Geometric phases in quantum information

The rise of quantum information science has opened up a new venue for applications of the geometric phase (GP), as well as triggered new insights into its physical, mathematical, and conceptual nat ...

[1]  Gabriele De Chiara,et al.  Berry phase for a spin 1/2 particle in a classical fluctuating field. , 2003, Physical review letters.

[2]  Ognyan Oreshkov Holonomic quantum computation in subsystems. , 2009, Physical review letters.

[3]  H. C. Longuet-Higgins,et al.  Studies of the Jahn-Teller effect .II. The dynamical problem , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  Vlatko Vedral,et al.  Anandan et al. Reply , 2002 .

[5]  S. Berger,et al.  Experimental realization of non-Abelian non-adiabatic geometric gates , 2013, Nature.

[6]  Tomita,et al.  Observation of Berry's topological phase by use of an optical fiber. , 1986, Physical review letters.

[7]  E. Sjoqvist,et al.  Hidden parameters in open-system evolution unveiled by geometric phase , 2010, 1005.2992.

[8]  C. Souza,et al.  Topological phase structure of vector vortex beams. , 2013, Journal of the Optical Society of America. A, Optics, image science, and vision.

[9]  S. Berger,et al.  Exploring the effect of noise on the Berry phase , 2013, 1302.3305.

[10]  A. Pati,et al.  Geometric aspects of noncyclic quantum evolutions. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[11]  Klaas Bergmann,et al.  LASER-DRIVEN POPULATION TRANSFER IN FOUR-LEVEL ATOMS : CONSEQUENCES OF NON-ABELIAN GEOMETRICAL ADIABATIC PHASE FACTORS , 1999 .

[12]  Fausto Rossi,et al.  Semiconductor-based geometrical quantum gates , 2002, quant-ph/0207019.

[13]  Vahid Azimi Mousolou,et al.  Universal non-adiabatic holonomic gates in quantum dots and single-molecule magnets , 2012, 1209.3645.

[14]  D. M. Tong,et al.  Robustness of nonadiabatic holonomic gates , 2012, 1204.5144.

[15]  Noncyclic geometric changes of quantum states , 2005, quant-ph/0512045.

[16]  J. Pekola,et al.  Non-Abelian geometric phases in ground-state Josephson devices , 2010, 1002.0957.

[17]  K. Mølmer,et al.  QUANTUM COMPUTATION WITH IONS IN THERMAL MOTION , 1998, quant-ph/9810039.

[18]  E. Sjoqvist,et al.  Noncyclic mixed state phase in SU(2) polarimetry , 2003, quant-ph/0305132.

[19]  D. Suter,et al.  Experimental observation of a topological phase in the maximally entangled state of a pair of qubits , 2007, 0705.3566.

[20]  D. M. Tong,et al.  Geometric phases for nondegenerate and degenerate mixed states , 2003, quant-ph/0304068.

[21]  A. Khoury,et al.  Non Abelian structures and the geometric phase of entangled qudits , 2014, 1406.4209.

[22]  R. Laflamme,et al.  Geometric phase with nonunitary evolution in the presence of a quantum critical bath. , 2010, Physical review letters.

[23]  W. Wootters,et al.  Distributed Entanglement , 1999, quant-ph/9907047.

[24]  Topological phase for entangled two-qubit states and the representation of the SO(3)group , 2005, quant-ph/0511034.

[25]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[26]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[27]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[28]  D. Arovas,et al.  Topological indices for open and thermal systems via Uhlmann's phase. , 2014, Physical review letters.

[29]  S. Urabe,et al.  Realization of holonomic single-qubit operations , 2013, 1304.6215.

[30]  S. Pancharatnam,et al.  Generalized theory of interference, and its applications , 1956 .

[31]  A. Khoury,et al.  Fractional topological phase for entangled qudits. , 2010, Physical review letters.

[32]  Frank Wilczek,et al.  Appearance of Gauge Structure in Simple Dynamical Systems , 1984 .

[33]  Jeeva Anandan,et al.  Non-adiabatic non-abelian geometric phase , 1988 .

[34]  D. Leibfried,et al.  Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate , 2003, Nature.

[35]  Aharonov,et al.  Phase change during a cyclic quantum evolution. , 1987, Physical review letters.

[36]  A. Khoury,et al.  Fractional topological phase on spatially encoded photonic qudits , 2013, 1301.5539.

[37]  A Fault-Tolerant Scheme of Holonomic Quantum Computation on Stabilizer Codes with Robustness to Low-weight Thermal Noise , 2013, 1309.1534.

[38]  Erik Sjöqvist,et al.  A new phase in quantum computation , 2008 .

[39]  P. Zanardi,et al.  Quantum computation in noiseless subsystems with fast non-Abelian holonomies , 2013, 1308.1919.

