Consider a depth-$D$ circuit whose gates have range at most $r$. Under backward Heisenberg evolution, a one-site observable has a <backward light cone of a local quantum circuit> of radius at most $rD$. Choose sites $i,j$ separated by $L>2rD$. Their two backward light cones are disjoint. Since the input is a <product state>, expectations factorize, and hence every <connected correlation function> between the two output observables vanishes.

For the <GHZ state>
$$
|\operatorname{GHZ}_N\rangle
=\frac{|0\rangle^{\otimes N}+|1\rangle^{\otimes N}}{\sqrt2},
$$
however,
$$
\langle Z_i\rangle=\langle Z_j\rangle=0,
\qquad
\langle Z_iZ_j\rangle=1,
\qquad
\langle Z_iZ_j\rangle_c=1
$$
at every separation. The light cones must therefore overlap, which forces
$$
D\geq\frac{L}{2r}.
$$
For opposite ends of a one-dimensional chain, $L=\Theta(N)$, so $D=\Omega(N)$ and no constant-depth local circuit can prepare the GHZ state. The continuous-time version follows directly from the <Lieb-Robinson bound>, with preparation time at least $L/(2v_{\rm LR})$ up to exponentially small tails.

This is the <GHZ-state circuit-depth lower bound>. <Finite-depth local quantum circuit>[Finite-depth local circuits] define equivalence within a gapped phase, so a state with this <long-range order> cannot lie in the same circuit phase as a product state. The GHZ state is the finite-size cat state associated with spontaneous symmetry breaking; it is not a unique short-range-entangled ground state. The persistent distant correlation is precisely the obstruction.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 343 / 3 / b / Solution

A depth-$D$ circuit of gates with range at most $r$ expands the support of a local observable by at most $rD$. States connected by a depth bounded independently of system size are regarded as belonging to the same short-range-entangled phase when no required symmetry is violated.


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