In this article we outline a method for generating linear optics circuits that encode quantum-error-correcting codes. Using this method we produce a single-error-correcting code encoding one wave packet over five which can be implemented using linear optics and feed-forward correction. This code improves on the capacity of the best known code that can be implemented using linear optics and saturates the lower bound for the number of carriers needed for a single-error-correcting code. Our code can correct arbitrary single errors that occur randomly on each wave packet corresponding to a non-Gaussian error model, thus circumventing the so-called no-go theorem for Gaussian quantum-error correction.In this article we outline a method for generating linear optics circuits that encode quantum-error-correcting codes. Using this method we produce a single-error-correcting code encoding one wave packet over five which can be implemented using linear optics and feed-forward correction. This code improves on the capacity of the best known code that can be implemented using linear optics and saturates the lower bound for the number of carriers needed for a single-error-correcting code. Our code can correct arbitrary single errors that occur randomly on each wave packet corresponding to a non-Gaussian error model, thus circumventing the so-called no-go theorem for Gaussian quantum-error correction.

Physical Review AJun 4, 20108 citations

Introduction to the Theory of Error‐Correcting Codes

Vera S. Pless

Wiley eBooksJun 18, 1998170 citations

Optimal Exponent of the Single-Error Correction Threshold with Fixed Redundancy for Analog Error-Correcting Codes

Zhengyi Jiang, Wenhao Liu, Zhongyi Huang +1 more

Analog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $Γ_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$ . Li et al. [2] recently established that for redundancy $r=2$ , every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $Γ_2(\mathcal C)\geq \csc^2(\fracπ{2n})$ , resolving an open problem in [1], and showed that, for every fixed $r \geq 2$ , there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $Γ_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$ . This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq rAnalog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $Γ_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$. Li et al. [2] recently established that for redundancy $r=2$, every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $Γ_2(\mathcal C)\geq \csc^2(\fracπ{2n})$, resolving an open problem in [1], and showed that, for every fixed $r \geq 2$, there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $Γ_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$. This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq r

arXiv:2608.08171Aug 8, 20260 citations

Practical error correction codes and near-term algorithms for quantum computation

Xiaosi Xu

The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.

Jan 1, 20200 citations

Error Correcting Ability of Linear Block Codes

辰一小島

群馬大学教育学部紀要. 自然科学編Jan 1, 19890 citations

Γ_2(\mathcal C)

for real

[n,k]

linear codes with fixed redundancy

r=n-k\geq 2

. Li et al. [2] recently established that for redundancy

r=2

, every real

[n,n-2]

linear code

\mathcal{C}

satisfies

Γ_2(\mathcal C)\geq \csc^2(\fracπ{2n})

, resolving an open problem in [1], and showed that, for every fixed

r \geq 2

, there exists a class of

[n,k]

linear code

\mathcal{C}

over

\mathbb{R}

such that

Γ_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})

. This paper proves the matching converse in [2]. For every

[n,k]

linear code

\mathcal{C}\subseteq \mathbb R^n

with fixed redundancy $2\leq r