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Simplified scalable construction of highly-conditional quantum gates

Svetoslav S. Ivanov, Nikolay V. Vitanov

Published Jun 1, 2011• 0 citations
OpenAlex

Abstract

We present a scalable uniform technique for construction of highly conditional C$^n$-NOT quantum gates of trapped ion qubits, such as the Toffoli gate, without using ancilla states and circuits of an exorbitant number of concatenated one- and two-qubit gates. Apart from the initial dressing of the internal qubit states with vibrational phonons and the final restoration of the phonon ground state, our technique requires the application of just a single composite pulse on the target qubit and is applicable both in and outside the Lamb-Dicke regime. We design special narrowband composite pulses, which suppress all transitions but the conditional transition of the target qubit; moreover, these composite pulses significantly improve the spatial addressing selectivity.

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We present a scalable uniform technique for construction of highly conditional Cn^nn-NOT quantum gates of trapped ion qubits, such as the Toffoli gate, without using ancilla states and circuits of an exorbitant number of concatenated one- and two-qubit gates. Apart from the initial dressing of the internal qubit states with vibrational phonons and the final restoration of the phonon ground state, our technique requires the application of just a single composite pulse on the target qubit and is applicable both in and outside the Lamb-Dicke regime. We design special narrowband composite pulses, which suppress all transitions but the conditional transition of the target qubit; moreover, these composite pulses significantly improve the spatial addressing selectivity.
We present a scalable uniform technique for the construction of highly conditional multiply-controlled-not quantum gates of trapped ion qubits, such as the Toffoli gate, without using ancilla states and circuits of an exorbitant number of concatenated one- and two-qubit gates. Apart from the initial dressing of the internal qubit states with vibrational phonons and the final restoration of the phonon ground state, our technique requires the application of just a single composite pulse on the target qubit and is applicable both in and outside the Lamb-Dicke regime. We design special narrowband composite pulses, which suppress all transitions but the conditional transition of the target qubit; moreover, these composite pulses significantly improve the spatial addressing selectivity.
We propose a fast and efficient technique to create classes of highly entangled states of trapped ions, such as arbitrary Dicke states and superpositions of them, e.g. NOON states. The ions are initialized in the phonon ground state and are addressed globally with a composite pulse that is resonant with the first motional sideband. The technique is fairly robust to parameter fluctuations and operates on comparatively short time scales, as resonant interactions allow one to use the minimum laser pulse area. The number of single pulses from the composite sequence is equal to the number of ions; thus the implementation complexity grows only linearly with the size of the system. The approach does not require individual addressing of the ions in the trap and can be applied both inside and outside the Lamb–Dicke regime.
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.
Conditional multiqubit gates are a key component for elaborate quantum algorithms. In a recent work, Rasmussen et al. [Phys. Rev. A 101, 022308 (2020)] proposed an efficient single-step method for a prototypical multiqubit gate, a Toffoli gate, based on a combination of Ising interactions between control qubits and an appropriate driving field on a target qubit. Trapped ions are a natural platform to implement this method, since Ising interactions mediated by phonons have been demonstrated in increasingly large ion crystals. However, the simultaneous application of these interactions and the driving field required for the gate results in undesired entanglement between the qubits and the motion of the ions, reducing the gate fidelity. In this work, we propose a solution based on adiabatic switching of these phonon-mediated Ising interactions. We study the effects of imperfect ground-state cooling and use spin-echo techniques to undo unwanted phase accumulation in the achievable fidelities. For gates coupling to all axial modes of a linear crystal, we calculate high-fidelity (>99(>99%)(>99 NNN-qubit rotations with N=3−−7N=3--7N=3−−7 ions cooled to their ground state of motion and a gate time below 1 ms. Finally, we study the effect of laser intensity fluctuations and find that the proposed gate requires intensity stabilization with subpercentage noise levels. The high fidelities obtained also for large crystals could make the gate competitive with gate-decomposed, multistep variants of the NNN-qubit Toffoli gate, at the expense of requiring ground-state cooling of the ion crystal.
We present an alternative scheme to exactly implement one-qubit and two-qubit quantum gates with a single trapped cold ion driven by a travelling laser field. The internal degree of freedom of the ion acts as the target qubit and the control qubit is encoded by two Fock states of the external vibration of the ion. The conditions to realize these operations, including the duration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approximation is required. The multiquantum transition between the internal and external degrees of freedom of the ion is considered.
An efficient implementation of the Toffoli gate is of conceptual importance for running various quantum algorithms, including Grover's search and Shor's integer factorization. However, direct implementation of the Toffoli gate either entails a prohibitive increase in the number of two-qubit gates or requires ancilla qubits, whereas both of these resources are limited in the current generation of noisy intermediate-scale quantum devices. Here, we experimentally demonstrate a scalable N-qubit Toffoli gate improvement using ^{171}Yb^{+} trapped-ion-based optical-metastable-ground encoded qutrits for the cases of up to N=10. With the use of the Mølmer-Sørensen gate as a basic entangling operation, we compare the standard qubit decomposition with the qutrit approach, where upper levels are used as ancillae. The presented decomposition requires only global control of the ancilla levels, which simplifies experimental implementation of the proposed approach. Using the example of a three-qubit Grover's search, we also demonstrate an increase in the algorithm's accuracy by monitoring the leakage from the qubit subspace during a qutrit-based Toffoli gate implementation.