[40]  G. Castagnoli,et al.  Geometric quantum computation with NMR , 1999, quant-ph/9910052.

[41]  N. J. Cerf,et al.  Multipartite nonlocality without entanglement in many dimensions , 2006 .

[42]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[43]  Á. Rivas,et al.  Two-dimensional density-matrix topological fermionic phases: topological Uhlmann numbers. , 2014, Physical review letters.

[44]  C. Mead The molecular Aharonov—Bohm effect in bound states , 1980 .

[45]  E. Sjöqvist Geometric phase for entangled spin pairs , 2000 .

[46]  E. Sjoqvist,et al.  Validity of the rotating-wave approximation in nonadiabatic holonomic quantum computation , 2013, 1307.1536.

[47]  M. Berry Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[48]  Samuel,et al.  General setting for Berry's phase. , 1988, Physical review letters.

[49]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[50]  F. Wilczek,et al.  Geometry of self-propulsion at low Reynolds number , 1989, Journal of Fluid Mechanics.

[51]  Mixed state geometric phases, entangled systems, and local unitary transformations. , 2002, Physical review letters.

[52]  Barry Simon,et al.  Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase , 1983 .

[53]  Dieter Suter,et al.  Berry's phase in magnetic resonance , 1987 .

[54]  Geometric phases in open systems: A model to study how they are corrected by decoherence (5 pages) , 2006, quant-ph/0606036.

[55]  P. Zanardi,et al.  Noiseless Quantum Codes , 1997, quant-ph/9705044.

[56]  Geometric phase in open systems. , 2003, Physical review letters.

[57]  C. Zu,et al.  Experimental realization of universal geometric quantum gates with solid-state spins , 2014, Nature.

[58]  H. C. Longuet-Higgins The intersection of potential energy surfaces in polyatomic molecules , 1975, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[59]  J. C. Loredo,et al.  Observation of entanglement-dependent two-particle holonomic phase. , 2013, Physical review letters.

[60]  Jens Siewert,et al.  Non-Abelian holonomies, charge pumping, and quantum computation with Josephson junctions. , 2003, Physical review letters.

[61]  I Fuentes-Guridi,et al.  Holonomic quantum computation in the presence of decoherence. , 2005, Physical review letters.

[62]  Observation of geometric phases for mixed states using NMR interferometry. , 2003, Physical review letters.

[63]  P Geltenbort,et al.  Experimental demonstration of the stability of Berry's phase for a spin-1/2 particle. , 2008, Physical review letters.

[64]  P'erola Milman Phase dynamics of entangled qubits , 2006 .

[65]  Peter Hänggi,et al.  Geometric phase as a determinant of a qubit– environment coupling , 2010, Quantum Inf. Process..

[66]  Apoorva G. Wagh,et al.  On measuring the Pancharatnam phase. I. Interferometry , 1995 .

[67]  V Vedral,et al.  Geometric phases for mixed states in interferometry. , 2000, Physical review letters.

[68]  Noncyclic Pancharatnam phase for mixed state SU(2) evolution in neutron polarimetry , 2005, quant-ph/0505209.

[69]  M. Nakahara,et al.  Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies , 2013, 1307.8001.

[70]  A. Khoury,et al.  Topological phase structure of entangled qudits , 2014, 1401.3536.

[71]  Vlatko Vedral,et al.  Composite geometric phase for multipartite entangled states , 2007, quant-ph/0702080.

[72]  Daniel A Lidar,et al.  Fault-tolerant holonomic quantum computation. , 2009, Physical review letters.

[73]  D A Lidar,et al.  Holonomic quantum computation in decoherence-free subspaces. , 2005, Physical review letters.

[74]  On polynomial invariants of several qubits , 2008, 0804.1661.

[75]  J. Luque,et al.  Polynomial invariants of four qubits , 2002, quant-ph/0212069.

[76]  P. Goldbart,et al.  Geometric measure of entanglement and applications to bipartite and multipartite quantum states , 2003, quant-ph/0307219.

[77]  A. Uhlmann A gauge field governing parallel transport along mixed states , 1991 .

[78]  Guilu Long,et al.  Protecting geometric gates by dynamical decoupling , 2014 .

[79]  Johan Åberg,et al.  Adiabatic approximation for weakly open systems , 2005 .

[80]  General setting for a geometric phase of mixed states under an arbitrary nonunitary evolution , 2005, quant-ph/0507280.

[81]  R. Srikanth,et al.  Geometric phase of a qubit interacting with a squeezed-thermal bath , 2006, quant-ph/0611161.

[82]  P. Milman,et al.  Topological phase for spin-orbit transformations on a laser beam , 2007, 2008 Conference on Lasers and Electro-Optics and 2008 Conference on Quantum Electronics and Laser Science.

