We are currently approaching the point where quantum systems with 15 or more qubits will be controllable with high levels of coherence over long timescales. One of the fundamental problems that has been identified is that, as the number of qubits increases to these levels, there is currently no clear way to use efficiently the information that can be obtained from such a system to make diagnostic inferences and to enable improvements in the underlying quantum gates. Even with systems of only a few bits the exponential scaling in resources required by techniques such as quantum tomography or gate-set tomography will render these techniques impractical. Randomized benchmarking (RB) is a technique that will scale in a practical way with these increased system sizes. Although RB provides only a partial characterization of the quantum system, recent advances in the protocol and the interpretation of the results of such experiments confirm the information obtained as helpful in improving the control and verification of such processes. This thesis examines and extends the techniques of RB including practical analysis of systems affected by low frequency noise, extending techniques to allow the anisotropy of noise to be isolated, and showing how additional gates required for universal computation can be added to the protocol and thus benchmarked. Finally, it begins to explore the use of machine learning to aid in the ability to characterize, verify and validate noise in such systems, demonstrating by way of example how machine learning can be used to explore the edge between quantum non-locality and realism.
We are currently approaching the point where quantum systems with 15 or more qubits will be controllable with high levels of coherence over long timescales. One of the fundamental problems that has been identified is that, as the number of qubits increases to these levels, there is currently no clear way to use efficiently the information that can be obtained from such a system to make diagnostic inferences and to enable improvements in the underlying quantum gates. Even with systems of only a few bits the exponential scaling in resources required by techniques such as quantum tomography or gate-set tomography will render these techniques impractical. Randomized benchmarking (RB) is a technique that will scale in a practical way with these increased system sizes. Although RB provides only a partial characterization of the quantum system, recent advances in the protocol and the interpretation of the results of such experiments confirm the information obtained as helpful in improving the control and verification of such processes. This thesis examines and extends the techniques of RB including practical analysis of systems affected by low frequency noise, extending techniques to allow the anisotropy of noise to be isolated, and showing how additional gates required for universal computation can be added to the protocol and thus benchmarked. Finally, it begins to explore the use of machine learning to aid in the ability to characterize, verify and validate noise in such systems, demonstrating by way of example how machine learning can be used to explore the edge between quantum non-locality and realism.
Quantum computers have the potential to bring about a new age of technology, but in order for them to become a reality, the elephant in the room must be addressed, quantum errors. The complexity of quantum errors grows exponentially as the system size is increased; scalable characterization protocols which work under physically realistic noise are therefore needed to assess and optimize control. \n \nRandomized Benchmarking (RB) is a family of characterization protocols with the requisite qualities. RB protocols are independent of state preparation and measurement (SPAM) errors, and because they extract partial information about the error profiles in the form of figures of merit, they are scalable in the number of qubits. In this thesis, we study these figures of merit under various error models, gate-sets, and sample distributions, to further our understanding of their physical meaning, and to fine-tune the best practice for diagnosing errors in basic gate-sets. \n \nThe first study presented in this thesis looks at the physical meaning of the unitarity under physically realistic gate-dependent noise. Numerically, we demonstrate that under these realistic noise models, the unitarity obtained via Extended RB (XRB) overlaps strongly with the average of the unitarity of the errors over the set of individual operations. \n \nThe second study numerically and analytically proves that the Standard RB (SRB) decay parameter p coincides with the decay parameter of the novel figure of merit gate-set circuit fidelity. We also prove that, in the limit of high-fidelity single-qubit experiments, the disconnect between the average gate fidelity and SRB experimental results is caused by a basis mismatch between the gates and the state-preparation and measurement procedures. \n \nIn the third part of this thesis, we compare the results of two variations of the SRB protocol, one that samples uniformly (SRB) and one that samples non-uniformly (NIST RB) from the Clifford group. We prove that NIST RB produces an exponential fidelity curve that gives different infidelity results than SRB. \n \nThe fourth and final study gives a standard for assessing basic gate-sets. Experimental results are given from the assessment run on super-conducting and ion-trap devices. We also experimentally show how the systematic error bounds can be tightened by accounting for the error's degree of coherence, as well as using weaker randomizing sets.
