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Tradeoffs on the volume of fault-tolerant circuits

Anirudh Krishna, Gilles Zémor

TL;DR

The paper studies the tradeoffs between code rate, distance, and computability in fault-tolerant circuits by analyzing encoded CNOT gates implemented with constant-depth gadgets on binary linear codes. It introduces a model with sparse, encoded operations, an erasure adversary, and a robustness notion, proving that short-depth gadgets force the code to behave as a $(q,r)$-local code with $q=O(1)$ and $r= heta(d)$, which in turn implies a capacity-like bound $k=O(n/d^{1/q})$. Consequently, achieving good rate and distance cannot be done without accepting very deep circuits, highlighting fundamental limits on the volume overhead of fault-tolerant computation. The results connect classical locally decodable code theory to encoded fault-tolerance and have implications for the design of fault-tolerant architectures and potentially quantum fault-tolerance, where encoded access to information constrains feasible code choices. Overall, the work clarifies intrinsic tradeoffs between encoding efficiency and computability in fault-tolerant schemes, guiding code selection and circuit design in architectures with constrained depth.

Abstract

Dating back to the seminal work of von Neumann [von Neumann, Automata Studies, 1956], it is known that error correcting codes can overcome faulty circuit components to enable robust computation. Choosing an appropriate code is non-trivial as it must balance several requirements. Increasing the rate of the code reduces the relative number of redundant bits used in the fault-tolerant circuit, while increasing the distance of the code ensures robustness against faults. If the rate and distance were the only concerns, we could use asymptotically optimal codes as is done in communication settings. However, choosing a code for computation is challenging due to an additional requirement: The code needs to facilitate accessibility of encoded information to enable computation on encoded data. This seems to conflict with having large rate and distance. We prove that this is indeed the case, namely that a code family cannot simultaneously have constant rate, growing distance and short-depth gadgets to perform encoded CNOT gates. As a consequence, achieving good rate and distance may necessarily entail accepting very deep circuits, an undesirable trade-off in certain architectures and applications.

Tradeoffs on the volume of fault-tolerant circuits

TL;DR

The paper studies the tradeoffs between code rate, distance, and computability in fault-tolerant circuits by analyzing encoded CNOT gates implemented with constant-depth gadgets on binary linear codes. It introduces a model with sparse, encoded operations, an erasure adversary, and a robustness notion, proving that short-depth gadgets force the code to behave as a -local code with and , which in turn implies a capacity-like bound . Consequently, achieving good rate and distance cannot be done without accepting very deep circuits, highlighting fundamental limits on the volume overhead of fault-tolerant computation. The results connect classical locally decodable code theory to encoded fault-tolerance and have implications for the design of fault-tolerant architectures and potentially quantum fault-tolerance, where encoded access to information constrains feasible code choices. Overall, the work clarifies intrinsic tradeoffs between encoding efficiency and computability in fault-tolerant schemes, guiding code selection and circuit design in architectures with constrained depth.

Abstract

Dating back to the seminal work of von Neumann [von Neumann, Automata Studies, 1956], it is known that error correcting codes can overcome faulty circuit components to enable robust computation. Choosing an appropriate code is non-trivial as it must balance several requirements. Increasing the rate of the code reduces the relative number of redundant bits used in the fault-tolerant circuit, while increasing the distance of the code ensures robustness against faults. If the rate and distance were the only concerns, we could use asymptotically optimal codes as is done in communication settings. However, choosing a code for computation is challenging due to an additional requirement: The code needs to facilitate accessibility of encoded information to enable computation on encoded data. This seems to conflict with having large rate and distance. We prove that this is indeed the case, namely that a code family cannot simultaneously have constant rate, growing distance and short-depth gadgets to perform encoded CNOT gates. As a consequence, achieving good rate and distance may necessarily entail accepting very deep circuits, an undesirable trade-off in certain architectures and applications.

Paper Structure

This paper contains 18 sections, 11 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Our Model
  4. Our Main Result
  5. Related work
  6. Acknowledgements
  7. Background & Notation
  8. Basic Definitions
  9. Locality
  10. Gates & Circuits
  11. Main result
  12. Short-Depth Circuits are Robust to Erasures
  13. The Doubled Gadget
  14. Influence
  15. Robustness Implies Disjoint Influence Sets
  16. ...and 3 more sections

Key Result

Theorem 1

Consider an infinite family of $[n,k(n),d(n)]$ codes $\mathscr{C} = \{C : \mathbb{F}_2^{k} \to \mathbb{F}_2^{n}\}$ such that for all codes $C : \mathbb{F}_2^{k} \to \mathbb{F}_2^{n}$, for all $i \in [k]$, there exists $j \in [k]$, $j \neq i$, and a gadget $\widehat{G}_i$ such that Then we have the relationship where $1 < q \leq 2^{\ell}$ is a constant.

Figures (1)

  • Figure 1: Schematic illustrating the conversion of a sparse circuit $\Lambda$, on the left, into a fault-tolerant circuit $\widehat{\Lambda}$, on the right. In the $t$th time step in $\Lambda$, there is one elementary gate $G_t$ that acts on at most $2$ bits. In $\widehat{\Lambda}$, the gate $G_t$ is simulated using the gadget $\widehat{G}_t$ that acts directly on encoded information. It is then followed by a round of error correction.

Theorems & Definitions (27)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Theorem 5
  • Lemma 6
  • proof
  • Claim 7
  • proof
  • ...and 17 more