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The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes

Michael E. Beverland, Aleksander Kubica, Krysta M. Svore

arXiv:2101.02211v1quant-ph

TL;DR

Two-dimensional topological codes lack a natural transversal non-Clifford gate, leaving the overhead of alternative T-gate implementations an important question. The paper simulates state distillation and code switching under circuit noise and finds that code switching provides no substantial space or space-time savings over state distillation.

  • Problem

    Because 2D topological codes cannot implement non-Clifford gates transversally, the overhead trade-off between state distillation and code switching to a 3D code needs direct comparison.

    Source: A. Motivation and summary of results · pp. 3–4

  • Method

    The authors simulate state distillation and an optimized code-switching procedure for preparing high-fidelity T states in 2D color codes under circuit noise.

    Source: A. Motivation and summary of results · p. 6

  • Results

    Code switching offers no substantial space or space-time savings over state distillation; its T-gate circuit-noise threshold is 0.07(1)%, versus 0.37(1)% for state distillation.

    Source: A. Motivation and summary of results · pp. 4, 6

  • Takeaways & Limitations

    The comparison informs choices of fault-tolerant schemes and hardware designs; the authors expect the overhead finding to hold for other topological codes such as the toric code.

    Source: A. Motivation and summary of results · p. 6

Abstract

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Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and color codes which have high thresholds but lack a natural implementation of a non-Clifford gate. In this work, we directly compare two leading FT implementations of the T gate in 2D color codes under circuit noise across a wide range of parameters in regimes of practical interest. We report that implementing the T gate via code switching to a 3D color code does not offer substantial savings over state distillation in terms of either space or space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting. To arrive at these results, we provide and simulate an optimized code switching procedure, and bound the effect of various conceivable improvements. Many intermediate results in our analysis may be of independent interest. For example, we optimize the 2D color code for circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D color code with perfect measurements under Z noise. Our work provides a much-needed direct comparison of the overhead of state distillation and code switching, and sheds light on the choice of future FT schemes and hardware designs.

I. INTRODUCTION

The introduction frames implementing non-Clifford gates in 2D topological codes as a challenge and compares state distillation with code switching for preparing high-fidelity T states. Its main finding is that code switching offers no substantial space or space-time overhead savings over state distillation.

A. Motivation and summary of results

  • 2D topological codes lack transversal non-Clifford gates, motivating a comparison of magic-state distillation with switching to a 3D code that supports transversal T.The study assumes geometrically local noisy quantum operations in 3D and globally available, perfect classical operations.Source: A. Motivation and summary of results · pp. 3–4
  • Code switching offers no substantial savings in qubit or space-time overhead; state distillation outperforms it across most circuit-noise rates 10^-4 ≤ p ≤ 10^-3 and target infidelities 10^-20 ≤ pfin ≤ 10^-4.At the smallest physical error rates, code switching slightly outperforms state distillation; more optimized distillation schemes could further favor distillation.Source: A. Motivation and summary of results · p. 4
  • The 2D color-code circuit-noise threshold exceeds 0.37(1)%, the highest reported for that code in the study.The authors adapt the projection decoder for boundaries and imperfect syndrome extraction, optimize stabilizer-extraction circuits, and analyze noise equilibration during logical operations.Source: A. Motivation and summary of results · p. 5
  • The optimized restriction decoder for the 3D color code with a boundary achieves a threshold of 0.80(5)% and performs better for small system sizes.Source: A. Motivation and summary of results · p. 6
  • 0.07(1)% is the code-switching T-gate circuit-noise threshold, compared with 0.37(1)% for state distillation.The code-switching protocol’s overhead was estimated numerically and bounded under conceivable improvements, including optimal measurement circuits and classical algorithms.Source: A. Motivation and summary of results · p. 6

B. Noise and simulation

  • Circuit-noise simulations apply depolarizing noise to operations and allow preparation or measurement faults with probability p; every elementary operation takes one time unit.The paper also considers depolarizing and phenomenological noise models.Source: B. Noise and simulation · p. 7

C. Basics of 2D and 3D color codes

The paper uses triangular 2D and tetrahedral 3D color-code lattices, whose logical structure supports Clifford operations and, in the 3D stabilizer code, a transversal T gate. For computation with 2D color codes, it implements T by teleportation using an encoded T state and performs continuous error correction.

