There is increasing interest in the development of gate-based quantum circuits for the training of machine learning models. Yet, little is understood concerning the parameters of circuit design, and the effects of noise and other measurement errors on the performance of quantum machine learning models. In this paper, we explore the practical implications of key circuit design parameters (number of qubits, depth etc.) using several standard machine learning datasets and IBM's Qiskit simulator. In total we evaluate over 6500 unique circuits with $n \approx 120700$ individual runs. We find that in general shallow (low depth) wide (more qubits) circuit topologies tend to outperform deeper ones in settings without noise. We also explore the implications and effects of different notions of noise and discuss circuit topologies that are more / less robust to noise for classification machine learning tasks. Based on the findings we define guidelines for circuit topologies that show near-term promise for the realisation of quantum machine learning algorithms using gate-based NISQ quantum computer.
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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 1995 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.
In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called lazy training, and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the tangent kernel, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of geometrically local parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.
In this paper, we study quantum algorithms for computing the exact value of the treewidth of a graph. Our algorithms are based on the classical algorithm by Fomin and Villanger (Combinatorica 32, 2012) that uses $O(2.616^n)$ time and polynomial space. We show three quantum algorithms with the following complexity, using QRAM in both exponential space algorithms: $\bullet$ $O(1.618^n)$ time and polynomial space; $\bullet$ $O(1.554^n)$ time and $O(1.452^n)$ space; $\bullet$ $O(1.538^n)$ time and space. In contrast, the fastest known classical algorithm for treewidth uses $O(1.755^n)$ time and space. The first two speed-ups are obtained in a fairly straightforward way. The first version uses additionally only Grover's search and provides a quadratic speedup. The second speedup is more time-efficient and uses both Grover's search and the quantum exponential dynamic programming by Ambainis et al. (SODA '19). The third version uses the specific properties of the classical algorithm and treewidth, with a modified version of the quantum dynamic programming on the hypercube. Lastly, as a small side result, we also give a new classical time-space tradeoff for computing treewidth in $O^*(2^n)$ time and $O^*(\sqrt{2^n})$ space.
We initiate the study of parameterized complexity of $\textsf{QMA}$ problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + $t$ $T$-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most $t$ qubits (independent of the system size). Furthermore, we derive new lower bounds on the $T$-count of circuit satisfiability instances and the $T$-count of the $W$-state assuming the classical exponential time hypothesis ($\textsf{ETH}$). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.
We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the $p$-faces of the $n$-cube (for $n>p$) and stabilizer constraints with faces of dimension $(p\pm1)$. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical qubits and displays local testability with a soundness of $\Omega(1/\log(N))$ beating the current state-of-the-art of $1/\log^{2}(N)$ due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the $n$-cube by arbitrary linear classical codes of length $n$. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary $n$-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.
In the classical world, the existence of commitments is equivalent to the existence of one-way functions. In the quantum setting, on the other hand, commitments are not known to imply one-way functions, but all known constructions of quantum commitments use at least one-way functions. Are one-way functions really necessary for commitments in the quantum world? In this work, we show that non-interactive quantum commitments (for classical messages) with computational hiding and statistical binding exist if pseudorandom quantum states exist. Pseudorandom quantum states are sets of quantum states that are efficiently generated but their polynomially many copies are computationally indistinguishable from the same number of copies of Haar random states [Ji, Liu, and Song, CRYPTO 2018]. It is known that pseudorandom quantum states exist even if $\BQP=\QMA$ (relative to a quantum oracle) [Kretschmer, TQC 2021], which means that pseudorandom quantum states can exist even if no quantum-secure classical cryptographic primitive exists. Our result therefore shows that quantum commitments can exist even if no quantum-secure classical cryptographic primitive exists. In particular, quantum commitments can exist even if no quantum-secure one-way function exists. In this work, we also consider digital signatures, which are other fundamental primitives in cryptography. We show that one-time secure digital signatures with quantum public keys exist if pseudorandom quantum states exist. In the classical setting, the existence of digital signatures is equivalent to the existence of one-way functions. Our result, on the other hand, shows that quantum signatures can exist even if no quantum-secure classical cryptographic primitive (including quantum-secure one-way functions) exists.
Research in cognitive science has provided extensive evidence on human cognitive ability in performing physical reasoning of objects from noisy perceptual inputs. Such a cognitive ability is commonly known as intuitive physics. With the advancements in deep learning, there is an increasing interest in building intelligent systems that are capable of performing physical reasoning from a given scene for the purpose of advancing fluid and building safer AI systems. As a result, many of the contemporary approaches in modelling intuitive physics for machine cognition have been inspired by literature from cognitive science. Despite the wide range of work in physical reasoning for machine cognition, there is a scarcity of reviews that organize and group these deep learning approaches. Especially at the intersection of intuitive physics and artificial intelligence, there is a need to make sense of the diverse range of ideas and approaches. Therefore, this paper presents a comprehensive survey of recent advances and techniques in intuitive physics-inspired deep learning approaches for physical reasoning. The survey will first categorize existing deep learning approaches into three facets of physical reasoning before organizing them into three general technical approaches and propose six categorical tasks of the field. Finally, we highlight the challenges of the current field and present some future research directions.
In this expository article we present an overview of the current state-of-the-art in post-quantum group-based cryptography. We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. We then, describe some applications of combinatorail algebra in fully homomorphic encryption. In the end we discussing several open problems in this direction.
Quantum computing systems rely on the principles of quantum mechanics to perform a multitude of computationally challenging tasks more efficiently than their classical counterparts. The architecture of software-intensive systems can empower architects who can leverage architecture-centric processes, practices, description languages, etc., to model, develop, and evolve quantum computing software (quantum software for short) at higher abstraction levels. We conducted a systematic literature review (SLR) to investigate (i) architectural process, (ii) modeling notations, (iii) architecture design patterns, (iv) tool support, and (iv) challenging factors for quantum software architecture. Results of the SLR indicate that quantum software represents a new genre of software-intensive systems; however, existing processes and notations can be tailored to derive the architecting activities and develop modeling languages for quantum software. Quantum bits (Qubits) mapped to Quantum gates (Qugates) can be represented as architectural components and connectors that implement quantum software. Tool-chains can incorporate reusable knowledge and human roles (e.g., quantum domain engineers, quantum code developers) to automate and customize the architectural process. Results of this SLR can facilitate researchers and practitioners to develop new hypotheses to be tested, derive reference architectures, and leverage architecture-centric principles and practices to engineer emerging and next generations of quantum software.
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.
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