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Universal quantum computing with twist-free and temporally encoded lattice surgery

Christopher Chamberland, Earl T. Campbell

arXiv:2109.02746v3quant-ph

TL;DR

The paper addresses the challenges of measuring arbitrary Pauli operators and reducing lattice-surgery runtime within qubit and routing constraints. It introduces a decoder, twist-free surgery, temporal encoding, and a core-cache layout. The reported protocols include a 2× runtime reduction from temporal encoding without additional qubit overhead, while twist-free surgery incurs an approximate 2× slowdown.

  • Problem

    In surface codes, measuring Pauli operators containing Y terms is more difficult and has been addressed with twist defects, while prior time-optimal approaches can require large qubit-cost increases.

    Source: §IV · p. 10, A. Parallelizable Pauli measurements · p. 12

  • Method

    The paper develops a lattice-surgery decoder, a twist-free protocol for arbitrary Pauli measurements, temporal encoding of measurement sequences, and a core-cache architecture.

    Source: §VII · p. 19, Appendix C · p. 22

  • Results

    Temporal encoding achieves a 2× runtime reduction for practical-scale algorithms without additional qubit overhead, while twist-free surgery incurs a 2× runtime slowdown.

    Source: §VII · p. 19

  • Takeaways & Limitations

    Temporal encoding can further improve runtime for more parallelizable or larger algorithms through larger classical code distances and other code families.

    Source: §VII · p. 19

    Whether twist-based surgery can outperform the twist-free protocol remains unknown without careful numerical analysis of twists.

    Source: §VII · p. 19

Abstract

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Lattice surgery protocols allow for the efficient implementation of universal gate sets with two-dimensional topological codes where qubits are constrained to interact with one another locally. In this work, we first introduce a decoder capable of correcting spacelike and timelike errors during lattice surgery protocols. Afterwards, we compute logical failure rates of a lattice surgery protocol for a biased circuit-level noise model. We then provide a new protocol for performing twist-free lattice surgery, where we avoid twist defects in the bulk of the lattice. Our twist-free protocol eliminates the extra circuit components and gate scheduling complexities associated with the measurement of higher weight stabilizers when using twist defects. We also provide a protocol for temporally encoded lattice surgery that can be used to reduce both runtimes and the total space-time costs of quantum algorithms. Lastly, we propose a layout for a quantum processor that is more efficient for rectangular surface codes exploiting noise bias, and which is compatible with the other techniques mentioned above.

I. INTRODUCTION

The introduction presents lattice-surgery decoding, twist-free measurements, temporal encoding, and a core-cache architecture; the review frames these methods in Pauli-based computation and its routing and runtime constraints.

I. INTRODUCTION

  • The proposed decoder corrects spacelike and timelike lattice-surgery errors, and simulations evaluate an X ⊗X measurement under biased circuit-level noise.The introduction also says the decoder supports numerical failure-rate analysis for that measurement.Source: §I · p. 1
  • The twist-free protocol measures arbitrary Pauli operators while avoiding extra circuits and scheduling complexity, at an approximate 2× runtime cost.The relative performance of twist-based surgery remains unquantified.Source: §I · p. 1
  • Temporal encoding protects fast, noisier lattice-surgery measurements by encoding their sequence in a classical error-correcting code, yielding a 2.2× runtime improvement in a practical-scale regime.It adds no direct qubit overhead and improves overall space-time complexity.Source: §I · p. 1
  • The core-cache architecture uses thin rectangular surface-code strips under large noise bias, with routing adding 1.5× resource costs versus 2× for Litinski’s fast data access structures.Compact caching can reduce routing overhead at the cost of additional read/write time.Source: §I · p. 1

II. BRIEF REVIEW OF UNIVERSAL QUANTUM COMPUTING VIA LATTICE SURGERY

  • Pauli-based computation drives a computation through sequential multi-qubit Pauli measurements, with later measurements conditioned on earlier outcomes and Clifford corrections deferred to the end.The review also describes lattice surgery as the means of measuring these operators in a two-dimensional layout.Source: A. Overview of Pauli based computation · p. 2, §II · p. 2

