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Supervised learning for quantum maximum entropy estimation

Ningping Cao, Jie Xie, Aonan Zhang, Shi‐Yao Hou, Lijian Zhang, Bei Zeng

Published May 4, 2020• 1 citation
OpenAlex

Abstract

For an unknown quantum state $\rho$, a set of measurement outcomes provides partial information. The principe of maximum entropy states that the density matrix which best represents the current knowlege is the one with the maximum entropy. However, the maximum entropy estimation (MEE) is hard to calculate in general. In this work, we develop a method that outputs an approximation to the MEE from limited measurement results by using supervised learning techniques. The method estimates the MEE of theoretical data of several different quantum systems with high fidelities. We also test our method experimentally with a photonic set-up, which exhibits high fidelities (all greater than 99.7%) and robustness against experimental errors. In comparison with previous known optimization algorithms for MEE, our method provides adequate fidelities with high efficiency.

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Machine learning is widely regarded as a promising application in which quantum computing could offer a computational advantage. However, general quantum machine learning models with provable guarantees on performance, trainability, and efficient inference remain limited. In this work, we introduce quantum Hamiltonian embeddings, a supervised learning framework in which a classical training dataset is encoded into a single effective, parameterized Hamiltonian. Training optimizes the parameters of this data-dependent Hamiltonian, whose approximate ground state defines the classification rule. Once the ground state has been approximated using a quantum algorithm, inference can be performed either classically using a classical representation of the learned state, or with a quantum representation of the state. We establish efficient methods for gradient estimation and prove favorable generalization guarantees. Furthermore, under standard cryptographic assumptions, we show that Hamiltonian embeddings can realize hypothesis functions beyond the reach of efficient classical algorithms. We then demonstrate the framework at scale on IBM Heron processors using a standard benchmark dataset. Together, these results establish Hamiltonian embeddings as a new framework for quantum machine learning, combining theoretical guarantees with a path for experimental investigations at scale on quantum hardware.