C. Basics of 2D and 3D color codes

  • The studied lattice families are a triangular 2D lattice with triangular boundary and a bcc 3D lattice with tetrahedral boundary.Their code distances are d = 3, 5, 7, . . .; the 2D lattice can be derived from the 3D lattice by retaining vertices connected to a boundary vertex and their simplices.Source: C. Basics of 2D and 3D color codes · p. 9
  • The 2D color code has one logical qubit, string-like logical operators, and transversal implementation of the full Clifford group.The 3D subsystem color code also implements the full Clifford group transversally, with sheet-like bare and string-like dressed logical operators.Source: C. Basics of 2D and 3D color codes · pp. 9–10
  • A transversal T gate is implemented in the 3D stabilizer color code by applying T and T^-1 to two differently colored groups of tetrahedra.The groups contain (n + 1)/2 white and (n − 1)/2 black tetrahedra, with no same-colored pair sharing a face.Source: C. Basics of 2D and 3D color codes · p. 10

D. Fault-tolerant computation with 2D color codes

  • A QEC cycle is a full stabilizer-extraction round that produces a syndrome; with faulty measurements, decoding uses the history of syndromes to find a correction.The decoder fails when the correction combined with the accumulated errors is a nontrivial logical operator.Source: D. Fault-tolerant computation with 2D… · pp. 10–11
  • In the 2D color-code stack, adjacent patches support geometrically local transversal CNOT and SWAP operations, while an encoded T state enables T-gate teleportation using Clifford operations.The paper also notes that a planar lattice-surgery layout is possible, but some elementary operations such as CNOT would be slower than in a stack.Source: D. Fault-tolerant computation with 2D… · pp. 10, 12, Figure 5 · p. 11

E. State distillation

State distillation uses Clifford operations and postselection to turn noisy resource states into fewer, less noisy states. The section reviews three protocol classes and the Haah-Hastings family, while the overhead analysis focuses on the standard 15-to-1 protocol.

E. State distillation

  • State distillation uses Clifford operations and postselection to convert noisy resource states into fewer, less noisy states.The paper's overhead analysis considers the standard 15-to-1 protocol, while assuming perfect Clifford operations in this background review.Source: E. State distillation · p. 12
  • The three protocol classes use a transversal resource rotator before code projection, a resource stabilizer before code projection, or a resource stabilizer measurement after encoding.The first class suppresses infidelity from q to O(q^d); the second can postselect unsuccessfully even for perfect resource states.Source: E. State distillation · pp. 12–13
  • 15 input T states of infidelity q yield an output of infidelity 35q^3 with acceptance probability 1 − 15q in the standard 15-to-1 protocol.Knill's protocol and the first Bravyi-Kitaev scheme have these parameters.Source: E. State distillation · p. 13
  • High-rate k/n schemes produce multiple outputs but tend to have large Clifford circuits that may hinder practicality when Clifford noise is included.The reviewed Haah-Hastings family includes the 15-to-1 protocol analyzed in detail in the paper.Source: E. State distillation · pp. 13–14
  • Haah-Hastings protocols prepare a Reed-Muller state with a depth-(3r + 1) Clifford circuit on 2^(3r + 1) qubits before puncturing and distillation.The resulting scheme uses n − k encoded T states and outputs k states with infidelity A q^dZ.Source: E. State distillation · pp. 13–14

II. 2D COLOR CODE ANALYSIS

This section adapts the 2D color-code projection decoder to boundaries and faulty syndrome measurements, then evaluates its performance under depolarizing and phenomenological noise. It also shows why finite-size performance and long-time fault tolerance require care when interpreting decoder results.

II. 2D COLOR CODE ANALYSIS

A. Projection decoder with boundaries

  • The boundary-aware projection decoder pairs highlighted syndrome vertices by color pair, then corrects the smaller region bounded by the combined pairings.Boundary vertices are added when a color-pair vertex set has odd size.Source: A. Projection decoder with boundaries · p. 16, Figure 8 · p. 17
  • 12.45(4)% is the conservative threshold estimate under depolarizing noise, taken from the maximum observed crossing at d = 29.The extrapolated crossing and pseudo-threshold intercepts differ, cautioning against naive linear extrapolation from small systems.Source: A. Projection decoder with boundaries · p. 17

