Magic transition in measurement-only circuits

Abstract

Magic, also known as nonstabilizerness, quantifies the distance of a quantum state to the set of stabilizer states, and it serves as a necessary resource for potential quantum advantage over classical computing. In this work, we study magic in a measurement-only quantum circuit with competing types of Clifford and non-Clifford measurements, where magic is injected through the non-Clifford measurements. This circuit can be mapped to a classical model that can be simulated efficiently, and the magic can be characterized using any magic measure that is additive for tensor product of single-qubit states. Leveraging this observation, we study the magic transition in this circuit in both one- and two-dimensional lattices using large-scale numerical simulations. Our results demonstrate the presence of a magic transition between two different phases with extensive magic scaling, separated by a critical point in which the mutual magic exhibits scaling behavior analogous to entanglement. We further show that these two distinct phases can be distinguished by the topological magic. In a different regime, with a vanishing rate of non-Clifford measurements, we find that the magic saturates in both phases. Our work sheds light on the behavior of magic and its linear combinations in quantum circuits, employing genuine magic measures.

Figures (9)

• Figure 1: Measurement-only quantum circuit with two types of competing measurements in (a) one-dimensional and (b) two-dimensional lattices. Gray boxes on edges denote measurements $M_{zz}$ on adjacent spins and violet circles denote measurements $M_x$ on a single spin. Each time step comprises one row of $M_{zz}$ measurements followed by a row of $M_x$ measurements.
• Figure 2: (a) Sketch of rotated Bell cluster. (b) Schematics of partitions: in the left part we show the partition for an open chain for the calculation of topological magic in Eq. \ref{['eq:topo_magic']}. In the right part we show the partition for periodic boundary condition for the calculation of mutual magic in Eq. \ref{['eq:mutual_magic']}.
• Figure 3: (a) Magic density $\frac{\mathcal{M}/\mathcal{M}_T}{L}$ and (b) mutual magic of half subsystem $I_\mathcal{M}(L/2)/\mathcal{M}_T$ with periodic boundary condition.
• Figure 4: (a) Mutual magic $(I_\mathcal{M}(\ell) - I_\mathcal{M}(L/2))/\mathcal{M}_T$ at $p=p_c$ and $L=2048$ and (b) the dynamics of $I_\mathcal{M}(L/2)/\mathcal{M}_T$ as a function of time $t$ for different system sizes with periodic boundary condition. The black line denotes a linear fit.
• Figure 5: (a) The topological magic $\mathcal{M}^t_{\mathrm{topo}}/\mathcal{M}_T$ with open boundary condition. (b) Data collapse for the topological magic, showing excellent agreement with the percolation value $p_c=0.5$ and $\nu=4/3$.