Sonika Johri

Image-based data is a popular arena for testing quantum machine learning algorithms. A crucial factor in realizing quantum advantage for these applications is the ability to efficiently represent images as quantum states. Here we present a novel method for creating quantum states that approximately encode images as amplitudes, based on recently proposed techniques that convert matrix product states to quantum circuits. The numbers of gates and qubits in our method scale logarithmically in the number of pixels given a desired accuracy, which make it suitable for near term quantum computers. Finally, we experimentally demonstrate our technique on 8 qubits of a trapped ion quantum computer for complex images of road scenes, making this the first large instance of full amplitude encoding of an image in a quantum state.

State preparation is a necessary component of many quantum algorithms. In this work, we combine a method for efficiently representing smooth differentiable probability distributions using matrix product states with recently discovered techniques for initializing quantum states to approximate matrix product states. Using this, we generate quantum states encoding a class of normal probability distributions in a trapped ion quantum computer for up to 20 qubits. We provide an in depth analysis of the different sources of error which contribute to the overall fidelity of this state preparation procedure. Our work provides a study in quantum hardware for scalable distribution loading, which is the basis of a wide range of algorithms that provide quantum advantage.

Quantum computers are emerging as a viable alternative to tackle certain computational problems that are challenging for classical computers. With the rapid development of quantum hardware such as those based on trapped ions, there is practical motivation for identifying risk management problems that are efficiently solvable with these systems. Here we focus on network partitioning as a means for analyzing risk in critical infrastructures and present a quantum approach for its implementation. It is based on the potential speedup quantum computers can provide in the identification of eigenvalues and eigenvectors of sparse graph Laplacians, a procedure which is constrained by time and memory on classical computers.

Variational quantum eigensolvers (VQE) are among the most promising approaches for solving electronic structure problems on near-term quantum computers. A critical challenge for VQE in practice is that one needs to strike a balance between the expressivity of the VQE ansatz versus the number of quantum gates required to implement the ansatz, given the reality of noisy quantum operations on near-term quantum computers. In this work, we consider an orbital-optimized pair-correlated approximation to the unitary coupled cluster with singles and doubles (uCCSD) ansatz and report a highly efficient quantum circuit implementation for trapped-ion architectures. We show that orbital optimization can recover significant additional electron correlation energy without sacrificing efficiency through measurements of low-order reduced density matrices (RDMs). In the dissociation of small molecules, the method gives qualitatively accurate predictions in the strongly-correlated regime when running on noise-free quantum simulators. On IonQ's Harmony and Aria trapped-ion quantum computers, we run end-to-end VQE algorithms with up to 12 qubits and 72 variational parameters - the largest full VQE simulation with a correlated wave function on quantum hardware. We find that even without error mitigation techniques, the predicted relative energies across different molecular geometries are in excellent agreement with noise-free simulators.

Copulas are mathematical tools for modeling joint probability distributions. Since copulas enable one to conveniently treat the marginal distribution of each variable and the interdependencies among variables separately, in the past 60 years they have become an essential analysis tool on classical computers in various fields ranging from quantitative finance and civil engineering to signal processing and medicine. The recent finding that copulas can be expressed as maximally entangled quantum states has revealed a promising approach to practical quantum advantages: performing tasks faster, requiring less memory, or, as we show, yielding better predictions. Studying the scalability of this quantum approach as both the precision and the number of modeled variables increase is crucial for its adoption in real-world applications. In this paper, we successfully apply a Quantum Circuit Born Machine (QCBM) based approach to modeling 3- and 4-variable copulas on trapped ion quantum computers. We study the training of QCBMs with different levels of precision and circuit design on a simulator and a state-of-the-art trapped ion quantum computer. We observe decreased training efficacy due to the increased complexity in parameter optimization as the models scale up. To address this challenge, we introduce an annealing-inspired strategy that dramatically improves the training results. In our end-to-end tests, various configurations of the quantum models make a comparable or better prediction in risk aggregation tasks than the standard classical models. Our detailed study of the copula paradigm using quantum computing opens opportunities for its deployment in various industries.

Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms. Here, we report the results of an experimental demonstration of amplitude estimation on a state-of-the-art trapped ion quantum computer. The amplitude estimation algorithms were used to estimate the inner product of randomly chosen four-dimensional unit vectors, and were based on the maximum likelihood estimation (MLE) and the Chinese remainder theorem (CRT) techniques. Significant improvements in accuracy were observed for the MLE based approach when deeper quantum circuits were taken into account, including circuits with more than ninety two-qubit gates and depth sixty, achieving a mean additive estimation error on the order of $10^{-2}$. The CRT based approach was found to provide accurate estimates for many of the data points but was less robust against noise on average. Last, we analyze two more amplitude estimation algorithms that take into account the specifics of the hardware noise to further improve the results.

Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this work, we design quantum machine learning algorithms to model copulas. We show that any copula can be naturally mapped to a multipartite maximally entangled state. A variational ansatz we christen as a `qopula' creates arbitrary correlations between variables while maintaining the copula structure starting from a set of Bell pairs for two variables, or GHZ states for multiple variables. As an application, we train a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables for historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to 8 qubits and show that our results outperform those obtained through equivalent classical generative learning. Further, we present theoretical arguments for exponential advantage in our model's expressivity over classical models based on communication and computational complexity arguments.

Quantum computers have the potential to advance material design and drug discovery by performing costly electronic structure calculations. A critical aspect of this application requires optimizing the limited resources of the quantum hardware. Here, we experimentally demonstrate an end-to-end pipeline that focuses on minimizing quantum resources while maintaining accuracy. Using density matrix embedding theory as a problem decomposition technique, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms without freezing any electrons. The originally 20-qubit system is decomposed into 10 two-qubit problems, making it amenable to currently available hardware. Combining this decomposition with a qubit coupled cluster circuit ansatz, circuit optimization, and density matrix purification, we accurately reproduce the potential energy curve in agreement with the full configuration interaction energy in the minimal basis set. Our experimental results are an early demonstration of the potential for problem decomposition to accurately simulate large molecules on quantum hardware.

Quantum machine learning has seen considerable theoretical and practical developments in recent years and has become a promising area for finding real world applications of quantum computers. In pursuit of this goal, here we combine state-of-the-art algorithms and quantum hardware to provide an experimental demonstration of a quantum machine learning application with provable guarantees for its performance and efficiency. In particular, we design a quantum Nearest Centroid classifier, using techniques for efficiently loading classical data into quantum states and performing distance estimations, and experimentally demonstrate it on a 11-qubit trapped-ion quantum machine, matching the accuracy of classical nearest centroid classifiers for the MNIST handwritten digits dataset and achieving up to 100% accuracy for 8-dimensional synthetic data.

Generating high-quality data (e.g. images or video) is one of the most exciting and challenging frontiers in unsupervised machine learning. Utilizing quantum computers in such tasks to potentially enhance conventional machine learning algorithms has emerged as a promising application, but poses big challenges due to the limited number of qubits and the level of gate noise in available devices. In this work, we provide the first practical and experimental implementation of a quantum-classical generative algorithm capable of generating high-resolution images of handwritten digits with state-of-the-art gate-based quantum computers. In our quantum-assisted machine learning framework, we implement a quantum-circuit based generative model to learn and sample the prior distribution of a Generative Adversarial Network. We introduce a multi-basis technique that leverages the unique possibility of measuring quantum states in different bases, hence enhancing the expressivity of the prior distribution. We train this hybrid algorithm on an ion-trap device based on $^{171}$Yb$^{+}$ ion qubits to generate high-quality images and quantitatively outperform comparable classical Generative Adversarial Networks trained on the popular MNIST data set for handwritten digits.

Interacting quantum systems illustrate complex phenomena including phase transitions to novel ordered phases. The universal nature of critical phenomena reduces their description to determining only the transition temperature and the critical exponents. Numerically calculating these results for systems in new universality classes is complicated due to critical slowing down, requiring increasing resources near the critical point. An alternative approach analytically continues the calculation onto the complex plane and determines the partition function via its zeros. Here we show how to robustly perform this analysis on noisy intermediate scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the determination of critical phenomena for systems that cannot be solved otherwise.

The efficient simulation of correlated quantum systems is the most promising near-term application of quantum computers. Here, we present a measurement of the second Renyi entropy of the ground state of the two-site Fermi-Hubbard model on a $5$-qubit programmable quantum computer based on trapped ions. Our work illustrates the extraction of a non-linear characteristic of a quantum state using a controlled-swap gate acting on two copies of the state. This scalable measurement of entanglement on a universal quantum computer will, with more qubits, provide insights into many-body quantum systems that are impossible to simulate on classical computers.