Construction Of Two Quantum Stabilizer Codes And Related Fault-Tolerant Universal Quantum Gate Sets

Quantum error-correcting codes(QECCs)play a key role in reliable quantum communication and quantum computation,within which,binary standard stabilizer codes have been the hottest topic and are widely applied in almost all aspects in the field of quantum error correction(QEC).Recently,two significant extension of binary stabilizer codes – entanglement-assisted(EA)quantum stabilizer codes and quantum synchronizable codes,have developed rapidly and raised broad attention of scholars.In the quantum communication area,EA codes allow the sender to exploit the error-correcting properties of an arbitrary set of Pauli operators,not merely the abelian ones,with the help of pre-shared pairs of maximally entangled qubits.For the other,quantum synchronizable codes deal with both the best-studied Pauli error model,and misalignment with respect to the block structure of a qubit stream,which can,accordingly,relax the requirements on external mechanisms.In the quantum computation area,construction of fault-tolerant universal quantum gate sets on general stabilizer codes,with EA stabilizer codes and quantum synchronizable codes involved,is also worthy of research.Despite the prevalence of binary quantum codes,their supremacy is questionable.Since quantum systems of a higher dimension actually provide richer-structured state spaces,and there indeed exist some applications in which non-binary QECCs would be more useful,this thesis bases main results on non-binary fields,including the construction of two above-mentioned quantum stabilizer codes and related fault-tolerant universal quantum gate sets:(1)Given an arbitrary non-abelian stabilizer over a prime field,this thesis proposes a detailed algorithm to determine the encoding and decoding circuits and computes the optimal number of required maximally entangled pairs.Several bounds,such as the BCH bound,the Gilbert-Varshamov bound and the Linear Programming bound are also discussed.Over a non-prime finite field,this thesis proves that the former construction method,that is,to directly encode the canonical code with a unitary operation,applies only when the non-commuting stabilizer satisfies a sophisticated limitation.Several influencing factors of the limitation and certain cases when the unitaries don't exist have been provided.(2)With the help of classical non-binary repeated-root cyclic codes and cyclic product codes,this thesis enriches the variety of available quantum synchronizable codes and broadens the range of optional code lengths.The exact minimum distances of non-binary repeated-root cylic codes are computed in this thesis.And compared with those from non-binary BCH codes,the constructed quantum synchronizable codes are shown to possess good error-correcting capabilities towards Pauli errors.Furthermore,the thesis proves that quanum synchronizable codes that are produced by these two classical cyclic codes,with well-chosen code parameters,can obtain the best attainable tolerance towards misalignment.(3)This thesis proves that if n?2,then an arbitrary non-binary n-qudit Clifford group can be constructed by a combination of ADD gates and unitary operators from a non-binary single-qudit Clifford group.Based on this,this thesis presents a non-binary universal quantum gate set,which is generated by a two-qudit Clifford group and the HORNER gate.Combined with some fault-tolerant measurements,this thesis proposes a fault-tolerant construction of this universal gate set on general stabilizer codes over prime fields.The infeasibility of generalizing the above construction process to non-prime finite fields is also illustrated in this thesis.