Entanglement-magic separation in hybrid quantum circuits

Abstract

Magic describes the distance of a quantum state to its closest stabilizer state. It is -- like entanglement -- a necessary resource for a potential quantum advantage over classical computing. We study magic, quantified by stabilizer entropy, in a hybrid quantum circuit with projective measurements and a controlled injection of non-Clifford resources. We discover a phase transition between a (sub)-extensive and area law scaling of magic controlled by the rate of measurements. The same circuit also exhibits a phase transition in entanglement that appears, however, at a different critical measurement rate. This mechanism shows how, from the viewpoint of a potential quantum advantage, hybrid circuits can host multiple distinct transitions where not only entanglement, but also other non-linear properties of the density matrix come into play.

Figures (8)

• Figure 1: Phase transitions in entanglement (a) compared to the possible phase transitions in entanglement and magic (b) in one--dimensional quantum circuits. Quantum circuits in phase I may lead to an advantage over classical computing, while circuits in phases II and III are amenable to simulation through tensor network methods, and circuits in phases III and IV can be tackled efficiently with the stabilizer formalism.
• Figure 2: Hybrid quantum circuit with a brickwork structure of random 2--site Clifford gates interspersed with measurements in the computational bases and T--gates for $N=8$ qubits and 4 time steps. The measurements and T--gates appear randomly with probability $p$ and $q(N)=\eta/N^\beta$, respectively, where $\beta$ determines the scaling of the T--gate density with prefactor $\eta$.
• Figure 3: Numerical results for the average steady state entanglement (a) and magic (b) versus system size $N$ for several different measurement rates $p$ and T--gate density $q(N) = \eta/N^\beta$ with $\eta=2.0$ and $\beta=1$. Note that for the measurement rate $p=0.18$ (marked with $\bigstar$) the lines bend downwards in panel (a) but upwards in panel (b), which indicates an area law in entanglement but a (sub)--extensive law in magic.
• Figure 4: Phase diagram ($\eta$ vs. $p$) extracted from the numerical results for $\beta=1$ and $\eta=\{0.5,\ldots,8.0\}$. The bullets and crosses show the position of the entanglement and magic phase transition, respectively. We present the details of the numerical error estimation (displayed as error bars) in the SM sm_note. The colored regions resulting from this data correspond to the phases I, II, and III shown in Fig. \ref{['fig:magic-and-entanglement']}. For low measurement rates we find that both entanglement and magic follow a (sub)--extensive scaling (phase I), while for high measurement rates both follow an area law (phase III). However, we also find an extended region in between in which entanglement scales as an area law, but magic scales (sub)--extensively (phase II). The dotted line corresponds to the data shown in Fig. \ref{['fig:entropies-vs-size']}.
• Figure 5: Phase diagram ($\beta$ vs. $p$) of magic for the separable hybrid quantum circuit (a) and a sketch of the conjectured phase diagram for the full hybrid quantum circuit (b). While preliminary numerical results for the full circuit indicate that most of the region below $\beta<1$ follows a (sub)--extensive law it is still unclear whether there exists a transition to an area scaling with $p_c^\mathrm{magic}<1$ in that region.