Fault-tolerant interfaces for quantum LDPC codes

TL;DR

This work develops constant-overhead fault-tolerant interfaces for quantum LDPC codes to enable fault-tolerant quantum state preparation with only constant qubit overhead. It introduces partial decoding interfaces that map encoded data across a hierarchy of QLDPC code levels, while maintaining a local stochastic error model and leveraging efficient QLDPC decoders. The authors prove a main theorem asserting the existence of decoding interfaces with constant overhead and controlled output errors, and they analyze error propagation via a sophisticated block-error-pattern framework mapped onto a binary tree. They further apply these interfaces to achieve fault-tolerant state preparation and discuss downstream benefits for fault-tolerant quantum computation and communication, highlighting potential practical reductions in resource overhead for large-scale quantum tasks.

Abstract

The preparation of a quantum state using a noisy quantum computer (gate noise strength $δ$), will necessarily affect an O($δ$)-fraction of the qubits, no matter which protocol is used. Here, we show that fault-tolerant quantum state preparation can be achieved with constant space overhead improving on previous constructions requiring polylogarithmic overhead. To achieve this, we add to the toolbox of fault-tolerant schemes for circuits with quantum input and output. More specifically, we construct fault-tolerant interfaces that decrease the level of protection for quantum low-density parity-check (LDPC) codes. When information is encoded in multiple code blocks, our interfaces have constant space overhead. In our decoder construction that change the level of protection by an arbitrary amount, we circumvent bottlenecks to error pileup and overhead by gradual lowering of the level of encoding at the same time as we increase the number of blocks on which decoding is carried out simultaneously.

Figures (8)

• Figure 1: Teleportation-based decoding interface. A code state is teleported out of the code space by performing a logical Bell measurement between the input state $\ket{\psi_r}$ and one half of the entangled resource state $\ket{\Psi_r}$ as defined in Eq. \ref{['eq:ent-state-int']}. Based on the two classical outcomes of the Bell measurement a Pauli correction $P$ is applied to the remaining half of $\ket{\Psi_r}$.
• Figure 2: The figure represents the construction of a constant overhead interface using sequential implementation of a non-constant overhead interface $\Gamma_r$. Each horizontal wire represents either a block of $n_r$ qubits or a block of $m_r$ qubits as written on top of the wire. In the first layer, $\Gamma_r$ is applied on the first wire mapping $n_r$ qubits to $m_r$ qubits. While $\Gamma_r$ is being applied on the first wire, error correction steps are applied on the remaining wires. On the second layer, $\Gamma_r$ is applied on the second wire, while on the first wire multiple layers of idle gate is applied, and on the wires $i = 3, 4, \dots$ correction steps are applied. Similarly, on $j^{th}$ layer for $j \leq h$, where $h$ is the total number of wires, $\Gamma_r$ is applied on the $j^{th}$ wire, and on all the wires before it $i = 1, \dots, j-1$, idle gates are applied and on all the wires after it $i = j+1, \dots, h$ error correction steps are applied.
• Figure 3: Constant overhead interface: sequential application of the partial interfaces $\Gamma_{r-y,r-y-1}$ for $y = 0, 1, 2$ and $h = 4$. As the level decreases from $r$ to $r-1$ to $r-2$, the number of blocks increases $(4, 8, 16)$, and the fraction of blocks on which the interface is applied in parallel (shaded) also grows $(1/4, 1/2, 3/4)$.
• Figure 4: The quantum circuit $\Gamma_{r,r'}$. Here $s_1, s_2 = O(\mathop{\mathrm{poly}}\nolimits(m_r))$.
• Figure 5: Construction of the interface $\Xi^{[h]}_{r}$: for a fixed $r'$, we first apply $\Xi^{[h]}_{r,r'}$ and then apply $\Gamma_{r',1}$ to each $\mathcal{C}_{r'}$ block.

Theorems & Definitions (55)

• Theorem 1: Informal version of Theorem \ref{['thm:main-stprep']}
• Theorem 2: Informal version of Theorem \ref{['thm:ft-cons-int-main']}
• Definition 3: Weight of an operator
• Definition 4: Weight of a quantum channel
• Definition 5: Diamond norm
• Definition 6: Stabilizer reduced error weight of a Pauli operator gu2024single
• Definition 7: Stabilizer reduced error weight of a linear operator christandl2025fault
• Definition 8: Stabilizer reduced error weight of a superoperator christandl2025fault
• Definition 9: Quantum circuit
• Definition 10: Stochastic circuit-level noise