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It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit.
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Indeed, if d ≥ 2 d\geq 2 then all logical states must have the same single-qubit marginals. However, any state sharing the same single-qubit marginals with an unentangled (product) state must be equal to the unentangled state itself. Hence, a code with k ≥ 1 k\geq 1 and d ≥ 2 d\geq 2 cannot have an unentangled logical state
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Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians
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NLTS hamiltonians from good quantum codes
Anurag Anshu, Nikolas P Breuckmann, and Chinmay Nirkhe · 2023
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Classical simulation of peaked shallow quantum circuits
Sergey Bravyi, David Gosset, and Yinchen Liu · 2024
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How much entanglement is needed for topological codes and mixed states with anomalous symmetry?
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