What Functions can Quantum Models Learn?

Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

Quantum Machine Learning was born during the past decade as the intersection of Quantum Computing and Machine Learning. Today, advances in quantum computer hardware and the design of simulation frameworks able to run quantum algorithms in classic computers make it possible to extend classic artificial intelligence models to a quantum environment. Despite these achievements, several questions regarding the whole quantum machine learning pipeline remain unanswered, for instance the problem of classical data representation on quantum hardware, or the methodologies for designing and evaluating quantum models for common learning tasks such as classification, function approximation, clustering, etc. These problems become even more difficult to solve in the case of Time Series processing, where the context of past historical data may influence the behavior of the decision-making model. In this piece of research, we address the problem of Time Series classification using quantum models, and propose an efficient and compact representation of time series in quantum data using amplitude embedding. The proposal is capable of representing a time series of length n in log2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$log_2(n)$$\\end{document} computational units, and experiments conducted on benchmark time series classification problems show that quantum models designed for classification can also outperform the accuracy of classic methods.

We describe a method for multivariate function approximation which combines neural network learning, clustering and multiple regression. Neural networks with a single hidden layer are universal function approximators. However, due to the complexity of the network topology and the nonlinear transfer function used in computing the activation of the hidden units, the predictions of a trained network are difficult to comprehend. On the other hand, predictions from a multiple linear regression equation are easy to understand but not accurate when the underlying relationship between the input variables and the output variable as nonlinear. The method presented in this paper generates a set of multiple linear regression equations using neural networks. The number of regression equations as determined by clustering the weighted input variables. The predictions for samples in the same cluster are computed by the same regression equation. Experimental results on real-world data demonstrate that the new method generates relatively few regression equations from the training data samples. The errors an prediction using these equations are comparable to or lower than those achieved by existing function approximation methods.

Considering nonlinear control, identification of nonlinearities has become essential. Thus, a universal function approximator has to be found which is able to approximate an unknown static1 nonlinear function. For this objective, various neural network and also fuzzy approaches are offered.

Measurement-Based Quantum Computing (MBQC), proposed in 2001 is a model of quantum computing that achieves quantum computation by performing a series of adaptive single-qubit measurements on an entangled cluster state. Our project is aimed at introducing MBQC to a wide audience ranging from high school students to quantum computing researchers through a Tangram puzzle with a modified set of rules played on an applet. The player is provided a quantum circuit which they have to map to MBQC using polyominos. Polyominos, the building blocks of our game, consist of square tiles joined edge-to-edge to form different shapes. Each tile represents a measurement basis, differentiated by its color. Polyominos rest on a square-grid playing board, which signifies a cluster state. We show that mapping a quantum circuit to MBQC is equivalent to arranging a set of polyominos—each corresponding to a gate in the circuit—on the playing board, subject to certain rules. We state the rules in simple terms with no reference to quantum computing. One such rule describes ways to deform a polyomino while it still correctly realizes a given quantum gate. The player has to place polyominos on the playing board conforming to the rules. Any correct solution creates a valid realization of the quantum circuit in MBQC. A higher-scoring correct solution fills up less space on the board, resulting in a lower-overhead embedding of the circuit in MBQC, a challenging research problem.

Quantum computing has the potential to outperform classical computers and is expected to play an active role in various fields. In quantum machine learning, a quantum computer has been found useful for enhanced feature representation and high-dimensional state or function approximation. Quantum–classical hybrid algorithms have been proposed in recent years for this purpose under the noisy intermediate-scale quantum computer (NISQ) environment. Under this scheme, the role played by the classical computer is the parameter tuning, parameter optimization, and parameter update for the quantum circuit. In this paper, we propose a gradient descent-based backpropagation algorithm that can efficiently calculate the gradient in parameter optimization and update the parameter for quantum circuit learning, which outperforms the current parameter search algorithms in terms of computing speed while presenting the same or even higher test accuracy. Meanwhile, the proposed theoretical scheme was successfully implemented on the 20-qubit quantum computer of IBM Q, ibmq_johannesburg. The experimental results reveal that the gate error, especially the CNOT gate error, strongly affects the derived gradient accuracy. The regression accuracy performed on the IBM Q becomes lower with the increase in the number of measurement shot times due to the accumulated gate noise error.

