Tough error models – Open Quantum Problems

Cite this problem as Problem 14.

Problem

An error model $E$ is an $e$ -dimensional vector space of operators acting on an $n$ -dimensional Hilbert space $H$ . A quantum code is a subspace $C\subset H$ , and is said to correct $E$ , if the projector $P_C$ onto $C$ satisfies $P_C A^{*} B P_C = \lambda(A,B) P_C$ for all $A,B\in E$ , and suitable scalars $\lambda(A,B)$ .

*Given $e$ and $n$ , find the largest $c=c(e,n)$ such that we can assert the existence of a code $C$ of dimension $c$ without further information about $E$ .

*Find “tough error models” for which this bound is (nearly) tight.

Background

For an introduction to quantum error-correction see, for example, [1].

See [2], where a lower bound of $c(e,n) > n/(e^{2} (e^{2}+1))$ is given.

A trivial upper bound on $c(e,n)$ comes from taking orthogonal projections of roughly equal dimension $n/e$ as the error model. Since the channel with these Kraus operators (a Lüders-von Neumann projective measurement) has capacity at most $n/e$ , it is impossible to find larger code spaces. Hence $c(e,n)\leq \lceil n/e\rceil$ .

References

[1] E. Knill, R. Laflamme, A. Ashikhmin, H. Barnum, L. Viola, and W. H. Zurek, Introduction to Quantum Error Correction, quant-ph/0207170 (2002).

[2] E. Knill, R. Laflamme, and L. Viola, Theory of Quantum Error Correction for General Noise, Phys. Rev. Lett. 84, 2525 (2000) and quant-ph/9908066.