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Critical faults of leakage errors on the surface code
Natalie C. Brown, Andrew W. Cross, Kenneth R. Brown
TL;DR
Leakage can spread errors and corrupt syndrome measurements, and data and ancilla leakage affect the surface code differently. The paper analyzes depolarizing leakage by qubit type and circuit location, and finds that effective distance can be maintained by controlling data leakage or critical ancilla leakage.
Problem
Leakage can spread errors, corrupt syndrome measurements, and accumulate, while its effects on data and ancilla qubits may differ.
Source: §I · p. 1Method
The paper studies depolarizing leakage separately on data and ancilla qubits, isolates critical circuit locations, and develops leakage-handling schemes.
Source: §I · p. 1, §IV · p. 5, §V · pp. 5–6Results
Ancilla leakage is more damaging than data leakage; effective distance is maintained with data-confined leakage and an LRC, or by eliminating ancilla leakage at critical locations.
Source: §III · p. 3, §VI · p. 7Takeaways & Limitations
The analysis identifies data-leakage removal each syndrome round and control of ancilla leakage after initialization and the first CNOT as conditions for maintaining effective distance.
Source: §VI · p. 7
Abstract
from arXiv · showhide
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 show 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 aniclla qubit leakage at the critical fault location. We also introduce new techniques for handling leakage by using gates with one-sided leakage and by mixing two types of leakage reducing circuits: one to handle data leakage and one to handle ancilla leakage.
I. INTRODUCTION
Leakage leaves the computational subspace and can spread errors, corrupt measurements, and accumulate in surface codes. The paper focuses on depolarizing leakage, separates data from ancilla leakage, and investigates where leakage must be controlled.
I. INTRODUCTION
- Leakage occurs when a qubit leaves the computational states; unlike erasure or qubit loss, its location is unknown.The paper notes that leakage occurs in several architectures, including trapped ions, quantum dots, superconducting qubits, and anyons [13] [16] [23] [24].Source: §I · p. 1
- Leakage can spread errors to other qubits, corrupt measurements, and accumulate unless removed, although extra circuits allow a surface-code threshold.Source: §I · p. 1
- Resource-efficient leakage-reducing circuits can suppress effective code distance, making it important to locate critical leakage faults while minimizing overhead.Source: §I · p. 1
- Interacting leakage can induce errors on an unleaked qubit, while non-interacting leakage does not change its state through the two-qubit interaction.The depolarizing interacting model suppresses surface-code distance; non-interacting models tend to be less damaging, though single-qubit gates may still affect unleaked qubits.Source: §I · p. 1
- The study examines depolarizing leakage by separating data and ancilla leakage, isolating critical circuit locations, and constructing fault-tolerant schemes.Source: §I · p. 1
II. TOPOLOGICAL SURFACE CODES AND LEAKAGE REDUCING CIRCUITS
Surface codes use repeated stabilizer measurements to detect errors, but leakage can reduce their effective distance. Leakage-reducing circuits remove leakage by exchanging leaked qubits with reinitialized qubits, with different designs trading overhead against leakage control.
II. TOPOLOGICAL SURFACE CODES AND LEAKAGE REDUCING CIRCUITS
- A single leakage error in the depolarizing model can produce a two-qubit error chain, halving effective distance and limiting guaranteed correction to ⌊(d−1)/4⌋ physical errors.The distance suppression is a known effect of leakage on the surface code.Source: §II · p. 2
- Surface-code syndrome extraction initializes ancillas, applies four CNOTs to neighboring data qubits, then measures the ancillas.Source: §II · p. 2
- Because surface codes are not designed to detect or correct leakage, leakage must be actively removed; at the logical level, LRCs convert it into detectable and correctable Pauli-type errors.Most proposed LRCs swap newly prepared qubits for leaked ones; they differ in how much swapping is done and the overhead of additional qubits.Source: §II · p. 2
- The Full LRC swaps in new qubits for data and ancilla after every gate and doubles qubit count, while the circuit LRC swaps data only at cycle end and adds half as many qubits.Source: §II · p. 2
- The SWAP LRC exchanges data and ancilla at each syndrome-extraction cycle, using ancilla measurement and reset to remove leakage without extra qubits.It adds one gate and has the lowest overhead among circuit-level leakage-reduction techniques.Source: §II · p. 2, Figure 2 · p. 2
III. ANCILLA LEAKAGE VS DATA LEAKAGE
The comparison finds ancilla leakage substantially more damaging than data leakage. Effective distance is maintained when leakage is confined to data qubits and removed with an LRC.
