Low-overhead fault-tolerant quantum computing using long-range connectivity

TL;DR

The paper tackles the large resource overhead of fault-tolerant quantum computation by leveraging quantum LDPC codes with long-range connectivity. It develops a code-deformation framework that enables fault-tolerant Clifford operations via logical Pauli measurements, while preserving the LDPC properties and code distance. The results indicate substantial overhead reductions—potentially an order of magnitude for hundreds of logical qubits—alongside practical pathways using magic-state distillation and decoders, with a favorable outlook for realizing thousands of physical qubits at realistic error rates. The approach is supported by detailed constructions (CSS and non-CSS measurements, simultaneous measurements), distance-preservation proofs, and discussions on ancilla sizing, parallelism, and decoding, suggesting a viable route to scalable, low-overhead quantum computing.

Abstract

Vast numbers of qubits will be needed for large-scale quantum computing due to the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes as well as the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around one hundred logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.

Figures (8)

• Figure 1: Parallelism. An example circuit consisting of Pauli measurements on an architecture with a parallelism of $6$. This circuit contains $4$ rounds of error corrected logical measurements. In each round, at most $6$ logical qubits in total can be involved in logical measurements.
• Figure 2: Measurement of a logical $\tilde{X}$ operator of the code $\mathcal{C}$.(a) Bipartite subgraph $\mathcal{G}_{\tilde{X}}$ of the Tanner graph of $\mathcal{C}$ on the support of $\tilde{X}$. Black nodes are the variable nodes corresponding to qubits in the support of $\tilde{X}$. Red nodes are the check nodes corresponding to $Z$-type stabilizers in $\mathcal{C}$ that act on qubits in the support of $\tilde{X}$. (b) The dual graph $\mathcal{G}^T = (V^T_{\tilde{X}}, C^T_{\tilde{X}}, E^T_{\tilde{X}})$ of the logical $\tilde{X}$ in (a). There is a one-to-one mapping between the $X$-type generators and the qubits in (a), and the qubits and the $Z$-type generators in (a). (c) Measurement of $\tilde{X}$ using the ancilla system $\mathcal{G}_{\textrm{anc}} = \mathcal{G}_{\textrm{merged}} \backslash \mathcal{G}$. The Tanner graph $\mathcal{G}_{\textrm{anc}}$ is constructed by taking alternating layers of the subgraph $\mathcal{G}^{T}_{\tilde{X}}$ in (b) and the subgraph $\mathcal{G}_{\tilde{X}}$ in (a). The vertical edges are the set $E_{\textrm{extra}}$, which connect adjacent layers. The product of the $X$ generators gives the logical $\tilde{X}$ and hence the product of the measurement results for each $X$ generator gives the measurement result of the logical $\tilde{X}$. After merging the codes and measuring $\tilde{X}$ we then split the codes by measuring the stabilizers for $\mathcal{C}$ and measuring the qubits in $\mathcal{C}_{\textrm{anc}}$ in the $Z$ basis, returning us to the original code space.
• Figure 3: Measurement of the logical operator $\tilde{X}_1 \tilde{X}_2$. First, ancilla systems for the logical operators $\tilde{X}_1$ and $\tilde{X}_2$ are constructed as in Fig. 2. These ancilla systems are connected together as highlighted (green box). This ancilla system is then connected to the logical $\tilde{X}_1 \tilde{X}_2$ as previously. Observe that the product of the $X$ stabilizer generators in the ancilla system gives the logical $X_1X_2$. Furthermore, if we take the product of the $X$ stabilizers on the left ancilla system we do not obtain $X_1$, since this product will include qubits in the highlighted (green) region. The same holds for the stabilizers in the right ancilla system. Hence the product of the measurement results for these generators gives the measurement result for $X_1X_2$. After obtaining the measurement result, we can again measure the original stabilizers to return to the code space.
• Figure 4: Measurement of a logical $\tilde{Y} = i\tilde{X}\tilde{Z}$ operator. This measurement closely follows that of logical $\tilde{X}_1\tilde{X}_2$ shown in Fig. 3, with the key difference being that ancilla systems for the logical operators $X_1$ and $Z_2$ are connected using non-CSS generators. Observe that the product of the $X$ stabilizer generators on the left, the $Z$ stabilizer generators on the right, and the mixed stabilizer generators in the dual layers gives the logical $\tilde{Y}$.
• Figure 5: Measurement of the logical operator $\tilde{X}_1\tilde{Z}_2$. This measurement closely follows that of logical $\tilde{X}_1\tilde{X}_2$ shown in Fig. 3, with the key difference being that ancilla systems for the logical operators $X_1$ and $Z_2$ are connected using non-CSS generators (highlighted green box). Observe that the product of the $X$ stabilizer generators on the left, the $Z$ stabilizer generators on the right, and the mixed stabilizer generators in the highlighted region gives the logical $\tilde{X}_1\tilde{Z}_2$.

Theorems & Definitions (9)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• proof
• Theorem 1
• proof