A. Overview of Pauli based computation

  • Commuting Pauli operators may still require sequential measurement when they compete for routing space, giving seqPBC runtime proportional to (dm + 1)µ.Here, dm is the number of syndrome rounds and µ is the number of sequential Pauli measurements.Source: A. Overview of Pauli based computation · p. 2
  • For Clifford plus T and Toffoli computations, the measurement count is µ = NA + NT + 3NTOF + NunTOF, commonly approximated as NT + 4NTOF.The approximation applies when Toffolis appear in compute/uncompute pairs and algorithmic readouts are few.Source: A. Overview of Pauli based computation · p. 3
  • Algorithm runtime is max{Tmagic, TPBC}; increasing magic-state factories can reduce Tmagic but raises qubit costs, motivating approaches that beat the Pauli-measurement bottleneck.The review introduces temporal encoding as an alternative to prior speedups that use additional teleportation gadgets.Source: A. Overview of Pauli based computation · p. 3

B. Overview of lattice surgery

Lattice surgery merges surface-code patches through routing-space gauge fixing to measure multi-qubit Pauli operators, but twist defects used for Y terms introduce demanding stabilizers. The section motivates a decoder that handles both spacelike and timelike failures, including measurement-parity errors and logical-qubit errors.

B. Overview of lattice surgery

  • Lattice surgery merges surface-code patches through routing-space gauge fixing, reads the measured Pauli parity from stabilizers, then splits the patches by measuring routing-space qubits.For Z-type measurements, domain walls prevent a reduction in minimum-weight representatives of logical Z operators.Source: B. Overview of lattice surgery · p. 3
  • Twist-based Y measurements extend patches and use weight-five stabilizers, while twist defects add hardware difficulty beyond elongated stabilizers.The figure identifies long-range gates or merged ancillas as implementation challenges for elongated stabilizers, and weight-five measurement as an additional twist-defect challenge.Source: B. Overview of lattice surgery · p. 4, Figure 3 · p. 4
  • Timelike measurement failures are exponentially suppressed as measurement rounds dm increase, until dm ≫ O(dz, dx), when logical Pauli errors dominate.Here dm is the measurement distance; dz and dx are the minimum weights of logical Z- and X-type operators.Source: B. Overview of lattice surgery · p. 4

III. DECODING TIMELIKE ERRORS DURING LATTICE SURGERY

  • The numerical study reports multi-qubit Pauli measurement failure versus syndrome-round count, alongside logical-qubit failure rates..Source: §III · p. 5
  • Earlier decoder work did not specify boundary vertices for spacelike and timelike failures together, and prior circuit-level simulations did not model both realistically.The cited comparison describes Ref. [33] as using idealized boundaries that excluded logical spacelike errors.Source: §III · p. 5

A. The decoding algorithm

  • The MWPM decoder adds past and future ancilla vertices at the merge boundaries, enabling correction of timelike failures alongside spacelike errors.Past vertical edges have zero weight, while future vertical-edge weights are derived from timelike failure processes; sufficiently long consecutive parity-measurement errors are needed for a timelike failure.Source: A. The decoding algorithm · pp. 5–6
  • Transition vertices track pre-merge boundaries that become parity vertices, so highlighted corrections at those vertices can require flipping the measured Pauli parity.The algorithm also combines data-qubit corrections from highlighted edges using modulo-two arithmetic.Source: A. The decoding algorithm · pp. 6–7

B. Decoder simplifications

The decoder can be simplified by removing a set of never-highlighted vertices and consolidating boundary vertices. The simulation methodology uses a biased circuit-level noise model, classifies failure mechanisms, and applies MWPM to the full syndrome history.

B. Decoder simplifications

  • Vertices in V(r+1) are never highlighted, so they can be removed while their weights are assigned to past vertical edges; boundary vertices can also be merged into one.In that representation, each matching-graph edge encodes both a spacelike and a timelike correction.Source: B. Decoder simplifications · p. 7

C. Noise model and simulation methodology

  • The simulations use a biased circuit-level noise model with η = 100 and assume equal durations for CNOT gates and ancilla measurement and re-initialization.The model also includes a single idle location on data qubits during ancilla measurement and re-initialization.Source: C. Noise model and simulation… · pp. 7–8
  • For dx = 9, dz = 11, r = dz, and ℓ = 5, Figure 9 compares the fitted P(0,1,0,0) polynomial with Monte Carlo data across dm values.Translucent data points were omitted when fitting the polynomial.Source: Figure 9 · p. 9
  • The authors label 15 failure mechanisms by four bits for left-patch Z error, timelike failure, right-patch Z error, and X error.They report polynomials for three mechanisms and choose dx so that μP(0,0,0,1) ≤ δ, given the high noise bias.Source: C. Noise model and simulation… · p. 9
  • Simulations use rounds 1 through r + dm and a final perfect-error-correction round, decoding the full syndrome history without perfect correction between rounds r and r + 1..Source: C. Noise model and simulation… · p. 9

D. Simulation results and conclusions

The authors fit failure-rate polynomials to lattice-surgery simulation data and compare the timelike-failure fit with Monte Carlo results. The fitted rates show exponential suppression of purely timelike errors with measurement distance, and the polynomials are used for algorithm resource estimates.

