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Register a new userRealizing quantum linear regression with auxiliary qumodes
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In order to exploit quantum advantages, quantum algorithms are indispensable for operating machine learning with quantum computers. We here propose an intriguing hybrid approach of quantum information processing for quantum linear regression, which utilizes both discrete and continuous quantum variables, in contrast to existing wisdoms based solely upon discrete qubits. In our framework, data information is encoded in a qubit system, while information processing is tackled using auxiliary continuous qumodes via qubit-qumode interactions. Moreover, it is also elaborated that finite squeezing is quite helpful for efficiently running the quantum algorithms in realistic setup. Comparing with an all-qubit approach, the present hybrid approach is more efficient and feasible for implementing quantum algorithms, still retaining exponential quantum speed-up.
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In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that $ell_2$-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.
With the rise of quantum technologies, it is necessary to have practical and preferably non-destructive methods to measure and read-out from such devices. A current line of research towards this has focussed on the use of ancilla systems which couple to the system under investigation, and through their interaction, enable properties of the primary system to be imprinted onto and inferred from the ancillae. We propose the use of continuous variable qumodes as ancillary probes, and show that the interaction Hamiltonian can be fully characterised and directly sampled from measurements of the qumode alone. We suggest how such probes may also be used to determine thermodynamical properties, including reconstruction of the partition function. We show that the method is robust to realistic experimental imperfections such as finite-sized measurement bins and squeezing, and discuss how such probes are already feasible with current experimental setups.
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In many predictive decision-making scenarios, such as credit scoring and academic testing, a decision-maker must construct a model that accounts for agents propensity to game the decision rule by changing their features so as to receive better decisions. Whereas the strategic classification literature has previously assumed that agents outcomes are not causally affected by their features (and thus that strategic agents goal is deceiving the decision-maker), we join concurrent work in modeling agents outcomes as a function of their changeable attributes. As our main contribution, we provide efficient algorithms for learning decision rules that optimize three distinct decision-maker objectives in a realizable linear setting: accurately predicting agents post-gaming outcomes (prediction risk minimization), incentivizing agents to improve these outcomes (agent outcome maximization), and estimating the coefficients of the true underlying model (parameter estimation). Our algorithms circumvent a hardness result of Miller et al. (2020) by allowing the decision maker to test a sequence of decision rules and observe agents responses, in effect performing causal interventions through the decision rules.
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