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There has been a large body of work investigating potential advantages of quantum computational algorithms over their classical counterparts, with the ultimate proof being an experimental verification of a "quantum advantage”. Some of the classical problems that are being targeted include factorization of large integers and unstructured database searching. While the advances in both experimental hardware and software are driving the field slowly but steadily towards quantum supremacy, more efforts are still required in both fields. Some of the most promising algorithms for potential near-term quantum advantages include the class of variational quantum algorithms (VQAs). VQAs have been applied to many scientific domains, including molecular dynamical studies, and quantum optimization problems. VQAs are also studied for various quantum machine learning (QML) applications such as regression, classification, generative modeling, deep reinforcement learning, sequence modeling, speech recognition, metric and embedding learning, transfer learning, and federated learning.

In addition to VQAs being applied as quantum implementations of classical machine learning paradigms, conversely VQAs may also themselves benefit from various machine learning paradigms, with one of the most popular being Reinforcement Learning (RL). RL has been utilized to assist in several problems in quantum information processing, such as decoding errors, quantum feedback, and adaptive code design. While so-far such schemes are envisioned with a classical computer as a RL co-processor, implementing “quantum” RL using quantum computers has been shown to make the decision-making process for RL agents quadratically faster than on classical hardware.

In this work, the authors present a new quantum architecture search framework that includes an RL agent interacting with a quantum computer or quantum simulator powered by deep reinforcement learning (DRL). The objective is to investigate the potential of training an RL agent to search for a quantum circuit architecture for generating a desired quantum state.

The proposed framework consists of two major components; a quantum computer or quantum simulator and an RL agent hosted on a classical computer that interacts with the quantum computer or quantum simulator. In each time step, the RL agent chooses an action from the possible set of actions consisting of different quantum operations (one- and two-qubit gates) thereby updating the quantum circuit. After each update, the quantum simulator executes the new circuit and calculates the fidelity to the given target state. The fidelity of the quantum circuit is then evaluated to determine the reward to be sent back to the agent; positive in case the fidelity reaches a pre-defined threshold, else negative.

The authors use the set of single-qubit Pauli measurements as the “states” or observations which the environment returns to the RL agent. The RL agent is then iteratively updated based on this information. The procedure continues until the agent reaches either the desired threshold or the maximum allowed steps. In this work, RL algorithms like A2C and PPO were used to optimize the agent. The results demonstrate that given the same neural network architecture, PPO performs significantly better than the A2C in terms of convergence speed and stability for both the case of noise-free and noisy environments. The result is shown to be consistent with the 2-qubit case.

In this work, the simulation of quantum circuits in both noise-free and noisy environments is implemented via Qiskit software from IBM. The efficiency of such an approach when it comes to large quantum circuits is quite low as the complexity scales exponentially with the number of qubits. Theoretically, a quantum circuit may approximate any quantum state (up to an error tolerance) using a finite number of gates, given a universal set of one and two-qubit quantum gates. Hence, in principle, the RL approach is valid for arbitrary large qubit size but is extremely hard to implement computationally. It would also require thousands of training episodes to verify this kind of experiment on real quantum computers. One can expect significant development in the future when quantum computing resources are more accessible. Finally, another interesting scope will be to investigate the quantum architecture search problem with different target quantum states and different noise configurations.
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Enhancing machine learning applications with quantum computing is currently being massively investigated, since it might prove quantum advantage during the NISQ era. Enhancement is typically performed through improvement of the training process of existing classical models or enhancing inference in graphical models. Another way is through the construction of quantum models that generate correlations between variables that are inefficient to represent through classical computation (e.g. quantum neural networks). If the model leverages a quantum circuit that is hard to sample results from classically, then there is potential for a quantum advantage.

The main message of this work is to show quantitatively that when classical models are provided with some training data, even if those were obtained from a quantum model that cannot be computed classically, they can reach similar performance as the quantum model. The authors provide rigorous prediction error bounds for training classical and quantum ML methods based on kernel functions in order to learn quantum mechanical models. It has been proven that kernel methods provide provable guarantees, but are also very flexible in the functions they can learn.

The main contribution of the author’s work lies in the development of quantum kernels and the implementation of a guidebook that generates ML problems which give a large separation between quantum and classical models. The use of prediction error bounds quantifies the separation between prediction errors of quantum and classical ML models for a fixed amount of training data. Typically, the comparison is done based on the geometric difference defined by the closest efficient classical ML model, however, in practice, one should consider the geometric difference with respect to a suite of optimized classical ML models. When the geometric difference is small then a classical ML method is guaranteed to provide similar or better performance in prediction on the data set, independent of the function values or labels. When the geometry differs greatly, there exists a data set that exhibits large prediction advantage using the quantum ML model.

