Quantum learning: optimal classification of qubit states

Abstract

Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d. training set $(X_1,Y_1),...(X_n,Y_n)$ where $X_i$ represents a feature and $Y_i\in \{0,1\}$ is a label attached to that feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can learn about it from the training set and we aim at devising low error classifiers $f:X\to Y$ used to predict the label of new incoming features. Here we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown qubit states. Given a number of `training' copies from each of the states, we would like to `learn' about them by performing a measurement on the training set. The outcome is then used to design mesurements for the classification of future systems with unknown labels. We find the asymptotically optimal classification strategy and show that typically, it performs strictly better than a plug-in strategy based on state estimation. The figure of merit is the excess risk which is the difference between the probability of error and the probability of error of the optimal measurement when the states are known, that is the Helstrom measurement. We show that the excess risk has rate $n^{-1}$ and compute the exact constant of the rate.

24 pages, 4 figures

References in corpus (20)

• Scalable multi-particle entanglement of trapped ions
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Fast learning rates for plug-in classifiers
• Classical randomness in quantum measurements
• Optimal quantum learning of a unitary transformation
• Local asymptotic normality for finite dimensional quantum systems
• Quantum learning and universal quantum matching machine
• A universal programmable quantum state discriminator that is optimal for unambiguously distinguishing between unknown states
• Optimal full estimation of qubit mixed states
• Local asymptotic normality for qubit states
• Local asymptotic normality in quantum statistics
• Quantum pure-state identification
• Separable Measurement Estimation of Density Matrices and its Fidelity Gap with Collective Protocols
• Unambiguous pure state identification without classical knowledge
• Collective vs local measurements in qubit mixed state estimation
• Optimal estimation of qubit states with continuous time measurements
• Quantum learning by measurement and feedback
• Conciliation of Bayes and Pointwise Quantum State Estimation: Asymptotic information bounds in quantum statistics
• Quantum teleportation benchmarks for independent and identically-distributed spin states and displaced thermal states

Cited by in corpus (20)

• An introduction to quantum machine learning
• Machine learning for discriminating quantum measurement trajectories and improving readout
• Learning in Quantum Control: High-Dimensional Global Optimization for Noisy Quantum Dynamics
• Designing High-Fidelity Single-Shot Three-Qubit Gates: A Machine Learning Approach
• Machine learning \& artificial intelligence in the quantum domain
• Quantum Ensemble Classification: A Sampling-based Learning Control Approach
• Quantum learning of coherent states
• Unsupervised classification of quantum data
• Real-time calibration of coherent-state receivers: learning by trial and error
• Quantum-enhanced learning of rotations about an unknown direction
• An agnostic-Dolinar receiver for coherent states classification
• Asymptotically optimal purification and dilution of mixed qubit and Gaussian states
• Optimal quantum discrimination of single-qubit unitary gates between two candidates
• Framework for Learning and Control in the Classical and Quantum Domains
• Universal algorithms for quantum data learning
• Fidelity-based supervised and unsupervised learning for binary classification of quantum states
• Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states
• Constructing a qubit POVM from quantum data
• State learning from pairs of states
• Measurement disturbance tradeoffs in three-qubit unsupervised quantum classification