The quality of quantum information processing devices has been improving at an unprecedented speed. How to faithfully represent the quality of these devices has become an increasingly imminent problem. In this thesis we focus on two aspects in representing and characterizing quantum devices. First, we discuss why most conventional quality metrics are not in principle appropriate to quantify experimentally-determined representations of gate-set elements, due to a gauge degree of freedom in quantum experiments. We then propose an operational quality measure for a gate-set and discuss its usefulness in representing degree of errors and improving experimental control. Second, we develop a protocol that separately and unambiguously characterizes state and measurement errors, relying on high-quality quantum gates. By integrating a method called randomized compiling, we derive a favorable upper bound for the effects of gate errors on the estimated parameters, and numerically demonstrate its performance in the presence of an adversarial gate error.
The quality of quantum information processing devices has been improving at an unprecedented speed. How to faithfully represent the quality of these devices has become an increasingly imminent problem. In this thesis we focus on two aspects in representing and characterizing quantum devices. First, we discuss why most conventional quality metrics are not in principle appropriate to quantify experimentally-determined representations of gate-set elements, due to a gauge degree of freedom in quantum experiments. We then propose an operational quality measure for a gate-set and discuss its usefulness in representing degree of errors and improving experimental control. Second, we develop a protocol that separately and unambiguously characterizes state and measurement errors, relying on high-quality quantum gates. By integrating a method called randomized compiling, we derive a favorable upper bound for the effects of gate errors on the estimated parameters, and numerically demonstrate its performance in the presence of an adversarial gate error.
We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, and a protocol for measuring this figure.
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.
Correctly characterizing state preparation and measurement (SPAM) processes is a necessary step towards building reliable quantum processing units (QPUs). In this work, we discuss the subtleties behind separately measuring SPAM errors. We propose a protocol that can separately estimate SPAM errors, in the case where quantum gates are ideal. In the case where the quantum gates are imperfect, we derive bounds on the estimated SPAM error rates, based on gate error measures, which can be estimated independently of SPAM processes. Our method shows that the gauge ambiguity in characterizing SPAM operations can be resolved by assuming that there exists one qubit whose initial state is uncorrelated with other qubits in a QPU. We test the protocol on a publicly available five-qubit QPU and demonstrate its validity by comparing our results with simulations.
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.
In this thesis, we are concerned with the problem of characterizing noise associated with implementations of quantum circuits. We first explore the notion of error rates of quantum circuits and argue that the semantic distinction between process fidelity as “average error rate” and diamond distance as “worst-case error rate” is wrong. To this end, we analyze their worst-case and average analogies, and show they are proportional to their original measures. We show that the non-unital contributions to the diamond distance are negligible and the discrepancy between the process fidelity and diamond distance is primarily due to unitary errors. \n \nWe then turn to a new analysis of cycle benchmarking, a randomized benchmarking-like protocol to estimate the process fidelity of a cycle as engineered by randomized compiling. Using this approach, we prove that gate-dependent noise on the randomized gates is de- scribed by a single perturbation term that decays rapidly as long as the implementation is close to a representation. We also comment on how the protocol can be extended to qudits, what the cycle benchmarking decay actually measures, and how our analysis is amenable to the Fourier analysis of randomized benchmarking. We end with a discussion of how the gate-dependent cycle benchmarking process fidelity relates to gate-dependent randomized compiling.