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arXiv:0903.0543 · Physical Review A

Optimal quantum learning of a unitary transformation

Alessandro Bisio, Giulio Chiribella, Giacomo Mauro D’Ariano, Stefano Facchini, Paolo Perinotti

Published Mar 25, 2010• 148 citations
arXivHTMLPDF

Abstract

We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary with maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine) and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a ``measure-and-rotate'' retrieving. Differing from the case of quantum cloning, where the incoherent ``measure-and-prepare'' strategies are typically suboptimal, in the case of learning the ``measure-and-rotate'' strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the ``measure-and-rotate'' strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.

DOI
10.1103/physreva.81.032324

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We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary with maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine) and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a measure-and-rotate'' retrieving. Differing from the case of quantum cloning, where the incoherent measure-and-prepare'' strategies are typically suboptimal, in the case of learning the measure-and-rotate'' strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the measure-and-rotate'' strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.
We address the task of converting n uses of an unknown unitary transformation into a quantum state (i.e., storage) and later retrieval of the transformation. Specifically, we consider the case where the unknown unitary is selected with equal prior probability from two options. First, we prove that the optimal storage strategy involves the sequential application of the n uses of the unknown unitary, and it produces the optimal state for discrimination between the two possible unitaries. Next, we show that incoherent “measure-and-prepare” retrieval achieves the maximum fidelity between the retrieved operation and the original (qubit) unitary. We then identify the retrieval strategy that maximizes the probability of successfully and perfectly retrieving the unknown transformation. In the regime in which the fidelity between the two possible unitaries is large the probability of success scales as P succ = 1 − O ( n − 2 ) , which is a quadratic improvement with respect to the case in which the unitaries are drawn from the entire unitary group U ( d ) with uniform prior probability. Finally, we present an optical experiment for this approach and assess the storage and retrieval quality using quantum tomography of states and processes. The results are discussed in relation to nonoptimal measure-and-prepare strategy, highlighting the advantages of our protocol.
We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about $1\ensuremath{\rightarrow}2$ learning of the measurement, otherwise the task is called $1\ensuremath{\rightarrow}2$ cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension ddd of the Hilbert space. For $1\ensuremath{\rightarrow}2$ cloning we also propose a simple quantum network that achieves the optimal fidelity. The optimal fidelity for $1\ensuremath{\rightarrow}2$ learning just slightly outperforms the estimate and prepare strategy in which one first estimates the unknown measurement and depending on the result suitably prepares the duplicate.
Abstract. Quantum learning of a unitary transformation estimates a quantum channel in a process similar to process tomography. The classical counterpart of this goal, finding an unknown function, is regression, although the methodology hardly resembles the outline of classical algorithms. To gain a better understanding what such a methodology means to learning theory, we anchor it to the familiar concepts of active learning and transduction. Learning the unitary translates to storing it in quantum memory, but the procedure also requires an optimal maximally entangled input state; this resembles active learning. The retrieval strategy is a blend of inductive and transductive learning.
We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ that transforms a given unknown state $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ to a known fiducial state $|f\ensuremath{\rangle}$. Then, after completion of the learning process, we can estimate and reproduce $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ based on the learned $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ and $|f\ensuremath{\rangle}$. 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 finite number, say NNN, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large NNN, namely $\ensuremath{\simeq}O({N}^{\ensuremath{-}1})$ scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.
We develop a quantum learning scheme for binary discrimination of coherent states of light. This is a problem of technological relevance for the reading of information stored in a digital memory. In our setting, a coherent light source is used to illuminate a memory cell and retrieve its encoded bit by determining the quantum state of the reflected signal. We consider a situation where the amplitude of the states produced by the source is not fully known, but instead this information is encoded in a large training set comprising many copies of the same coherent state. We show that an optimal global measurement, performed jointly over the signal and the training set, provides higher successful identification rates than any learning strategy based on first estimating the unknown amplitude by means of Gaussian measurements on the training set, followed by an adaptive discrimination procedure on the signal. By considering a simplified variant of the problem, we argue that this is the case even for non-Gaussian estimation measurements. Our results show that, even in absence of entanglement, collective quantum measurements yield an enhancement in the readout of classical information, which is particularly relevant in the operating regime of low-energy signals.
Recent results have established dramatic advantages in learning properties of quantum states when a quantum computer is available to process or jointly measure multiple copies of the unknown quantum state. Learning tasks can be accomplished with exponentially fewer copies of the state when compared to optimized classical learning strategies that are restricted to measuring one copy of the state at a time. While these results were established in abstract settings and for artificial learning tasks, they motivate the application of quantum computers to imaging and sensing of weak electromagnetic fields since these settings are ultimately concerned with the learning of unknown quantum states. In this work we apply these new results in quantum learning to the problem of learning Gaussian states of the electromagnetic field, which are germane since they describe most fields used in imaging and sensing. In order to connect with quantum learning theory, we consider the transduction of an nnn-mode Gaussian state into a register of qubits on a quantum computer followed by optimized measurements on these qubits to extract the parameters defining the original Gaussian state. We rigorously bound the number of copies of the Gaussian state required to achieve worst-case additive error in parameter estimates. The scaling of this bound with nnn is exponentially better than naïve strategies for characterizing Gaussian states and matches recently derived bounds for characterization of Gaussian states using continuous-variable (CV) classical shadows. In addition, our bound has a polynomially better dependence on the energy of the multimode Gaussian state compared to the CV shadows protocol.