Fault-tolerant structures for measurement-based quantum computation on a network
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2025-05-05 |
| Journal | Quantum |
| Authors | Yves van Montfort, SĂ©bastian de Bone, David Elkouss |
| Institutions | Centrum Wiskunde & Informatica, Delft University of Technology |
| Citations | 2 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive Summaryâ- Novel Architecture Development: Introduced a systematic method using Z2 chain complexes and splitting operations (cell-vertex, face-edge) to construct and distribute fault-tolerant 3D cluster states for Measurement-Based Quantum Computation (MBQC).
- Performance Superiority: Numerical simulations confirm that non-foliated lattices, specifically the diamond lattice, exhibit higher fault-tolerant error thresholds than the conventional cubic lattice under both monolithic (circuit-level) and distributed (network) noise models.
- Threshold Quantification: Monolithic diamond architectures achieved a circuit-level noise threshold (Po,th) of 0.631 percent, significantly outperforming the cubic lattice (0.521 percent) for the tested gate orderings.
- Distributed Scalability: Distributed diamond architectures maintain fault tolerance with network link error probabilities (Pn) up to approximately 2.0 percent, corresponding to a required Bell state fidelity greater than 98 percent.
- Erasure Resilience: The high erasure thresholds observed in non-cubic lattices (up to 55.15 percent for triamond) enable a critical trade-off: entanglement distillation can be used to convert high network infidelity (Pn) into manageable erasure errors (Pe), boosting overall fault tolerance.
- Simulation Methodology: Thresholds were determined using an efficient stabilizer simulator combined with the Union-Find decoder, modeling realistic circuit-level noise (depolarizing gates, faulty measurements) and network noise (depolarized entangled states).
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| Diamond Lattice Bit-Flip Threshold (Pm,th) | 5.32 | percent | Phenomenological noise, pure bit-flip. |
| Triamond Lattice Erasure Threshold (Pe,th) | 55.15 | percent | Phenomenological noise, pure erasure. |
| Monolithic Circuit Threshold (Po,th) | 0.631 | percent | Highest threshold found for Diamond lattice (specific CZ gate ordering). |
| Monolithic Circuit Threshold (Po,th) | 0.521 | percent | Highest threshold found for Cubic lattice (specific CZ gate ordering). |
| Distributed Network Threshold (Pn,th) | ~2.0 | percent | Maximum network error rate for fault tolerance (Diamond 4-ring/7-node architecture). |
| Required Bell State Fidelity (1 - Pn) | > 98 | percent | Minimum fidelity required for entangled links in distributed architectures. |
| Cluster State Valency (z) | 4 | N/A | Cubic lattice valency. |
| Cluster State Valency (z) | 6 | N/A | Diamond lattice valency. |
| Cluster State Valency (z) | 10 | N/A | Triamond lattice valency. |
| Sub-threshold Error Rate Scaling Impact | 1.5 - 2 | Factor | Penalty factor on error rate scaling when boundaries are introduced (compared to periodic conditions). |
Key Methodologies
Section titled âKey Methodologiesâ-
Lattice Definition via Z2 Chain Complex:
- Used Z2 chain complexes to mathematically describe the 3D lattices (Cubic, Diamond, Double-Edge Cubic, Triamond).
- Employed Unit Cell Complexes and Miller index notation to incorporate translational symmetry, allowing the description of large crystalline structures from small basis elements.
-
Cluster State Distribution via Splitting:
- Cell-Vertex Splitting: Used to transform the lattice geometry, generating non-foliated structures like the diamond lattice.
- Face-Edge Splitting: Introduced to replace monolithic cluster state qubits with entangled resource states (Bell pairs or GHZn states), creating distinct, networked nodes.
-
Circuit and Noise Modeling:
- Monolithic Noise: Modeled using standard circuit-level noise: noisy state preparation (Pp), two-qubit depolarizing noise after every CZ gate (Pg), and classical bit-flips during Pauli-X measurement (Pm).
- Distributed Network Noise: Assumed nodes share depolarized Werner states or GHZ states with network error probability Pn.
- Stabilizer Formalism: Used to efficiently model and transfer Pauli errors through the Clifford circuits, enabling large-scale simulation.
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Decoding and Threshold Calculation:
- Decoder: Employed the Union-Find decoder, capable of jointly correcting both Pauli errors and erasure errors.
- Threshold Estimation: Calculated fault-tolerant thresholds (Pth) by fitting the logical error probability (PL) as a function of error rate (P) and lattice size (L) using a second-order polynomial model.
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Entanglement Distillation Analysis:
- Investigated the use of distillation protocols (e.g., concatenated DEJMPS, 5-to-1) to improve initial link fidelity (1 - Pn).
- Distillation failures were modeled as erasure errors (Pe) on the corresponding qubits, allowing the trade-off between Pn and Pe to be quantified against the fault-tolerant region.
Commercial Applications
Section titled âCommercial Applicationsâ- Modular Quantum Computing Systems: Provides validated architectural designs (especially diamond lattice networks) for constructing large-scale quantum processors by linking smaller, high-fidelity quantum modules.
- Distributed Quantum Networks: Relevant for designing robust quantum internet infrastructure where entanglement must be shared across noisy, long-distance links, leveraging high erasure thresholds for error resilience.
- Fault-Tolerant Quantum Memory: The analysis of logical error suppression directly informs the design of memory protocols that protect stored quantum information against depolarizing and erasure noise over time.
- Fusion-Based Quantum Computing (FBQC): The splitting methodology generates the necessary fusion networks, offering a pathway to low circuit depth quantum computation architectures.
- Topological Code Implementation: Offers performance benchmarks for implementing non-foliated 3D topological codes, which may provide higher error resilience than traditional 2D surface codes in specific hardware environments.
- Quantum Hardware Optimization: Results guide hardware engineers in optimizing the connectivity (valency) and gate ordering of physical qubits to maximize fault-tolerant thresholds under realistic noise constraints.
View Original Abstract
In this work, we introduce a method to construct fault-tolerant <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ><mml:mrow class=âMJX-TeXAtom-ORDâ><mml:mtext class=âMJX-tex-mathitâ mathvariant=âitalicâ>measurement-based quantum computation</mml:mtext></mml:mrow></mml:math> (MBQC) architectures and numerically estimate their performance over various types of networks. A possible application of such a paradigm is distributed quantum computation, where separate computing nodes work together on a fault-tolerant computation through entanglement. We gauge error thresholds of the architectures with an efficient stabilizer simulator to investigate the resilience against both circuit-level and network noise. We show that, for both monolithic (i.e., non-distributed) and distributed implementations, an architecture based on the diamond lattice may outperform the conventional cubic lattice. Moreover, the high erasure thresholds of non-cubic lattices may be exploited further in a distributed context, as their performance may be boosted through <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ><mml:mrow class=âMJX-TeXAtom-ORDâ><mml:mtext class=âMJX-tex-mathitâ mathvariant=âitalicâ>entanglement distillation</mml:mtext></mml:mrow></mml:math> by trading in entanglement success rates against erasure errors during the error-decoding process. These results highlight the significance of lattice geometry in the design of fault-tolerant measurement-based quantum computing on a network, emphasizing the potential for constructing robust and scalable distributed quantum computers.