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Ben-Aroya, Avraham; Schwartz, Oded; Ta-Shma, Amnon

doi:10.4086/toc.2010.v006a003

Cited by 53 publications

(8 citation statements)

References 16 publications

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“…One consideration is that the von-Neumann entropy [8] of the density operator representing the ensemble {p i , |ψ i AB } limits the number of qubits to which it can be noiselessly compressed. However, finding the entropy of this density operator is not trivial -in fact, given a circuit that constructs a density operator ρ, it is known that, in general, even estimating the entropy of ρ is QSZK-complete [37]. This then opens the possibility for quantum autoencoders to efficiently give some estimate of the entropy of a density operator.…”

Section: Discussionmentioning

confidence: 99%

Quantum autoencoders for efficient compression of quantum data

Romero¹,

Olson²,

Aspuru‐Guzik³

2017

Quantum Sci. Technol.

487

424

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Classical autoencoders are neural networks that can learn efficient low dimensional representations of data in higher dimensional space. The task of an autoencoder is, given an input x, is to map x to a lower dimensional point y such that x can likely be recovered from y. The structure of the underlying autoencoder network can be chosen to represent the data on a smaller dimension, effectively compressing the input. Inspired by this idea, we introduce the model of a quantum autoencoder to perform similar tasks on quantum data. The quantum autoencoder is trained to compress a particular dataset of quantum states, where a classical compression algorithm cannot be employed. The parameters of the quantum autoencoder are trained using classical optimization algorithms. We show an example of a simple programmable circuit that can be trained as an efficient autoencoder. We apply our model in the context of quantum simulation to compress ground states of the Hubbard model and molecular Hamiltonians. * Corresponding author: aspuru@chemistry.harvard.edu Figure 1. a) A graphical representation of a 6-bit autoencoder with a 3-bit latent space. The map E encodes a 6-bit input (red dots) into a 3-bit intermediate state (yellow dots), after which the decoder D attempts to reconstruct the input bits at the output (green dots). b) Circuit implementation of a 6-3-6 quantum autoencoder. without exponentially costly classical memory, for instance, in dimension reduction of quantum data. A related work proposing a quantum autoencoder model establishes a formal connection between classical and quantum feedforward neural networks where a particular setting of parameters in the quantum network reduces to a classical neural network exactly [7]. In this work, we provide a simpler model which we believe more easily captures the essence of an autoencoder, and apply it to ground states of the Hubbard model and molecular Hamiltonians.

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Section: Discussionmentioning

confidence: 99%

Quantum autoencoders for efficient compression of quantum data

Romero¹,

Olson²,

Aspuru‐Guzik³

2017

Quantum Sci. Technol.

487

424

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“…The entropy can thus provide another way to identify a critical point. Note that diverging entanglement and diverging correlation length do not necessarily occur together; on the one hand, there exist states with all correlation functions short-range but with arbitrarily large entanglement [3,4], while on the other hand there are critical points in two dimensions where the correlation functions become long-range but the area law is still obeyed [5].…”

Section: Introductionmentioning

confidence: 99%

Entanglement versus gap for one-dimensional spin systems

Gottesman

Hastings

2010

New J. Phys.

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We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap ∆ is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small ∆. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in

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“…In independent work, Hastings [18] and Ben-Aroya, Schwartz and Ta-Schma [3] introduced quantum expanders as a special class of quantum channels defined analogously to spectral expanders. For a superoperator Φ, the expansion parameter is given by…”

Section: Quasirandomness In Quantum Information Theorymentioning

confidence: 99%

“…Furthermore, for a linear map A : C(S) → C(S) define A π by A π f := (Af π ) π −1 and say that A is transitive covariant with respect to Γ if for any π ∈ Γ we have A π = A. 3 We sometimes omit the group and simply say A is transitive covariant if such a group Γ exists. In [9], the following result is proved (over the real numbers) for the case S = [n], in which case transitive covariant linear maps A are simply n × n matrices which commute with the permutation matrices of a transitive subgroup Γ of S n .…”

Section: Commutative Casementioning

confidence: 99%

Quasirandom quantum channels

Bannink

Briët

Labib

et al. 2020

Quantum

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Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or `quantum', case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.

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Cited by 53 publications

References 16 publications

Quantum autoencoders for efficient compression of quantum data

Quantum autoencoders for efficient compression of quantum data

Entanglement versus gap for one-dimensional spin systems

Quasirandom quantum channels

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