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Fault-Tolerant Quantum LDPC Encoders

Abhi Kumar Sharma, Shayan Srinivasa Garani

TL;DR

Fault-tolerant encoders for quantum LDPC codes are developed by partitioning qubits into blocks and using preshared entanglement to enable transversal encoding, reducing error propagation under depolarizing noise. The framework covers both entanglement-unassisted CSS-based QLDPCs and entanglement-assisted variants, employing row-echelon transformations and local block encoders to realize encoders. The fault-tolerant construction provides explicit bounds on error propagation and demonstrates a practical example that yields a transversal, block-local encoding with improved reliability over non-ft designs. Together, these contributions offer a scalable route to practical QLDPC encoding in quantum information processing.

Abstract

We propose fault-tolerant encoders for quantum low-density parity check (LDPC) codes. By grouping qubits within a quantum code over contiguous blocks and applying preshared entanglement across these blocks, we show how transversal implementation can be realized. The proposed encoder reduces the error propagation while using multi-qubit gates and is applicable for both entanglement-unassisted and entanglement-assisted quantum LDPC codes.

Fault-Tolerant Quantum LDPC Encoders

TL;DR

Fault-tolerant encoders for quantum LDPC codes are developed by partitioning qubits into blocks and using preshared entanglement to enable transversal encoding, reducing error propagation under depolarizing noise. The framework covers both entanglement-unassisted CSS-based QLDPCs and entanglement-assisted variants, employing row-echelon transformations and local block encoders to realize encoders. The fault-tolerant construction provides explicit bounds on error propagation and demonstrates a practical example that yields a transversal, block-local encoding with improved reliability over non-ft designs. Together, these contributions offer a scalable route to practical QLDPC encoding in quantum information processing.

Abstract

We propose fault-tolerant encoders for quantum low-density parity check (LDPC) codes. By grouping qubits within a quantum code over contiguous blocks and applying preshared entanglement across these blocks, we show how transversal implementation can be realized. The proposed encoder reduces the error propagation while using multi-qubit gates and is applicable for both entanglement-unassisted and entanglement-assisted quantum LDPC codes.
Paper Structure (10 sections, 2 theorems, 36 equations, 3 figures)

This paper contains 10 sections, 2 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Non-fault-tolerant Encoders
  4. Transformation of X stabilizers
  5. Transformation of Z stabilizers
  6. Error Propagation Analysis
  7. Fault-tolerant encoders
  8. Error Propagation Analysis
  9. Conclusions
  10. Proofs of the results in the main text

Key Result

Lemma 1

For a $[[n,k,d]]_2$ CSS code with X and Z stabilizer matrices $\Tilde{H}^{(x)}_{\rho_1 \times n}$ and $\Tilde{H}^{(z)}_{\rho_2 \times n}$, the probability of error propagation using CNOT gates for a non-fault tolerant encoder is upper bounded as $P_{NF} \leq 1 -P_{NF_1} P_{NF_2}$, where $p$ is the probability of depolarizing noise channel $\pi(p)$, $w^{(AB)}_i$$\left(\mathrm{or}\, w^{(A)}_i\right

Figures (3)

  • Figure 1: (a) shows the error propagation of an X error to the target qubit when the control qubit is affected with an X error. (b) shows the error propagation of the Z error to the control qubit when the target qubit is affected by the Z error.
  • Figure 2: (a) shows the encoding of a non-fault-tolerant quantum code $Q=[[n,k,d]]_2$ such that the operator $\mathcal{E}$ are applied on all $n$-qubits to generate a codeword. (b) shows the fault-tolerant encoding of an modified quantum code $Q_F=[[n+g\rho,k+g\rho,d]]_2$ such that the qubits are divided into the blocks $B_i$ for all $i\in\{1,2,\cdots,g\}$. For every block $B_i$, an $i^{\mathrm{th}}$ entangled qubit of each $\ket{\Phi}^{(i)}_g$ state is associated to design a local encoder $\mathcal{E}_i$ and produces a $n+g\rho$ qubits codeword such that any encoder $\mathcal{E}_i$ does not connect with the $\mathcal{E}_j$ for $i\neq j$ through a multi-qubit gate, so there is no error propagation between any local encoder $\mathcal{E}_i$.
  • Figure 3: The fault-tolerant encoding circuit of a $[[15,10,2;1]]_2$ entanglement-assisted quantum code designed from the $[[9,4,2;1]]_2$ entanglement assisted quantum code is shown. The information state $\ket{\Psi}$ and the four Bell pairs. The reader must note that corresponding to the three stabilizers, we have three $\ket{\Phi}_2$ preshared entangled states across the blocks of the quantum code that are of the form $\ket{\Phi^{+}}$. The other Bell pair $\ket{\Phi^{+}}$ is shared across the transceiver towards achieving entanglement assistance. For convenience, we show all these entangled states as $\ket{\Phi}_2$ in the architecture. $\ket{\Psi}$ is encoded using a series of CNOT gates, where three $\ket{\Phi}_2$s are used to make encoder fault-tolerant and the last $\ket{\Phi}_2$ is used for the entanglement assistance of the code. The horizontal dashed line separates qubits into two different blocks of qubit where CNOT gates are applied locally.

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2