Learning unknown pure quantum states

Abstract

We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\hat{U}$ that transforms a given unknown state $|{\psi }_{\tau }⟩$ to a known fiducial state $|f⟩$. Then, after completion of the learning process, we can estimate and reproduce $|{\psi }_{\tau }⟩$ based on the learned $\hat{U}$ and $|f⟩$. To realize this idea, we cast a random-based learning algorithm, called `single-shot measurement learning,' in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a {\em finite} number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $\simeq O(N^{-1})$ scales of average infidelity. This result is comparable to those yielded from the standard quantum tomographic method in the case where additional information is available. It highlights a non-trivial message, that is, a random-based adaptive strategy can potentially be as accurate as other standard statistical approaches.

References in corpus (5)

• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Self-guided quantum tomography
• Knowledge and ignorance in incomplete quantum state tomography
• Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion
• Adaptive estimation of qubits by symmetry measurements

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• Neural-network approach for identifying nonclassicality from click-counting data
• Variational Quantum Algorithms for Trace Distance and Fidelity Estimation
• Unified multivariate trace estimation and quantum error mitigation
• Experimental Learning of Pure Quantum States using Sequential Single-Shot Measurement Outcomes
• Assisted quantum simulation of open quantum systems
• Reinforcement learning to learn quantum states for Heisenberg scaling accuracy
• Run-length certificates in quantum learning: sample complexity and noise thresholds
• Quantum Machine Learning for State Tomography Using Classical Data