constructed cNOT gate by performing two-qubit quantum process tomography (QPT). Finally, we present the experimental implementation of the Deutsch-Jozsa algorithm using the cNOT gate.
We examine the realization of a quantum CNOT gate by adiabatic operations.The principles of such systems and their analysis are briefly discussed and a model consisting of two weakly coupled double- potential well qubits is studied numerically. Regions of the parameter space with suitable well-defined sets of wavefunctions are found, in which then an adiabatic sweep of an external bias produces the switching behavior of CNOT. Results are presented on the adiabatic condition and the identification with the parameters of a flux-coupled two-SQUID system is given.For typical parameters adiabatic times in the nanosecond regime are obtained.