Quantum error correction through the lens of multipartite entanglement

Sudevan, Sowrabh (2026) Quantum error correction through the lens of multipartite entanglement. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Sowrabh Sudevan (18IP003)) 18IP003.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

n-qubit, m-uniform states are pure states such that all of their marginals or reduced density matrices on m-qubits are maximally mixed. Any set of m-qubits is maximally entangled with the rest and therefore these states are genuinely multipartite entangled (GME) states. This thesis explores how such states can be found within a class of graph states and how the combination of their entanglement and stabilizer structure makes them useful for quantum error correction. We show that a class of graph states defined on regular lattices called cluster states are m-uniform. In fact, we find constructions of m-uniform states for all integer values of m among such graph states. Individual m-uniform graph states are merely codewords of m+1 distance non-degenerate codes, i.e., they can uniquely identify the action of up to $\lfloor m/2 \rfloor$ Pauli operators, but they cannot encode any logical information. Despite the lack of space to store information, we found a novel protocol that uses such states for benchmarking the error rates on a quantum computer. We also demonstrate this benchmarking protocol on the IBM manilla quantum computer. Next, we explore a novel measurement-based protocol that we have called “tent-peg protocol” to encode logical qubits on to qubits hosting an m-uniform graph state, i.e., build non-trivial codes from trivial codes or codewords. The codes we have built belong to the codeword stabilized codes (CWS) framework. Our contributions to this framework are: • We have explored the CWS framework where the initial input graph state is also m-uniform and found the necessary and sufficient mathematical condition for proper encoding. • As opposed to earlier extensive numerical searches we have explicit constructions with proofs for codes which can store an extensive number of logical qubits. • As opposed to earlier circuit based encoding protocols, we present a measurement-based protocol where we emphasize its useful features such as the ability to sequentially encode or partial decode logical qubits. We conclude with discussions of how generalizing m-uniformity can result in new types of codes tailor-made for specific types of correlated noise.

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