B. Noisy-syndrome projection decoder with boundaries

  • For faulty measurements, the decoder matches syndrome flips in a space-time lattice and locally flattens each connected pairing component to obtain corrections.The lattice extends the 2D code by a time dimension representing QEC cycles.Source: B. Noisy-syndrome projection decoder… · pp. 18–19
  • 4.38(3)% is the linearly extrapolated intercept for the long-time pseudo-threshold under phenomenological noise.The estimate uses data for d ≥7.Source: Figure 11 · p. 20, B. Noisy-syndrome projection decoder… · p. 20
  • Unlike local flattening, Stephens’ global flattening has vanishing success probability after sufficiently many cycles at finite noise and therefore lacks a finite error-correction threshold.The authors report that its time-dependent pseudo-threshold fails to stabilize at large cycle counts.Source: B. Noisy-syndrome projection decoder… · pp. 20–21

C. Optimizing stabilizer extraction and circuit noise analysis

The authors optimize the CNOT schedule used to extract 2D color-code stabilizers under circuit noise. Their best schedule yields a 0.37(1)% threshold, while schedules of the same length can have substantially different failure rates.

C. Optimizing stabilizer extraction and circuit noise analysis

  • The search retained the 234 valid schedules with the fewest time units, after requiring schedules to meet concurrency and correct-syndrome-extraction conditions.No six-time-unit schedule met the conditions; 234, 4854, and 39160 valid schedules were found for 7, 8, and 9 time units, respectively.Source: C. Optimizing stabilizer extraction… · pp. 21–22
  • 12.8(1)% was the best schedule’s logical failure probability, versus 21.1(1)% for the worst length-7 schedule.The best schedule’s QEC cycle takes 8 time units; the comparison used d = 7 and seven cycles at p = 0.0035.Source: C. Optimizing stabilizer extraction… · p. 23, Figure 13 · p. 23
  • 0.37(1)% is the estimated circuit-noise threshold for the best-performing CNOT schedule.The long-time pseudo-threshold differs by almost a factor of two between the best- and worst-performing length-7 schedules at d = 13.Source: Figure 13 · p. 23, C. Optimizing stabilizer extraction… · p. 23

D. Modelling noise in logical operations

The effective noise model estimates logical-operation failure rates under circuit noise after accounting for QEC equilibration. It treats operations as independent and conservatively models their nontrivial logical Pauli errors.

D. Modelling noise in logical operations

  • The effective model assigns failure probabilities to preparation, idle, CNOT, SWAP, and measurement operations, with logical Pauli measurements assumed perfect.It is designed to overestimate each nontrivial logical Pauli and assumes operations fail independently.Source: D. Modelling noise in logical operations · pp. 23–24
  • The model uses conservative idle Pauli probabilities, while SWAP is treated as two idle qubits despite potentially significant propagated errors.The authors note that large-distance failure estimates are loose and could be tightened.Source: D. Modelling noise in logical operations · pp. 23–24
  • d QEC cycles before an operation are used to reach equilibrium, where failure probability grows linearly with cycle count.The simulations use the best-performing CNOT schedule and a distance-d patch requires (3d^2 −1)/2 qubits.Source: D. Modelling noise in logical operations · pp. 24, 26
  • The idle failure rate pidle(p, d) is estimated from the slope of failure probability versus QEC cycles after equilibration.The initial perfect state produces an artificially reduced early failure probability and a negative fit intercept.Source: D. Modelling noise in logical operations · pp. 25–26
  • For transversal CNOT, two post-gate QEC cycles suffice for equilibration, and propagated errors make ZI and IX the dominant failures.The CNOT propagates X errors from control to target and Z errors from target to control.Source: D. Modelling noise in logical operations · pp. 26–27
  • Readout noise is orders of magnitude below other logical-operation noise, so it is neglected in the effective model.Measurement failure probabilities are reported only for p = 10^-3 because lower-noise estimates require prohibitively many samples.Source: Table I · p. 24, D. Modelling noise in logical operations · p. 28, Figure 18 · p. 28

III. STATE DISTILLATION ANALYSIS

This section analyzes the performance and estimates the overhead of the standard 15-to-1 protocol for distilling T states.

III. STATE DISTILLATION ANALYSIS

  • The section analyzes the performance of T-state distillation using the standard 15-to-1 scheme.The section also estimates the scheme’s overhead.Source: §III · p. 29
  • The analysis concerns state distillation of T states.This section’s stated scope is the standard 15-to-1 scheme.Source: §III · p. 29
  • The section estimates the overhead associated with the standard 15-to-1 T-state distillation scheme.Source: §III · p. 29

A. Creating the T state via state distillation in 3 steps

The procedure initializes encoded T states in small 2D color-code patches, expands and moves the patches between rounds, then distills 15 inputs into one higher-fidelity output. These stages are repeated at increasing code distances to reach the target quality.

