Complexity of quantum circuits via sensitivity, magic, and coherence

Abstract

Quantum circuit complexity-a measure of the minimum number of gates needed to implement a given unitary transformation-is a fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this work, we study the complexity of quantum circuits using the notions of sensitivity, average sensitivity (also called influence), magic, and coherence. We characterize the set of unitaries with vanishing sensitivity and show that it coincides with the family of matchgates. Since matchgates are tractable quantum circuits, we have proved that sensitivity is necessary for a quantum speedup. As magic is another measure to quantify quantum advantage, it is interesting to understand the relation between magic and sensitivity. We do this by introducing a quantum version of the Fourier entropy-influence relation. Our results are pivotal for understanding the role of sensitivity, magic, and coherence in quantum computation.

42 pages

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Cited by in corpus (14)

• Phase transition in magic with random quantum circuits
• Entanglement-magic separation in hybrid quantum circuits
• Quantum complexity in gravity, quantum field theory, and quantum information science
• Magic Resources of the Heisenberg Picture
• Pseudorandom unitaries are neither real nor sparse nor noise-robust
• Magic Resource Can Enhance the Quantum Capacity of Channels
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Bridging Entanglement and Magic Resources within Operator Space
• Lower bound for simulation cost of open quantum systems: Lipschitz continuity approach
• Coherence and Imaginarity as Resources in Quantum Circuit Complexity
• Wasserstein Complexity of Quantum Circuits
• Resource-Dependent Complexity of Quantum Channels
• On the Hardness of Measuring Magic
• Decoded Quantum Interferometry Under Noise