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Leakage and Paralysis in Ancilla-assisted Qubit Measurement: Consequences for Topological Error Correction in Superconducting Architectures

Joydip Ghosh, Austin G. Fowler, John M. Martinis, Michael R. Geller

Published Jun 4, 2013• 1 citation
OpenAlex

Abstract

Abstract : Although topological error-correcting codes offer a promising paradigm for fault- tolerant quantum computation, their robustness in the presence of leakage to non- computational states is unclear. Here we explore the signature and consequences of leakage errors on ancilla-assisted Pauli operator measurement in superconducting devices. We consider a realistic coupled-qutrit model and simulate the repeated measurement of a single sigma z operator. Typically, a data-qubit leakage event manifests itself by producing a noisy ancilla qubit that randomly reads /0/ or /1/ from cycle to cycle. Although the measurement operation is compromised, the presence of the leakage event is apparent and detectable. However, there is also the possibility of a less typical but more dangerous type of leakage event, where the ancilla becomes paralyzed rendering it oblivious to data-qubit errors for many consecutive measurement cycles and compromising the fault-tolerance. Certain dynamical phases associated with the entangling gate determine which type of leakage event will occur in practice. Leakage errors occur in most qubit realizations and our model and results are relevant for many stabilizer-based error correction protocols.

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arXiv:1410.8562•Quantum Information and Computation•Sep 1, 2015•27 citations
Abstract : Although topological error-correcting codes offer a promising paradigm for fault- tolerant quantum computation, their robustness in the presence of leakage to non- computational states is unclear. Here we explore the signature and consequences of leakage errors on ancilla-assisted Pauli operator measurement in superconducting devices. We consider a realistic coupled-qutrit model and simulate the repeated measurement of a single sigma z operator. Typically, a data-qubit leakage event manifests itself by producing a noisy ancilla qubit that randomly reads /0/ or /1/ from cycle to cycle. Although the measurement operation is compromised, the presence of the leakage event is apparent and detectable. However, there is also the possibility of a less typical but more dangerous type of leakage event, where the ancilla becomes paralyzed rendering it oblivious to data-qubit errors for many consecutive measurement cycles and compromising the fault-tolerance. Certain dynamical phases associated with the entangling gate determine which type of leakage event will occur in practice. Leakage errors occur in most qubit realizations and our model and results are relevant for many stabilizer-based error correction protocols.
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.
Superconducting qubits, while promising for scalability and long coherence times, contain more than two energy levels, and therefore are susceptible to errors generated by the leakage of population outside of the computational subspace. Such leakage errors constitute a prominent roadblock towards fault-tolerant quantum computing (FTQC) with superconducting qubits. FTQC using topological codes is based on sequential measurements of multiqubit stabilizer operators. Here, we first propose a leakage-resilient procedure to perform repetitive measurements of multiqubit stabilizer operators, and then use this scheme as an ingredient to develop a leakage-resilient approach for surface code quantum error correction with superconducting circuits. Our protocol is based on swap operations between data and ancilla qubits at the end of every cycle, requiring read-out and reset operations on every physical qubit in the system, and thereby preventing persistent leakage errors from occurring.
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.
Abstract Fault-tolerant quantum computing relies on the ability to detect and correct errors, which in quantum error correction codes is typically achieved by projectively measuring multi-qubit parity operators and by conditioning operations on the observed error syndromes. Here, we experimentally demonstrate the use of an ancillary qubit to repeatedly measure the ZZ and XX parity operators of two data qubits and to thereby project their joint state into the respective parity subspaces. By applying feedback operations conditioned on the outcomes of individual parity measurements, we demonstrate the real-time stabilization of a Bell state with a fidelity of F ≈ 74% in up to 12 cycles of the feedback loop. We also perform the protocol using Pauli frame updating and, in contrast to the case of real-time stabilization, observe a steady decrease in fidelity from cycle to cycle. The ability to stabilize parity over multiple feedback rounds with no further reduction in fidelity provides strong evidence for the feasibility of executing stabilizer codes on timescales much longer than the intrinsic coherence times of the constituent qubits.
Fault tolerant quantum error correction is a procedure which satisfies the feature that if one of the gates in the procedure has failed then the failure causes at most one error in the output qubits of the encoded block. Quantum computer is based on the idea of two quantum state systems (Qubits). However, the majority of systems are constructed from higher than two- level subspace. Bad control and environmental interactions in these systems lead to leakage fault. Leakage errors are errors that couple the states inside a code subspace to the states outside a code subspace. One example for leakage fault is loss errors. Since the fault tolerant procedure may be unable to recognize the leakage fault because it was designed to deal with Pauli errors. In that case a single leakage fault might disrupt the fault tolerant technique. In this thesis we investigate the effect of ancilla losses on fault-tolerant quantum error correction in the [[7,1,3]] Steane code. We proved that both Shor and Steane methods are still fault tolerant if loss errors occur.
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.