2. Expansion and movement of patches

  • Between distillation rounds, patches are enlarged to higher code distance and moved to their next-round locations.The analysis neglects expansion and movement noise and time overhead, assuming these fit within the needed QEC rounds.Source: A. Creating the T state via state… · p. 29, §2 · p. 31

3. 15-to-1 state distillation circuit

  • Each 15-to-1 round uses distance-d 2D color-code patches stacked in 3D and nearest-neighbor logical Clifford operations to produce one higher-fidelity T state.The circuit analysis keeps errors through order q^3 and p, where q is input-state infidelity and p is effective circuit-noise strength.Source: A. Creating the T state via state… · p. 29, §3 · p. 31, Figure 21 · p. 32

1. T state initialization

  • Initialization prepares a T state in a small patch, measures each stabilizer twice, rejects inconsistent syndromes, and applies a Pauli correction.The CNOT order is optimized; code expansion adds negligible noise under the stated assumption that its errors are suppressed like error-correction errors.Source: §1 · p. 30, §2 · p. 31
  • The best d = 3 initialization CNOT schedule has a lowest-order failure probability more than twofold below the worst schedule.The authors searched 234 valid seven-time-unit schedules.Source: §1 · p. 30

3. 15-to-1 state distillation circuit

  • The analyzed distillation round has rejection probability 15q + 12.3 pprep + (466 + 4.13d) pidle + 51.7 pCNOT.The contributions combine input-state errors with preparation, idle, and CNOT faults in the effective noise model.Source: §3 · pp. 31, 34

B. State distillation overhead

The overhead analysis searches round counts and code distances to meet a target output infidelity. Under the assumed perfect Clifford operations, the estimated state-distillation threshold is set by 2D color-code error correction.

B. State distillation overhead

  • Expected qubit overhead accounts for round acceptance probabilities and is set by the most qubit-expensive distillation round.The analysis considers average overhead across many output states.Source: B. State distillation overhead · p. 34
  • The overhead search varies the number of rounds from one to three and the code distance at each round to meet target infidelity.It compares choices that minimize either qubit or space-time overhead.Source: B. State distillation overhead · pp. 34–35, Figure 23 · p. 35
  • 0.37(1)% is the estimated threshold for 15-to-1 state distillation with the 2D color code, with error correction identified as the bottleneck.The first distillation round’s critical input error rate is estimated at about 4%, above the color-code circuit-noise threshold.Source: B. State distillation overhead · p. 35

IV. FURTHER INSIGHTS INTO 3D COLOR CODES

The 3D subsystem color code provides a common framework for switching between 2D and 3D stabilizer color codes through gauge fixing. Its gauge flux is constrained by a local Gauss law, and linked flux components organize the relevant excitation configurations.

A. A simple way to switch between 2D and 3D color codes

  • Gauge fixing switches the 3D subsystem color code to either the 3D stabilizer code or a 2D color code with spherical codes in the interior.The two stabilizer-code choices share bare logical operators with the subsystem code, allowing a transfer that preserves logical information.Source: A. A simple way to switch between 2D… · pp. 36–37, Figure 24 · p. 37

B. Physics of the gauge flux in 3D color codes

C. Restriction decoder for 3D color codes with boundaries

The adapted restriction decoder pairs restricted syndromes, lifts the resulting edges to local corrections, and handles the boundary with a 2D decoder. A color-repeated variant improves the measured threshold and weight-2 correction range; the code-switching protocol then uses gauge fixing to transfer a Bell-linked state for transversal T application.

C. Restriction decoder for 3D color codes with boundaries

  • The decoder pairs syndrome vertices in restricted lattices, lifts those pairings locally, and uses a 2D projection decoder to lift the boundary vertex.The boundary lifting problem is equivalent to decoding the 2D color code on the nearby tetrahedral-lattice facet.Source: C. Restriction decoder for 3D color… · pp. 39–40
  • 0.80(5)% is the threshold after repeating the restriction decoder across four colors and selecting the lowest-weight correction, up from 0.55(5)%.The modification reduces the lowest distance that corrects all weight-2 errors from d = 11 to d = 7.Source: C. Restriction decoder for 3D color… · p. 40, Figure 27 · p. 40

A. Creating the T state via code switching in 6 steps

  • Code switching prepares a Bell state across two 2D patches, switches one patch to a 3D stabilizer code, applies transversal T, and measures the 3D data to leave an encoded T state.The protocol avoids switching back from 3D to 2D; decoding the 3D measurements determines whether a logical Z correction is needed.Source: A. Creating the T state via code… · pp. 41–42, Figure 28 · p. 41

1. Preparing the Bell state in 2D codes

The code-switching analysis estimates the effects of the Bell-pair preparation, 3D-interior preparation, gauge measurements, and gauge fixing. Optimistic substitutions for multiple steps suggest a threshold ceiling of 0.22(5)% if all considered improvements are achievable.

