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Scientific Reports

An Exact Approach to Elimination of Leakage in a Qubit Embedded in a Three-level System

Yifan Sun, Jun-Yi Zhang, Lian-Ao Wu

Published Jul 30, 2019• 1 citation
Publisher

Abstract

Leakage errors damage a qubit by coupling it to other levels. Over the years, several theoretical approaches to dealing with such errors have been developed based on perturbation arguments. Here we propose a different strategy: we use a sequence of finite rotation gates to exactly eliminate leakage errors. The strategy is illustrated by the recently proposed charge quadrupole qubit in a triple quantum dot, where there are two logical states to encode the qubit and one leakage state. We found an su(2) subalgebra in the three-level system, and by using the subalgebra we show that ideal Pauli x and z rotations, which are universal for single-qubit gates, can be generated by two or three propagators. In addition, the magnitude of detuning fluctuation can be estimated based on the exact solution.

DOI
10.1038/s41598-019-47479-9

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Leakage errors damage a qubit by coupling it to other levels. Over the years, several theoretical approaches to dealing with such errors have been developed based on perturbation arguments. Here we propose a different strategy: we use a sequence of finite rotation gates to exactly eliminate leakage errors. The strategy is illustrated by the recently proposed charge quadrupole qubit in a triple quantum dot, where there are two logical states to encode the qubit and one leakage state. We found an su(2) subalgebra in the three-level system, and by using the subalgebra we show that ideal Pauli x and z rotations, which are universal for single-qubit gates, can be generated by two or three propagators. In addition, the magnitude of detuning fluctuation can be estimated based on the exact solution.
Leakage errors damage a qubit by coupling it to other levels. Over the years, several theoretical approaches to dealing with such errors have been developed based on perturbation arguments. Here we propose a different strategy: we use a sequence of finite rotation gates to exactly eliminate leakage errors. The strategy is illustrated by the recently proposed charge quadrupole qubit in a triple quantum dot, where there are two logical states to support the qubit and one leakage state. We have found an su(2) subalgebra in the three-level system, and by using the subalgebra we show that ideal Pauli x and z rotations, which are universal for single-qubit gates, can be generated by two or three propagators of experimentally-available Hamiltonians. The proposed strategy does not require additional pulses, is independent of error magnitude, and potentially reduces experimental overheads. In addition, the magnitude of detuning fluctuation can be estimated based on the exact solution.
Many realizations of solid-state qubits involve couplings to leakage states lying outside the computational subspace, posing a threat to high-fidelity quantum gate operations. Mitigating leakage errors is especially challenging when the coupling strength is unknown, e.g., when it is caused by noise. Here we show that simple pulse sequences can be used to strongly suppress leakage errors for a qubit embedded in a three-level system. As an example, we apply our scheme to the recently proposed charge quadrupole (CQ) qubit for quantum dots. These results provide a solution to a key challenge for fault-tolerant quantum computing with solid-state elements.
Coherent spatial transport or shuttling of a single electron spin through semiconductor nanostructures is an important ingredient in many spintronic and quantum computing applications. In this work we analyze the possible errors in solid-state quantum computation due to leakage in transporting a single-spin qubit through a semiconductor double quantum dot. In particular, we consider three possible sources of leakage errors associated with such transport: finite ramping times, spin-dependent tunneling rates between quantum dots induced by finite spin-orbit couplings, and the presence of multiple valley states. In each case we present quantitative estimates of the leakage errors, and discuss how they can be minimized. The emphasis of this work is on how to deal with the errors intrinsic to the ideal semiconductor structure, such as leakage due to spin-orbit couplings, rather than on errors due to defects or noise sources. In particular, we show that in order to minimize leakage errors induced by spin-dependent tunnelings, it is necessary to apply pulses to perform certain carefully designed spin rotations. We further develop a formalism that allows one to systematically derive constraints on the pulse shapes and present a few examples to highlight the advantage of such an approach.
A triple-quantum-dot system can be operated as either an exchange-only qubit or a resonant-exchange qubit. While it is generally believed that the decisive advantage of the resonant-exchange qubit is the suppression of charge noise because it is operated at a sweet spot, we show that the leakage is also an important factor. Through molecular-orbital-theoretic calculations, we show that when the system is operated in the exchange-only scheme, the leakage to states with double electron occupancy in quantum dots is severe when rotations around the axis ${120}^{\ensuremath{\circ}}$ from $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ is performed. While this leakage can be reduced by either shrinking the dots or separating them further, the exchange interactions are also suppressed at the same time, making the gate operations unfavorably slow. When the system is operated as a resonant-exchange qubit, the leakage is three to five orders of magnitude smaller. We have also calculated the optimal detuning point which minimizes the leakage for the resonant-exchange qubit, and have found that although it does not coincide with the double sweet spot for the charge noise, they are rather close. Our results suggest that the resonant-exchange qubit has another advantage, that leakage can be greatly suppressed compared to the exchange-only qubit, and operating at the double sweet spot point should be optimal both for reducing charge noise and suppressing leakage.
``Leakage'' errors are particularly serious errors which couple states within a code subspace to states outside of that subspace, thus destroying the error protection benefit afforded by an encoded state. We generalize an earlier method for producing leakage elimination decoupling operations and examine the effects of the leakage eliminating operations on decoherence-free or noiseless subsystems which encode one logical, or protected qubit into three or four qubits. We find that by eliminating a large class of leakage errors, under some circumstances, we can create the conditions for a decoherence-free evolution. In other cases we identify a combined decoherence-free and quantum error correcting code which could eliminate errors in solid-state qubits with anisotropic exchange interaction Hamiltonians and enable universal quantum computing with only these interactions.