Data is often loadable in short depth: Quantum circuits from tensor networks for finance, images, fluids, and proteins

Though there has been substantial progress in developing quantum algorithms to study classical datasets, the cost of simply loading classical data is an obstacle to quantum advantage. When the amplitude encoding is used, loading an arbitrary classical vector requires up to exponential circuit depths with respect to the number of qubits. Here, we address this ``input problem'' with two contributions. First, we introduce a circuit compilation method based on tensor network (TN) theory. Our method -- AMLET (Automatic Multi-layer Loader Exploiting TNs) -- proceeds via careful construction of a specific TN topology and can be tailored to arbitrary circuit depths. Second, we perform numerical experiments on real-world classical data from four distinct areas: finance, images, fluid mechanics, and proteins. To the best of our knowledge, this is the broadest numerical analysis to date of loading classical data into a quantum computer. Consistent with other recent work in this area, the required circuit depths are often several orders of magnitude lower than the exponentially-scaling general loading algorithm would require. Besides introducing a more efficient loading algorithm, this work demonstrates that many classical datasets are loadable in depths that are much shorter than previously expected, which has positive implications for speeding up classical workloads on quantum computers.Comment: 10 pages, 3 figure

Impact of Data Augmentation on QCNNs

In recent years, Classical Convolutional Neural Networks (CNNs) have been applied for image recognition successfully. Quantum Convolutional Neural Networks (QCNNs) are proposed as a novel generalization to CNNs by using quantum mechanisms. The quantum mechanisms lead to an efficient training process in QCNNs by reducing the size of input from $N$ to $log_2N$. This paper implements and compares both CNNs and QCNNs by testing losses and prediction accuracy on three commonly used datasets. The datasets include the MNIST hand-written digits, Fashion MNIST and cat/dog face images. Additionally, data augmentation (DA), a technique commonly used in CNNs to improve the performance of classification by generating similar images based on original inputs, is also implemented in QCNNs. Surprisingly, the results showed that data augmentation didn't improve QCNNs performance. The reasons and logic behind this result are discussed, hoping to expand our understanding of Quantum machine learning theory.Comment: 14 pages, 9 figure

Krylov Spread Complexity of Quantum-Walks

Given the recent advances in quantum technology, the complexity of quantum states is an important notion. The idea of the Krylov spread complexity has come into focus recently with the goal of capturing this in a quantitative way. The present paper sheds new light on the Krylov complexity measure by exploring it in the context of continuous-time quantum-walks on graphs. A close relationship between Krylov spread complexity and the concept of limiting-distributions for quantum-walks is established. Moreover, using a graph optimization algorithm, quantum-walk graphs are constructed that have minimal and maximal long-time average Krylov $\bar C$-complexity. This reveals an empirical upper bound for the $\bar C$-complexity as a function of Hilbert space dimension and an exact lower bound.Comment: 9 pages, 6 figure

Linear decomposition of approximate multi-controlled single qubit gates

We provide a method for compiling approximate multi-controlled single qubit gates into quantum circuits without ancilla qubits. The total number of elementary gates to decompose an n-qubit multi-controlled gate is proportional to 32n, and the previous best approximate approach without auxiliary qubits requires 32nk elementary operations, where k is a function that depends on the error threshold. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multi-target and multi-controlled CNOT and SU(2) gates. Computational experiments show the reduction in the number of CNOT gates to apply multi-controlled U(2) gates. As multi-controlled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications

Automated Function Implementation via Conditional Parameterized Quantum Circuits with Applications to Finance

Classical Monte Carlo algorithms can theoretically be sped up on a quantum computer by employing amplitude estimation (AE). To realize this, an efficient implementation of state-dependent functions is crucial. We develop a straightforward approach based on pre-training parameterized quantum circuits, and show how they can be transformed into their conditional variant, making them usable as a subroutine in an AE algorithm. To identify a suitable circuit, we propose a genetic optimization approach that combines variable ansatzes and data encoding. We apply our algorithm to the problem of pricing financial derivatives. At the expense of a costly pre-training process, this results in a quantum circuit implementing the derivatives' payoff function more efficiently than previously existing quantum algorithms. In particular, we compare the performance for European vanilla and basket options.Comment: 10 pages, 12 figures, 2 algorithm