Simulation of the dynamics of quantum materials is emerging as a promising scientific application for noisy intermediate-scale quantum (NISQ) computers. Due to their high gate-error rates and short decoherence times, however, NISQ computers can only produce high-fidelity results for those quantum circuits smaller than some given circuit size. Dynamic simulations, therefore, pose a challenge as current algorithms produce circuits that grow in size with each subsequent time-step of the simulation. This underscores the crucial role of quantum circuit compilers to produce executable quantum circuits of minimal size, thereby maximizing the range of physical phenomena that can be studied within the NISQ fidelity budget. Here, we present two domain-specific (DS) quantum circuit compilers for the Rigetti and IBM quantum computers, specifically designed to compile circuits simulating dynamics under a special class of time-dependent Hamiltonians. The compilers outperform state-of-the-art general-purpose compilers in terms of circuit size reduction by around 25%–30% as well as wall-clock compilation time by around 40% (dependent on system size and simulation time-step). Drawing on heuristic techniques commonly used in artificial intelligence, both compilers scale well with simulation time-step and system size. Code for both compilers is open-source and packaged into a full-stack quantum simulation software with tutorials included for ease of use for future researchers wishing to perform dynamic simulations of quantum materials on quantum computers. As our DS compilers provide significant improvements in both compilation time and simulation fidelity, they provide a building block for accelerating progress toward physical quantum supremacy.

Quantum Computer: Quantum Model and Reality

While current quantum computers, referred to as Noisy Intermediate-Scale Quantum (NISQ) computers, are expected to be beneficial for different applications, they are prone to different types of errors. In order to enhance the reliability of quantum systems, noise-aware quantum compilers are used to generate physical quantum circuits to be executed on NISQ computers. The quantum hardware is calibrated very frequently and its error rates are computed accordingly. Based on the hardware error rates, a quantum compiler allocates physical qubits and schedules quantum operations. However, error rates may change post-calibration. To incorporate dynamic error rates into quantum circuit compilation with minimum cost, we propose a Machine Learning (ML)-based scheme to detect the incorrect output of the quantum circuit and predict the Probability of Successful Trials (PST) with high accuracy. Our approach can verify the error rates of the quantum hardware and validate the correctness of the extracted quantum circuit output. We provide a case study of our ML-based reliability models using IBM Q16 Melbourne quantum computer. Our results show that the proposed scheme achieves a very high prediction accuracy.

Quantum computing simulation platform can simulate the computation results of the quantum computer based on traditional computers, which is an effective way to promote the development of quantum computing software, algorithms and hardware at the current immature stage of the real quantum computer. Since quantum computers have exponential calculation acceleration compared with traditional computers, the main problems in implementing quantum computing simulation on traditional computers are low computational efficiency and long time-consuming. A quantum circuit which is a sequence of quantum gates acting on a collection of qubits is the general quantum computing model. So by the means of quantum circuit optimization, the calculation speed can be significantly increased while keeping the calculation result unchanged. The existing empirical rules of quantum circuit optimization methods have limitations and there is no common and automatic quantum circuit optimization method. In this paper, a general and automatic quantum circuit optimization method based on the genetic algorithm is proposed, by which the equivalent optimal quantum circuit is obtained through a finite number of searching in a large searching space. This method is not limited by the hardware of the quantum computing simulation and the composition of the quantum circuit. The experimental results show that for the QFT algorithm of 29 qubits, the running time can be shortened by 41.4% and for the variational circuit of 6 qubits, the running time can be shortened by 18.8% compared with the state-of-the-art quantum circuit optimization method. So this method can improve the quantum computing simulation capability and operating efficiency and provide a rapid development way for quantum algorithms and applications.

The best design for practical quantum computing is one that emulates the multi-agent quantum logic function of natural biological systems. Such systems are theorized to be based upon a quantum gate formed by a nucleic acid Szilard engine (NASE) that converts Shannon entropy of encountered molecules into useful work of nucleic acid geometric reconfiguration. This theoretical mechanism is logically and thermodynamically reversible in this special case because it is literally constructed out of the (nucleic acid) information necessary for its function, thereby allowing the nucleic acid Szilard engine to function reversibly because, since the information by which it functions exists on both sides of the theoretical mechanism simultaneously, there would be no build-up of information within the theoretical mechanism, and therefore no irreversible thermodynamic energy cost would be necessary to erase information inside the mechanism. This symmetry breaking Szilard engine function is associated with emission and/or absorption of entangled photons that can provide quantum synchronization of other nucleic acid segments within and between cells. In this manner nucleic acids can be considered as a natural model of topological quantum computing in which the nonabelian interaction of genes can be represented within quantum knot/braid theory as anyon crosses determined by entropic loss or gain that leads to changes in nucleic acid covalent bond angles. This naturally occurring biological form of topological quantum computing can serve as a model for workable man-made multi-agent quantum computing systems.

