Fault-Tolerant Quantum Computation with Constant Overhead

TL;DR

This work shows that fault-tolerant quantum computation can be achieved with constant qubit overhead by using quantum LDPC stabilizer codes with fixed locality properties, in a model that allows long-range gates and fast classical processing. The author proves a threshold theorem (Thm main) for such codes under local stochastic noise, with the asymptotic overhead scaling as the inverse code rate $1/R$ and overhead bounded by $rac{ ext{const} imes k}{R}$ for any desired factor $rac{1}{ ext{const}}{>1}$. The paper analyzes candidate LDPC code families (e.g., hypergraph product and hyperbolic codes), their decoding challenges, and the trade-offs between distance, rate, and decodability, while providing schemes for fault-tolerant error correction, gate teleportation, and ancilla preparation that keep overhead sublinear in the code block size. It also discusses depth implications and practical limitations, emphasizing that finding LDPC codes with exponential error suppression and efficient decoding remains a central open problem for realistic large-scale quantum computing.

Abstract

What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.

Figures (3)

• Figure 1: An example of a syndrome adjacency graph. Layers of the original code adjacency graph alternate with layers representing the syndrome bits. Different layers represent different times. The layers surrounded with parallelograms represent copies of the original adjacency graph; the other layers represent syndrome bits. Blue edges are the new ones not present in the original graph.
• Figure 2: The Shor fault-tolerant error-correction procedure applied to measure the syndrome bit for the generator $X \otimes Z \otimes Z \otimes X$. First build and test the cat state, then interact it transversally with the codeword.
• Figure 3: A technique to fault-tolerantly create any ancilla state $|\Psi\rangle$: Encode a concatenated code; each level uses the encoding circuit $\mathcal{E}$. Using a fault-tolerant protocol for the concatenated code, run a fault-tolerant version $\mathcal{\tilde{D}}$ of a circuit $\mathcal{D}$ which creates $|\Psi\rangle$. Then decode the concatenated code.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• Lemma 2
• Theorem 3
• proof
• Theorem 4
• proof
• proof : Proof of Thm. \ref{['thm:main']}