Quantum Graph Learning : Frontiers and Outlook

Quantum theory has shown its superiority in enhancing machine learning. However, facilitating quantum theory to enhance graph learning is in its infancy. This survey investigates the current advances in quantum graph learning (QGL) from three perspectives, i.e., underlying theories, methods, and prospects. We first look at QGL and discuss the mutualism of quantum theory and graph learning, the specificity of graph-structured data, and the bottleneck of graph learning, respectively. A new taxonomy of QGL is presented, i.e., quantum computing on graphs, quantum graph representation, and quantum circuits for graph neural networks. Pitfall traps are then highlighted and explained. This survey aims to provide a brief but insightful introduction to this emerging field, along with a detailed discussion of frontiers and outlook yet to be investigated

Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging

The controlled-SWAP and controlled-controlled-NOT gates are at the heart of the original proposal of reversible classical computation by Fredkin and Toffoli. Their widespread use in quantum computation, both in the implementation of classical logic subroutines of quantum algorithms and in quantum schemes with no direct classical counterparts, have made it imperative early on to pursue their efficient decomposition in terms of the lower-level gate sets native to different physical platforms. Here, we add to this body of literature by providing several logically equivalent CNOT-count-optimal circuits for the Toffoli and Fredkin gates under all-to-all and linear qubit connectivity, the latter with two different routings for control and target qubits. We then demonstrate how these decompositions can be employed on near-term quantum computers to mitigate coherent errors via equivalent circuit averaging. We also consider the case where the three qubits on which the Toffoli or Fredkin gates act nontrivially are not adjacent, proposing a novel scheme to reorder them that saves one CNOT for every SWAP. This scheme also finds use in the shallow implementation of long-range CNOTs. Our results highlight the importance of considering different entanglement structures and connectivity constraints when designing efficient quantum circuits.Comment: Main text: 10 pages, 8 figures. Appendix: 4 sections, 5 figures. QASM files will be made available in open-source online platform upon next update of preprin

Thermality Versus Objectivity: Can They Peacefully Coexist?

Under the influence of external environments, quantum systems can undergo various different processes, including decoherence and equilibration. We observe that macroscopic objects are both objective and thermal, thus leading to the expectation that both objectivity and thermalisation can peacefully coexist on the quantum regime too. Crucially, however, objectivity relies on distributed classical information that could conflict with thermalisation. Here, we examine the overlap between thermal and objective states. We find that in general, one cannot exist when the other is present. However, there are certain regimes where thermality and objectivity are more likely to coexist: in the high temperature limit, at the non-degenerate low temperature limit, and when the environment is large. This is consistent with our experiences that everyday-sized objects can be both thermal and objective

Quantum Solutions for Training a Single Layer Binary Neural Network

Quantum computing is a relatively new field starting in the early 1980s when a physicist named Paul Benioff proposed a quantum mechanical model of the Turing machine, introducing quantum computers. Previously, the focus of most quantum computers was in the study of quantum applications instead of broad applications due to the fact that quantum technology is a newer field with many technology constraints, such as limited qubits and noisy environments. However, quantum computers are still capable of using quantum mechanics to solve specific algorithms with an exponential speed-up in comparison to their classical counterparts. One key algorithm is the HHL algorithm proposed by Harrow, Hassidim and Lloyd in 2009 [1]. This algorithm outlines a quantum approach to solve a linear systems of equations with a best case time complexity of O(poly(log N )), in comparison to the best case time complexity for classical algorithms of O(N^3 ). The HHL algorithm outlines a use for quantum circuits outside of quantum applications. One such application is in machine learning, as many networks use linear regression in their training algorithm. Currently it is not feasible to solve for weight vectors of floating point precision on a quantum computer, but if the weight vector is constrained to binary values 0 or 1 then the problem becomes small enough to implement even on current noisy quantum computers. This work outlines two different circuit designs to solve for 2 × 2 and 4 × 4 systems of equations, so long as the matrices follow the eigenvalue constraint of having eigenvalues be powers of 2. In addition, the problem of reading data from the quantum state to classical data is addressed through the use of a swap test between the solution state |x\u3e and an test state |test\u3e. By using a swap test vector of all 1s, it is shown it is possible to find how many ones lay in the solution vector; thus reducing the number of possible solution states without performing quantum tomography. While it is not possible to beat classical algorithms with the noise on current quantum circuits, this work shows it is possible to implement quantum algorithms for non-quantum applications establishing potential for future hybrid approaches