D. Simulation results and conclusions

  • Purely timelike failure probabilities decrease exponentially with dm, and the Monte Carlo data agree well with the best-fit polynomials.The comparison is for P(0,1,0,0), with the figure caption specifying dx = 9, dz = 11, r = dz, and ℓ = 5.Source: D. Simulation results and conclusions · p. 9
  • The failure-rate polynomials give error rates per syndrome measurement round, with dz referring to the merged patch in Eq. (6) and to individual patches in Eqs. (4) and (5)..Source: D. Simulation results and conclusions · p. 9
  • The authors use the fitted logical-failure polynomials to estimate quantum-algorithm implementation overheads..Source: D. Simulation results and conclusions · p. 10
  • For a measurement of k logical patches, the failure labels generalize to k + 2 bits representing patch errors, parity-measurement failure, and an error affecting all merged qubits..Source: D. Simulation results and conclusions · p. 10

IV. PROTOCOL FOR TWIST-FREE LATTICE SURGERY

The twist-free protocol measures Pauli operators containing Y terms using XZ-Pauli lattice surgery, avoiding twist defects at the cost of roughly 2× runtime and up to two additional logical ancillas.

IV. PROTOCOL FOR TWIST-FREE LATTICE SURGERY

  • Twist defects make Y-containing Pauli measurements difficult because each requires a weight-five stabilizer, which can add ancillas, failures, scheduling burden, and correlated errors.The authors also cite reduced effective code distance for twist-based surface codes with an MWPM decoder.Source: §IV · p. 10
  • XZ-Pauli measurements use only weight-four stabilizers, whereas Pauli operators containing Y terms require additional support in standard surface-code lattice surgery.The paper defines an XZ-Pauli as a tensor product of {1l, X, Z}.Source: §IV · p. 10
  • The protocol avoids twist defects’ extra circuit components and gate-scheduling complexities, while the runtime cost of twist-based measurements remains unquantified.The authors note that the relative performance of the two approaches is an open question because twist performance lacks data.Source: §IV · p. 10
  • For even numbers of Y terms, the protocol decomposes P into commuting X[u] and Z[v] operators and masks their individual outcomes to measure P.The resulting measurement outcome is mx ⊕ mz ⊕ c, where c depends on u and v.Source: §IV · p. 10, Figure 10 · p. 11
  • For odd numbers of Y terms, a reusable |Y⟩ ancilla enables measurement of Y ⊗ P, which has an even number of Y terms.The ancilla is unaffected by this measurement, so its preparation cost is paid once per algorithm.Source: §IV · pp. 10, 12
  • The protocol adds up to two logical ancillas and doubles runtime relative to measuring a Pauli operator without Y terms.If steps 2 and 3 each fail with probability P, the protocol failure probability is P′ = 2P(1 − P) ≈ 2P.Source: §IV · pp. 10, 12
  • The 2D layout maps the protocol to XZ-Pauli measurements, using routing space for a temporary |0⟩ ancilla and a designated region for elongated stabilizers.The elongated checks may use adapted hardware or a modified measurement procedure, such as GHZ ancillas or a flag qubit, with possible additional failure or scheduling costs.Source: Figure 11 · p. 11, §IV · p. 12

V. TEMPORAL ENCODING FOR FAST LATTICE SURGERY

Temporal encoding uses fast, noisy lattice-surgery operations and encodes the measured Pauli sequence in a classical error-correcting code. This corrects errors from the extra noise without direct qubit overhead, though magic-state injection can add a small qubit cost.