1. Preparing the Bell state in 2D codes

  • The encoded Bell state is prepared fault-tolerantly in two distance-d 2D color codes, taking 8d time units.The protocol finds a marginal benefit from using the patch initialized in |+⟩ for code switching.Source: §1 · pp. 42–43
  • 0.22(5)% is the estimated threshold if all four considered code-switching steps are replaced by optimistic versions.Improving interior preparation and developing a better decoder offer the most potential, according to the analysis.Source: §1 · p. 43, Figure 29 · p. 43

2. Preparing the 3D interior

  • The 3D interior is prepared as spherical 2D color-code states, with invalid syndromes triggering repetition and, if still invalid, a randomly flipped outcome.Because the spherical codes encode no logical qubits, this step does not require the distance-proportional QEC cycles used for logical-state preparation.Source: §2 · p. 44

3. Measuring gauge operators

  • Measuring the selected Z-edge gauge operators takes 8 time units with one ancilla per edge, though ancilla preparation can overlap the final interior-preparation time unit.The resulting restricted noisy gauge flux may violate the Gauss law because of measurement faults.Source: §3 · p. 44

4. Gauge fixing

  • Gauge fixing validates each noisy restricted flux by pairing violation points, then applies an X-type gauge operator consistent with the validated flux.The procedure assumes the restricted flux can be treated as arising from a flux without branching points plus measurement errors; the optimistic bound instead corrects prior X errors and identifies the gauge exactly.Source: §4 · p. 45

5. Applying T and measuring the 3D code’s data qubits

The transversal T operation and subsequent X-basis measurements are difficult to simulate because residual X errors do not commute with T. The simulation models the measurement syndrome using residual-error flux structure, while omitting a possible logical Z contribution and linking charge.

5. Applying T and measuring the 3D code’s data qubits

  • Residual X errors do not commute with the logical T gate, and their flux can affect measurement outcomes by flipping outcomes along the Z-gauge flux.The syndrome model samples excitation configurations uniformly from linked flux components.Source: §5 · p. 46
  • The simulation ignores the possibility of introducing a logical Z at this stage, which the authors say underestimates code-switching failure.This caveat applies to the simulated T-and-measurement step.Source: §5 · p. 46
  • The simulation samples excitation configurations without linking charge rather than from the full configuration collection.The authors do not expect this simplifying assumption to substantially affect performance.Source: §5 · p. 46

6. Decoding Z errors in the 3D color code

The section describes the modified decoder used to recover the output T state, estimates code-switching overhead, and compares it with state distillation. Its results show that code switching is generally more costly except near the lowest studied physical error rates, while asymptotic scaling favors distillation in a specified regime.

6. Decoding Z errors in the 3D color code

  • The modified restriction decoder adapts Ref. [31] to boundaries and selects the lowest-weight correction from four independently generated candidates.The final decoder uses the X-measurement outcomes to produce a Z-type correction and read off the logical X measurement.Source: §6 · p. 47
  • 1.9% is the known threshold of the optimal decoder for the 3D stabilizer color code under iid Z noise.The paper uses this benchmark when estimating the effect of possible decoder improvements on code switching.Source: §6 · p. 47

B. Code switching overhead

  • The overhead estimate selects the smallest odd code distance meeting the target infidelity, then calculates qubit and space-time costs for baseline and optimistic cases.The plotted overheads vary with output infidelity and physical error rate; no curve is shown for p = 0.001 because that rate exceeds the observed code-switching threshold.Source: B. Code switching overhead · pp. 47–48

