[Paper Review] How Much Entanglement Does a Quantum Code Need?

Gaojun Luo, Martianus Frederic Ezerman+2 more|arXiv (Cornell University)|Jul 12, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates the trade-off between entanglement cost and error-correcting performance in entanglement-assisted quantum error-correcting codes (EAQECCs). By leveraging classical coding theory, the authors introduce three new propagation rules to systematically vary entanglement resource usage, enabling the construction of optimal qubit and qutrit EAQECCs with improved rates and error-handling capabilities, as validated through comprehensive tables of best-known parameters for lengths up to 64 qubits and 36 qutrits.

ABSTRACT

In the setting of entanglement-assisted quantum error-correcting codes (EAQECCs), the sender and the receiver have access to pre-shared entanglement. Such codes promise better information rates or improved error handling properties. Entanglement incurs costs and must be judiciously calibrated in designing quantum codes with good performance, relative to their deployment parameters. Revisiting known constructions, we devise tools from classical coding theory to better understand how the amount of entanglement can be varied. We present three new propagation rules and discuss how each of them affects the error handling. Tables listing the parameters of the best performing qubit and qutrit EAQECCs that we can explicitly construct are supplied for reference and comparison.

Motivation & Objective

• To understand how entanglement resources can be systematically varied in EAQECCs without sacrificing error-correcting capability.
• To develop new propagation rules that allow for fine-tuning of entanglement usage in quantum code design.
• To construct and tabulate the best-performing qubit and qutrit EAQECCs for practical deployment, minimizing entanglement cost while maximizing rate and distance.
• To provide a reference framework for comparing EAQECCs using net rate and minimum distance as key performance metrics.
• To identify open problems in bounding EAQECC parameters and optimizing hull dimensions and dual distances.

Proposed method

• Proposes three new propagation rules to modify EAQECC parameters by adjusting entanglement (c), length (n), dimension (κ), and minimum distance (δ).
• Applies classical coding theory techniques, including subcode, puncturing, and shortening operations, to generate new EAQECCs from existing ones.
• Uses randomized and exhaustive search methods to determine the minimum number of maximally entangled pairs (cmin) required for qutrit codes.
• Employs Theorem 14 and Theorem 16 to construct codes with improved parameters, particularly when d′ = d + 1.
• Compresses and lists best-known parameters using eight propagation rules, including length extension, subcode, puncturing, and entanglement-increasing operations.
• Validates results by comparing constructed codes against known bounds and providing online records of parameters for qubit (n ≤ 64) and qutrit (n ≤ 36) codes.

Experimental results

Research questions

• RQ1 How can the amount of pre-shared entanglement in EAQECCs be minimized while preserving or improving error-correcting performance?
• RQ2 What are the optimal trade-offs between code rate, minimum distance, and entanglement cost in qubit and qutrit codes?
• RQ3 Which classical coding constructions yield the most efficient EAQECCs under entanglement constraints?
• RQ4 Can propagation rules be systematically derived and applied to generate new EAQECCs with improved parameters?
• RQ5 What are the tightest theoretical bounds on the parameters of EAQECCs, especially for small-length codes?

Key findings

• The authors construct the best-known qubit EAQECCs for lengths 3 ≤ n ≤ 64 and qutrit EAQECCs for 3 ≤ n ≤ 36, with parameters listed in Tables 1 and 2.
• The new propagation rules enable systematic construction of EAQECCs with reduced entanglement cost while maintaining or improving code distance and rate.
• For qutrit codes, a randomized search strategy yields cmin values that outperform standard constructions, especially when d′ = d + 1.
• The net rate ¯ρ(Q) can be positive, negative, or zero; even negative-rate codes are useful if entanglement is pre-shared and stored for later use.
• The study identifies that using a matrix of rank s instead of a diagonal matrix in Theorem 16 can improve the efficiency of randomized code construction.
• The results are publicly available in an online record of bounds on minimum distance for entanglement-assisted quantum codes, facilitating benchmarking and further research.