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Leakage Suppression in the Toric Code

Martin Suchara, Andrew W. Cross, Jay M. Gambetta

arXiv:1410.8562v1quant-ph

TL;DR

Leakage faults take qubits outside the computational subspace, and their impact and mitigation costs in topological codes remain unclear. This paper models leakage, simulates leakage-reducing circuits and decoders, and reports that thresholds depend on leakage rates and decoding information.

  • Problem

    Leakage faults are excluded from common depolarizing-noise estimates, and their impact and mitigation cost in topological codes remain unclear.

    Source: §1 · pp. 1–2

  • Method

    The authors develop a stochastic leakage model and use Monte Carlo simulations to compare leakage-reducing circuits and syndrome-processing strategies.

    Source: §3 · p. 5, §8 · p. 18

  • Results

    Thresholds decrease monotonically as relative leakage increases across the studied circuits and decoders, while three-outcome measurements could in principle improve on the idealized-decoder threshold.

    Source: §7 · pp. 16–17

  • Takeaways & Limitations

    For r = s = 1, the Full-LRU circuit has the lowest threshold among the compared circuits and decoders, while its success rate improves faster at low physical error rates than Quick with the Standard decoder.

    Source: §7 · pp. 17–18

    The two-qubit gate model destroys correlations from leaked-qubit interactions and is not a worst-case stochastic model.

    Source: §3 · p. 6

Abstract

from arXiv · show

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 accuracy threshold and can increase overhead and experimental complexity, and these costs have not been thoroughly understood. Our work explores a variety of techniques for leakage-resilient, fault-tolerant error correction in the context of topological codes. Our contributions are threefold. First, we develop a leakage model that differs in critical details from earlier models. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform three-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulation results show that simple circuits with one extra CNOT per qubit 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 x 10^-4, placing LRUs after every gate may achieve the lowest logical error rates of all of the circuits we considered. We expect the closely related planar and rotated codes to exhibit the same accuracy thresholds and that the ideas may generalize naturally to other topological codes.

1 Introduction

Leakage faults take qubits outside the computational subspace, a fault type absent from common depolarizing-noise threshold estimates. The paper addresses the unclear impact and cost of leakage mitigation in topological codes by studying leakage-reducing circuits and syndrome processing.

1 Introduction

  • Leakage faults move qubit states outside the computational subspace, which standard depolarizing-noise threshold estimates do not model.Many physical qubits are multilevel systems expected to experience leakage.Source: §1 · p. 1
  • Leakage-detection gadgets flag leakage as located loss, whereas LRUs convert leakage into regular errors without indicating its occurrence.Aliferis and Terhal showed an accuracy threshold for leakage in concatenated codes using LRUs.Source: §1 · p. 2
  • The impact of leakage on topological-code accuracy thresholds and the cost of mitigating it remain unclear.Earlier work addresses detected loss and leakage in repetition codes or particular superconducting gates, but leaves this broader impact unresolved.Source: §1 · p. 2
  • The paper systematically studies leakage-reducing circuits and develops a syndrome-processing strategy for topological codes.Its organization includes a leakage model, circuit and decoder designs, three-outcome measurements, simulations, and numerical results.Source: §1 · p. 2

2 Toric Code

The toric code places data qubits on a periodic square lattice and uses vertex and face checks to encode two logical qubits. Fault-tolerant correction combines repeated stabilizer measurements with decoding that matches syndrome defects.

2 Toric Code

  • The toric code has n = 2d^2 data qubits on the edges of a periodic d by d square array and encodes two logical qubits.Its check operators are attached to vertices and faces.Source: §2 · pp. 2–3
  • Each vertex star is an X check and each face plaquette is a Z check, with each check acting on four data qubits.The checks commute because each star and face share zero or two edges.Source: §2 · p. 3
  • Each cycle measures all checks with ancillas prepared and measured in the appropriate bases, using four CNOT steps between ancillas and neighboring data qubits.One cycle yields 2d^2 noisy syndrome bits.Source: §2 · pp. 3–4, Figure 1 · p. 3
  • Fault-tolerant correction uses O(d) syndrome cycles, then independently decodes bit- and phase-flip syndromes by minimum-weight matching.Matching connects pairs of defects on decoding graphs whose edges represent circuit error events; edge weights approximate negative log error probabilities.Source: §2 · pp. 4–5

3 Leakage Errors

The paper models leakage as stochastic qutrit faults that remain compatible with stabilizer-based simulation. Its simplifying assumptions make the model tractable but exclude some correlations that could arise around leaked qubits.