V. TEMPORAL ENCODING FOR FAST LATTICE SURGERY

  • Temporal encoding uses fast, noisier lattice-surgery operations and corrects their extra errors by encoding the sequence of measured Pauli operators in a classical error-correcting code.The encoding takes place in the time domain.Source: §V · p. 12
  • The time-domain encoding does not directly add qubit overhead, but magic-state injection can require a small additive qubit cost because states are stored longer.Source: §V · p. 12
  • Temporal encoding is presented as a way to run algorithms at faster clock speeds than standard lattice surgery.Source: §V · p. 12

A. Parallelizable Pauli measurements

The section defines parallelizable Pauli sets and motivates temporal encoding as a way to exploit them without the qubit overhead of time-optimal approaches. It formalizes redundant measurements as a classical measurement code whose distance controls undetected timelike errors.

A. Parallelizable Pauli measurements

  • A parallelizable set contains commuting Pauli measurements whose Clifford corrections can move to the sequence’s end; a circuit with μ T-gates and T-depth γ yields γ sets averaging k = μ/γ.The paper gives magic-state implementation of T^⊗k as an example.Source: A. Parallelizable Pauli measurements · p. 12
  • Time-optimal PBC reduces an n-qubit computation of T-depth γ to O(n + γ) runtime, but does not compress space-time volume; Litinski’s examples require over 6× qubit cost for 56.5% runtime and over 20× for 11%.The authors note that qubit limits matter for early fault-tolerant computers; the estimates may rise further with code-distance overhead.Source: A. Parallelizable Pauli measurements · p. 12

B. Encodings and code parameter proofs

  • For {P1, P2, P1P2}, a product mismatch flags a lattice-surgery failure and permits remeasurement, adding a measurement while tolerating one failure.The passage contrasts this with measuring P1 and P2 alone, which takes 2(dm + 1) error-correction cycles.Source: B. Encodings and code parameter proofs · p. 13
  • TELS represents an overcomplete generating set with a full-rank [n, k, d] measurement-code generator, where n counts physical measurements and d is the minimum row-span weight.The number k is the number of unencoded Pauli operators in the generating set.Source: B. Encodings and code parameter proofs · p. 13
  • A measurement code of distance d detects timelike-error patterns below weight d, with error-detection failure probability O(P^d) for single-error probability P.The distance is the lowest nonzero weight in the row-span of the generator matrix.Source: B. Encodings and code parameter proofs · p. 14
  • The authors say this is the first use of overcomplete stabilizer-measurement dependencies to improve lattice-surgery runtimes, despite related concepts appearing as measurement codes, metachecks, symmetries, and syndrome measurement codes.They qualify the novelty claim as being to the best of their knowledge.Source: B. Encodings and code parameter proofs · p. 14

C. Examples and numerics

The examples compare measurement-code choices and error-handling strategies for temporally encoded lattice surgery (TELS). Extended Hamming codes reduce runtime in the illustrated practical regime, while classical-code properties imply a stronger asymptotic scaling advantage.

C. Examples and numerics

  • Extended Hamming measurement codes detect up to three errors, while concatenated single-error-detection codes have parameters [(α+1)^m, α^m, 2^m].The [8, 4, 4] code is the m = 3 member of the extended-Hamming family.Source: C. Examples and numerics · p. 14
  • The detect/remeasure strategy uses a distance-d measurement code for dm rounds, then remeasures the original Pauli set for ⌈qdm⌉ rounds after a detected error.The remeasure round uses the original set P rather than the overcomplete set S.Source: C. Examples and numerics · p. 14
  • Choosing q = d makes the total failure probability O(p^d d_m/2), although constant factors may shift the numerically optimal q.A detected error adds a kqdm remeasurement delay, but the passage says its effect on expected runtime is minimal because p_d is small.Source: C. Examples and numerics · p. 14
  • Error correction is less favorable than detect/remeasure because it roughly doubles the leading runtime term while removing a smaller term.The comparison attributes the smaller term to the low probability p_d.Source: C. Examples and numerics · p. 14
  • For parallelizable gates, extended-Hamming TELS runs at 46% of standard seqPBC runtime, implying about 2.2× speedup on practical problems with average k between 9 and 14.The comparison assumes δ = 10^-15; TELS has no multiplicative qubit overhead, but requires storing all k magic states for the protocol’s full duration.Source: C. Examples and numerics · p. 15, Figure 13 · p. 15
  • For large k, good classical codes imply TELS can measure k parallelizable Paulis in O(k) time at error δ = O(p^αk), versus O(k polylog(k)) for unencoded seqPBC.The passage says higher-distance codes may help at lower error rates or under a changed noise model.Source: C. Examples and numerics · p. 15

VI. THE CORE-CACHE ARCHITECTURE AND ROUTING OVERHEADS

The architecture combines a quantum core with surface-code patches and routing space, with a cache option for reducing routing overhead. Thin rectangular patches are particularly relevant when noise is strongly biased.

