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Entanglement-magic separation in hybrid quantum circuits (2312.02039v2)

Published 4 Dec 2023 in quant-ph and cond-mat.stat-mech

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.

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Citations (19)

Summary

  • The paper shows that hybrid quantum circuits undergo distinct phase transitions in magic, quantified via stabilizer Rényi entropy, separate from entanglement transitions.
  • It employs a framework of random two-site Clifford gates with controlled T-gate injections and projective measurements to probe these phase shifts.
  • The findings imply new strategies for designing fault-tolerant, resource-efficient quantum architectures by independently harnessing quantum entanglement and magic.

Entanglement–Magic Separation in Hybrid Quantum Circuits

The paper "Entanglement–magic separation in hybrid quantum circuits" investigates the roles of entanglement and quantum 'magic' in the context of hybrid quantum circuits, thereby providing insights into the preconditions necessary for quantum computational advantage over classical counterparts. This paper diverges from conventional focuses solely on entanglement by demonstrating how 'magic,' quantified via stabilizer entropy, manifests phase transitions independent of entanglement within specifically engineered quantum circuits. The implications of this separation present new potential pathways for the architecture and understanding of quantum computation, particularly in fault-tolerant and resource-efficient setups.

Quantifying Magic and Its Importance

In quantum mechanics, stabilizer states form a class of states that can be efficiently simulated on classical computers. Magic, defined as the nonstabilizerness of quantum states, is a requisite for potential quantum speedup, along with entanglement. Quantum circuits that incorporate non-Clifford gates—beyond the universal Hadamard, phase, and controlled-not (CNOT) gates—can achieve this transition into nonstabilizerness. Specifically, 'magic' is evaluated using stabilizer R\'enyi entropy, an efficient measure applicable to many-body systems.

Main Findings

The paper introduces a hybrid quantum circuit that utilizes a blend of projective measurements and controlled injections of T-gates (non-Clifford resources) into a random two-site Clifford gate scaffold. The numerical exploration reveals multiple distinct phase transitions characterized by:

  • A transition in magic between sub-extensive and area-law scaling at critical measurement rates differing from those governing entanglement transitions.
  • Identification of a regime in which entanglement follows an area law, while magic maintains a (sub)-extensive behavior, indicating that the manipulations alter the density matrices' non-linear properties beyond entanglement alone.

This separation suggests that non-Clifford resources introduce a novel form of criticality in quantum circuits, highlighting a new dimension of resource theories needed for universal quantum computing. The measurements utilized in the circuit play a vital role in driving these transitions, which could independently influence entanglement and magic.

Practical and Theoretical Implications

Understanding independent transitions of magic provides practical insights into designing quantum architectures that ensure robust quantum error correction while maintaining advantageous computational properties. The independent phase transitions found in magic, in combination with entanglement, imply more complex dynamics in quantum state evolution than previously known, influencing how quantum information is preserved or destroyed in computational processes.

The theoretical implications are twofold. Firstly, they necessitate a broader view of quantum phase transitions in hybrid circuits, opening avenues for further exploration of state properties like magic and their role in different quantum systems. Secondly, they offer a novel perspective on building quantum devices, potentially optimizing quantum gates' implementation and resource allocation in fault-tolerant frameworks by minimizing non-Clifford resource overhead.

Concluding Remarks and Future Directions

The paper's results underscore the need for further investigation into the nature of non-linear properties in density matrices and their implications for computational advancements. Future research could explore different non-Clifford resources or hybrid circuits with alternative measurement configurations. Such explorations might lead to practical methodologies for leveraging independent phase transitions to optimize quantum algorithms and error correction codes effectively. Understanding these dynamics will be crucial as the field progresses toward realizing scalable, efficient, and powerful quantum computing.

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