Abstract
The classical simulation of highly entangling quantum dynamics is conjectured to be generically hard. Thus, recently discovered measurement-induced transitions between highly entangling and low-entanglement dynamics are phase transitions in classical simulability. Here, we study simulability transitions beyond entanglement: noting that some highly entangling dynamics (e.g., integrable systems or Clifford circuits) are easy to classically simulate, thus requiring “magic”—a subtle form of quantum resource—to achieve computational hardness, we ask how the dynamics of magic competes with measurements. We study the resulting “dynamical magic transitions” focusing on random monitored Clifford circuits doped by gates (injecting magic). We identify dynamical “stabilizer purification”—the collapse of a superposition of stabilizer states by measurements—as the mechanism driving this transition. We find cases where transitions in magic and entanglement coincide, but also others with a magic and simulability transition in a highly (volume-law) entangled phase. In establishing our results, we use Pauli-based computation, a scheme distilling the quantum essence of the dynamics to a magic state register subject to mutually commuting measurements. We link stabilizer purification to “magic fragmentation” wherein these measurements separate into disjoint, -weight blocks, and relate this to the spread of magic in the original circuit becoming arrested.
- Received 18 December 2023
- Revised 23 May 2024
- Accepted 3 June 2024
DOI:https://doi.org/10.1103/PRXQuantum.5.030332
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
What makes quantum mechanics truly quantum? A possible answer lies in the complexity of simulating quantum mechanics on a classical computer. By exploiting the structure of quantum states, one can sometimes efficiently classically simulate certain corners of the Hilbert space. Two prominent classes of classically tractable states are product and stabilizer states. These states reveal two indicators of complexity: entanglement and “magic,” lacking from product and stabilizer states, respectively. “True” complexity requires both. Here we study how these ingredients impact the simulability of quantum dynamics.
Concretely, we examine how locally injecting magic into stabilizer dynamics can result in magic being scrambled to spread across an entire system and lead to complexity and how local measurements can combat this by arresting the spread of magic. While previous research considered the competition between the generation and destruction of entanglement, here, by considering separate entangling and magic-injecting operations, we reveal a new dimension to how classically tractable versus complex dynamics can emerge.
The three mechanisms—magic injection, scrambling, and measurements—compete highly nontrivially. We find sharp transitions between a phase in which magic survives for long times and spreads and a phase in which it is short lived and localized. In the former phase, classically simulating the system can be hard, while it is easy in the latter phase. Remarkably, despite the fundamental difference between magic and entanglement, this transition can robustly coincide with the previously known transition for entanglement. However, our model also has a regime where a highly entangled and magic-rich, and thus complex, phase gives way to a classically simulable phase by a transition in magic, while persisting in the generation of large entanglement.
This work highlights the central importance of magic spreading in generating complex quantum dynamics. Indeed, we show that it can be the driving force behind a transition from classically easily simulable to “truly quantum” behavior.