3 Leakage Errors

  • Each qubit has a computational subspace HC[j] and auxiliary leakage space HL[j]; leakage and relaxation occur with probabilities p↑ and p↓.Leakage events occur independently on gate outputs, while relaxation is analogous to amplitude damping.Source: §3 · pp. 5–6
  • The stochastic leakage model serves as a tractable proxy for leakage phenomena that can be coherent and not accurately captured by Markovian noise.It assumes ideal gates are sealed and each subsystem is a qutrit.Source: §3 · p. 5
  • When a leaked qubit decays, it is replaced by a maximally mixed contained qubit, and the model preserves mixtures of leaked and contained states.Such states are called |2⟩-dephased.Source: §3 · p. 6
  • Sealed two-qubit gates depolarize a contained qubit interacting with a leaked qubit, while gates with both inputs leaked act as identity in the simulation model.Noisy gates also apply independent excitation and relaxation maps to their output qutrits.Source: §3 · pp. 6–8
  • The two-qubit gate model destroys correlations among contained qubits interacting with leakage, so it is not a worst-case stochastic model.The effect of a worst-case stochastic model is left for future work; leaked qubits interact with few neighbors and persist only a few syndrome cycles before reset.Source: §3 · p. 6
  • The simulator tracks each qubit with a Pauli-error or leakage label instead of retaining the full stabilizer state.The model's depolarization and resets remove correlations needed to predict syndrome history from a full stabilizer description.Source: §3 · pp. 6–7
  • The model sets measurement-error probability q = p and initializes data qubits at a steady-state leakage rate determined by excitation and relaxation.For p↑≈p↓, the stated equilibrium leakage fraction is 40%, beyond which the threshold is far exceeded without leakage reduction.Source: §3 · pp. 7–8

4 Leakage Reduction

The section introduces leakage reduction units (LRUs) and compares error-correction circuits that trade leakage suppression against circuit overhead. It also describes how decoding-graph weights are calculated for these circuits.

4 Leakage Reduction

  • An LRU converts leakage into a regular error; in this model, one-bit teleportation implements one using sealed two-qubit gates.The LRU acts as identity on computational-space inputs and outputs a computational-space state for leakage-space inputs.Source: §4 · pp. 8–9
  • Full-LRU applies an LRU after every gate, using 16 additional gates in four LRUs per data and ancilla qubit per cycle.No LRU is applied to ancillas before measurement.Source: §4 · p. 9
  • Partial-LRU applies one LRU to each data qubit per cycle, adding four gates per data qubit without additional ancilla gates.Unlike Full-LRU, it does not attempt to reduce leakage on ancillas before their periodic measurement and reinitialization.Source: §4 · p. 9
  • Quick adds one CNOT per data qubit, swaps each data qubit with an ancilla at cycle end, and resets each physical qubit every other cycle.The toric-code implementation requires no extra ancillas.Source: §4 · pp. 9–10
  • The leakage-reduction circuits add no new decoding-graph edges; Full-LRU weights on edges e and f are four times those of the other circuits.The Standard decoder uses these weights but does not account for leakage, so the authors expect it to be suboptimal.Source: §4 · p. 11

5 Heralded Leakage Reduction

The section uses three-outcome measurements to identify leakage events and adapt syndrome decoding without changing the quantum circuits. The decoder conditions graph weights on the observed events while accounting for uncertainty about when leakage occurred.

5 Heralded Leakage Reduction

  • Three-outcome measurements let the decoders use leakage information without circuit changes, improving logical error rates and threshold in the studied approach.The “L” outcome is assumed to occur if and only if the measured qubit is leaked, but it does not reveal the exact leakage location.Source: §5 · p. 11
  • The HL decoder builds a conditional graph with edge weights based on fault probabilities given the “L” events; the graph can be non-translation-invariant and include new edges.The decoder uses local prior probabilities because an “L” event only locates leakage between initialization and measurement.Source: §5 · pp. 11–12
  • For syndrome checks missing during consecutive “L” measurements, the decoder assigns zero weight to the corresponding time edges and places a defect if the endpoint syndrome bits differ.The defect is placed at the earliest vertex in the boundary of the first such edge.Source: §5 · p. 12
  • For an “L” event, the decoder considers possible leakage locations between initialization and measurement, then adjusts edge probabilities for the resulting X and Z errors.It estimates leakage at the i-th location as i/n and assigns each interacting qubit an X or Z error probability of i/2n.Source: §5 · p. 13
  • In the Quick circuit example, an ancilla leak can cause a long X-error edge between defects α and δ with conditional probability approximately 1/22.The added edge reflects an X error spreading from U through subsequent interactions and syndrome rounds.Source: §5 · p. 14
  • In the Partial-LRU example, an ancilla leak yields a Z error on R with probability 3/10, modifying a diagonal edge between defects at times t and t+1.The passage also describes conditional graph updates for leakage detected on data qubits by LRUs.Source: §5 · p. 15

6 Simulation Methods

The simulations compare leakage-reduction circuits and decoders on distance-d toric codes using the paper’s leakage and regular-error model. They process d noisy rounds alongside a final perfect syndrome round.

