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arXiv:1410.8562 · Quantum Information and Computation

Leakage suppression in the toric code

Martin Suchara, Andrew W. Cross, Jay Gambetta

Published Sep 1, 2015• 27 citations
arXivHTMLPDF

Abstract

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 $4$ when leakage and depolarizing noise rates are comparable. This becomes a factor of $2$ when the decoder uses 3-outcome measurements. Finally, when the physical error rate is less than $2\times 10^{-4}$, 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.

DOI
10.26421/qic15.11-12-8

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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.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [ ⁣[7,1,3] ⁣][\![7,1,3]\!][[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×\times× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−55.1 \times 10^{-5}5.1×10−5 under realistic noise conditions with physical error rates of 10−310^{-3}10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
In this thesis we present three main contributions to the field of topological quantum error correcting codes. We focus on some of the properties of such codes required for fault-tolerant quantum computation. Prior work has concentrated on determining error rate thresholds of particular models, but increasingly other parameters are gaining prominence. One of these is the overhead -- the quantity of a named resource required to achieve a desired level of performance from the code. We characterise the qubit overhead of the toric code in a fault-tolerant setting. These results provide a general framework for determining the overhead for other code constructions with more complicated noise models. Next we introduce a decoding algorithm, applicable to topological codes in a qudit architecture, specifically those where fault-tolerance is achieved through repeated syndrome measurements. It is computationally light and capable of decoding qudits of arbitrarily high dimension with negligible increase in its run time. The threshold of the decoder is limited by the percolation of the syndromes. Using local matching techniques we are able to overcome this limitation, increasing the threshold by almost a factor of two for high qudit dimensions. Finally, we turn our attention to a second family of topological quantum codes: the colour codes. In three and higher spatial dimensions such codes can support transversal non-Clifford gates. We show, using a technique that we call a star-bipartition of the vertices of the lattice, that any existing qubit colour code lattice can be used to define a qudit colour code. By generalising the notion of triorthogonal matrices we derive analogous transversality properties in the qudit codes.
Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC).We propose a scheme using two leakage-reduction units (LRUs) that mitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QECcycle time as in previous proposals.For data qubits, we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly disturbs the computational states for realistic system parameters.For ancilla qubits, we apply a |1 ↔ |2 π pulse conditioned on the measurement outcome.Using density-matrix simulations of the distance-3 surface code, we show that the average leakage lifetime is reduced to almost one QEC cycle, even when the LRUs are implemented with limited fidelity.Furthermore, we show that this leads to a significant reduction of the logical error rate.This LRU scheme opens the prospect for near-term scalable QEC demonstrations.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we’re trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor’s cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane’s encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM’s Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 ×10^−5 under realistic noise conditions with physical error rates of 10^−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.