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IEEE International Conference on Quantum Computing and Engineering (QCE)

Mitigating qubit leakage errors in quantum circuits with gadgets and post-selection

Karl Mayer

Published Sep 1, 2022• 0 citations
Publisher

Abstract

Leakage is a quantum error process in which a qubit ends up in a state outside of the computational subspace. In trapped-ion hyperfine qubits, leakage can occur as a result of spontaneous emission from an excited state during a two-qubit gate. This error can be detected by a circuit gadget that couples a data qubit to an ancilla qubit, followed by a measurement of the ancilla. The effects of leakage errors on the circuit can then be removed by post-selecting shots in which leakage was not detected. We first experimentally demonstrate improved two-qubit gate fidelity as measured by randomized benchmarking and cycle benchmarking. We then benchmark the use of leakage detection on system-level mirror benchmarking circuits. Finally, we present a protocol to measure the failure rate of the leakage detection gadget, defined as the probability that the gadget produces an error on the data qubit while not heralding the ancilla.

DOI
10.1109/qce53715.2022.00126

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Leakage is a quantum error process in which a qubit ends up in a state outside of the computational subspace. In trapped-ion hyperfine qubits, leakage can occur as a result of spontaneous emission from an excited state during a two-qubit gate. This error can be detected by a circuit gadget that couples a data qubit to an ancilla qubit, followed by a measurement of the ancilla. The effects of leakage errors on the circuit can then be removed by post-selecting shots in which leakage was not detected. We first experimentally demonstrate improved two-qubit gate fidelity as measured by randomized benchmarking and cycle benchmarking. We then benchmark the use of leakage detection on system-level mirror benchmarking circuits. Finally, we present a protocol to measure the failure rate of the leakage detection gadget, defined as the probability that the gadget produces an error on the data qubit while not heralding the ancilla.
The majority of quantum-error-detection and correction protocols assume that the population in a qubit does not leak outside of its computational subspace. For many existing approaches, however, the physical qubits do possess more than two energy levels and consequently are prone to such leakage events. Analyzing the effects of leakage is therefore essential to devise optimal protocols for quantum gates, measurement, and error correction. In this article, we present a detailed study of leakage in a two-qubit superconducting stabilizer measurement circuit. We simulate the repeated ancilla-assisted measurement of a single ${\ensuremath{\sigma}}^{z}$ operator for a data qubit, record the outcome at the end of each measurement cycle, and explore the signature of leakage events in the obtained readout statistics. An analytic model is also developed that closely approximates the results of our numerical simulations. We find that leakage leads to destructive features in the quantum-error-detection scheme, making additional hardware and software protocols necessary.
Leakage errors are unwanted transfer of population outside of a defined computational subspace and they occur in almost every platform for quantum computing. While prevalent, leakage is often overlooked when measuring and reporting the fidelity of quantum gates with standard methods. In fact, when leakage is substantial it can cause a large overestimation of fidelity from the typical method used to measure fidelity, randomized benchmarking. We provide several methods for properly estimating fidelity in the presence of leakage errors that are applicable in different error regimes with carefully chosen sequence lengths. Then, we numerically demonstrate the methods for two-qubit randomized benchmarking, which often have the largest errors. Finally, we reanalyze previously shared data from Quantinuum systems with some of the methods provided.
Population leakage outside the qubit subspace presents a particularly harmful source of error that cannot be handled by standard error correction methods. Using a trapped 171Yb+^{171}{\mathrm{Yb}}^{+}171Yb+ ion, we demonstrate an optical pumping scheme to suppress leakage errors in atomic hyperfine qubits. The selection rules and narrow linewidth of a quadrupole transition are used to selectively pump population out of leakage states and back into the qubit subspace. Each pumping cycle reduces the leakage population by a factor of $\ensuremath{\sim}3$, allowing for an exponential suppression in the number of cycles. We use interleaved randomized benchmarking on the qubit subspace to show that this pumping procedure has negligible side effects on the qubit subspace, bounding the induced qubit memory error by $\ensuremath{\le}2.0(8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{per}$ cycle, and qubit population decay to $\ensuremath{\le}1.4(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\text{ }\mathrm{per}$ cycle. These results clear a major obstacle for implementations of quantum error correction and error mitigation protocols.
Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits in the context of the toric code with a depolarizing leakage error model and found that for magnetic-field noise with a standard deviation less than 32 $\ensuremath{\mu}\mathrm{G}$ that the 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ Zeeman qubit outperforms the 171Yb+^{171}\mathrm{Yb}^{+}171Yb+ hyperfine qubit. Here we examine a physically motivated leakage error model based on ions interacting via the M\o{}lmer-S\o{}renson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic-field noise with a standard deviation less than 10 $\ensuremath{\mu}\mathrm{G}$. At these low magnetic fields, we find that the best choice is a mixed qubit scheme where the hyperfine qubits are the syndrome qubits and the leakage is handled without the need of an additional leakage reduction circuit.
Leakage is a particularly damaging error that occurs when a qubit leaves the defined computational subspace. Leakage errors limit the effectiveness of quantum error correcting codes by spreading additional errors to other qubits and corrupting syndrome measurements. The effects of leakage errors on the surface code has been studied in various contexts. However, the effects of a leaked data qubit versus a leaked ancilla qubit can be quite different. Here, we study the effects of data leakage and ancilla leakage separately. We find that data leakage is much less damaging. We show that the surface code maintains its distance in the presence of leakage by either confining leakage to data qubits or eliminating ancilla qubit leakage at the critical fault location. To demonstrate this, we introduce new techniques for handling leakage: taking advantage of gates with one-sided leakage, effectively isolating leakage events to data qubits alone and mixing two types of leakage reducing circuits, to handle data leakage and ancilla leakage separately.
Leakage errors occur when a quantum system leaves the two-level qubit subspace. Reducing these errors is critically important for quantum error correction to be viable. To quantify leakage errors, we use randomized benchmarking in conjunction with measurement of the leakage population. We characterize single qubit gates in a superconducting qubit, and by refining our use of derivative reduction by adiabatic gate pulse shaping along with detuning of the pulses, we obtain gate errors consistently below 10^{-3} and leakage rates at the 10^{-5} level. With the control optimized, we find that a significant portion of the remaining leakage is due to incoherent heating of the qubit.