III. ANCILLA LEAKAGE VS DATA LEAKAGE
- The simulations compare an ancilla-only leakage system without an LRC against a data-only leakage system using a SWAP LRC.The authors state that adding a SWAP LRC when leakage is limited to ancillas can let leakage persist longer and adds fault locations.Source: §III · p. 3
- Data leakage requires an LRC because data qubits are never reset, while ancilla leakage is more damaging in the reported comparison.Source: §III · p. 3
- Effective distance is maintained for data-confined leakage with an LRC, whereas ancilla-confined leakage suppresses code distance.The logical-error-rate comparison reports a substantial gain for data-confined leakage handled by a SWAP LRC versus ancilla-confined leakage; the SWAP LRC adds one gate.Source: Figure 3 · p. 3, §III · p. 3
A. Propagation of errors from data leakage
A leaked data qubit can induce an error on a stabilizer ancilla, which then spreads errors to other data qubits. Removing leakage every syndrome-extraction round is necessary for the reported distance-maintenance result.
A. Propagation of errors from data leakage
- One leaked data qubit induces an error on a stabilizer ancilla, which spreads three additional errors to data qubits.The passage contrasts this propagation with a Pauli error, which would not propagate in the same way.Source: A. Propagation of errors from data… · p. 3
- Without a SWAP LRC, leakage persists across syndrome rounds, so corrections based on compromised syndrome information do not remove the leakage.Adding the SWAP gate removes leakage and allows the proper correction to be applied.Source: A. Propagation of errors from data… · p. 3
- Simulations compare SWAP-LRC use every round with use every other round; the conclusion is that effective distance can be maintained when data leakage is removed every round.Source: A. Propagation of errors from data… · p. 3
B. Propagation of errors from ancilla leakage
Ancilla leakage can create malignant hook errors that run along logical operators, with leakage after the first CNOT producing the most damaging case. SWAP LRCs must be applied every syndrome-extraction round to prevent data-leakage errors from continuing to spread.
B. Propagation of errors from ancilla leakage
- A leaked ancilla after the first CNOT can produce four hook errors along the logical supports, the hook errors implicated in code-distance suppression.For the toric code, two different hook errors can occur because it encodes two logical qubits.Source: B. Propagation of errors from ancilla… · p. 3, Figure 6 · p. 4
- Hook errors parallel to a logical operator are malignant, while two-qubit errors perpendicular to it are benign.Clever gate scheduling can minimize harmful hooks but cannot eliminate them.Source: B. Propagation of errors from ancilla… · p. 3
- Without an LRC, a data leak can spread three or more errors across later syndrome-extraction rounds; SWAP LRC reduces the resulting configuration to one physical error.The data leak spreads a Z error through an ancilla to other data qubits.Source: Figure 4 · p. 4
- SWAP LRC must be applied every syndrome-extraction round to effectively mitigate errors spreading from data leakage.The figure compares applying it every round with applying it every other round.Source: Figure 5 · p. 4
- The findings motivate circuit designs that confine leakage to data qubits.Source: B. Propagation of errors from ancilla… · p. 4
IV. CONFINING LEAKAGE TO DATA
The gate-biased circuit uses one-sided leakage behavior and gate identities to confine leakage to data qubits. In simulations, it outperformed the standard circuit and preserved code distance, at the cost of 12 additional single-qubit gates.
IV. CONFINING LEAKAGE TO DATA
- A one-sided leakage model makes one qubit in a two-qubit gate much more likely to leak than the other.The comparison assumed only controls leak; because the standard circuit has controls on both data and ancilla qubits, it is a two-sided leakage model.Source: §IV · pp. 4–5
- The gate-biased circuit uses 12 additional H gates to confine leakage to data qubits, where SWAP LRC can remove it.Conjugating a CNOT with H gates reverses its direction.Source: §IV · p. 5, Figure 7 · p. 5
- With leakage-to-depolarizing gate noise at 1:1, the gate-biased circuit outperforms the standard circuit and preserves code distance despite 12 added single-qubit gates.Source: Figure 8 · p. 5, §IV · p. 5
V. CRITICAL LEAKAGE LOCATIONS
Leakage at the first CNOT is the critical source of hook errors and code-distance suppression; leakage at later CNOTs maintains the effective distance. Initialization leakage shares the first-CNOT hook risk and was excluded from the simulations.
