Research On Average Fidelity Of A Noisy Quantum Channel Using Unitary Designs

Within the last three decades,quantum information and computation have made tremendous progress.Quantum communication based on quantum key distribution has moved from laboratory to engineering application.Quantum computation based on quantum state superposition can dramatically speed up certain calculations,such as number factoring.In addition,IBM,Intel,Google,Microsoft,Huawei,Alibaba,Baidu and Tencent have joined the team of quantum computing research.From the current research,the development into a fully featured large quantum computer faces a scalability challenge which comprises of integrating a large number of qubits and correcting quantum errors.To address these challenges,a variety of fault-tolerant computing architectures have been proposed.In these frameworks,the accurate estimation of errors in quantum processes is required.A typical method is that randomized benchmarking is a promising tool for characterizing the noise in experimental implementations by performing random unitary operations in quantum systems.This method is widely used to measure the noise of a quantum channel or an error rate of a set of quantum gates.This dissertation is based on the research of randomized benchmarking via unitary t-design.An ?-approximate unitary t-design,constructed by local random circuits,is used to estimate the average fidelity of a quantum noisy channel.Moreover,it can also be used to tailor the noise of a quantum circuit for lower fidelity gates to achieve fault-tolerance of the quantum computation,which is conducive to the realization of scalable fault-tolerant quantum computing.The main research results are as follows:1.A criterion is proposed for deciding if a set of unitary operators construct a unitary t-design.A unitary t-design is a set of unitary matrices that mimic the properties of the Haar measure for polynomials of degree up to t in the entire unitary group.By studying the properties of t-frames,the relationship between the unitary t-designs and the spherical t-designs is obtained,that is,both associate to the local minimizer of the corresponding tframe potential under different constraints.A set of unitary operators based on a unitary t-design can be used to twirl a noisy quantum channel to obtain the corresponding average fidelity of the noise.2.The construction of an ?-approximate unitary t-design is proposed by applying local random circuits.The length of local random circuits to construct ?-approximate unitary tdesign is obtained by giving a tighter lower bound of the spectral gap of the Hamiltonian about the local random circuits.The parameter in the local random quantum circuit can be better used to construct an ?-approximate unitary t-design with high confidence.However,the complexity of the quantum process in the experiment is increased due to the length of the local random circuits.Therefore,compared with the unitary designs chosen from Clifford group,the construction of local random circuits is suitable for the average fidelity estimation of many-body large-scale quantum noise channel with complexity construction.3.The average fidelity estimation of a cascade quantum noise channel is proposed by using unitary 2t-design.We extend the method to the average fidelity estimation of cascade quantum circuits.We analyze the average fidelity for tree noise characteristics: gateindependent and time-independent noise,gate-dependent and time-independent noise,and gate-independent and time-dependent noise.Since the unitary error has a great influence on the result of the average fidelity.We further analyze the unitary error caused by state preparation and measurement and give a lower bound on the unitary error.4.Noise tailoring for quantum circuits is proposed using unitary 2t-designs.The method randomizes the noise of the quantum gate in a quantum circuit.Given such tailored noise,quantum gates with lower fidelity are sufficient to achieve fault-tolerant quantum computation.Compared with the compiled quantum circuits based on easy and hard gates twirled by Pauli operators,our method does not need to recognize the specific structure of quantum circuits and is suitable for the tailoring noise of large-scale quantum circuits in the future.