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 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.
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.
As quantum devices make steady progress towards intermediate scale and fault-tolerant quantum computing, it is essential to develop rigorous and efficient measurement protocols that account for known sources of noise. Most existing quantum characterization protocols such as gate-set tomography and randomized benchmarking assume the noise acting on the qubits is Markovian. However, this assumption is often not valid, as for the case of 1/f charge noise or hyperfine nuclear spin noise. Here, we present a general framework for quantum process tomography (QPT) in the presence of time-correlated noise. We further introduce fidelity benchmarks that quantify the relative strength of different sources of Markovian and non-Markovian noise. As an application of our method, we perform a comparative theoretical and experimental analysis of silicon spin qubits. We first develop a detailed noise model that accounts for the dominant sources of noise and validate the model against experimental data. Applying our framework for time-correlated QPT, we find that the number of independent parameters needed to characterize one- and two-qubit gates can be compressed by 10× and 100× , respectively, when compared to the fully generic case. These compressions reduce the amount of tomographic measurements needed in experiment, while also significantly speeding up numerical simulations of noisy quantum circuit dynamics compared to time-dependent Hamiltonian simulation. Using this compressed noise model, we find good agreement between our theoretically predicted process fidelities and two-qubit interleaved randomized benchmarking fidelities of 99.8% measured in recent experiments on silicon spin qubits. More broadly, our formalism can be directly extended to develop efficient and scalable tuning protocols for high-fidelity control of large arrays of quantum devices with non-Markovian noise. Published by the American Physical Society 2024
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.
This research paper provides an in-depth examination of verification and testing approaches for quantum circuits, focusing on methodologies essential for ensuring the correctness, reliability and operational efficiency of quantum computing systems. Since quantum circuits function on the fundamental principles of superposition and entanglement, they present verification challenges that cannot be effectively addressed using conventional classical testing techniques. As a result, the development of specialized quantum-specific verification strategies has become critically important. The study highlights simulation-based verification as a foundational technique, wherein the outcomes of quantum circuit simulations are systematically compared with results obtained from physical quantum hardware. This process enables the identification of functional inconsistencies and supports iterative refinement of circuit designs. In addition, randomized benchmarking is examined as a key evaluation method for estimating circuit fidelity and quantifying error rates through the application of random gate sequences and performance comparisons against established benchmarks. To further improve system robustness, the research investigates fault-tolerant mechanisms, particularly quantum error correction codes, which play a vital role in minimizing the effects of noise, decoherence and hardware imperfections. The integration of advanced quantum metrology and characterization techniques is also explored, enabling precise measurement and detailed analysis of quantum system behavior. These techniques provide deeper insights into circuit performance and contribute significantly to validation processes. By combining simulation-based verification, randomized benchmarking, fault-tolerant protocols and quantum metrology, this study proposes a comprehensive and reliable framework for quantum circuit verification and testing. The findings contribute to the advancement of dependable quantum technologies and support their growing applications in areas such as cryptography, materials science and other emerging domains.
Abstract Noise characterization methods such as randomized benchmarking (RB) are critical for the development of scalable quantum computers. Modern RB protocols for multiqubit systems extract physically relevant error rates by exploiting the structure of the group representation generated by the set of benchmarked operations. However, existing techniques become prohibitively inefficient for representations that are highly reducible yet decompose into irreducible subspaces of high dimension. These situations prevail when benchmarking high-dimensional systems such as qudits or bosonic modes, where experimental control is limited to implementing a small subset of all possible unitary operations. We introduce a broad framework for enhancing the sample efficiency of RB that is sufficiently powerful to extend the practical reach of RB beyond the multiqubit setting. Our strategy, which applies to any benchmarking group, uses ‘synthetic’ quantum circuits with classical post-processing of both input and output data to leverage the full structure of reducible superoperator representations. To demonstrate the efficacy of our approach, we develop a detailed theory of RB for systems with rotational symmetry. Such systems carry a natural action of the group SU ( 2 ) , and they form the basis for several novel quantum error-correcting codes. We show that, for experimentally accessible high-spin systems, synthetic RB protocols can reduce the complexity of measuring rotationally invariant error rates by two orders of magnitude relative to standard approaches such as character RB.