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We prove that given the ability to make entangled measurements on at most $k$ replicas of an $n$-qubit state $\rho$ simultaneously, there is a property of $\rho$ which requires at least order $2^n$ measurements to learn. However, the same property only requires one measurement to learn if we can make an entangled measurement over a number of replicas polynomial in $k, n$. Because the above holds for each positive integer $k$, we obtain a hierarchy of tasks necessitating progressively more replicas to be performed efficiently. We introduce a powerful proof technique to establish our results, and also use this to provide new bounds for testing the mixedness of a quantum state.

Quantum noise is the key challenge in Noisy Intermediate-Scale Quantum (NISQ) computers. Previous work for mitigating noise has primarily focused on gate-level or pulse-level noise-adaptive compilation. However, limited research efforts have explored a higher level of optimization by making the quantum circuits themselves resilient to noise. We propose QuantumNAS, a comprehensive framework for noise-adaptive co-search of the variational circuit and qubit mapping. Variational quantum circuits are a promising approach for constructing QML and quantum simulation. However, finding the best variational circuit and its optimal parameters is challenging due to the large design space and parameter training cost. We propose to decouple the circuit search and parameter training by introducing a novel SuperCircuit. The SuperCircuit is constructed with multiple layers of pre-defined parameterized gates and trained by iteratively sampling and updating the parameter subsets (SubCircuits) of it. It provides an accurate estimation of SubCircuits performance trained from scratch. Then we perform an evolutionary co-search of SubCircuit and its qubit mapping. The SubCircuit performance is estimated with parameters inherited from SuperCircuit and simulated with real device noise models. Finally, we perform iterative gate pruning and finetuning to remove redundant gates. Extensively evaluated with 12 QML and VQE benchmarks on 10 quantum comput, QuantumNAS significantly outperforms baselines. For QML, QuantumNAS is the first to demonstrate over 95% 2-class, 85% 4-class, and 32% 10-class classification accuracy on real QC. It also achieves the lowest eigenvalue for VQE tasks on H2, H2O, LiH, CH4, BeH2 compared with UCCSD. We also open-source QuantumEngine (//github.com/mit-han-lab/pytorch-quantum) for fast training of parameterized quantum circuits to facilitate future research.

Recently Implicit Neural Representations (INRs) gained attention as a novel and effective representation for various data types. Thus far, prior work mostly focused on optimizing their reconstruction performance. This work investigates INRs from a novel perspective, i.e., as a tool for image compression. To this end, we propose the first comprehensive compression pipeline based on INRs including quantization, quantization-aware retraining and entropy coding. Encoding with INRs, i.e. overfitting to a data sample, is typically orders of magnitude slower. To mitigate this drawback, we leverage meta-learned initializations based on MAML to reach the encoding in fewer gradient updates which also generally improves rate-distortion performance of INRs. We find that our approach to source compression with INRs vastly outperforms similar prior work, is competitive with common compression algorithms designed specifically for images and closes the gap to state-of-the-art learned approaches based on Rate-Distortion Autoencoders. Moreover, we provide an extensive ablation study on the importance of individual components of our method which we hope facilitates future research on this novel approach to image compression.

We present data structures and algorithms for native implementations of discrete convolution operators over Adaptive Particle Representations (APR) of images on parallel computer architectures. The APR is a content-adaptive image representation that locally adapts the sampling resolution to the image signal. It has been developed as an alternative to pixel representations for large, sparse images as they typically occur in fluorescence microscopy. It has been shown to reduce the memory and runtime costs of storing, visualizing, and processing such images. This, however, requires that image processing natively operates on APRs, without intermediately reverting to pixels. Designing efficient and scalable APR-native image processing primitives, however, is complicated by the APR's irregular memory structure. Here, we provide the algorithmic building blocks required to efficiently and natively process APR images using a wide range of algorithms that can be formulated in terms of discrete convolutions. We show that APR convolution naturally leads to scale-adaptive algorithms that efficiently parallelize on multi-core CPU and GPU architectures. We quantify the speedups in comparison to pixel-based algorithms and convolutions on evenly sampled data. We achieve pixel-equivalent throughputs of up to 1 TB/s on a single Nvidia GeForce RTX 2080 gaming GPU, requiring up to two orders of magnitude less memory than a pixel-based implementation.