The small geometric difference is a consequence of the exponentially large Hilbert space employed by existing quantum models, where all inputs are too far apart. To circumvent the setback, the authors propose an improvement which enlarges the geometric difference by projecting quantum states embedded from classical data back to approximate classical representation. This proposal allows for the construction of a data set that demonstrates large prediction advantage over common classical ML models in numerical experiments up to 30 qubits. In that way, one can use a small quantum computer to generate efficiently verifiable ML problems that could be challenging for classical ML models.
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There is currently a big effort in proving the existence of quantum advantage in NISQ devices, with typical applications including machine learning due to its wide applicability. However, an argument has been raised stipulating whether the advantage being proven in such applications arises from the nature of quantum computing or the way the data is provided. Actually, there are recent works which have shown that, when classical algorithms have an analogous sampling access to data, then they can reach similar performance as their quantum counterparts. Therefore, a fair study concerning quantum advantage in such a setting should only assume classical access to data for both the classical and quantum algorithm.

Most quantum algorithms being explored at the moment are of the variational nature, where a circuit is selected from a parametrized circuit family via classical optimization. However, such algorithms are heuristic and there is no formal evidence that they exhibit a genuine quantum advantage. In their work, the authors describe an exponential quantum speedup via the use of a quantum-enhanced feature space, where each data point is mapped non-linearly to a quantum state and then classified by a linear classifier in the high dimensional Hilbert space. The classification is achieved through a support vector machine (SVM), whose kernel matrix is obtained by measuring the pairwise inner product of the feature states, known as quantum kernel estimation (QKE). This kernel matrix is then given to a classical optimizer that efficiently finds the linear classifier that optimally separates the training data in feature space by running a convex quadratic program.

The advantage of such a quantum learner comes from the ability to recognize classically intractable complex patterns, using a feature map. This implementation combines the ideas of the generalization of soft margin classifiers and rigorously bounding the misclassification error of the SVM-QKE algorithm, thereby providing quantum advantage for the quantum classifier with guaranteed high accuracy. Furthermore, it is proven that this SVM classifier can learn the optimal separator in the exponentially large feature space, while also making it robust against additive sampling errors due to the error mitigation techniques that were used.

Finally, the classification problem showing the exponential quantum speed-up was based on the discrete logarithm problem (DLP) and it is proven that no efficient classical algorithm can achieve an accuracy that is inverse-polynomially better than random guessing, assuming the widely-believed classical hardness of DLP.

The results from this paper are relevant to a broader class of quantum machine learning algorithms exploiting feature maps and which aim to avoid the input-output problem. Although this particular problem is not practically motivated, these results set a positive theoretical foundation for the search of practical quantum advantage in machine learning.
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A prospective application of quantum computing is solving quantum chemistry problems, however, obtaining exact solutions is difficult due to the lack of general method of obtaining such solutions. Typically the solution lies in finding the ground state energy, even though the energy is not descriptive enough to fully characterize all desired properties of a system. In order to find such properties, many measurements of the wavefunction are required. These measurements are expensive, because the wavefunction cannot be copied and must often be re-prepared before a second measurement is performed. Finding the full wavefunction would require exponentially many measurements, so one option would be to encode many solutions into one measurement by using a machine learned (ML) model. Training of the ML model requires finding exact quantities at several different external potentials. Besides that, one can use density functional theory (DFT), to replace the wavefunction with the one-body density, n(r), which has fewer variables. When DFT is used, instead of the Hamiltonian, the universal functional, F[n], must be found. The quantities required for the classical user to find self-consistent solutions are the exact functional (determining the energy) and the functional derivative. So, in addition to finding F[n], one also must find some other quantity such as the density, n(r), or the Kohn-Sham(KS) potential, vs(r). With these components, one can fully characterize a quantum ground state and solve for other measurable quantities.

The authors propose an algorithm that finds the ML model for F[n] on the quantum computer if a ground-state wavefunction is available. The algorithm leaves the wavefunction largely undisturbed so it can be used as the starting state for another system, greatly reducing the pre-factor required to solve other systems through a quantum counting algorithm to extract descriptive quantities such as the density. Most of this algorithm is kept entirely on the quantum computer to motivate future improvements for speed, but the counting algorithm does allow for information to be output classically. Moreover, the authors demonstrate that the exact Kohn-Sham potential can be solved in a faster way with a gradient evaluated on a cost function for the Kohn-Sham system.