VI. THE CORE-CACHE ARCHITECTURE AND ROUTING OVERHEADS

  • The architecture uses rectangular surface-code patches of size dz by dx with qubits between them reserved as a lattice-surgery bus or routing space.Fast access means every patch’s logical X and Z operators are adjacent to that routing space.Source: §VI · p. 15
  • The proposed quantum core provides a fast data-access layout without requiring Litinski’s two-tile, two-qubit hexon patches.The cited work describes those patches as one option in Litinski’s data-access structures.Source: §VI · p. 15
  • Thin rectangular patches (dz ≫ dx) can lower routing overhead in the high-noise-bias setting, and a core-cache model can further reduce routing costs by temporarily storing logical qubits.The cache introduces time costs when logical qubits are swapped in and out.Source: §VI · p. 15

A. A quantum computer core

The core groups surface-code patches into routed unit cells, with boundary padding and patch spacing chosen to support fast Pauli measurements without bulk twists. The cache lowers routing footprint by temporarily storing qubits, with access and swap costs shaping when it is useful.

A. A quantum computer core

  • A rectangular device of height h tiles and width w tiles has T = wh tiles and 2T + w + h + 1 qubits; a surface-code patch uses d_x d_z tiles and has 2d_xd_z qubits.The patch actively uses 2d_xd_z − 1 data and ancilla qubits, so the tightest possible 2D packing contains one idling qubit per patch.Source: A. A quantum computer core · p. 16
  • Four surface-code patches form a unit cell, and repeated unit cells form a padded core with each patch’s logical X and Z boundaries connected to routing space.The unused spacing permits lattice surgery without twists, unlike data-access structures described as assuming liberal use of twists.Source: A. A quantum computer core · p. 16, Figure 14 · p. 16
  • In the large-bias limit dz ≫ dx, the core’s routing overhead is 1.5×, compared with 2× for Litinski’s fast data blocks.For unbiased noise with dz = dx, the proposed scheme’s asymptotic overhead is 9/4; it has a routing advantage when dx < (2/3)dz.Source: A. A quantum computer core · p. 17

B. A quantum computer cache

  • The core-cache design stores some logical qubits in a cache to reduce routing footprint, then swaps them into the core when needed.The layout combines the core with a cache and magic-state factories.Source: B. A quantum computer cache · p. 17
  • Cached qubits support single-qubit X or Z measurements and multi-qubit measurements with cache-side X operators; accessing their Z logical operators requires swapping them into the core.Cache-side multi-qubit measurements take dm time, while single-qubit measurements take zero time.Source: B. A quantum computer cache · p. 17
  • The cache reduces routing overhead with mild runtime impact when cache swaps are infrequent, and can be combined with twist-free or temporally encoded lattice surgery.More general swaps may require up to six lattice-surgery operations when measurements are not XZ-type and twists are avoided.Source: B. A quantum computer cache · pp. 17–18
  • Write-to-cache and read-from-cache are teleportation protocols using Clifford-frame-adapted Pauli measurements; together they swap a core and cache qubit using three multi-qubit measurements.Write-to-cache requires two multi-qubit Pauli measurements and a cache |0⟩ ancilla; read-from-cache uses one multi-qubit and one zero-time single-qubit measurement.Source: B. A quantum computer cache · p. 18

VII. CONCLUSION

The conclusion brings together a lattice-surgery decoder, twist-free measurements, temporal encoding, and a core-cache architecture. It reports runtime and routing-cost outcomes while identifying twist benchmarking and alternative codes as future directions.

VII. CONCLUSION

  • The lattice-surgery decoder and failure-rate polynomials support choosing code distances d_x and d_z and syndrome-round count dm for algorithm implementations.The analysis numerically treats dominant failure mechanisms of an X ⊗ X Pauli measurement.Source: §VII · p. 19
  • The twist-free protocol measures arbitrary Pauli operators with a 2× runtime slowdown, avoiding higher-weight stabilizers and added gate-scheduling complexity associated with twists.Whether twist-based surgery can outperform that cost remains unresolved without careful numerical analysis.Source: §VII · p. 19
  • Temporal encoding reduces practical-scale algorithm runtime by 2× without additional qubit overhead; larger code distances or other code families may improve runtime further for larger or more parallel algorithms.The conclusion notes that temporal encoding may also reduce runtime in magic-state distillation factories.Source: §VII · p. 19
  • The core-cache architecture reduces routing footprint by caching some data qubits and adds a 1.5× routing factor to total lattice-surgery resources.The conclusion identifies analyzing such architectures with twists as future work.Source: §VII · p. 19
  • Future work includes applying the methods to other codes, including XZZX surface codes, high-threshold subsystem codes, and high-rate hyperbolic surface codes.These are proposed candidates for potentially lower resource costs.Source: §VII · p. 19

