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Encoding and Node Choices in Transversal Fault-Tolerant Distributed Quantum Computations: An Initial Study

Seng W. Loke

arXiv:2609.37210v1quant-phcs.DC

TL;DR

This paper asks how encoding and node choices affect logical error rates in distributed quantum operations, where lower ebit consumption can trade off against code distance. It compares BB-code and node configurations for distributed CNOT and GCZ operations, concluding that ebit use alone is not a sufficient selection criterion and that choices depend on the circuit and architecture.

  • Problem

    In distributed computations, larger BB blocks can offer greater code distance but require more inter-node ebits, leaving the effect of encoding and node choices on logical error rate to be assessed.

    Source: §I · pp. 1–2

  • Method

    The paper studies encoding and node configurations for distributed circuits and frames their selection for heterogeneous QPUs as a constrained optimization over logical-qubit allocation, encodings, and block decomposition.

    Source: §V · p. 7

  • Results

    In the cases studied, larger blocks and fewer nodes can reduce inter-node operations and yield lower LER for the noise models examined, so fewer ebits do not necessarily mean lower LER.

    Source: §IV · pp. 6–7

  • Takeaways & Limitations

    Encoding and node selection should account for their combined effect on the LER of the actual distributed circuit rather than ebit consumption, n, d, or k alone.

    Source: §V · p. 7

    The study is limited to BB codes and a limited set of transversal operations; whether its findings hold for other circuits requires further study.

    Source: §IV · p. 7

Abstract

from arXiv · show

We compare and study different Bivariate-Bicycle (BB) encodings and node choices for distributed quantum operations such as transversal non-local CNOTs. We observe that while some encodings have more physical qubits requiring more ebits for a distributed computation, reducing ebit consumption alone might not be the best criterion for selecting an encoding for the logical qubits, if the goal is to reduce the logical error rate of the distributed computation; for example, one should also consider encodings with a larger distance, which using more physical qubits can provide. We consider distributed, or non-local, CNOTs and computation of the global gate (GCZ) over distributed logical qubits as examples. The choice of encoding enables particular concurrent operations; e.g., we show that a transversal physical GCZ on a self-dual $[[120,8,12]] BB$ code can realize eight concurrent logical GCZ operations after accounting for the logical permutation induced by transversal Hadamard.

I. INTRODUCTION

The paper studies how BB-code and node choices affect distributed logical operations, where reducing ebit use can trade off against code distance and logical error rate. It frames encoding and node selection as choices for heterogeneous QPUs and examines their effects on distributed CNOTs and GCZ.

I. INTRODUCTION

  • Distributed quantum computing can combine connected quantum and classical nodes, and prior work has considered several error-correcting encodings and architectures.The paper builds on prior distributed transversal CNOT studies using BB codes and examines encoding and node choices.Source: §I · p. 1
  • For a BB code [[n, k, d]], n is the number of physical qubits, while the code encodes k logical qubits with distance d.For the studied BB codes, transversal operations enact k logical operations through n physical operations; larger n can provide larger d but require more inter-node ebits.Source: §I · p. 1
  • The study assumes a choice among potentially heterogeneous QPUs for hosting circuits and examines both encoding selection and node combinations.It analyzes transversal CNOTs first and then distributed GCZ, assuming nodes have computation qubits, including ancillas, and communication qubits for ebits.Source: §I · pp. 1–2
  • Choosing between one [[90, 8, 10]] block per node and two [[36, 4, 6]] blocks trades distance 10 for 6 against ebit use of 90 versus 72.The paper asks how this trade-off affects logical error rate and notes that smaller blocks may ease operations on subsets of qubits.Source: §I · pp. 1–2

II. ENCODING AND NODE CHOICES FOR THE NON-LOCAL TRANSVERSAL CNOT

The section compares two ways to encode 16 logical qubits across two nodes for an inter-node transversal CNOT. Simulations find lower logical error rates for the larger [[90, 8, 10]] blocks in the studied cases, despite their greater ebit use.

II. ENCODING AND NODE CHOICES FOR THE NON-LOCAL TRANSVERSAL CNOT

  • The comparison encodes 16 logical qubits as one [[90, 8, 10]] block per node or two [[36, 4, 6]] blocks per node, then compares CNOT LER.Both arrangements perform inter-node CNOTs between the same eight logical-qubit pairs.Source: §II · p. 2
  • Each physical non-local CNOT uses a distributed CNOT gadget, and the one-block arrangement performs eight inter-node logical CNOTs concurrently.The physical CNOTs pair corresponding qubits across the nodes; Figure 2 depicts the one-block and two-block arrangements.Source: §II · p. 2, Figure 2 · p. 3
  • The one-block arrangement uses 90 physical ebits, while the two-block arrangement uses 72, reducing ebit use by 18 at the cost of lowering distance from 10 to 6.The authors note that reduced ebit use and reduced error-correction distance may affect LER in opposite directions.Source: §II · p. 2
  • The LER study uses TMCBS code for simulating non-local transversal CNOTs, with stim and sinter for Monte Carlo sampling.The supplied results compare cases with ebit error probabilities pebit = 10p and pebit = 50p.Source: §II · pp. 2–3
  • A factor of 140 lower LER is reported for [[90, 8, 10]] than for 2×[[36, 4, 6]] at p = 2.71 × 10^-4 and pebit = 10p.The larger-block scheme also has lower LER when pebit = 50p, despite using more ebits.Source: §II · p. 3

III. ENCODING AND NODE CHOICES FOR GCZ

The section compares BB encodings and node configurations for distributed global gates, where ebit cost and logical error rate do not necessarily favor the same choice. It describes implementations using inter-node transversal CNOTs and local fanouts, and shows that a transversal GCZ on the self-dual [[120, 8, 12]] code can realize eight concurrent logical GCZs up to a logical permutation.

