Quantum error correction will be integral in developing full scale quantum computers, but as of yet beneficial quantum error correction has not been demonstrated experimentally. An important question is therefore what prerequisites need to be met to achieve this. Results of density matrix simulations of the performance of the seven qubit Steane code in a quantum computing setting are presented. The full density matrix was kept throughout the whole simulations, which means that all errors can be accounted for. In particular, the importance of the circuit depth, i.e. the number of gates in series before error correction is applied, for the overall performance was investigated. It was found that the depth of the circuit has a large impact on the threshold error rate for which error correction becomes beneficial. A gain parameter was defined, which describes the largest constant factor by which errors can be suppressed. It was shown that there is a trade-off between the threshold error and the gain; The highest threshold value was found to be around p_th~10^-4, which is in line with other estimates, but the maximum gain for this value was only 3. To achieve a gain of 100, an error rate of p_err~2x10^-9 is required. In addition, performance statistics such as run-time as a function of circuit depth and width for the matlab code used for simulations are presented.
Quantum error correction will be integral in developing full scale quantum computers, but as of yet beneficial quantum error correction has not been demonstrated experimentally. An important question is therefore what prerequisites need to be met to achieve this. Results of density matrix simulations of the performance of the seven qubit Steane code in a quantum computing setting are presented. The full density matrix was kept throughout the whole simulations, which means that all errors can be accounted for. In particular, the importance of the circuit depth, i.e. the number of gates in series before error correction is applied, for the overall performance was investigated. It was found that the depth of the circuit has a large impact on the threshold error rate for which error correction becomes beneficial. A gain parameter was defined, which describes the largest constant factor by which errors can be suppressed. It was shown that there is a trade-off between the threshold error and the gain; The highest threshold value was found to be around p_th10^-4, which is in line with other estimates, but the maximum gain for this value was only 3. To achieve a gain of 100, an error rate of p_err2x10^-9 is required. In addition, performance statistics such as run-time as a function of circuit depth and width for the matlab code used for simulations are presented.
Quantum computers hold the promise to solve some hard problems that are intractable for even the most powerful current supercomputers. One of the most famous examples is Shor's algorithm for factorizing large numbers, which has exponential speedup compared to its best classical counterparts. However, running such an algorithm will require to build a large-scale quantum computer consisting of thousands or even millions of qubits that include quantum error correction (QEC) and fault-tolerant (FT) mechanisms. Quantum computing is already a reality with the so-called Noisy Intermediate-Scale Quantum (NISQ) processors, some of them available in the cloud. Noisy refers to the imperfect control over the qubits and intermediate-scale to the relatively low number of quits (from fifty to a few hundred). Although current and near-term quantum devices will not have enough qubits for implementing large and fully corrected quantum computations, the use of small quantum error correction codes may extend the computation lifetime of NISQ devices. In this context and as a first step, it is important to test and demonstrate the fault-tolerance of these QEC codes. In this thesis, we explore the fault-tolerance of two small quantum correction codes that are good candidates to be applied to NISQ processors, the [[4,2,2]] code and the [[7,1,3]] Steane code. To this purpose, by following the FT criterion prosed by Daniel Gottesman in 2016, we tested both codes using two simulators, the stabilizer formalism simulator that includes quite simple error models and a full density matrix simulator called quantumsim, which includes more realistic noise. The simulations are performed under reasonable noise parameter values. For the [[4,2,2]] code, 235 circuits are tested based on the two simulators. The results show that in the stabilizer formalism simulation, the FT criterion is satised for all circuits, while not fully satisfied in the full density matrix simulation. For the [[7,1,3]] Steane code, we use a parallel-flag error correction implementation which is tested using the full density matrix simulator. Our results show that without applying any QEC cycle, for all circuits (84 circuits for 1 logical qubit simulation and 452 circuits for 2 logical qubits simulation), the error rate of the encoded circuits is lower than the unencoded ones. Adding a quantum error correction (QEC) cycle will in general increase the error rate of the computation.
One of the most important challenges in constructing large-scale quantum computers is handling the high error rates in quantum systems. Using error correction codes and frequently applying error correction on qubits is a common approach to compete with this challenge. Quantum error analysis techniques are required to compare the effectiveness of error correction codes and also to efficiently insert the error correction procedures (ECPs) in a quantum circuit. However, most existing assessment approaches are either inaccurate and inefficient or are slow and time-consuming. To address these problems, in this paper, a fast analytical approach is proposed that analyzes errors and smartly inserts ECPs into a quantum circuit and finally estimates the failure probability of the circuit. The approach is also able to estimate the pseudo-threshold of quantum error correction codes. Experimental results show that the proposed method improves the number of ECPs and the latency of the circuit encoded by the Steane’s [[7, 1, 3]] code on average by about 52.6% and 29.4% for the one-level one and by about 79.4% and 48.5% for the two-level concatenated code respectively compared with a related method in the literature. Moreover, the proposed method takes only a few seconds to run on even our relatively large benchmark circuits.
Abstract Fault-tolerant quantum computing requires many qubits with long lifetimes and accurate quantum gate operations. However, external noise limits the computing time and hampers accurate quantum gate operations. Quantum error correction (QEC) codes may extend such limits, but imperfect gate operations during QEC cause errors, which could cancel out QEC. We used density matrix simulations to examine the performance of QEC codes with five qubits. In current quantum devices, less than ten qubits are needed to conduct sufficient gate operations within their lifetime so that it is feasible to implement QEC codes. We analyzed the maximum tolerable error rate and error correction effect of individual QEC codes according to the qubit arrangement and gate accuracy. Assuming a 0.1% gate error probability, a logical ∣ 1 〉 state encoded by a five-qubit QEC code is expected to have a 0.25 higher fidelity than its physical counterpart.
Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweeps like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.
Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweep like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.
The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.