Source-linked AI summary
Transversal Fanout for Fault Tolerant Distributed Quantum Computing: Analysis and Application
Seng W. Loke
TL;DR
Distributed logical fanout can require substantial non-local communication between encoded blocks. This paper constructs transversal fanout for BB-encoded qubits and applies concurrent logical fanouts to distributed global gates; simulations report lower LER for one-shot GHZ fanout in the low-PER regime, contingent on low-noise GHZ generation.
Problem
Implementing logical fanout directly between encoded blocks can require substantial non-local communication in distributed quantum computing.
Source: §I · p. 1Method
The paper combines transversal BB-code logical CNOTs into coordinate-wise physical fanouts, implementing multiple logical fanouts concurrently.
Source: §III · p. 2Results
At low p, 1-shot GHZ fanout improved LER by up to around 2.3× over Bell-pair CNOTs for 8 blocks; at p=10^-4, LER was ≈10^-6, below PER.
Source: §IV · pp. 4–5Takeaways & Limitations
Concurrent logical fanouts from transversal operations can construct distributed high-fanout CZ/CNOT circuits, including GCZ implementations.
Source: §V · pp. 5–6, §VI · p. 7The advantages depend on forming distributed GHZ states efficiently and with sufficiently low noise.
Source: §VI · p. 6
Abstract
from arXiv · showhide
We study a resource-efficient approach for implementing logical fanout operations in fault-tolerant distributed quantum computing using transversal operations on quantum error-correcting code blocks. Logical fanout, comprising multiple controlled-NOT operations from a common control qubit to target qubits located at remote nodes, is an important primitive for distributed quantum computation but can require substantial non-local communication when implemented directly between encoded blocks. We exploit the structure of encoded blocks and the availability of transversal logical operations to construct distributed fanout circuits that reduce the required non-local operations while preserving the logical action of the fanout operation. The construction is developed for encoded quantum information and illustrated using Bivariate Bicycle (BB)-code blocks. We analyze the resulting physical gate, entanglement, circuit-depth, and ancilla requirements. The approach provides a systematic method for implementing large logical fanout operations across distributed error-corrected quantum processors. Also, we study a distributed implementation of the global gate GCZ involving logical qubits (encoded using BB-code blocks), exploiting the concurrency in transversal distributed fanouts.
I. INTRODUCTION
Distributed quantum computing uses non-local gates to connect separate processors, while fault-tolerant implementations use encoded blocks such as BB codes. This paper develops transversal distributed fanout for BB-encoded qubits and explores its use in distributed global gates.
I. INTRODUCTION
- Non-local CNOT gates, realized using Bell pairs, connect disparate quantum processing units and, with single-qubit gates, provide a universal gate set.Experimental realizations of this connectivity have been reported [15].Source: §I · p. 1
- Fault-tolerant distributed computing has considered surface codes, BB codes, and Floquet codes, including transversal non-local CNOTs and inter-module operations for BB codes.The cited examples include surface codes [4], [16], [18], BB-code operations [21],, and Floquet codes [22].Source: §I · p. 1
- Fanout uses one control qubit for multiple targets and has been proposed for reduced-depth computation in some hardware and through distributed GHZ states.The cited work includes trapped-ion hardware [6], and distributed GHZ-state proposals and investigations [25], [12],.Source: §I · p. 1
- The paper develops transversal distributed fanout for Bivariate-Bicycle (BB)-encoded qubits and explores its use for distributed global gates.The global gates include distributed versions of GMS(θ).Source: §I · p. 1
II. BRIEF REVIEW OF DISTRIBUTED FAN-OUT OPERATIONS
A distributed fanout applies one control qubit to multiple targets located on different nodes. It can arise from a circuit’s structure or from decomposing a multi-qubit controlled operation across nodes.
II. BRIEF REVIEW OF DISTRIBUTED FAN-OUT OPERATIONS
- Distributed fanout is an operation in which the same control qubit is used for multiple target qubits on different nodes.It can arise from a circuit’s structure or when a control-U has a unitary spanning multiple qubits that is decomposed.Source: §II · p. 1
- A multi-target control-U becomes distributed fanout when its separate target operations act on qubits at different nodes.The paper focuses on the case Ui = X unless otherwise noted.Source: §II · p. 2
- The illustrated three-target fanout uses a four-node GHZ state across the communication qubits, including the control node and each target node.The state is 1/√2(|0000⟩+|1111⟩) across nodes A′, A, B, and C.Source: §II · p. 2
III. TRANSVERSAL NON-LOCAL FANOUT
BB-code transversal logical CNOTs can be combined to implement logical fanout across encoded blocks. The construction performs one physical fanout per code coordinate, concurrently realizing k logical fanouts.
III. TRANSVERSAL NON-LOCAL FANOUT
- For BB-encoded qubits, logical CNOTs are transversal, and their commuting product can be reordered into independent physical fanouts, one per code coordinate.The resulting logical fanout gadget is transversal because it is composed from transversal logical CNOT gadgets, [21].Source: §III · p. 2
- With a [[n, k, d]] BB encoding, the construction performs n physical fanouts, one for each coordinate, to implement logical fanout on corresponding encoded qubits.Each control and target block encodes k logical qubits.Source: §III · p. 2
- Each transversal fanout operates on k logical qubits simultaneously, yielding k concurrent logical fanouts across the target blocks.The physical fanout pairs corresponding qubits across the blocks without coupling qubits within a code block.Source: §III · p. 2
A. Comparison with Pairwise Distributed CNOTs
The section compares GHZ-based distributed fanout with an equivalent sequence of pairwise distributed CNOTs. The GHZ-based approach may reduce error sources and depth, but depends on generating sufficiently low-error distributed GHZ states.
