Learning Quantum States and Unitaries of Bounded Gate Complexity

While quantum state tomography is notoriously hard, most states hold little interest to practically minded tomographers. Given that states and unitaries appearing in nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>G</a:mi></a:math> two-qubit gates to a small trace distance, a sample complexity scaling linearly in <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>G</d:mi></d:math> is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>G</g:mi></g:math> gates to a small average-case error scales linearly in <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><j:mi>G</j:mi></j:math>. While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><m:mi>G</m:mi></m:math> must scale exponentially in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:mi>G</p:mi></p:math>. We illustrate how these results establish fundamental limitations on the expressivity of quantum machine-learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries. Published by the American Physical Society 2024