The novelty of the proposed algorithm suggested in this work is the limitation of the number of measurements and re-preparations of the wavefunction especially in the case of time-dependent quantities. Since there is no algorithm for the general case that is exponentially better than the proposed algorithm, limiting the number of measurements and re-preparations of the wavefunction is as best as one can achieve. The proposed algorithm is a combination of several known algorithms including quantum phase estimation, quantum amplitude estimation, and quantum gradient methods that are iteratively used to train a machine learned model.
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Quantum computers are envisioned to have significantly higher computational capabilities compared to their classical counterparts, especially for optimization and machine learning problems that involve a large classical data set. However, existing quantum algorithms use the trivial methods of turning large classical datasets into either quantum oracles or quantum states which are so expensive that negate any possible quantum advantage. Such quantum algorithms focus at problems in which classical runtime scales rapidly with the input size, perhaps exponentially. To be able to achieve quantum speedup with algorithms like Grover search, a “quantum RAM” is proposed, which is a large classical memory that can be queried in superposition. Although quantum RAMs do not yet exist and creating one might encounter the same challenges that quantum computer hardware faces, it has the potential to provide significant speedup to applications like the k-means clustering, logistical regression, zero-sum games and boosting.

This paper introduces hybrid classical-quantum algorithms for problems involving a large classical data set X and a space of models Y such that a quantum computer has superposition access to Y but not X. Then a data reduction technique is used to construct a weighted subset of X called a coreset that yields approximately the same loss for each model. The coreset can be constructed either by a classical computer or by the combination of classical – quantum computer by utilization of quantum measurements.

The main message of this work is that in order to avoid losing quantum speedup for ‘big-data’ applications, techniques such as data reduction are required, so that the time for loading and storing the data set is limited. Also, non-fault tolerant quantum algorithms should be designed in a way that does not require an excessive amount of gates, so that the algorithm can run before qubits lose their coherence and invalidate the result. The goal of the paper is to draw attention to problems that arise from such actions like testing for quantum advantage when data reduction is used, explore general data reduction techniques and investigate new hybrid classical-quantum algorithms.

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Clustering is a form of unsupervised machine learning, where instances are organized into groups whose members share similarities. The assignments are, in contrast to classification, not known a priori, but generated by the algorithm. In this paper, Neukart et al.  present an algorithm for quantum-assisted cluster analysis (QACA) that makes use of the topological properties of a D-Wave 2000Q quantum processing unit (QPU). They explain how the problem can be expressed as a quadratic unconstrained binary optimization (QUBO) problem, and show that the introduced quantum-assisted clustering algorithm is, regarding accuracy, equivalent to commonly used classical clustering algorithms.

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In this paper, Dunjko et al. provide a comprehensive review of the current (Sept 2017) state of quantum machine learning, including quantum providing speed-ups or enhancing classical ML and classical classical ML being used for quantum-control or to design quantum-circuits

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02 March 2018

Quantum circuit learning

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Quantum machine learning (QML) algorithms based on the Harrow-Hassidim- Lloyd (HHL) algorithm rely on quantum phase estimation which requires high circuit-depth. To allow QML on current noisy intermediate scale quantum (NISQ) devices classical-quantum hybrid algorithms have been suggested applying low-depth circuits like quantum variational eigensolvers and quantum approximate optimization. Such hybrid algorithms typically divide the ML problem into two parts, each part to be performed either classically or on a quantum-computer. In this paper, Mitarai et al. present a new hybrid framework, called quantum circuit learning (QCL), which is easily realizable on current NISQ devices. Under QCL a circuit learns by providing input data, while iteratively tuning the circuit parameters to give the desired output. They show that QCL is able to learn nonlinear functions and perform simple classification tasks. They also show that a 6-qubit circuit is capable of learning dynamics of a 10-spin system with a fully connected Ising Hamiltonian, implying that QCL could be well suited for learning complex many-body systems.

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The perceptron algorithm dates back to the late 1950s and is an algorithm for supervised learning of binary classifiers. In a 2016 paper, Wiebe et al. proposed a quantum algorithm (based on Grover’s quantum-search approach), which can quadratically speed-up the training of a perceptron. In this paper, Zheng et al. describe their design for a quantum-circuit to implement the training-algorithm of Wiebe et al. They also analyze the resource requirements (qubits and gates) and demonstrate the feasibility of their quantum-circuit by testing it on the ibmqx5 (a 16 qubit universal gate quantum processor developed by IBM)

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22 September 2017

Quantum reinforcement learning

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Reinforcement learning6 differs from supervised and unsupervised learning in that it takes into account a scalar parameter (reward) to evaluate the input-output relation in a trial and error way. In this paper, Cardenas-Lopez et al. propose a protocol to perform generalized quantum reinforcement learning. They consider diverse possible scenarios for an agent, an environment, and a register that connects them, involving multi-qubit and multi-level systems, as well as open-system dynamics and they propose possible implementations of this protocol in trapped ions and superconducting circuits. 

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