A. Comparison with Pairwise Distributed CNOTs
- GHZ-based fanout could reduce error sources and circuit depth relative to equivalent pairwise CNOTs, provided distributed GHZ states can be generated with low error rates.The comparison considers one control block and m−1 remote target blocks over m nodes.Source: A. Comparison with Pairwise… · p. 2
- A transversal fanout across code blocks acts on k groups of target logical qubits simultaneously.This follows from a transversal CNOT between code blocks acting on k target logical qubits at once.Source: A. Comparison with Pairwise… · p. 2
- For m nodes, transversal fanout uses n non-local physical FANOUTs and n m-qubit GHZ states; the resource comparison treats GHZ states and Bell pairs as available.The architecture-dependent costs of generating those entanglement resources are not included.Source: A. Comparison with Pairwise… · p. 3, Table I · p. 3
IV. SIMULATION STUDY OF NON-LOCAL FANOUTS
The study compares full-circuit logical error rates for four- and eight-node non-local fanouts under simulated noise, including GHZ-assisted and sequential CNOT implementations. It reports low-error advantages for one-shot GHZ fanout and evaluates circuit-level distance.
IV. SIMULATION STUDY OF NON-LOCAL FANOUTS
- Transversality does not by itself establish full circuit-level fault tolerance, which also depends on syndrome extraction and the decoder.The simulations therefore evaluate the logical error rate of the complete noisy circuit.Source: §IV · p. 3
- Simulations extend TMCBS with Stim and Tesseract; 3cnot-seq adds syndrome extraction between CNOTs, while GHZ fanout treats the operation as one logical step.GHZ states are built from a root Hadamard and non-local CNOTs; the one-shot simulation counts their construction as one step.Source: §IV · pp. 3–4, Table II · p. 4
- For four nodes with p = pebit = pghz, 1-shot GHZ generally has the lowest LER at low PER, while the GHZ-implementation ordering can cross at intermediate rates.LER falls below PER when p is approximately 5 × 10^-4 or lower.Source: §IV · p. 4, Figure 3 · p. 4
- With entanglement error set to 10 times the qubit PER, the four-node results resemble the equal-error case, but 1-shot GHZ's differences from alternatives are slightly larger.The authors attribute its relative efficiency to the higher entanglement costs.Source: §IV · p. 4, Figure 4 · p. 5
- At p = 10^-4, LER is about 10^-6 and below PER across cases; for eight nodes, 1-shot GHZ improves on Bell-pair sequencing by up to 2.3× at low p.The eight-node 1-shot GHZ fanout also has lower LER than the four-node case.Source: §IV · pp. 4–5
- Circuit-level distance is 6 for both four- and eight-node fanouts, with no undetectable logical error found below fault weight 6.The search evaluated both sequential non-local CNOTs and 1-shot GHZ fanout using STIM's undetectable-error search heuristic.Source: §IV · p. 5
V. DISTRIBUTED GCZ - AN APPLICATION OF TRANSVERSAL NON-LOCAL FANOUTS
The section applies concurrent transversal fanouts to distributed implementations of global CZ gates. For a BB encoding, the construction decomposes the gate into local and distributed operations, with GHZ-state requirements that scale with the circuit size under a stated condition.
V. DISTRIBUTED GCZ - AN APPLICATION OF TRANSVERSAL NON-LOCAL FANOUTS
- Transversal fanout applies k logical fanout operations concurrently because each block encodes k logical qubits.The section uses this concurrency to motivate the construction of distributed global gates.Source: §V · p. 5
- In the 16-logical-qubit, four-node example, GCZ16 has 120 CZ operations, including 96 inter-node CZs.The first 54 CZs are implemented using ancilla blocks on target nodes, with control-block CZs batched before the inter-node CZs.Source: §V · p. 6
- The first 54 CZs are expressed as local and distributed operations, with the rightmost operations executing first.The construction combines local GCZ on the control node, distributed transversal fanout to target-node ancillas, and local fanout operations on each target node.Source: §V · p. 6
- For a [[n, k, d]]BB encoding across m nodes, GCZ_N=km uses 2(m−1) distributed transversal fanouts; when n/k is constant, the GHZ-state requirement is O(N).Each distributed fanout involving m′ blocks requires n physical m′-qubit GHZ states; a one-control, one-target fanout instead uses Bell pairs.Source: §V · p. 6
VI. CONCLUSION AND FUTURE WORK
The paper demonstrates distributed transversal fanout for BB-encoded logical qubits and applies concurrent logical fanouts to distributed high-fanout CZ/CNOT circuits, including GMS/GCZ constructions. Its demonstrated one-shot fanout advantage depends on efficiently formed, sufficiently low-noise distributed GHZ states; further analysis could consider other GHZ noise models and larger circuits.
VI. CONCLUSION AND FUTURE WORK
- The demonstrated advantage of one-shot fanout depends on forming distributed GHZ states efficiently and with sufficiently low noise.The authors identify investigation of other GHZ-state noise models as future work.Source: §VI · p. 6
- Concurrent logical fanouts from transversal operations are used to construct distributed high-fanout CZ/CNOT circuits, including GMS/GCZ constructions.The paper demonstrates distributed transversal fanout for BB-encoded logical qubits.Source: §VI · pp. 6–7
- The authors identify analysis of much larger circuits as future work.The conclusion also notes that other noise models for GHZ states could be investigated.Source: §VI · p. 7