Appendix A: Twist-free proof

The appendix proves the twist-free protocol’s measurement behavior by analyzing its two possible step-5 outcomes. The resulting Pauli-measurement outcome and correction depend on the branch and on u·v modulo 4.

Appendix A: Twist-free proof

  • For the q = 0 branch with even u·v, mx ⊕ mz is the outcome when u·v = 0 (mod 4), while mx ⊕ mz ⊕ 1 is the outcome when u·v = 2 (mod 4).The projector identities support the branch analysis; for u·v = 2 (mod 4), adding a Y operator to P and using a |Y⟩ ancilla gives c = 1.Source: Appendix A · pp. 19–20
  • For the q = 1 branch, M−1 differs from M+1 by a Z[v] correction applied in step 6; the global phase (−1)^mz is unimportant.The branch projects onto |1⟩, and the stated projector identities give the relation used to identify the correction.Source: Appendix A · p. 19
  • For u·v = 3 (mod 4), the proof sets c = 0 after relating the measurement to the case u·v = 2 (mod 4) with an additional Y operator.The passage also states that for u·v = 2 (mod 4), c = 1.Source: Appendix A · p. 20

Appendix B: Treating space-time correlated edges incident to parity vertices

The appendix specifies how the decoder treats spacetime-correlated edges incident to parity vertices during the first merged-patch round. Their weights and inclusion depend on whether they can affect the X ⊗ X measurement outcome.

Appendix B: Treating space-time correlated edges incident to parity vertices

  • Purple edges incident to routing-space vertices in rounds r + 1 and r + 2 are removed because random routing-space stabilizer outcomes cannot be highlighted, while such edges in later rounds have finite weights.The edges are assigned infinite weight in those early rounds.Source: Appendix B · p. 20
  • Red edges can arise from gate failures whose errors flip only parity vertices in that round, and v2 must count red edges incident to transition vertices because they can flip the X ⊗ X measurement parity.The errors are detected in the next round by X-stabilizers in both logical patches and the routing region.Source: Appendix B · p. 20
  • Black edges have the same correction effect as green logical-patch edges because failures producing them cannot flip the X ⊗ X measurement parity.Black edges are incident to parity vertices in rounds j ≥ r + 2.Source: Appendix B · p. 20
  • A round-r + 1 failure producing a red edge highlights only one later logical-patch vertex, and the left-right edge-count asymmetry yields an asymmetry in P(1,0,0,0) and P(0,0,1,0).This single-vertex behavior assumes no other failures occur.Source: Appendix B · p. 20

Appendix C: Resource cost analysis of the Hubbard model

Appendix C estimates Hubbard-model resource costs for core-cache layouts. It computes code distances and reports physical-qubit and routing overheads, while noting that the core/cache allocation also affects runtime.

Appendix C: Resource cost analysis of the Hubbard model

  • Table I reports minimum dx, dz, and dm at δ ∼ 1%, physical-qubit totals, core and cache logical-qubit counts, and a routing-cost multiplier, excluding magic-state-factory costs.The distances are selected by solving Eqs. (C7) to (C9).Source: Table I · p. 21, Appendix C · p. 21
  • The estimates pessimistically assume the full core routing space is used for each magic-state injection and use the full core area for FA, while ignoring higher-order lattice-surgery contributions.For PZL, the analysis uses P(1,1,0,0), treating its difference from P(0,1,1,0) as negligible.Source: Appendix C · p. 21
  • Layouts with more logical qubits in the core relative to the cache can substantially increase routing overhead.The table’s last column gives the multiplicative routing factor added to physical-qubit overhead.Source: Table I · p. 21, Appendix C · p. 22
  • Runtime depends partly on the cache-to-core logical-qubit ratio, which affects how often the algorithm reads from and writes to the cache.The magic-state injection time is Tb = µ(dm + 1)Tsurf, with µ = 4NTOF + NT.Source: Appendix C · pp. 21–22
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