Our aim in this paper is to present experimental evidence of a side-channel attack on superconducting cloud-based quantum computers, demonstrating that a quantum circuit can be analyzed indirectly using data gathered before and after its execution on a quantum computer. We hope that this can motivate further research into hardware and system stability of quantum computers as well as into more-nuanced systems-level attacks. Although much work has been done on establishing error correcting codes to address issues of decoherence, and schemes can be utilized to leverage quantum effects for the sake of intrusion detection over quantum communication channels, there is a lack of literature in the field regarding side-channel attacks on transpiled quantum circuits, particularly those which leverage quantum effects to gather information about quantum circuits running in the cloud. We explored issues of decoherence by examining quantum circuit behavior both before and after a specified circuit was executed in order to ascertain decalibration rates and potential sources of decoherence: our data clearly shows that information on an interim circuit can be acquired in this manner. Using convenient data classification techniques and various circuit diagrams consisting of one quantum gate and several measurements, we were able to distinguish between these various circuits when executed at position n in the interim phase by training our classifier on data from probing circuits executed at queue positions n-1 and n+1. This shows that there is sufficient information residing in the data before and after a circuit is executed on a cloud-based superconducting quantum computer to allow an attacker to ascertain information about an otherwise unknown third-party circuit and shows potential to form the basis of a side-channel attack in cases where queue positions can be examined to target a third party’s transpiled circuit during experimentation. This side channel attack merits further examination and extrapolation to identify more-complex circuits, and deeper statistical consideration to determine potential sources of information leakage and mitigation strategies.

Chapter 5 - Quantum Algorithms and Methods

The current advancement in quantum computers has been focusing on increasing the number of qubits and enhancing their fidelity. However, the available quantum devices, known as Intermediate Scale Quantum (NISQ) computers, still suffer from different sources of noise that impact their reliability. Thus, practical noise modeling is of great importance in the development of quantum error mitigation approaches. In this paper, we propose a Machine Learning (ML)-based scheme to predict the output fidelity of the quantum circuit executed on NISQ devices. We show the benefit of using Graph Neural Network (GNN)-based models compared to traditional ML-based models in capturing the quantum circuit structure in addition to its gates’ features, which enable characterizing unpredicted quantum circuit errors. We use different metrics to measure the fidelity of the quantum circuit output. Our experimental results using different quantum algorithms executed on IBM Q Guadalupe quantum computer show the high prediction accuracy of our ML reliability models. Our results also show that our models can guide the single-qubit gate rescheduling to improve the output fidelity of the quantum circuit without the need for prior execution of dedicated calibration circuits.

An efficient quantum circuit (program) compiler aims to minimize the gate-count - through efficient instruction translation, routing, gate, and cancellation - to improve run-time and noise. Therefore, a high-efficiency compiler is paramount to enable the game-changing promises of quantum computers. To date, the quantum computing hardware providers are offering a software stack supporting their hardware. However, several third-party software toolchains, including compilers, are emerging. They support hardware from different vendors and potentially offer better efficiency. As the quantum computing ecosystem becomes more popular and practical, it is only prudent to assume that more companies will start offering software-as-a-service for quantum computers, including high-performance compilers. With the emergence of third-party compilers, the security and privacy issues of quantum intellectual properties (IPs) will follow. A quantum circuit can include sensitive information such as critical financial analysis and proprietary algorithms. Therefore, submitting quantum circuits to untrusted compilers creates opportunities for adversaries to steal IPs. In this paper, we present a split compilation methodology to secure IPs from untrusted compilers while taking advantage of their optimizations. In this methodology, a quantum circuit is split into multiple parts that are sent to a single compiler at different times or to multiple compilers. In this way, the adversary has access to partial information. With analysis of over 152 circuits on three IBM hardware architectures, we demonstrate the split compilation methodology can completely secure IPs (when multiple compilers are used) or can introduce factorial time reconstruction complexity while incurring a modest overhead (~ 3% to ~ 6% on average).