State complexity and phase identification in adaptive quantum circuits

• URL: http://arxiv.org/abs/2509.17014v1
• Date: Sun, 21 Sep 2025 10:03:05 GMT
• Title: State complexity and phase identification in adaptive quantum circuits
• Authors: Guoding Liu, Junjie Chen, Xiongfeng Ma,
• Abstract summary: We show that states within the same quantum phase, defined by a set of quantum states connected with constant-depth circuits, must share the same scaling of weight or anti-shallowness.<n>This establishes these quantities as indicators of quantum phases and their essential roles in many-body physics.
• Score: 6.041950339728775
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Adaptive quantum circuits, leveraging measurements and classical feedback, significantly expand the landscape of realizable quantum states compared to their non-adaptive counterparts, enabling the preparation of long-range entangled states and topological phases at constant depths. However, the ancilla overhead for preparing arbitrary states can be prohibitive, raising a fundamental question: which states can be efficiently realized with limited ancilla and low depth? Addressing this question requires a rigorous quantitative characterization of state complexity, or the minimum depth and ancillas, to realize a state in adaptive circuits. In this work, we tackle this problem by introducing two properties of quantum states: state weight and anti-shallowness, connected to the correlation range and correlation strength within a state, respectively. We prove that these quantities are bounded under limited circuit resources, thereby providing rigorous bounds on the approximate complexity of state preparation and gate implementation. Illustrative examples include the GHZ state, W state, QLDPC code states, and the Toffoli gate. Besides complexity, we show that states within the same quantum phase, defined by a set of quantum states connected with constant-depth circuits, must share the same scaling of weight or anti-shallowness. This establishes these quantities as indicators of quantum phases and their essential roles in many-body physics.

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