Scalable Neural Network Decoders for Higher Dimensional Quantum Codes

Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around $7.1\%$ when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method

Lossy compression of discrete sources via Viterbi algorithm

We present a new lossy compressor for discrete-valued sources. For coding a sequence $x^n$, the encoder starts by assigning a certain cost to each possible reconstruction sequence. It then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor. The cost of each sequence is a linear combination of its distance from the sequence $x^n$ and a linear function of its $k^{\rm th}$ order empirical distribution. The structure of the cost function allows the encoder to employ the Viterbi algorithm to recover the minimizer of the cost. We identify a choice of the coefficients comprising the linear function of the empirical distribution used in the cost function which ensures that the algorithm universally achieves the optimum rate-distortion performance of any stationary ergodic source in the limit of large $n$, provided that $k$ diverges as $o(\log n)$. Iterative techniques for approximating the coefficients, which alleviate the computational burden of finding the optimal coefficients, are proposed and studied.Comment: 26 pages, 6 figures, Submitted to IEEE Transactions on Information Theor

Power law violation of the area law in quantum spin chains

The sub-volume scaling of the entanglement entropy with the system's size, $n$, has been a subject of vigorous study in the last decade [1]. The area law provably holds for gapped one dimensional systems [2] and it was believed to be violated by at most a factor of $\log\left(n\right)$ in physically reasonable models such as critical systems. In this paper, we generalize the spin$-1$ model of Bravyi et al [3] to all integer spin-$s$ chains, whereby we introduce a class of exactly solvable models that are physical and exhibit signatures of criticality, yet violate the area law by a power law. The proposed Hamiltonian is local and translationally invariant in the bulk. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as $n^{-c}$, where using the theory of Brownian excursions, we prove $c\ge2$. This rules out the possibility of these models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with $n$ and that the half-chain entanglement entropy to the leading order scales as $\sqrt{n}$ (Eq. 16). Geometrically, the ground state is seen as a uniform superposition of all $s-$colored Motzkin walks. Lastly, we introduce an external field which allows us to remove the boundary terms yet retain the desired properties of the model. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest.Comment: v3: 10+33 pages. In the PNAS publication, the abstract was rewritten and title changed to "Supercritical entanglement in local systems: Counterexample to the area law for quantum matter". The content is same otherwise. v2: a section was added with an external field to include a model with no boundary terms (open and closed chain). Asymptotic technique is improved. v1:37 pages, 10 figures. Proc. Natl. Acad. Sci. USA, (Nov. 2016

Computational complexity of the landscape I

We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the Bousso-Polchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure

Coding Strategies for Genetic Algorithms and Neural Nets

The interaction between coding and learning rules in neural nets (NNs), and between coding and genetic operators in genetic algorithms (GAs) is discussed. The underlying principle advocated is that similar things in "the world" should have similar codes. Similarity metrics are suggested for the coding of images and numerical quantities in neural nets, and for the coding of neural network structures in genetic algorithms. A principal component analysis of natural images yields receptive fields resembling horizontal and vertical edge and bar detectors. The orientation sensitivity of the "bar detector" components is found to match a psychophysical model, suggesting that the brain may make some use of principal components in its visual processing. Experiments are reported on the effects of different input and output codings on the accuracy of neural nets handling numeric data. It is found that simple analogue and interpolation codes are most successful. Experiments on the coding of image data demonstrate the sensitivity of final performance to the internal structure of the net. The interaction between the coding of the target problem and reproduction operators of mutation and recombination in GAs are discussed and illustrated. The possibilities for using GAs to adapt aspects of NNs are considered. The permutation problem, which affects attempts to use GAs both to train net weights and adapt net structures, is illustrated and methods to reduce it suggested. Empirical tests using a simulated net design problem to reduce evaluation times indicate that the permutation problem may not be as severe as has been thought, but suggest the utility of a sorting recombination operator, that matches hidden units according to the number of connections they have in common. A number of experiments using GAs to design network structures are reported, both to specify a net to be trained from random weights, and to prune a pre-trained net. Three different coding methods are tried, and various sorting recombination operators evaluated. The results indicate that appropriate sorting can be beneficial, but the effects are problem-dependent. It is shown that the GA tends to overfit the net to the particular set of test criteria, to the possible detriment of wider generalisation ability. A method of testing the ability of a GA to make progress in the presence of noise, by adding a penalty flag, is described

Literary review of content-based music recognition paradigms

During the last few decades, a need for novel retrieval strategies for large audio databases emerged as millions of digital audio documents became accessible for everyone through the Internet. It became essential that the users could search for songs that they had no prior information about using only the content of the audio as a query. In practice this means that when a user hears an unknown song coming out of the radio and wants to get more information about it, he or she can simply record a sample of the song with a mobile device and send it to a music recognition application as a query. Query results would then be presented on the screen with all the necessary meta data, such as the song name and artist. The retrieval systems are expected to perform quickly and accurately against large databases that may contain millions of songs, which poses lots of challenges for the researchers. This thesis is a literature review which will go through some audio retrieval paradigms that allow querying for songs using only their audio content, such as audio fingerprinting. It will also address the typical problems and challenges of audio retrieval and compare how each of these proposed paradigms performs in these challenging scenarios

The White-Box Adversarial Data Stream Model

We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $\Omega(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $\Omega(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)Comment: PODS 202