III. ENCODING AND NODE CHOICES FOR GCZ

  • For four nodes, GCZL16 comprises 120 CZLs, including 96 inter-node CZLs; the first 54 can be reordered so 48 inter-node interactions use three transversal CNOTs.The described full four-node scheme uses six inter-node transversal CNOTs, excluding uncomputation, with local fanouts from ancilla blocks.Source: §III · pp. 3–4
  • The [[54, 4, 8]] encoding can yield lower GCZ LER than [[36, 4, 6]] despite requiring 108 more ebits, assuming similar local-operation LERs and pebit = 10p.The estimate compares four nodes with four logical qubits per node; symmetry suggests the inequality could still hold with uncomputation.Source: §III · pp. 4–5
  • For two nodes, GCZL16’s 64 inter-node CZLs can be implemented with an inter-node transversal CNOT followed by local fanouts.The nodes can each use a block encoding eight logical qubits.Source: §III · p. 5, Figure 8 · p. 5
  • For the 4+12 node configuration, [[90, 8, 10]] in an 8+8 configuration has lower LER than the compared [[36, 4, 6]] and [[54, 4, 8]] ancilla-block options.The comparison assumes similar local-operation LERs; [[54, 4, 8]] comes closer to the [[90, 8, 10]] result.Source: §III · p. 5
  • A 12+12 configuration using [[144, 12, 12]] can have similar or slightly lower LER than an 8+16 configuration using [[90, 8, 10]], despite 54 additional ebits.The [[120, 8, 12]] option has the same distance with 24 fewer ebits than [[144, 12, 12]], but a lower LER is only suggested for sufficiently low physical error rates as future work.Source: §III · pp. 5–6
  • A transversal physical GCZ on the self-dual [[120, 8, 12]] code realizes eight concurrent logical GCZs up to the logical permutation associated with transversal H.The permutation can be a relabelling of the logical qubits.Source: §III · p. 6

IV. DISCUSSION

The discussion argues that ebit minimization alone is not a reliable encoding criterion: larger blocks can improve LER through distance and efficiency, while block size also affects concurrency and fit to sparse interactions.

IV. DISCUSSION

  • Larger blocks can yield lower LER through higher distance even when they require more inter-node ebits, so ebit reduction is not necessarily the dominant choice criterion.The discussion also suggests higher-n encodings at the same k, including [[54, 4, 8]] instead of [[36, 4, 6]] and [[120, 8, 12]] instead of [[90, 8, 10]].Source: §IV · p. 6
  • The 8+8 configuration has lower LER than the 2x4+2x4 configuration even with pebit = 50p, as larger blocks can offer higher kd^2/n efficiency and distance.A typical transversal CNOT may require many more ebits and ancilla qubits when performed in one step with a large block.Source: §IV · p. 6
  • Larger blocks can enact more concurrent logical operations in one transversal step, which can be useful for distributed computations requiring that concurrency.A transversal CNOT on a block with n physical qubits performs k logical CNOTs at the same time.Source: §IV · p. 6
  • Smaller blocks might suit sparsely connected circuits involving only selected cross-node interactions, but this trades off distance.Source: §IV · p. 7
  • The findings are limited to a small set of BB codes and transversal operations; other circuits and codes, code conversions, and nonuniform encodings remain unexplored.The study assumes uniform encodings for blocks involved in inter-node transversal CNOTs.Source: §IV · p. 7

V. CONCLUSION

Encoding and node selection should account for their combined effect on the logical error rate of the distributed circuit, rather than relying on ebit consumption or a single code parameter. For heterogeneous QPUs, this selection can be formulated as a constrained optimization problem and depends on the circuit and architecture.

V. CONCLUSION

  • Encoding and node configuration should be assessed by their combined effect on the distributed circuit’s logical error rate, not by ebit consumption or n, d, or k alone.The conclusion frames this as a consideration for evaluating efficiency.Source: §V · p. 7
  • Encoding and node selection for a distributed circuit on heterogeneous QPUs can be formulated as a constrained optimization problem.The variables are P, the partition/allocation of logical qubits across nodes, and E, the QEC encoding and block decomposition choices.Source: §V · p. 7
  • The appropriate encoding and node configuration is circuit- and architecture-dependent.Source: §V · p. 7

APPENDIX A

Appendix A describes a partial distributed implementation of 54 CZL operations using ancilla qubits. Local CZLs are batched before inter-node operations, which can be reordered as transversal CNOTs; ancilla uncomputation requires another sequence of transversal CNOTs.

APPENDIX A

  • The 54 CZLs are partially implemented across nodes using ancilla qubits, with local CZLs batched before the remaining inter-node operations.The implementation is adapted from [12].Source: Figure 9 · p. 8
  • The inter-node operations in the four red boxes can be reordered and implemented via transversal CNOTs.Source: Appendix A · p. 7, Figure 9 · p. 8
  • Ancilla initialization effectively copies logical-qubit information across nodes so related CZL operations can be carried out locally; uncomputation requires another sequence of transversal CNOTs.The uncomputation sequence is not shown.Source: Appendix A · p. 8, Figure 9 · p. 8
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