Quantum Circuit Depth Lower Bounds For Homological Codes

Understand

We provide an $\Omega(log(n))$ lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code.

• No circuit-depth lower bound was known before on this code in the general case where the gates can connect qubits even if they are far away; To the best of our knowledge, this is the first time a quantum circuit-depth lower bound is given for unique ground state of a {\it gapped} local Hamiltonian.
• Providing a lower bound in this case seems more challenging, since such systems exhibit exponential decay of correlations and standard lower bound techniques do not apply.
• We prove our lower bound by introducing the new notion of $\gamma$-separation, and analyzing its behavior using algebraic topology arguments.