Fermions are fundamental particles which obey seemingly bizarre quantum-mechanical principles, yet constitute all the ordinary matter that we inhabit. As such, their study is heavily motivated from both fundamental and practical incentives. In this dissertation, we will explore how the tools of quantum information and computation can assist us on both of these fronts. We primarily do so through the task of partial state learning: tomographic protocols for acquiring a reduced, but sufficient, classical description of a quantum system. Developing fast methods for partial tomography addresses a critical bottleneck in quantum simulation algorithms, which is a particularly pressing issue for currently available, imperfect quantum machines. At the same time, in the search for such protocols, we also refine our notion of what it means to learn quantum states. One important example is the ability to articulate, from a computational perspective, how the learning of fermions contrasts with other types of particles.
Fermions are fundamental particles which obey seemingly bizarre quantum-mechanical principles, yet constitute all the ordinary matter that we inhabit. As such, their study is heavily motivated from both fundamental and practical incentives. In this dissertation, we will explore how the tools of quantum information and computation can assist us on both of these fronts. We primarily do so through the task of partial state learning: tomographic protocols for acquiring a reduced, but sufficient, classical description of a quantum system. Developing fast methods for partial tomography addresses a critical bottleneck in quantum simulation algorithms, which is a particularly pressing issue for currently available, imperfect quantum machines. At the same time, in the search for such protocols, we also refine our notion of what it means to learn quantum states. One important example is the ability to articulate, from a computational perspective, how the learning of fermions contrasts with other types of particles.
Quantum computation is an exciting and rapidly developing paradigm of computing, which has the potential to improve or accelerate numerous important applications ranging from cryptography to drug discovery. Inspired by these prospects, researchers and engineers across the world have, in recent years, made tremendous advances in building programmable quantum devices. Nevertheless, the size and error rates in state-of-the-art quantum computers are still several orders of magnitude away from those required to achieve computational speedups using existing quantum algorithms. These developments, along with their limitations, motivate us to consider two important questions: First, are there any useful applications in which current or near-term quantum devices can outperform classical computers? Second, can we develop efficient error-correction schemes to suppress the error rates in near-term quantum devices? In this dissertation, we present novel solutions to both of these questions. To address the first question, we identify an important class of problems in condensed-matter physics, namely the identification and characterization of quantum phases of matter, as one promising application of current and near-term quantum information processors. We develop new algorithms for using such quantum devices to solve these problems, and we demonstrate how our methods substantially outperform existing approaches. In addition, we respond to the second question by developing hardware-efficient quantum error correction protocols. By leveraging the distinctive properties and advantages of a given experimental setup to overcome its particular limitations, our proposals significantly reduce the number of quantum bits and operations required for performing quantum error correction and fault-tolerant quantum computation. Together, the works in this dissertation thus pave the way for the near-term realization of quantum computational advantages.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
The analysis of quantum state properties is of fundamental importance in quantum physics and serves as a primary motivation for the development of quantum computing. In particular, low-temperature and ground states are of significant interest, as they most prominently exhibit phenomena unique to quantum mechanics. However, extracting such properties typically requires costly numerical simulations or laboratory experiments, while simulation on quantum hardware remains limited by the capabilities of current devices. This thesis proposes two complementary approaches to address these challenges.The first contribution presents an algorithm for predicting ground-state properties across an entire family of quantum states within the same phase of matter, based on training data. Under slightly stronger assumptions, we improve upon the sample complexity of existing methods, prove the same rigorous guarantees for a deep neural network-based approach, and demonstrate its practical advantages through extensive numerical experiments.The second contribution addresses the challenge of preparing quantum states of interest using Variational Quantum Algorithms (VQAs), a framework based on the classical optimization of parametrized quantum circuits. We introduce multi-armed stochastic bandits into this setting, propose an instance-optimal algorithm for continuous single-parameter optimization, and and motivate further exploration of the framework we introduced with an array of experiments.
We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods.Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states.By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques.We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements.Quantum information processing (QIP) promises advantages in a wide range of different contexts, including machine learning [2-4], chemistry simulation [5][6][7], and number theory [8,9].As such, the experimental effort to build useful QIP devices has exploded in recent years.In the course of this effort, quantum tomography is a valuable tool for diagnosing and debugging small quantum devices, and has subsequently seen a variety of different advances.In particular, Bayesian approaches to tomography which are especially well suited to utilizing prior information and adapting to changing experimental conditions have developed significantly in recent years [10-13], presenting a useful experimental tool [14][15][16].In this paper we demonstrate the efficiency and accuracy of an adaptive tomography protocol that we call PAQT: practical adaptive quantum tomography.PAQT intelligently selects new measurements based on the outcomes of previous ones [10,[17][18][19][20][21].Adaptivity has been experimentally demonstrated [14,15,[22][23][24][25], but is not currently standard practice.Though adaptivity increases accuracy, the computational costs incurred outweigh that of simply repeating standard measurements many times.The PAQT approach employs a simple heuristic that can be efficiently computed between measurements, even with embedded hardware [26][27][28][29].The algorithm we propose is therefore compatible with modern experimental design and avoids an important limitation of previous approaches.We base our algorithm off of self-guided quantum tomography (SGQT), which treats adaptive tomography as a direct optimization problem rather than a new optimization problem between each measurement [30].Though this affords an efficient and easy to implement adaptive heuristic, SGQT is not without its limitations.It requires assuming that the target state is pure, and it does not return rich region estimates for a state.What PAQT achieves is to effectively combine SGQT with conventional and easily-implemented tomographic estimators, such as the Bayesian particle filter or least-squares fit (LSF) estimators.Under this approach, an experimentalist can collect data using SGQT (even if its assumptions are not met), and then post-process this data using particle filtering or LSF.The benefit of PAQT is two-fold.(1) From the point of view of traditional tomography, it gives an adaptive tomography protocol requiring only modest computational resources, as the bulk of the computational cost is offloaded to post-processing.(2) From the point of view of simulation-based optimization tomography (such as SGQT), it effectively augments the output with region estimation providing a statistically robust quantification of uncertainty.Thus, while we do not explicitly demonstrate that the improved scaling of Ferrie [30] remains in the more general case considered here, PAQT does provide a practical and efficient procedure for performing adaptive quantum tomography with rigorous statistical principles.
This thesis is a collection of theoretical works aiming at adjusting quantum algorithms to the hardware of quantum computers. The overarching topic of these efforts is to enable digital quantum simulation, the process of approximating the ground state of an arbitrary physical system with elementary operations of a quantum computer. For fermionic systems, a class including molecules and materials, the impact of quantum computing would be undoubtedly high, and algorithms exist for their simulation. However, there is a certain gap between the requirements of those algorithms and what actual quantum devices can provide: it seems that our expectations of a fully-fledged quantum computer still exceed our capabilities to build it. To make quantum simulation feasible, we seek to adapt quantum algorithms to three different types of device limitations within this thesis.
Quantum computers are expected to become a powerful tool for studying physical quantum systems. Consequently, a number of quantum algorithms for studying the physical properties of such systems have been developed. While qubit-based quantum computers are naturally suited to the study of spin-1/2 systems, systems containing other degrees of freedom must first be encoded into qubits. Transformations to and from fermionic degrees of freedom have long been an important tool in physics and, now the simulation of fermionic systems on quantum computers based on qubits provides yet another application. In this perspective, we review methods for encoding fermionic degrees of freedom into qubits and attempt to dispel the persistent notion that fermionic systems beyond one dimension are fundamentally more difficult to deal with.