6 Simulation Methods

  • Each simulation propagates regular and leakage errors for d rounds, then adds a perfect syndrome round and processes all d+1 syndromes together.A failure is registered when the corrected code state anticommutes with a logical operator.Source: §6 · p. 15
  • The study evaluates No LRU, Partial-LRU, Full-LRU, and Quick; it tests the HL decoder for Partial-LRU and Quick, and the Standard decoder for all four circuits.Source: §6 · p. 15
  • Persistent qubits begin in leakage with the model’s equilibrium probability, including data qubits that are never reinitialized in No LRU.Source: §6 · p. 15
  • Because the HL graph is not translation-invariant, the simulation builds the full graph and uses shortest paths to determine distances between defects.The Standard graph instead allows distance calculation directly from defect coordinates.Source: §6 · p. 15
  • Monte Carlo runs continued to at least 10,000 iterations and 1,000 failures per configuration, using about 30,000 CPU-hours.Source: §6 · pp. 15–16

7 Results

The results compare toric-code thresholds across leakage rates and circuit/decoder choices, finding that leakage lowers thresholds while the HL decoder improves them at substantial leakage. Full-LRU has the lowest threshold in the plotted r = s = 1 comparison, but may yield better logical-error performance sufficiently below threshold.

7 Results

  • Threshold estimates use crossovers between failure rates for distances d = 7 and d = 9, with r varied in 0.1 increments and p in 0.01% increments.The simulations set s = 1.Source: §7 · p. 16
  • With no leakage, No LRU’s threshold is about 0.70%, versus about 0.22% for Full-LRU; Quick and Partial-LRU are nearly identical except for r < 0.3.The authors relate the similar Quick and Partial-LRU thresholds to a balance between fewer gates and qubits versus more frequent reinitialization and leakage suppression.Source: §7 · p. 16, Figure 11 · p. 16
  • When r exceeds about 0.3, the HL decoder significantly improves the threshold over the Standard decoder.Figure 11(a) summarizes thresholds for both decoders.Source: §7 · p. 16, Figure 11 · p. 16
  • Thresholds decrease monotonically as leakage increases; in an idealized replacement model, leakage contributes to an effective error rate of (1 + βr)p with β = 3/4.The model treats a leaked qubit as replaced by a completely depolarized one.Source: §7 · p. 16
  • The approximate threshold decay is α/(1 + βr), where α is the no-leakage threshold; the Standard decoder does not exceed the idealized-decoder threshold.A decoder using three-outcome measurements could in principle exceed that idealized threshold.Source: §7 · p. 17
  • For Quick with HL, the threshold changes little with the relative relaxation rate s, which the authors attribute to slower relaxation than circuit leakage reduction.The plot uses the threshold-recording method described for Figure 11.Source: §7 · p. 17
  • At r = s = 1, Full-LRU has the lowest threshold, followed by Quick with Standard and then Quick with HL; Partial-LRU is indistinguishable from Quick.The thresholds appear as crossing points of the failure-rate curves.Source: §7 · p. 17, Figure 12 · p. 17
  • Well below threshold, Full-LRU’s success rate improves faster than Quick with the Standard decoder, suggesting it may perform better at sufficiently low physical error rates.The authors leave lower-error-rate exploration for future work because such simulations are inefficient and may require different techniques.Source: Figure 12 · p. 17, §7 · p. 18

8 Conclusion

The conclusion reports that simpler leakage-mitigation circuits preserve toric-code thresholds relatively well, while three-outcome decoding improves thresholds and Full-LRU may perform best at low physical error rates. It identifies higher distances and lower error rates as areas for future study.

8 Conclusion

  • Below a physical error rate of 2×10^-4, Full-LRU may achieve the lowest logical error rates among the circuits considered.The authors suggest further study at higher distances and far below threshold, potentially using the splitting method [23].Source: §8 · p. 18
  • In a realistic scenario where leakage was added to depolarizing noise, Partial-LRU and Quick reduce the toric-code accuracy threshold by less than a factor of 4; HL decoding improves threshold by a factor of 2 over Standard decoding.The HL Decoder uses information about leakage-detection events.Source: §8 · p. 18
  • The study combines a leakage model with Monte Carlo comparisons of toric-code circuits and optimized decoders for measurements with and without leakage detection.The circuits range from adding one CNOT per syndrome extraction to reducing leakage after every gate.Source: §8 · p. 18
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