V. CRITICAL LEAKAGE LOCATIONS
- Only leakage at the first CNOT produces hook errors and suppresses effective code distance; leakage at the second, third, or fourth CNOT maintains it.After the second CNOT, leakage causes at most two additional physical errors along the stabilizer support.Source: §V · p. 5
- Initialization leakage can produce the same hook error as first-CNOT leakage and is more damaging because it permits more hook-error combinations.Source: §V · p. 5
- The first H gate in the X-stabilizer extraction circuit is also critical; minimizing leakage at initialization and the first CNOT addresses this location as well.Source: §V · pp. 5–6
- The distance-3 toric-code simulations show a significant logical-error-rate improvement when leakage from the first CNOT is disallowed.Only one CNOT caused leakage in these simulations, and initialization leakage was excluded.Source: Figure 9 · p. 6
A. Optimized gate biased circuit
The optimized gate-biased circuit targets early critical leakage locations rather than fully confining leakage to data qubits. It maintains code distance with four additional single-qubit gates and offers lower circuit depth than the 12-gate gate-biased circuit.
A. Optimized gate biased circuit
- The optimized design needs four rather than 12 additional single-qubit gates because leakage only needs to be eliminated at early critical locations.Source: A. Optimized gate biased circuit · p. 6, Figure 10 · p. 6
- Flipping the second CNOT removes leakage at the critical location, but this scheme does not eliminate initialization leakage.Changing only the first CNOT leaves the circuit open to single-qubit gate leakage at a critical location.Source: A. Optimized gate biased circuit · p. 6
- The optimized circuit maintains code distance and improves on both the standard and gate-biased circuits, with lower depth than the gate-biased circuit.Its advantage over the gate-biased circuit is not substantial.Source: Figure 11 · p. 6, A. Optimized gate biased circuit · p. 6
B. Mixed LRC
The mixed LRC combines ancilla replacement at a critical point with a final SWAP LRC for data leakage. Its effective distance is maintained, with doubled ancilla count.
B. Mixed LRC
- The gate-biased and optimized circuits provide fault-tolerant solutions only under a one-sided leakage model.The paper identifies critical leakage locations as relevant to fault-tolerant schemes for a two-sided model.Source: B. Mixed LRC · p. 6
- Replacing the ancilla after the second CNOT handles critical leakage from initialization and the first CNOT; a final SWAP LRC handles data leakage.Swapping in a new qubit mid-circuit eliminates hook errors that could arise from ancilla leakage.Source: B. Mixed LRC · pp. 6–7, Figure 12 · p. 7
- The ancilla SWAP must follow the second CNOT: an earlier swap may miss critical leakage or induce leakage at a critical location.After the second CNOT, any newly introduced leakage is not at a critical fault location.Source: B. Mixed LRC · p. 7
- The new LRC maintains the surface code’s effective distance while handling initialization leakage at a 1:1 leakage-to-gate-noise ratio.The authors describe it as an effective means of fault-tolerantly handling leakage.Source: B. Mixed LRC · p. 7
- The mixed LRC doubles the ancilla count, which the authors consider the most efficient fault-tolerant LRC for a two-sided leakage model.Their efficiency comparison concerns qubit count and logical error rate.Source: B. Mixed LRC · p. 7
VI. CONCLUSIONS
Ancilla leakage is more damaging than data leakage, but the surface code can maintain its effective distance under specified leakage-handling conditions. The conclusions point to critical fault locations and possible extensions to detection, physical mitigation, and other codes.
VI. CONCLUSIONS
- A single ancilla leakage fault can cause a hook error and a logical error in one syndrome-extraction round; data leakage requires failure to remove it each round.If leakage is confined to data qubits and efficiently removed every round, the effective distance can be maintained.Source: §VI · p. 7
- With a 1:1 leakage-to-depolarizing-noise ratio, the mixed LRC maintains effective distance while handling initialization and first-CNOT leakage.The mixed LRC combines two circuits to address both error types at the first CNOT.Source: Figure 13 · p. 7
- After scheduling minimizes but does not eliminate hooks, the remaining critical faults are after initialization and after the first CNOT.Eliminating ancilla leakage at those locations and efficiently removing data leakage preserves effective distance.Source: §VI · p. 7
- Leakage detection and flag-qubit schemes could help identify critical faults and inform decoding strategies.The authors propose these as future directions.Source: §VI · p. 7
- Physical leakage-elimination techniques at critical locations, potentially mixed with LRCs, could help minimize overhead.The authors identify investigation of different physical mechanisms as future work.Source: §VI · p. 7
- The authors hope the analysis informs leakage handling in other codes, particularly the triangular color code, and future large-scale architectures.They state leakage will not limit current devices.Source: §VI · p. 7