VI. DISCUSSION AND ACKNOWLEDGMENTS

  • 0.07(1)% is code switching’s observed threshold versus 0.37(1)% for state distillation; switching is not viable at 10^-3 but slightly outperforms near 10^-4.Near 5·10^-4, code switching is viable but requires considerably more overhead; even a highly optimistic implementation provides no substantial savings in the studied regimes.Source: §VI · p. 49
  • For log pfin / log p ≫ 1, the asymptotic analysis concludes that state distillation outperforms code switching.The analysis gives ΓCS = 3 and ΓSD = max(2, log_F R); for 15-to-1 distillation, log_F R ≈ 2.46, while extrapolation to more modest ratios predicts a possible crossover whose location is implementation-sensitive.Source: §VI · p. 49
  • The authors identify improved preparation, syndrome extraction, and decoding, as well as directly applying the T gate, as future research directions.They also suggest exploring code switching at very low error rates and broadening searches for distillation schemes beyond those with very low log_F R.Source: §VI · p. 49

APPENDICES

The appendices provide supporting lattice specifications and simulation analyses, then detail distillation initialization and preparation of the 3D interior used for code switching. They report that syndrome-repeat and syndrome-flip strategies improve code-switching performance.

A. Lattice parameters and specifications

B. Supplementary details for 2D color code simulations

  • The fitted decay parameters stabilize rapidly with distance, indicating residual noise reaches equilibrium over a time independent of system size.For the optimized circuit, the long-time pseudo-threshold is 0.193(2)%, compared with 0.126(3)% for the poor circuit.Source: B. Supplementary details for 2D color… · pp. 52, 54

C. Supplementary details for distillation analysis

  • Puncturing qubit three gives the smallest leading-order failure probability when pCNOT ≥ pprep ≥ pidle.The initialization simulation can start from a perfect code state, apply a syndrome-dependent Pauli operator, run noisy circuits, correct the observed syndrome, and test whether the net error is correctable.Source: C. Supplementary details for… · pp. 55–56
  • Across 234 valid CNOT schedules, the analysis identifies a lowest-failure schedule with pinit_fail = 6.07p.This schedule is used in the Reed–Muller-state preparation analysis for the 15-to-1 distillation scheme.Source: C. Supplementary details for… · pp. 55–56

D. Preparing the 3D interior for code switching

E. Choices of basis for code switching preparation

The section selects preparation bases for code switching, specifies gauge-measurement circuits, and estimates how much better an idealized 3D decoder might perform. The tested basis choices have little effect, while correlated and uncorrelated Z-error cases show similar decoder performance.

E. Choices of basis for code switching preparation

  • The tested choices of 2D-patch and 3D-interior preparation basis make very little difference to performance; the optimized protocol uses |+⟩ for both.The basis choice matters because the preparation can produce asymmetric X- and Z-type noise, which later protocol steps treat differently.Source: E. Choices of basis for code switching… · p. 60

F. Gauge measurement circuits for code switching

G. Optimistic improvements for the 3D color code decoder

  • The optimistic decoder estimate assumes the error-weight proxy and iid-Z-based failure behavior, while acknowledging that correlation-aware decoders can outperform uncorrelated-noise decoders.The authors regard the assumptions as approximate and support them by comparing decoder performance on correlated and uncorrelated errors.Source: G. Optimistic improvements for the 3D… · p. 63
  • At distance d = 9, the modified restriction decoder performs very similarly on correlated protocol-generated and uncorrelated fixed-weight Z errors.The correlated errors’ stabilizer-reduced weight is estimated using the smaller of the error’s weight and its correction’s weight; the two cases are similar but not identical.Source: G. Optimistic improvements for the 3D… · p. 63, Figure 41 · p. 64

H. Asymptotic overhead analysis

The asymptotic analysis estimates space and space-time overheads for code switching and iterative state distillation under small physical error and large target-string ratios. It predicts distillation dominance in that regime, with the first or last distillation round determining space overhead depending on log_F R.

H. Asymptotic overhead analysis

  • The analysis assumes p ≪ 1 and log pfin / log p ≫ 1, so code distances and the number of distillation rounds are large.For the code-switching implementation, leading-order resource factors are N3D = 13/12 qubits and τ3D = 8 time units per distance.Source: H. Asymptotic overhead analysis · p. 65
  • For state distillation, the first round dominates space overhead when log_F R > 2; otherwise, the last round dominates.The modeled 15-to-1 scheme uses R = 15 and F = 3, while the analysis assumes Clifford errors are negligible in each round.Source: H. Asymptotic overhead analysis · pp. 65–66
  • The asymptotic expressions are expected to apply generally to 2D-to-3D code switching and to state distillation using 2D codes.For the stated large-ratio regime, the discussion concludes that state distillation outperforms code switching.Source: H. Asymptotic overhead analysis · pp. 65–66
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