Entanglement Phase Transition with Spin Glass Criticality

We define an ensemble of random Clifford quantum circuits whose output state undergoes an entanglement phase transition between two volume-law phases as a function of measurement rate. Our setup maps exactly the output state to the ground space of a spin glass model. We identify the entanglement phases using an order parameter that is accessible on a quantum chip. We locate the transition point and evaluate a critical exponent, revealing spin glass criticality. Our Letter establishes an exact statistical mechanics theory of an entanglement phase transition.

Figure 1

The spin glass model in its various forms. (a) A graph of spins and interactions, as in Eq. (1). (b) The matrix $B$ of the graph, as defined in Eq. (2), where each column is a spin and each row is an interaction. (c) The quantum circuit built from $B$. A 1 corresponds to the rectangular gate shown in the inset, composed of a CNOT, then a swap, whereas a 0 is a swap gate. The initial quantum state is $|\psi {⟩}_{\mathrm{in}}$, the quantum state after the applying the gates in $B$ is $|\psi {⟩}_{\mathrm{out}}$ in Eq. (5), and after measuring a region $A$ of the interaction qubits, the output state is $|\psi {⟩}_{\mathrm{out},A}$ in Eq. (7). The labels for the qubits in the quantum circuit specify the spins and interactions.

Figure 2

Finite-size scaling of the averaged entanglement susceptibility $⟨{\chi }_S⟩$, in units of $\log 2$. We average over 20 000 samples for each system size, and error bars indicate standard error of the mean. We calculate the exact expression by taking a finite derivative of Eq. (10). The inset gives the collapsed version of the main plot within the region ${\alpha }_c\pm 0.05$.

Figure 3

Overlap $q(\alpha )$ of the ground states. We average over 1000 samples per system size. We sample $\min (100,{\mathcal{N}}_{\mathrm{GS}})$ ground states, and use the unique ones to calculate the overlap in Eq. (14). Error bars indicate standard error of the mean. We use either the $|\psi {⟩}_{\mathrm{in}}=|+{⟩}^{\otimes N}|0{⟩}^{\otimes M}$ input state (blue circles), or a random input state (red squares). The dashed line is the exact overlap in the thermodynamic limit [22].