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For an unknown quantum state ρ\rhoρ, a set of measurement outcomes provides partial information. The principe of maximum entropy states that the density matrix which best represents the current knowlege is the one with the maximum entropy. However, the maximum entropy estimation (MEE) is hard to calculate in general. In this work, we develop a method that outputs an approximation to the MEE from limited measurement results by using supervised learning techniques. The method estimates the MEE of theoretical data of several different quantum systems with high fidelities. We also test our method experimentally with a photonic set-up, which exhibits high fidelities (all greater than 99.7%) and robustness against experimental errors. In comparison with previous known optimization algorithms for MEE, our method provides adequate fidelities with high efficiency.
We explore a supervised machine-learning approach to estimate the entanglement entropy of multiqubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best-known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for nonunitary evolution by training our model on different noise strengths.
Abstract Quantum inverse problem (QIP) is the problem of estimating an unknown quantum system from a set of measurements, whereas the classical counterpart is the inverse problem of estimating a distribution from a set of observations. In this paper, we present a neural-network-based method for QIPs, which has been widely explored for its classical counterpart. The proposed method utilizes the quantumness of the QIPs and takes advantage of the computational power of neural networks to achieve remarkable efficiency for the quantum state estimation. We test the method on the problem of maximum entropy estimation of an unknown state ρ from partial information both numerically and experimentally. Our method yields high fidelity, efficiency and robustness for both numerical experiments and quantum optical experiments.
There is growing interest in constructing quantum devices that control increasingly large numbers of quantum systems. There is a tremendous need for methods that can measure the quality and other properties of these devices. Such measurements often amount to learning the expectation value of a quantum observable with respect to the quantum state of the device. Learning expectation values is also a key component in many well-known applications such as quantum machine learning and quantum optimization algorithms. Because of the large system sizes involved, it is essential to find methods for learning expectation values that are efficient with respect to the system size. Existing methods that have rigorous guarantees and are practical to implement are often not optimal for observables of interest. Other methods are heuristic, or require the use measurement protocols that are challenging to implement experimentally with current technology. In this study, we propose an estimation procedure, The Optimal Observable expectation value Learner or TOOL, that can learn the expectation values of observables using the outcomes of any given measurement protocol. We show that there is a seminorm on the set of all observables, which we call the minimax norm, that characterizes the smallest possible estimation error for learning a given observable using the outcomes of a given non-adaptive measurement protocol to a specified confidence level in the worst case over all states. We prove that TOOL is minimax optimal for every observable by showing that it can achieve an estimation error to within a small constant factor of the minimax norm. For many applications, one wishes to learn the expectation value of more than one observable from the same experiment. A popular method for learning the expectation values of one or many observables with rigorous guarantees is classical shadows. Classical shadows has near-optimal performance in the worst case over all observables. We prove that TOOL always performs at least as well as classical shadows. Moreover, we show by example that TOOL dramatically outperforms classical shadows for many observables of interest. This highlights the need to characterize the optimal performance for the task of simultaneously learning the expectation values of many observables. Under a mild assumption, we give such a characterization using the minimax norm and prove that TOOL is nearly minimax optimal for this task. We also study the applications of TOOL to fidelity estimation. Using experimental data from a trapped-ion quantum computer, we show that TOOL performs well in practice and matches the estimates obtained from Maximum Likelihood Estimation (MLE), but with rigorous guarantees on the estimation error unlike MLE. We also compare TOOL with another popular method called direct fidelity estimation, which estimates the fidelity by judiciously sampling Pauli observables and measuring them. We show that there is a different importance sampling scheme for Pauli measurements for which TOOL performs as well as, or better than, direct fidelity estimation. Since TOOL constructs an estimator using only the observable, the measurement protocol, and the confidence level, it provides the flexibility to perform estimation for experiments that have already been performed and experiments that will be performed in the future. Similarly, since the minimax norm can be computed beforehand, it can be used to compare the performance of different measurement protocols and allow minimax optimal design of experiments.
This paper reports a novel method for supervised machine learning based on the mathematical formalism that supports quantum mechanics. The method uses projective quantum measurement as a way of building a prediction function. Specifically, the relationship between input and output variables is represented as the state of a bipartite quantum system. The state is estimated from training samples through an averaging process that produces a density matrix. Prediction of the label for a new sample is made by performing a projective measurement on the bipartite system with an operator, prepared from the new input sample, and applying a partial trace to obtain the state of the subsystem representing the output. The method can be seen as a generalization of Bayesian inference classification and as a type of kernel-based learning method. One remarkable characteristic of the method is that it does not require learning any parameters through optimization. We illustrate the method with different 2-D classification benchmark problems and different quantum information encodings.
Impressive progress has been made in the past decade in the study of technological applications of varied types of quantum systems. With industry giants like IBM laying down their roadmap for scalable quantum devices with more than 1000-qubits by the end of 2023, efficient validation techniques are also being developed for testing quantum processing on these devices. The characterization of a quantum state is done by experimental measurements through the process called quantum state tomography (QST) which scales exponentially with the size of the system. However, QST performed using incomplete measurements is aptly suited for characterizing these quantum technologies especially with the current nature of noisy intermediate-scale quantum (NISQ) devices where not all mean measurements are available with high fidelity. We, hereby, propose an alternative approach to QST for the complete reconstruction of the density matrix of a quantum system in a pure state for any number of qubits by applying the maximal entropy formalism on the pairwise combinations of the known mean measurements. This approach provides the best estimate of the target state when we know the complete set of observables, which is the case of convergence of the reconstructed density matrix to a pure state. Our goal is to provide a practical inference of a quantum system in a pure state that can find its applications in the field of quantum error mitigation on a real quantum computer that we intend to investigate further.
Machine learning is widely regarded as a promising application in which quantum computing could offer a computational advantage. However, general quantum machine learning models with provable guarantees on performance, trainability, and efficient inference remain limited. In this work, we introduce quantum Hamiltonian embeddings, a supervised learning framework in which a classical training dataset is encoded into a single effective, parameterized Hamiltonian. Training optimizes the parameters of this data-dependent Hamiltonian, whose approximate ground state defines the classification rule. Once the ground state has been approximated using a quantum algorithm, inference can be performed either classically using a classical representation of the learned state, or with a quantum representation of the state. We establish efficient methods for gradient estimation and prove favorable generalization guarantees. Furthermore, under standard cryptographic assumptions, we show that Hamiltonian embeddings can realize hypothesis functions beyond the reach of efficient classical algorithms. We then demonstrate the framework at scale on IBM Heron processors using a standard benchmark dataset. Together, these results establish Hamiltonian embeddings as a new framework for quantum machine learning, combining theoretical guarantees with a path for experimental investigations at scale on quantum hardware.