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arXiv:1903.03937 · New Journal of Physics

Handling leakage with subsystem codes

Natalie C. Brown, Michael Newman, Kenneth R. Brown

Published Jul 1, 2019• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/ab3372

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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.
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 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.
Recently, significant effort has been devoted towards designing erasure qubits in which dominant physical noise excites detectable leakage states that can be returned to the qubit subspace. Interest in these erasure qubits has been driven by studies showing that the requirements for fault-tolerant quantum error correction are significantly relaxed when noise in every gate operation is dominated by erasures. However, these studies assume perfectly accurate erasure checks after every gate operation which generally come with undesirable time and hardware overhead costs. In this work, we investigate the consequences of using an imperfect but overhead-efficient erasure check for fault-tolerant quantum error correction with the surface code. We show that, under physically reasonable assumptions on the imperfect erasure checks, the threshold error rate is still at least over twice that for Pauli noise. We also study the impact of imperfect erasure checks on the effective error distance and find that it degrades the effective distance under a general error model in which a qubit suffers from depolarizing noise when interacting with a leaked qubit. We then identify a more restrictive but realistic noise model for a qubit that interacts with a leaked qubit, under which the effective error distance is twice that for Pauli noise. We apply our analysis to recently proposed superconducting dual-rail erasure qubits and show that achieving good-performance surface code quantum memories with relaxed system requirements is possible.
Abstract Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.
Abstract Leakage errors, in which a qubit is excited to a level outside the qubit subspace, represent a significant obstacle in the development of robust quantum computers. We present a computationally efficient simulation methodology for studying leakage errors in quantum error correcting codes (QECCs) using tensor network methods, specifically matrix product states. Our approach enables the simulation of various leakage processes, including thermal noise and coherent errors, without approximations (such as the Pauli twirling approximation) that can lead to errors in the estimation of the logical error rate. We apply our method to two QECCs: the one-dimensional repetition code and a thin 3 × d surface code. By leveraging the small amount of entanglement generated during the error correction process, we are able to study large systems, up to a few hundred qudits, over many code cycles. We consider a realistic noise model of leakage relevant to superconducting qubits to evaluate code performance and a variety of leakage removal strategies. Our numerical results suggest that appropriate leakage removal is crucial, especially when the code distance is large.
Abstract Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction 1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 × 10 −6 logical error per cycle floor set by a single high-energy event (1.6 × 10 −7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.