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Physical Review A

Phase-controlled two-qubit quantum gates

Vladimir S. Malinovsky, Ignacio R. Solá, Jiří Vala

Published Mar 3, 2014• 18 citations
Publisher

Abstract

We consider an implementation of a two-qubit entangling gate between trapped ions controlled by fully overlapping laser pulses. We particularly address the role of a relative phase between the pulses in the gate design. Though, in general, the relative phase determines the exact form of the two-qubit operation, it is shown that it affects only its single qubit part and has no effect on its entangling content. Based on a general canonical decomposition of the evolution operator we demonstrate that the controlled-not (cnot) gate can be decomposed in terms of a two-qubit entangled gate and single qubit phase gates. Our study shows that any relative phase errors in two-qubit operations related to phase fluctuations of the external fields can be successfully corrected locally by single-qubit operations, which are usually easy to implement.

DOI
10.1103/physreva.89.032301

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We consider an implementation of a two-qubit entangling gate between trapped ions controlled by fully overlapping laser pulses. We particularly address the role of a relative phase between the pulses in the gate design. Though, in general, the relative phase determines the exact form of the two-qubit operation, it is shown that it affects only its single qubit part and has no effect on its entangling content. Based on a general canonical decomposition of the evolution operator we demonstrate that the controlled-not (cnot) gate can be decomposed in terms of a two-qubit entangled gate and single qubit phase gates. Our study shows that any relative phase errors in two-qubit operations related to phase fluctuations of the external fields can be successfully corrected locally by single-qubit operations, which are usually easy to implement.
Working with trapped atoms at a close distance to each other, we show that one can implement entangling gates based on nonindependent qubits using a single pulse per qubit, or a single structured pulse. The optimal parameters depend on approximate solutions of Diophantine equations, causing the fidelity to never be exactly one, even under ideal conditions, although the errors can be made arbitrarily smaller at the cost of stronger fields. We fully characterize the mechanism by which the gates operate and study the effects of thermal motion and intensity fluctuations in the laser beams for different physical implementations of the gates. If instead of one pulse, we control the system with a two-pulse sequence, a plethora of mechanisms become possible where one can choose the optimal parameters from a wide range of values to achieve high-fidelity gates that are more protected from the effects of laser intensity fluctuations.
We introduce two types of two-qubit controlled-phase gates, which operate with trapped ions in a linear Paul trap. Each ion is addressed individually with two sequential bichromatic laser pulses, the two frequency components of which are tuned on exact resonance with the first blue and red vibrational sidebands of the qubit transition frequency. For suitably chosen pulse areas and relative laser phases the dependence on the phonon number vanishes and the net effect is a controlled-phase gate between the electronic qubit states. In contrast to earlier gates, in which the pulses addressing the two ions are separated in time, we use partly overlapping pulses, which make the present gates faster, e.g., by about 26% for rectangular and over 50% for sinusoidal pulses.
We show that, under certain conditions, the two-qubit controlled-NOT logic operations can be realized exactly using one laser pulse. The target qubit and the control qubit are encoded by the internal and external degrees of freedom of a single trapped ion. The multiquantum process of the interaction between the internal and external degrees of freedom of the ion is considered. This scheme neither requires the cooled ion in its vibrational ground state nor necessitates performance of the Lamb-Dicke approximation.
Abstract We present the experimental implementation of a two-qubit phase gate, using a radio frequency (RF) controlled trapped-ion quantum processor. The RF-driven gate is generated by a pulsed dynamical decoupling sequence applied to the ions’ carrier transitions only. It allows for a tunable phase shift with high-fidelity results. The conditional phase shift is measured using a Ramsey-type measurement with an inferred fringe contrast of up to 99 − 2 + 1 % . We also prepare a Bell state using this laser-free gate. The phase gate is robust against common sources of error. We investigate the effect of the excitation of the center-of-mass (COM) mode, errors in the axial trap frequency, pulse area errors and errors in sequence timing. The contrast of the phase gate is not significantly reduced up to a COM mode excitation < 20 phonons, trap frequency errors of +10%, and pulse area errors of −8%. The phase shift is not significantly affected up to < 10 phonons and pulse area errors of −2%. Both, contrast and phase shift are robust to timing errors up to −30% and +15%. The gate implementation is resource efficient, since only a single driving field is required per ion. Furthermore, it holds the potential for fast gate speeds (gate times on the order of 100 µ s) by using two axial motional modes of a two-ion crystal through improved setups.
We design composite controlled-phase gates, which compensate errors in the phase of a single gate. The errors can be of various natures, such as relative, absolute, or both. We present composite sequences which are robust to relative errors up to the sixth order with the number of the constituent gates growing just linearly with the desired accuracy, and we describe a method to achieve even higher accuracy. We show that the absolute error can be canceled entirely with only two gates. We describe an ion-trap implementation of our composite gates in which simultaneous cancellation of the error in both the pulse area and the detuning is achieved.
We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, iSWAP\sqrt{i\text{SWAP}}iSWAP​-based case the value of the detuning during the entangling operations must not exceed 2g2g2g, where ggg is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the ``reference'' qubit (with which it is in resonance), the detuning should not exceed ggg. If the drive is applied to the ``detuned'' qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.