Research problems on numerical ranges in quantum computing

References

• Bengtsson , I and Życzkowski , K . 2006 . Geometry of Quantum States: An Introduction to Quantum Entanglement , Cambridge, , UK : Cambridge University Press .
   Google Scholar
• Blum , K . 1996 . Density Matrix Theory and Applications , New York : Springer .
   Google Scholar
• Choi , MD , Kribs , D and Życzkowski , K . 2006 . Higher-rank numerical ranges and compression problems . Lin. Alg. Appl. , 418 : 828 – 839 .
   Web of Science ®Google Scholar
• Choi , MD , Kribs , DW and Życzkowski , K . 2006 . Quantum error correcting codes from the compression formalism . Rep. Math. Phys. , 58 : 77 – 91 .
   Web of Science ®Google Scholar
• Choi , MD , Holbrook , J , Kribs , D and Życzkowski , K . 2007 . Higher-rank numerical ranges of unitary and normal matrices . Oper. Matrices , 1 : 409 – 426 .
   Web of Science ®Google Scholar
• Dirr , G , Helmke , U , Kleinsteuber , M and Schulte-Herbruggen , T . 2008 . Relative C-numerical ranges for applications in quantum control and quantum information . Lin. Multilin. Alg. , 56 : 27 – 51 .
   Web of Science ®Google Scholar
• Duan , LM and Guo , GC . 1997 . Preserving coherence in quantum computation by pairing quantum bits . Phys. Rev. Lett. , 79 : 1953 – 1956 .
   Web of Science ®Google Scholar
• Fan , MKH and Tits , AL . 1988 . m-Form numerical range and the computation of the structured singular value . IEEE Trans. Automat. Contr. , 33 ( 3 ) : 284 – 289 .
   Web of Science ®Google Scholar
• Gilchrist , A , Langford , NK and Nielsen , MA . 2005 . Distance measures to compare real and ideal quantum processes . Phys. Rev. A , 71 : 062310 – 062324 .
   Web of Science ®Google Scholar
• Gottesman , D . 1996 . Class of quantum error-correcting codes saturating the quantum Hamming bound . Phys. Rev. A , 54 : 1862 – 1883 .
   PubMed Web of Science ®Google Scholar
• Knill , E and Laflamme , R . 1997 . Theory of quantum error-correcting codes . Phys. Rev. A , 55 : 900 – 905 .
   Web of Science ®Google Scholar
• Knill , E , Laflamme , R and Viola , L . 2000 . Theory of quantum error correction for general noise . Phys. Rev. Lett. , 84 : 2525 – 2528 .
   PubMed Web of Science ®Google Scholar
• Li , CK and Poon , YT . 1999 . Convexity of the joint numerical range . SIAM J. Matrix Anal. Appl. , 21 : 668 – 678 .
   Web of Science ®Google Scholar
• Li , CK and Sze , NS . 2008 . Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations . Proc. Amer. Math. Soc. , 136 : 3013 – 3023 .
   Web of Science ®Google Scholar
• Lidar , D , Chuang , I and Whaley , K . 1998 . Decoherence-free subspaces for quantum computation . Phys. Rev. Lett. , 81 : 2594 – 2597 .
   Web of Science ®Google Scholar
• Nebendahl , V , Haffner , H and Roos , CF . Optimal control of entangling operations for trapped ion quantum computing . preprint. Available at arXiv:0809.1414 (2008)
   Google Scholar
• Nielsen , MA and Chuang , IL . 2000 . Quantum Computation and Quantum Information , New York : Cambridge University Press .
   Google Scholar
• Poon , YT . 1994 . On the convex hull of the multiform numerical range . Lin. Multilin. Alg. , 37 : 221 – 223 .
   Google Scholar
• Ryan , CA , Negrevergne , C , Laforest , M , Knill , E and Laflamme , R . 2008 . Liquid-state nuclear magnetic resonance as a testbed for developing quantum control methods . Phys. Rev. A , 78 : 012328 – 012342 .
   Web of Science ®Google Scholar
• Schulte-Herbruggen , T , Sporl , A , Khaneja , N and Glaser , SJ . 2005 . Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity . Phys. Rev. A. , 72 : 042331 – 042337 .
   Web of Science ®Google Scholar
• Schulte-Herbruggen , T , Sporl , A , Khaneja , N and Glaser , SJ . Optimal control for generating quantum gates in open dissipative systems . preprint. Available at arXiv:quant-ph/0609037v1 (2006)
   Google Scholar
• Sporl , A , Schulte-Herbruggen , T , Glaser , SJ , Bergholm , V , Storcz , MJ , Ferber , J and Wilhelm , FK . 2007 . Optimal control of coupled Josephson qubits . Phys. Rev. A , 75 : 012302 – 012309 .
   Web of Science ®Google Scholar
• Stinespring , W . 1955 . Positive functions on C*-algebras . Proc. Amer. Math. Soc. , 6 : 211 – 216 .
   Google Scholar
• Woerdeman , HJ . 2008 . The higher rank numerical range is convex . Lin. Multilin. Alg. , 56 : 65 – 67 .
   Web of Science ®Google Scholar
• Zanardi , P . 2001 . Stabilizing quantum information . Phys. Rev. A , 63 : 12301 – 12305 .
   Web of Science ®Google Scholar
• Zanardi , P and Rasetti , M . 1997 . Noiseless quantum codes . Phys. Rev. Lett. , 79 : 3306 – 3309 .
   Web of Science ®Google Scholar