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arXiv:1904.10724 · Physical Review A

Leakage mitigation for quantum error correction using a mixed qubit scheme

Natalie C. Brown, Kenneth R. Brown

Published Sep 18, 2019• 31 citations
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Abstract

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 $^{174}\mathrm{Yb}^{+}$ Zeeman qubit outperforms the $^{171}\mathrm{Yb}^{+}$ 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.

DOI
10.1103/physreva.100.032325

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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 \ensuremathμG\ensuremath{\mu}\mathrm{G}\ensuremathμG 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 \ensuremathμG\ensuremath{\mu}\mathrm{G}\ensuremathμG. 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.
Many systems used for quantum computing possess additional states beyond those defining the qubit. Leakage out of the qubit subspace must be considered when designing quantum error correction codes. Here we consider trapped ion qubits manipulated by Raman transitions. Zeeman qubits do not suffer from leakage errors but are sensitive to magnetic fields to first order. Hyperfine qubits can be encoded in clock states that are insensitive to magnetic fields to first order, but spontaneous scattering during the Raman transition can lead to leakage. Here we compare a Zeeman qubit (174Yb+^{174}\mathrm{Yb}^{+}174Yb+) to a hyperfine qubit (171Yb+^{171}\mathrm{Yb}^{+}171Yb+) in the context of the surface code. We find that the number of physical qubits required to reach a specific logical qubit error can be reduced by using 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ if the magnetic field can be stabilized with fluctuations smaller than $10\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{G}$.
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.
Quantum codes excel at correcting local noise but fail to correct leakage faults that excite qubits to states outside the computational space. Aliferis and Terhal \cite{aliferis07} have shown that an accuracy threshold exists for leakage faults using gadgets called leakage reduction units (LRUs). However, these gadgets reduce the accuracy threshold and increase overhead and experimental complexity, and these costs have not been thoroughly understood. We explore a variety of techniques for leakage-resilient, fault-tolerant error correction in topological codes. Our contributions are threefold. First, we develop a leakage model that is physically motivated and efficient to simulate. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform 3-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulations show that simple circuits with one extra CNOT per check operator and no additional ancillas reduce the accuracy threshold by less than a factor of 444 when leakage and depolarizing noise rates are comparable. This becomes a factor of 222 when the decoder uses 3-outcome measurements. Finally, when the physical error rate is less than 2×10−42\times 10^{-4}2×10−4, placing LRUs after every gate may achieve the lowest logical error rates of all of the circuits we considered. We anticipate that the closely related planar codes might exhibit the same accuracy thresholds and that the ideas may generalize naturally to other topological codes.
Quantum codes excel at correcting local noise but fail to correct leakage faults that excite qubits to states outside the computational space. Aliferis and Terhal have shown that an accuracy threshold exists for leakage faults using gadgets called leakage reduction units (LRUs). However, these gadgets reduce the threshold and increase experimental complexity, and the costs have not been thoroughly understood. We explore a variety of techniques for leakage resilience in topological codes. Our contributions are threefold. First, we develop a leakage model that is physically motivated and efficient to simulate. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform 3-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulations show that simple circuits with one extra CNOT per qubit reduce the accuracy threshold by less than a factor of 4 when leakage and depolarizing noise rates are comparable compared to a scenario without leakage. This becomes a factor of 2 when the decoder uses 3-outcome measurements. Finally, we make the surprising observation that for physical error rates less than 2 × 10-4, placing LRUs after every gate may achieve the lowest logical error rate. We expect that the ideas may generalize to other topological codes.
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.
Abstract Leakage is a particularly damaging error that occurs when a qubit state falls out of its two-level computational subspace. Compared to independent depolarizing noise, leaked qubits may produce many more configurations of harmful correlated errors during error-correction. In this work, we investigate different local codes in the low-error regime of a leakage gate error model. When restricting to bare-ancilla extraction, we observe that subsystem codes are good candidates for handling leakage, as their locality can limit damaging correlated errors. As a case study, we compare subspace surface codes to the subsystem surface codes introduced by Bravyi et al. In contrast to depolarizing noise, subsystem surface codes outperform same-distance subspace surface codes below error rates as high as ⪅ 7.5 × 10−4 while offering better per-qubit distance protection. Furthermore, we show that at low to intermediate distances, Bacon–Shor codes offer better per-qubit error protection against leakage in an ion-trap motivated error model below error rates as high as ⪅ 1.2 × 10−3. For restricted leakage models, this advantage can be extended to higher distances by relaxing to unverified two-qubit cat state extraction in the surface code. These results highlight an intrinsic benefit of subsystem code locality to error-corrective performance.