Training deep quantum neural networks Beer, Kerstin; Bondarenko, Dmytro; Farrelly, Terry ...
Nature communications,
02/2020, Letnik:
11, Številka:
1
Journal Article
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Neural networks enjoy widespread success in both research and industry and, with the advent of quantum technology, it is a crucial challenge to design quantum neural networks for fully quantum ...learning tasks. Here we propose a truly quantum analogue of classical neurons, which form quantum feedforward neural networks capable of universal quantum computation. We describe the efficient training of these networks using the fidelity as a cost function, providing both classical and efficient quantum implementations. Our method allows for fast optimisation with reduced memory requirements: the number of qudits required scales with only the width, allowing deep-network optimisation. We benchmark our proposal for the quantum task of learning an unknown unitary and find remarkable generalisation behaviour and a striking robustness to noisy training data.
Abstract
We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e. sources that emit one of two possible quantum ...states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system
XA
, corresponding to an ensemble of two quantum states, with
X
being classical and
A
being quantum. We study the resource theory for two different classes of free operations: (i) CPTP
A
, which consists of quantum channels acting only on
A
, and (ii) conditional doubly stochastic maps acting on
XA
. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.
We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with ...asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.
We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For ...the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for open quantum systems in the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.
We derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical, leading to several new difficulties in comparison with the cubic ...NLS which emerges from Bose gases with pair-interactions. Our method is based on Bogoliubov’s approximation, which also provides the information on the fluctuations around the condensate in terms of a norm approximation for the
N
-body wave function.
Quantum privacy amplification is a central task in quantum cryptography. Given shared randomness, which is initially correlated with a quantum system held by an eavesdropper, the goal is to extract ...uniform randomness which is decoupled from the latter. The optimal rate for this task is known to satisfy the strong converse property and we provide a lower bound on the corresponding strong converse exponent. In the strong converse region, the distance of the final state of the protocol from the desired decoupled state converges exponentially fast to its maximal value, in the asymptotic limit. We show that this necessarily leads to totally insecure communication by establishing that the eavesdropper can infer any sent messages with certainty, when given very limited extra information. In fact, we prove that in the strong converse region, the eavesdropper has an exponential advantage in inferring the sent message correctly, compared to the achievability region. Additionally we establish the following technical result, which is central to our proofs, and is of independent interest: the smoothing parameter for the smoothed max-relative entropy satisfies the strong converse property.
The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say \(\rho\) ...and \(\sigma\). This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call \(s\)-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter \(s\). In particular, \(s\)-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of \(s\)-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as \(\xi_s(\rho\|\sigma)\), and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for \(s = 1\), and asymmetric hypothesis testing (the Umegaki relative entropy) for \(s = 0\).
Given a quantum channel and a state which satisfy a fixed point equation
approximately (say, up to an error $\varepsilon$), can one find a new channel
and a state, which are respectively close to the ...original ones, such that they
satisfy an exact fixed point equation? It is interesting to ask this question
for different choices of constraints on the structures of the original channel
and state, and requiring that these are also satisfied by the new channel and
state. We affirmatively answer the above question, under fairly general
assumptions on these structures, through a compactness argument. Additionally,
for channels and states satisfying certain specific structures, we find
explicit upper bounds on the distances between the pairs of channels (and
states) in question. When these distances decay quickly (in a particular,
desirable manner) as $\varepsilon\to 0$, we say that the original approximate
fixed point equation is rapidly fixable. We establish rapid fixability, not
only for general quantum channels, but also when the original and new channels
are both required to be unitary, mixed unitary or unital. In contrast, for the
case of bipartite quantum systems with channels acting trivially on one
subsystem, we prove that approximate fixed point equations are not rapidly
fixable. In this case, the distance to the closest channel (and state) which
satisfy an exact fixed point equation can depend on the dimension of the
quantum system in an undesirable way. We apply our results on approximate fixed
point equations to the question of robustness of quantum Markov chains (QMC)
and establish the following: For any tripartite quantum state, there exists a
dimension-dependent upper bound on its distance to the set of QMCs, which
decays to zero as the conditional mutual information of the state vanishes.
We study the problem of binary composite channel discrimination in the asymmetric setting, where the hypotheses are given by fairly arbitrary sets of channels, and samples do not have to be ...identically distributed. In the case of quantum channels we prove: (i) a characterization of the Stein exponent for parallel channel discrimination strategies and (ii) an upper bound on the Stein exponent for adaptive channel discrimination strategies. We further show that already for classical channels this upper bound can sometimes be achieved and be strictly larger than what is possible with parallel strategies. Hence, there can be an advantage of adaptive channel discrimination strategies with composite hypotheses for classical channels, unlike in the case of simple hypotheses. Moreover, we show that classically this advantage can only exist if the sets of channels corresponding to the hypotheses are non-convex. As a consequence of our more general treatment, which is not limited to the composite i.i.d. setting, we also obtain a generalization of previous composite state discrimination results.
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