Learning quantum states and unitaries of bounded gate complexity

TL;DR

This work proves that to learn a state generated by a quantum circuit with G two-qubit gates to a small trace distance, a sample complexity scaling linearly in G is necessary and sufficient and that the optimal query complexity to learn a unitary generated by G gates to a small average-case error scales linearly in G.

Abstract

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 $G$ two-qubit gates to a small trace distance, a sample complexity scaling linearly in $G$ is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by $G$ gates to a small average-case error scales linearly in $G$. 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 $G$ must scale exponentially in $G$. 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.

Figures (2)

• Figure 1: (a)-(c) Schematic overview of the learning models in this work. (a) Learning quantum states with bounded circuit complexity $G$. (b) Learning unitaries with bounded circuit complexity $G$. (c) Learning classical functions from quantum experiments with bounded circuit complexity $G$. (d) A conceptual depiction of the sample complexity of learning states in trace distance and unitaries in average-case distance scales linearly with circuit complexity, while that of learning unitaries in worst-case distance scales exponentially.
• Figure 2: Sample complexity $N$ of the learning algorithm with different gate numbers $G$ and reconstruction fidelity $F$. The unknown target states are pure states on $n=10000$ qubits generated from $G$ gates, either concentrated on the first $4$ qubits (left) or randomly placed (right). The contour plot represents the fidelity for different $G$ and $N$ averaged over many random instances. Sample complexities with average fidelity $F$ and median fidelity $F_\mathrm{med}$ are plotted in solid and dashed lines, respectively.

Theorems & Definitions (127)

• Theorem 1: State learning
• Theorem 2: State learning computational complexity
• Theorem 3: Worst-case unitary learning
• Theorem 4: Average-case unitary learning
• Theorem 5: Learning with classical descriptions
• Theorem 6: Unitary learning computational complexity
• Theorem 7: Approximating and learning with physical functions
• Lemma 1: Spectral and diamond distance of unitaries, variant of caro2022generalization
• proof
• Lemma 2: Subadditivity of diamond distance watrous2018book