Quantum computers promise to be a revolutionary new technology. However, in order to realise this promise many hurdles must first be overcome. In this thesis we investigate two such hurdles: the presence of noise in quantum computers and limitations on the connectivity and control in large scale quantum computing architectures.In order to combat noise in quantum devices we must first characterize this noise. To do this several diagnostic tools have been developed over the last two decades. The current industry standard for such a diagnostic tool is called randomized benchmarking. Randomized benchmarking doesn't give a full characterization of the noise afflicting the quantum device but rather attempts to give some indication of of the device's average behavior, captured in a quantity called the average fidelity. Because it does not endeavor to characterize every small detail of the noise it can be efficiently applied even to very large quantum devices. However, with this power also comes increased complexity. Randomized benchmarking has a lot of moving parts, and some fairly strong assumptions must be made in order to guarantee its correctness. In this thesis we attempt to justify these assumptions and if possible remove or weaken them, making randomized benchmarking a more robust and general tool. In chapter 6 of this thesis we investigate the finite statistics of randomized benchmarking. We prove strong bounds on the number of samples needed to perform rigorous randomized benchmarking. To do this we make use tools from representation theory. In particular we use a characterization of certain representations of the Clifford group, which we develop in chapter 5. In chapter 7 we re-use these tools to also bound the number of samples needed to perform rigorous unitarity randomized benchmarking, a newer variant of randomized benchmarking quickly gaining in popularity. These results retroactively justify the use of randomized benchmarking in an experimental setting and also provide guidance on optimal statistical practices in the context of randomized benchmarking. In chapter 8 we expand upon the standard randomized benchmarking protocol and formulate a new class of protocols which we call character randomized benchmarking. This new class of protocols removes a critical assumption made in standard randomized benchmarking, making character randomized benchmarking vastly more generally applicable. To show the advantages of character randomized benchmarking we implement it in an experiment characterizing the noise in a Si/SiGe quantum dot device. This experiment is detailed in chapter 9. Finally we deal with the second main topic of this thesis in chapter 10. Large scale quantum computer will, like classical computers, face limitations in the connectivity between different parts of the computer. This is due to a fundamental law in computer design called Rent's rule, which states that the number of wires connecting a (quantum) computer chip to the outside world is much smaller than the number of components in that chip. This means the individual components of the chip can not be controlled individually in parallel. Given that parallelism is absolutely critical for the functioning of quantum computers this is a serious problem for the development of large scale quantum computers. Luckily it is possible to organize quantum computing devices in such a way that they can be controlled using a relatively small amount of input wires. One example of such an organization is called a crossbar architecture. Recently a proposal was made for a crossbar architecture quantum computer in quantum dots, and in chapter 10 of this thesis we investigate in detail the advantages and disadvantages of such an architecture. We focus in particular on its effect on standard quantum error correction procedures, a key part of a functioning quantum computer, and one where parallel control of all parts of the quantum device is essential.
Characterization of experimental systems is an essential step in developing and improving quantum hardware. A collection of protocols known as Randomized Benchmarking (RB) was developed in the past decade, which provides an efficient way to measure error rates in quantum systems. In a recent paper (arxiv:2010.07974), a general framework for RB was proposed, which encompassed most of the known RB protocols and overcame the limitation on error models in previous works. However, even this general framework has a restriction: it can only be applied to a finite group of gates. This does not meet the need posed by experiments, in particular the demand for benchmarking non-Clifford gates and continuous gate sets on quantum devices. In this work we generalize the RB framework to continuous groups of gates and show that as long as the noise level is reasonably small, the output can be approximated as a linear combination of matrix exponential decays. As an application, we numerically study the fully randomized benchmarking protocol (i.e. RB with the entire unitary group as the gate set) enabled by our proof. This provides a unified way to estimate the gate fidelity for any quantum gate in an experiment.
The characterization of noisy quantum circuits is an important step in the development of large-scale quantum computers. As experimental quantum architectures approach the threshold for fault-tolerant quantum computing, obtaining a benchmark of process fidelities is needed to verify the integrity of the system and ensure functionality. Additionally, the ability to successfully implement a quantum code of choice may depend on the type of noise that is present in the system. It is therefore necessary to characterize not only the fidelity of the process, but also the structure of the noise. \n \nRandomized benchmarking is a characterization protocol which extracts partial information from a noisy quantum process. This thesis gives a detailed review of the randomized benchmarking protocol, and presents the results of three collaborative projects which use techniques similar to randomized benchmarking to obtain previously hidden information. The first protocol is designed to obtain a benchmark on loss errors. This protocol also provides information on the detector efficiency and an additional constraint on estimated parameters from randomized benchmarking. The second protocol is an extension of the first, under the special case of coherent leakage errors, for which we obtain an estimate of the leakage rate. The third protocol provides a method of testing for spatial correlations in noisy processes. This test may be used to determine the correlation structure of errors, and the error correcting codes needed for successful experiments in a particular architecture. All protocols presented in this thesis are scalable, platform-independent and insensitive to state preparation and measurement errors. We also provide results of numerical tests of these protocols for simulated noise, which show their accuracy and robustness.
The construction of large scale quantum computing devices might be one of the most exciting \nand promising endeavors of the 21st century, but it also comes with many challenges. \nAs quantum computers are supplemented with more registers, their error profile generally \ngrows in complexity, rendering the enterprise of quantifying the reliability of quantum computations \nincreasingly difficult through naive characterization techniques. In the last decade, \na lot of efforts has been directed toward developing highly scalable benchmarking schemes. \nA leading family of characterization methods built upon scalable principles is known as randomized \nbenchmarking (RB). \nIn this thesis, many tools are presented with the objective of improving the scalability, \nand versatility of RB techniques, as well as demonstrating their reliability under various \nerror models. \nThe first part of this work investigates the connection between the error of individual \ncircuit components and the error of their composition. Before reasoning about intricate circuit \nconstructions, it is shown that there exists a well-motivated way to define decoherent \nquantum channels, and that every channel can be factorized into a unitary-decoherent composition. \nThis dichotomy carries to the circuit evolution of important error parameters by \nassuming realistic error scenarios. Those results are used to improve the confidence interval \nof RB diagnoses and to reconcile experimentally estimated parameters with physically and \noperationally meaningful quantities. \nIn the second part of this thesis, various RB schemes are either developed or more rigorously \nanalyzed. A first result consists of the introduction of “dihedral benchmarking”, a \ntechnique which, if performed in conjunction with standard RB protocols, enables the characterization \nof operations that form a universal gate-set. Finally, rigorous analysis tools are \nprovided to demonstrate the reliability of a highly scalable family of generator-based RB \nprotocols known as direct RB.
Randomized benchmarking refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates.These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors.Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing.In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions.Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays.We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data.We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations.In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols.These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking.Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity.On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.