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Quantum computing is evolving so quickly that forces us to revisit, rewrite, and update the basis of the theory. Basic Quantum Algorithms revisits the first quantum algorithms. It started in 1985 with Deutsch trying to evaluate a function at two domain points simultaneously. Then, Deutsch and Jozsa created in 1992 a quantum algorithm that determines whether a Boolean function is constant or balanced. In the next year, Bernstein and Vazirani realized that the same algorithm can be used to find a specific Boolean function in the set of linear Boolean functions. In 1994, Simon presented a new quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. In the same year, Shor created two new quantum algorithms for factoring integers and calculating discrete logarithms, threatening the cryptography methods widely used nowadays. In 1995, Kitaev described an alternative version for Shor's algorithms that proved useful in many other applications. In the following year, Grover created a quantum search algorithm quadratically faster than its classical counterpart. In this work, all those remarkable algorithms are described in detail with a focus on the circuit model.

The Quantum CONGEST model is a variant of the CONGEST model, where messages consist of $O(\log(n))$ qubits. We give a general framework for implementing quantum query algorithms in Quantum CONGEST, using the concept of parallel-queries. We apply our framework for distributed quantum queries in two settings: when data is distributed over the network, and graph theoretical problems where the network defines the input. The first is slightly unusual in CONGEST but our results follow almost directly. The second is more traditional for the CONGEST model but here we require some classical CONGEST steps to get our results. In the setting with distributed data, we show how a network can schedule a meeting in one of $k$ dates using $\tilde{O}(\sqrt{kD}+D)$ rounds, with $D$ the network diameter. We also give an efficient algorithm for element distinctness: if all nodes are given numbers, then the nodes can find any duplicates in $\tilde{O}(n^{2/3}D^{1/3})$ rounds. We also generalize the protocol for the distributed Deutsch-Jozsa problem from the two-party setting considered in [arXiv:quant-ph/9802040] to general networks, giving a novel separation between exact classical and exact quantum protocols in CONGEST. When the input is the network structure itself, we almost directly recover the $O(\sqrt{nD})$ round diameter computation algorithm of Le Gall and Magniez [arXiv:1804.02917]. We also compute the radius in the same number of rounds, and give an $\epsilon$-additive approximation of the average eccentricity in $\tilde{O}(D+D^{3/2}/\epsilon)$ rounds. Finally, we give quantum speedups for the problems of cycle detection and girth computation. We detect whether a graph has a cycle of length at most $k$ in $O(D+(Dn)^{1/2-1/\Theta(k)})$ rounds. We also give a $\tilde{O}(D+(Dn)^{1/2-1/\Theta(g)})$ round algorithm for finding the girth $g$, beating the known classical lower bound.

In this paper, we study the problem of learning an unknown quantum circuit of a certain structure. If the unknown target is an $n$-qubit Clifford circuit, we devise an efficient algorithm to reconstruct its circuit representation by using $O(n^2)$ queries to it. For decades, it has been unknown how to handle circuits beyond the Clifford group since the stabilizer formalism cannot be applied in this case. Herein, we study quantum circuits of $T$-depth one on the computational basis. We show that the output state of a $T$-depth one circuit can be represented by a stabilizer pseudomixture with a specific algebraic structure. Using Pauli and Bell measurements on copies of the output states, we can generate a hypothesis circuit that is equivalent to the unknown target circuit on computational basis states as input. If the number of $T$ gates of the target is of the order $O({{\log n}})$, our algorithm requires $O(n^2)$ queries to it and produces its equivalent circuit representation on the computational basis in time $O(n^3)$. Using further additional $O(4^{3n})$ classical computations, we can derive an exact description of the target for arbitrary input states. Our results greatly extend the previously known facts that stabilizer states can be efficiently identified based on the stabilizer formalism.

The use of function contracts to specify the behavior of functions often remains limited to the scope of a single function call. Relational properties link several function calls together within a single specification. They can express more advanced properties of a given function, such as non-interference, continuity, or monotonicity. They can also relate calls to different functions, for instance, to show that an optimized implementation is equivalent to its original counterpart. However, relational properties cannot be expressed and verified directly in the traditional setting of modular deductive verification. Self-composition has been proposed to overcome this limitation, but it requires complex transformations and additional separation hypotheses for real-life languages with pointers. We propose a novel approach that is not based on code transformation and avoids those drawbacks. It directly applies a verification condition generator to produce logical formulas that must be verified to ensure a given relational property. The approach has been fully formalized and proved sound in the Coq proof assistant.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from $m$ to $n$ qubits for any $m,n \geq 0$. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.

