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Transversal non-Clifford gates for quantum LDPC codes on sheaves

Ting-Chun Lin

TL;DR

A large family of quantum low-density parity-check codes that support transversal non-Clifford gates is provided, to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation.

Abstract

A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support transversal non-Clifford gates. In this work, we provide a large family of such codes. The key insight is to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation. At a more abstract level, this construction is based on defining the cup product on the chain complex induced from a sheaf.

Transversal non-Clifford gates for quantum LDPC codes on sheaves

TL;DR

A large family of quantum low-density parity-check codes that support transversal non-Clifford gates is provided, to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation.

Abstract

A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support transversal non-Clifford gates. In this work, we provide a large family of such codes. The key insight is to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation. At a more abstract level, this construction is based on defining the cup product on the chain complex induced from a sheaf.

Paper Structure

This paper contains 35 sections, 6 theorems, 94 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof overview
  4. Further directions
  5. Quantum LDPC codes with transversal non-Clifford gates and nearly linear distance.
  6. Fully-fledged fault-tolerant scheme.
  7. Quantum PCP conjecture.
  8. Preliminary
  9. Chain complex
  10. Quantum operators over finite fields
  11. Quantum CSS codes
  12. Cubical complexes and cubical codes
  13. Labeled cubical complexes
  14. Chain complexes from cubical complexes
  15. Cubical quantum codes
  16. ...and 20 more sections

Key Result

Lemma 6.2

For every cell $\sigma \in X$, the map $\mathcal{F}_{\sigma} \xrightarrow{\iota_\sigma} \prod_{\tau \in X_{\ge \sigma}(t)} \mathcal{F}_{\tau}$, which sends $c \mapsto \prod_{\tau \in X_{\ge \sigma}(t)} \mathop{\mathrm{res}}\nolimits_{\sigma, \tau}(c)$, is injective. Furthermore, the restriction map where $\mathop{\mathrm{res}}\nolimits_{X_{\ge \sigma}(t), X_{\ge \pi}(t)}$ is the map that restrict

Figures (11)

  • Figure 1: The X-logical operators of three 3D surface codes, each oriented along a different axis. These logical operators are represented by the corresponding surfaces. A crucial feature of these surfaces is that their triple intersection number is always $1$.
  • Figure 2: The figure illustrates the subdivision process for part of a quantum code.
  • Figure 3: The left figure shows part of a Z logical operator, which has a natural string structure illustrated on the right.
  • Figure 4: The figure shows three condensing patterns for the code example above.
  • Figure 5: Each edge has a local coefficient $\alpha(e) \in V_e$. Such a local coefficient induces a vector $h_i^T \alpha(e) \in \mathbb{F}_q^\Delta$. The vector $h_i^T \alpha(e)$ is then used to induce segments between the midpoints of the edges and the centers of the faces. Notice that because $\delta \alpha = 0$, the sum of the segment values on a face is $0$.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Remark 1.1
  • Definition 2.1: Chain complex
  • Definition 2.2: Chain map
  • Definition 2.3: Chain homotopy
  • Remark 3.1
  • Definition 3.2
  • Example 3.3: Triorthogonal code
  • Example 3.4: 3D surface code
  • Remark 3.5
  • Definition 3.6
  • ...and 21 more