Approximate Quantum Error Correction with 1D Log-Depth Circuits

TL;DR

This work rigorously proves through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance and establishes that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance.

Abstract

Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose decoupling theorems tailored for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.

Figures (10)

• Figure 1: (a) Diagram of the 1D brickwork circuit. The circuit is constructed by brickwise two-qudit gates. The total qudit number is $k$, and the circuit depth is $d$. When using this circuit for encoding, we set the local dimension as $q = 2^{\frac{n}{k}}$ while only one qubit of information is encoded in one qudit. In our work, $\frac{n}{k}$ is chosen as a constant, so $q$ is a constant. The encoding rate maintains $\frac{k}{n}$ in this scheme, and the total physical qubit number is $n$. Note that our results apply to arbitrary local dimensions, including 1D brickwork circuits composed of two-qubit gates acting on $n$ qubits, where each qubit encodes logical information with probability $\frac{k}{n}$. The Choi error still decays polynomially, though with a different scaling. Further discussions are provided in Appendix \ref{['appendssc:1DLRCQEC']}. (b) Diagram of the 1D double-layer blocked circuit. Two layers of blocked unitary gates construct the circuit. The term $U_C^{i,i+1}$ is a random gate from a unitary 2-design ensemble on regions $i$ and $i+1$. The circuit is divided into $2N$ regions, with each region having $\xi = n/2N$ qubits. When using this circuit for encoding, the logical qubits and ancillary qubits are evenly distributed across each region. Each region contains $\frac{k}{n}\xi$ logical qubits and $\frac{n-k}{n}\xi$ ancillary qubits.
• Figure 2: The encoding rates of different bounds. For Pauli noise, the encoding rates given by the smooth and non-smooth decoupling theorems are bounded by $1-h(\Vec{p})$ and $1-f(\Vec{p})$, respectively. They are depicted in blue with solid and dashed lines, respectively. For the strength-$p$ depolarizing noise, the four parameters of the Pauli noise are $\vec{p} = (1-3p/4,p/4,p/4,p/4)$. The hashing bound $1-h(\Vec{p})$ is higher than the bound $1-f(\Vec{p})$. For erasure errors, the encoding rates given by the smooth and non-smooth decoupling theorems are bounded by $1-2p$ and $1-\log(1+3p)$ and depicted in red with solid and dashed lines, respectively. The bound $1-2p$ is the quantum channel capacity of the erasure error, and it is higher than the bound $1-\log(1+3p)$. For amplitude damping noise, the encoding rates given by the smooth and non-smooth decoupling theorems are bounded by $h(\frac{1-p}{2})-h(\frac{p}{2})$ and $-\log(\frac{1}{2-p}+\sqrt{\frac{p}{2-p}})$ and depicted in orange with solid and dashed lines, respectively. For strength-$p$ nearest-neighbor correlated noise, only the non-smooth decoupling theorem is applicable. The achievable encoding rate is $1-2\log(\sqrt{1-p}+\sqrt{p})$, depicted in green.
• Figure 3: The diagram of the decoupling task. The process begins with an entangled state, $\rho_{SR}$, where $R$ is a reference system identical to $S$. The system $S$ undergoes an encoding unitary operation, $U_S$, followed by a channel $\mathcal{T}_{S\rightarrow E}$ that maps $S$ to the environment $E$. The resulting output state is $\sigma_{ER}$. The decoupling task examines whether $\sigma_{ER}$ is sufficiently close to a tensor-product state $\tau_E \otimes \rho_R$ for certain choices of $U_S$ and $\mathcal{T}_{S\rightarrow E}$. The role of $U_S$ is to spread information within $S$, making it difficult for the environment $E$ to extract information about $S$ and $R$ through $\mathcal{T}_{S\rightarrow E}$. When $E$ has no information about $SR$, it becomes decoupled from $R$.
• Figure 4: Diagram of the procedure of quantum error correction. The initial logical information is stored in a quantum state $\rho_L$. Before undergoing noise $\mathcal{N}$, $\rho_L$ is first encoded by $\mathcal{E}$ with $\mathcal{E}(\rho) = U\rho\otimes \ketbra{\mathbf{0}}U^{\dagger}$ for unitary encoding operation $U$. Finally, one applies decoding operation $\mathcal{D}$ to retrieve logical information. More generally, $\rho_L$ may be entangled with a reference system $R$.
• Figure 5: (a) Parameter regime where $c_{\mathrm{block}} > 0$. In this case, the double-layer blocked encoding scheme can make the Choi error decay, but the block-encoding scheme cannot. (b) Parameter regime where $c_{\mathrm{block}} > -\frac{\alpha}{4}$. In this case, the double-layer blocked encoding scheme outperforms the block-encoding scheme.

Theorems & Definitions (59)

• Lemma 1: Lemma 3.7 in Ref. Dupuis2014Decoupling
• Theorem 1: Non-smooth decoupling theorem for 1D double-layer blocked circuits
• proof : Proof sketch
• Theorem 2: Decoupling with 1D brickwork circuits
• proof : Proof sketch
• Theorem 3: Smooth decoupling theorem for 1D double-layer blocked circuits
• Corollary 1: AQEC performance of $\mathfrak{C}$ for Pauli noise
• proof : Proof sketch
• Corollary 2: Random codes from $\mathfrak{C}$ achieve the hashing bound for Pauli noise
• proof : Proof sketch