How are abstract concepts and musical themes recognized on the basis of some previous experience? It is interesting to compare the different behaviors of human and of artificial intelligences with respect to this problem. Generally, a human mind that abstracts a concept (say, table) from a given set of known examples creates a table-Gestalt: a kind of vague and out of focus image that does not fully correspond to a particular table with well determined features. A similar situation arises in the case of musical themes. Can the construction of a gestaltic pattern, which is so natural for human minds, be taught to an intelligent machine? This problem can be successfully discussed in the framework of a quantum approach to pattern recognition and to machine learning. The basic idea is replacing classical data sets with quantum data sets, where either objects or musical themes can be formally represented as pieces of quantum information, involving the uncertainties and the ambiguities that characterize the quantum world. In this framework, the intuitive concept of Gestalt can be simulated by the mathematical concept of positive centroid of a given quantum data set. Accordingly, the crucial problem "how can we classify a new object or a new musical theme (we have listened to) on the basis of a previous experience?" can be dealt with in terms of some special quantum similarity-relations. Although recognition procedures are different for human and for artificial intelligences, there is a common method of "facing the problems" that seems to work in both cases.

Knowledge graphs (KGs) capture knowledge in the form of head--relation--tail triples and are a crucial component in many AI systems. There are two important reasoning tasks on KGs: (1) single-hop knowledge graph completion, which involves predicting individual links in the KG; and (2), multi-hop reasoning, where the goal is to predict which KG entities satisfy a given logical query. Embedding-based methods solve both tasks by first computing an embedding for each entity and relation, then using them to form predictions. However, existing scalable KG embedding frameworks only support single-hop knowledge graph completion and cannot be applied to the more challenging multi-hop reasoning task. Here we present Scalable Multi-hOp REasoning (SMORE), the first general framework for both single-hop and multi-hop reasoning in KGs. Using a single machine SMORE can perform multi-hop reasoning in Freebase KG (86M entities, 338M edges), which is 1,500x larger than previously considered KGs. The key to SMORE's runtime performance is a novel bidirectional rejection sampling that achieves a square root reduction of the complexity of online training data generation. Furthermore, SMORE exploits asynchronous scheduling, overlapping CPU-based data sampling, GPU-based embedding computation, and frequent CPU--GPU IO. SMORE increases throughput (i.e., training speed) over prior multi-hop KG frameworks by 2.2x with minimal GPU memory requirements (2GB for training 400-dim embeddings on 86M-node Freebase) and achieves near linear speed-up with the number of GPUs. Moreover, on the simpler single-hop knowledge graph completion task SMORE achieves comparable or even better runtime performance to state-of-the-art frameworks on both single GPU and multi-GPU settings.

The recently introduced BERT model exhibits strong performance on several language understanding benchmarks. In this paper, we describe a simple re-implementation of BERT for commonsense reasoning. We show that the attentions produced by BERT can be directly utilized for tasks such as the Pronoun Disambiguation Problem and Winograd Schema Challenge. Our proposed attention-guided commonsense reasoning method is conceptually simple yet empirically powerful. Experimental analysis on multiple datasets demonstrates that our proposed system performs remarkably well on all cases while outperforming the previously reported state of the art by a margin. While results suggest that BERT seems to implicitly learn to establish complex relationships between entities, solving commonsense reasoning tasks might require more than unsupervised models learned from huge text corpora.

Quantum machine learning is expected to be one of the first potential general-purpose applications of near-term quantum devices. A major recent breakthrough in classical machine learning is the notion of generative adversarial training, where the gradients of a discriminator model are used to train a separate generative model. In this work and a companion paper, we extend adversarial training to the quantum domain and show how to construct generative adversarial networks using quantum circuits. Furthermore, we also show how to compute gradients -- a key element in generative adversarial network training -- using another quantum circuit. We give an example of a simple practical circuit ansatz to parametrize quantum machine learning models and perform a simple numerical experiment to demonstrate that quantum generative adversarial networks can be trained successfully.