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Coded Computing Meets Quantum Circuit Simulation: Coded Parallel Tensor Network Contraction Algorithm

Jin Lee, Sofia Gonzalez-Garcia, Zheng Zhang, Haewon Jeong

TL;DR

The paper addresses the challenge of reliably performing large-scale tensor-network contractions for quantum circuit simulation on massively parallel classical hardware, where node failures and stragglers impede computation. It extends matrix-multiplication coding ideas to the tensor-network setting by introducing polynomial-encoded schemes that slice contraction indices and allow recovery from up to f failed workers. Two main constructions are proposed: a practical 2-node code for quantum-simulation tensor networks and a general hyperedge code for broader tensor-network topologies, each with explicit f-resilience and gains and connections to MatDot/PolyCode. The work also analyzes master-node encoding/decoding cost, showing these are negligible compared to contraction, and discusses how these codes fit within and extend existing coded computing frameworks. This approach provides a scalable, fault-tolerant pathway to run beyond-classical-timings in quantum circuit simulations and may generalize to other high-dimensional linear-algebra tasks on HPC systems.

Abstract

Parallel tensor network contraction algorithms have emerged as the pivotal benchmarks for assessing the classical limits of computation, exemplified by Google's demonstration of quantum supremacy through random circuit sampling. However, the massive parallelization of the algorithm makes it vulnerable to computer node failures. In this work, we apply coded computing to a practical parallel tensor network contraction algorithm. To the best of our knowledge, this is the first attempt to code tensor network contractions. Inspired by matrix multiplication codes, we provide two coding schemes: 2-node code for practicality in quantum simulation and hyperedge code for generality. Our 2-node code successfully achieves significant gain for $f$-resilient number compared to naive replication, proportional to both the number of node failures and the dimension product of sliced indices. Our hyperedge code can cover tensor networks out of the scope of quantum, with degraded gain in the exchange of its generality.

Coded Computing Meets Quantum Circuit Simulation: Coded Parallel Tensor Network Contraction Algorithm

TL;DR

The paper addresses the challenge of reliably performing large-scale tensor-network contractions for quantum circuit simulation on massively parallel classical hardware, where node failures and stragglers impede computation. It extends matrix-multiplication coding ideas to the tensor-network setting by introducing polynomial-encoded schemes that slice contraction indices and allow recovery from up to f failed workers. Two main constructions are proposed: a practical 2-node code for quantum-simulation tensor networks and a general hyperedge code for broader tensor-network topologies, each with explicit f-resilience and gains and connections to MatDot/PolyCode. The work also analyzes master-node encoding/decoding cost, showing these are negligible compared to contraction, and discusses how these codes fit within and extend existing coded computing frameworks. This approach provides a scalable, fault-tolerant pathway to run beyond-classical-timings in quantum circuit simulations and may generalize to other high-dimensional linear-algebra tasks on HPC systems.

Abstract

Parallel tensor network contraction algorithms have emerged as the pivotal benchmarks for assessing the classical limits of computation, exemplified by Google's demonstration of quantum supremacy through random circuit sampling. However, the massive parallelization of the algorithm makes it vulnerable to computer node failures. In this work, we apply coded computing to a practical parallel tensor network contraction algorithm. To the best of our knowledge, this is the first attempt to code tensor network contractions. Inspired by matrix multiplication codes, we provide two coding schemes: 2-node code for practicality in quantum simulation and hyperedge code for generality. Our 2-node code successfully achieves significant gain for -resilient number compared to naive replication, proportional to both the number of node failures and the dimension product of sliced indices. Our hyperedge code can cover tensor networks out of the scope of quantum, with degraded gain in the exchange of its generality.
Paper Structure (13 sections, 3 theorems, 71 equations, 5 figures)

This paper contains 13 sections, 3 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. Tensor Networks and contraction
  4. Parallel tensor network contraction
  5. System Model and Problem Statement
  6. Main Results
  7. 2-node Code for Quantum Simulation
  8. Hyperedge Code for General Case
  9. Master Node Complexity
  10. Existing matrix codes under our framework
  11. Discussion and future works
  12. Proof of Theorem 1
  13. Proof of Theorem 2

Key Result

Theorem 1

For a tensor network contraction parallelized by slicing at $n$ non-adjacent indices, each of which is $2$-node and $L_{i}$-dimensional ($i={1,2,\ldots,n}$), $f$-resilient number of following is achievable: The gain compared to replication for $f$ failures is:

Figures (5)

  • Figure 1: Diagrammatic representation and corresponding array for ($\boldsymbol{a}$) one dimensional tensor $\boldsymbol{A}_{i}$ ($\boldsymbol{b}$) two dimensional tensor $\boldsymbol{B}_{i,j}$ ($\boldsymbol{c}$) three dimensional tensor $\boldsymbol{C}_{i,j,k}$
  • Figure 2: Diagrammatic representation of tensor network contraction for ($\boldsymbol{a}$) $\sum_{j}\boldsymbol{A}_{i,j}\boldsymbol{B}_{j,k}$ ($\boldsymbol{b}$) $\sum_{k,j,l}\boldsymbol{A}_{a,j,k}\boldsymbol{B}_{b,k,l}\boldsymbol{C}_{c,l,j}$ ($\boldsymbol{c}$) $\sum_{j}\boldsymbol{A}_{j,a}\boldsymbol{B}_{b,j}\boldsymbol{C}_{j,c}$
  • Figure 3: Parallelization scheme by slicing index $k$ for the sample tensor network given in \ref{['sec:parallel']}.
  • Figure 4: ($\boldsymbol{a}$) System model for distributed parallel tensor network contraction without coding. Failure on any computational node will result as a failure of entire system ($\boldsymbol{b}$) System model for distributed coded parallel tensor network contraction. Even with some node failures, as long as the number of successful computational nodes is over $f$-resilient number, desired final outcome will always be retrieved.
  • Figure 5: Diagrammatic representation of ($\boldsymbol{a}$) polynomial code and ($\boldsymbol{b}$) MatDot code in tensor network formalism.

Theorems & Definitions (9)

  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1
  • Example 1: 2-node code
  • Theorem 2
  • proof
  • Example 2: Hyperedge code
  • Remark 2