Cryptography with quantum states exhibits a number of surprising and counterintuitive features. In a 2002 work, Barnum et al. argue that these features imply that digital signatures for quantum states are impossible (Barnum et al., FOCS 2002). In this work, we ask: can all forms of signing quantum data, even in a possibly weak sense, be completely ruled out? We give two results which shed significant light on this basic question. First, we prove an impossibility result for digital signatures for quantum data, which extends the result of Barnum et al. Specifically, we show that no nontrivial combination of correctness and security requirements can be fulfilled, beyond what is achievable simply by measuring the quantum message and then signing the outcome. In other words, only classical signature schemes exist. We then show a positive result: a quantum state can be signed with the same security guarantees as classically, provided that it is also encrypted with the public key of the intended recipient. Following classical nomenclature, we call this notion quantum signcryption. Classically, signcryption is only interesting if it provides superior performance to encypt-then-sign. Quantumly, it is far more interesting: it is the only signing method available. We develop "as-strong-as-classical" security definitions for quantum signcryption and give secure constructions based on post-quantum public-key primitives. Along the way, we show that a natural hybrid method of combining classical and quantum schemes can be used to "upgrade" a secure classical scheme to the fully-quantum setting, in a wide range of cryptographic settings including signcryption, authenticated encryption, and CCA security.

Learning effective visual representations that generalize well without human supervision is a fundamental problem in order to apply Machine Learning to a wide variety of tasks. Recently, two families of self-supervised methods, contrastive learning and latent bootstrapping, exemplified by SimCLR and BYOL respectively, have made significant progress. In this work, we hypothesize that adding explicit information compression to these algorithms yields better and more robust representations. We verify this by developing SimCLR and BYOL formulations compatible with the Conditional Entropy Bottleneck (CEB) objective, allowing us to both measure and control the amount of compression in the learned representation, and observe their impact on downstream tasks. Furthermore, we explore the relationship between Lipschitz continuity and compression, showing a tractable lower bound on the Lipschitz constant of the encoders we learn. As Lipschitz continuity is closely related to robustness, this provides a new explanation for why compressed models are more robust. Our experiments confirm that adding compression to SimCLR and BYOL significantly improves linear evaluation accuracies and model robustness across a wide range of domain shifts. In particular, the compressed version of BYOL achieves 76.0% Top-1 linear evaluation accuracy on ImageNet with ResNet-50, and 78.8% with ResNet-50 2x.

The guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, our main result is an algorithm that, upon the input of any qubit ensemble over a discrete ring and with uniform probability distribution, after finitely many steps outputs the exact closed-form analytic expression of its guesswork. The complexity of our guesswork-computing algorithm is factorial in the number of states, with a more-than-quadratic speedup for symmetric ensembles. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.

Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a class of highly entangled many-body wavefunctions. It has been suggested that this approach can be certified with the Linear Cross-Entropy Benchmark (XEB). We critically examine this notion. First, in a "benign" setting where an honest implementation of noisy quantum circuits is assumed, we characterize the conditions under which the XEB approximates the fidelity. Second, in an "adversarial" setting where all possible classical algorithms are considered for comparison, we show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. We present an efficient classical algorithm that, with 1 GPU within 2s, yields high XEB values, namely 2-12% of those obtained in experiments. By identifying and exploiting several vulnerabilities of the XEB, we achieve high XEB values without full simulation of quantum circuits. Remarkably, our algorithm features better scaling with the system size than noisy quantum devices for commonly studied random circuit ensembles. To quantitatively explain the success of our algorithm and the limitations of the XEB, we use a theoretical framework in which the average XEB and fidelity are mapped to statistical models. We illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different gate choices, and in the presence of noise. Our results show that XEB's utility as a proxy for fidelity hinges on several conditions, which must be checked in the benign setting but cannot be assumed in the adversarial setting. Thus, the XEB alone has limited utility as a benchmark for quantum advantage. We discuss ways to overcome these limitations.

We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.

Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.