量子糾錯碼上量子邏輯位元的邏輯受控反閘電路合成

Logical CNOT circuit synthesis on logical qubits of quantum error correction codes

吳柏叡(Bo-Ruei Wu)

指導教授 ： 管希聖

國立臺灣大學/理學院/物理學系/碩士(2025年)

https://doi.org/10.6342/NTU202500417

摘要

量子計算對比於經典計算的有前途的方法，在化學模擬、金融計算和機器學習中找到了應用。 最近的進步導致了配備各種量子位元的量子計算機的發展，如超導、離子陷阱和中性原子。 每種型別的量子位元都有其優缺點。 然而，儘管量子計算方法各不相同，但所有型別的量子位元都面臨著一個共同的挑戰：量子位元的固有誤差。 這個錯誤非常嚴重，可能會導致資訊丟失，並損壞從量子計算機獲得的結果。 為了克服這個問題，量子糾錯在量子計算中至關重要。
為了實現量子糾錯，有必要將物理量子位元編碼為邏輯量子位元，並根據不同型別的量子糾錯碼執行邏輯運算子。 我們專注於一個特定的糾錯碼家族，透過在量子糾錯程式碼的幾何形狀上排列物理量子位元，可以在邏輯量子位元的同一程式碼塊中實現邏輯CNOT。 在將邏輯電路編譯為物理電路時，由於不同塊中邏輯量子位之間的邏輯CNOT閘，這種方法可能會造成相當大的代價。
在所介紹的背景下，我們設計了一種演算法來合成電路，將這種開銷從 $O(\\\\\\\\\\\\\\\\frac{Q^2}{\\\\\\\\\\\\\\\\log_2{Q}})$ 降至近乎 $O(Q)$, $Q$ 是邏輯量子位元的是邏輯量子位元的數量數量 ，並且可以在真正的量子計算機上執行。 與原始電路相比，合成電路的保真度更好。

關鍵字

量子電腦 ； 量子電路 ； 量子糾錯碼 ； 邏輯合成

並列摘要

Quantum computing, a promising alternative to classical computing, has found applications in chemistry simulation, financial calculations, and machine learning. Recent advancements have led to the development of quantum computers equipped with various types of qubits, such as superconducting, ion-trap, and neutral atoms. Each type of qubit has its advantages and disadvantages. However, despite the diverse approaches to quantum computing, all types of qubits face a common challenge: the inherent error in qubits. This error is so significant that it can lead to information loss and corrupt the results obtained from quantum computers. To overcome this problem, error correction is essential in quantum computing.
To achieve quantum error correction, encoding physical qubits to logical qubits and performing logical operators based on different types of quantum error correction codes is necessary. We focus on a specific family of codes that can implement logical CNOT in the same code block on logical qubits by permutating physical qubits on the geometry of the quantum error correction code and transversal CNOT between two code blocks. While compiling logical circuits to physical circuits, this approach may cause significant overhead due to the logical CNOT gates between qubits in different blocks which are implemented by transversal CNOT on physical qubits.
In the context presented, we introduce an algorithm to synthesize CNOT circuits that reduce the overhead of transversal CNOT from $O(\\\\\\\\\\\\\\\\frac{Q^2}{\\\\\\\\\\\\\\\\log_2{Q}})$ to nearly $O(Q)$, where $Q$ is the number of logical qubits encoded by a quantum error correction code and the synthesized circuit can be performed on real quantum computers. The synthesized circuits have better fidelity compared to the original circuits.

並列關鍵字

quanutm computing ； quantum circuit ； quantum error correction code ； circuit synthesis

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