[83]  Abelian and non-Abelian geometric phases in adiabatic open quantum systems , 2005, quant-ph/0507012.

[84]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[85]  T. Bitter,et al.  Manifestation of Berry's topological phase in neutron spin rotation. , 1987, Physical review letters.

[86]  A. Harrow,et al.  Practical scheme for quantum computation with any two-qubit entangling gate. , 2002, Physical Review Letters.

[87]  John Calsamiglia,et al.  Adiabatic markovian dynamics. , 2010, Physical review letters.

[88]  Herbert Walther,et al.  Quantum computation with trapped ions in an optical cavity. , 2002, Physical review letters.

[89]  R. J. Schoelkopf,et al.  Observation of Berry's Phase in a Solid-State Qubit , 2007, Science.

[90]  G. Marmo,et al.  Geometric phase for mixed states: a differential geometric approach , 2004 .

[91]  E. Knill Quantum computing with realistically noisy devices , 2005, Nature.

[92]  E. Sjöqvist On geometric phases for quantum trajectories , 2006 .

[93]  Effect of noise on geometric logic gates for quantum computation , 2001, quant-ph/0105006.

[94]  Geometrical phases for the G(4,2) Grassmannian manifold , 2003, quant-ph/0301140.

[95]  Erik Sjöqvist,et al.  Nonadiabatic holonomic quantum computation in decoherence-free subspaces. , 2012, Physical review letters.

[96]  I. B. Spielman,et al.  Synthetic magnetic fields for ultracold neutral atoms , 2009, Nature.

[97]  W. Wootters Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

[98]  B. Sanders,et al.  Geometric phase distributions for open quantum systems. , 2004, Physical review letters.

[99]  J. Renes,et al.  Holonomic quantum computing in symmetry-protected ground states of spin chains , 2011, 1103.5076.

[100]  Measurement of geometric phase for mixed states using single photon interferometry. , 2004, quant-ph/0412216.

[101]  B. D'espagnat Conceptual Foundations Of Quantum Mechanics , 1971 .

[102]  Berry phase in entangled systems: a proposed experiment with single neutrons , 2003, quant-ph/0309089.

[103]  Operational approach to the Uhlmann holonomy , 2006, quant-ph/0608185.

[104]  V. C. Rakhecha,et al.  On measuring the Pancharatnam phase. II. SU(2) polarimetry , 1995 .

[105]  W. J. Munro,et al.  Decoherence of geometric phase gates , 2002 .

[106]  Paolo Zanardi,et al.  QUANTUM HOLONOMIES FOR QUANTUM COMPUTING , 2000, quant-ph/0007110.

[107]  A. Shimony Degree of Entanglement a , 1995 .

[108]  Daniel A. Lidar,et al.  Decoherence-Free Subspaces for Quantum Computation , 1998, quant-ph/9807004.

[109]  Vlatko Vedral GEOMETRIC PHASES AND TOPOLOGICAL QUANTUM COMPUTATION , 2002 .

[110]  Matthias Hübner,et al.  Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space , 1993 .

[111]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[112]  R. Simon,et al.  Quantum Kinematic Approach to the Geometric Phase. I. General Formalism , 1993 .

[113]  Robustness of optimal working points for nonadiabatic holonomic quantum computation , 2006, quant-ph/0604180.

[114]  A. Uhlmann The "transition probability" in the state space of a ∗-algebra , 1976 .

[115]  E. Sjöqvist Entanglement-induced geometric phase of quantum states , 2010 .

[116]  Shi-Liang Zhu,et al.  Geometric quantum gates that are robust against stochastic control errors , 2005 .

[117]  A. Khoury,et al.  Three-qubit topological phase on entangled photon pairs , 2013, 1301.5538.

[118]  Z. D. Wang,et al.  Experimental implementation of high-fidelity unconventional geometric quantum gates using an NMR interferometer , 2006 .

[119]  L. Kwek,et al.  Geometric phase for entangled states of two spin-1/2 particles in rotating magnetic field , 2003 .

[120]  Guilu Long,et al.  Universal Nonadiabatic Geometric Gates in Two-Qubit Decoherence-Free Subspaces , 2014, Scientific reports.

[121]  P. Lévay The geometry of entanglement: metrics, connections and the geometric phase , 2003, quant-ph/0306115.

[122]  E. Knill,et al.  DYNAMICAL DECOUPLING OF OPEN QUANTUM SYSTEMS , 1998, quant-ph/9809071.

[123]  E. Sjöqvist,et al.  Correlation-induced non-Abelian quantum holonomies , 2010, 1011.5182.

[124]  Armin Uhlmann,et al.  Parallel transport and “quantum holonomy” along density operators , 1986 .

[125]  Experimental demonstration of a unified framework for mixed-state geometric phases , 2011 .

[126]  Robust gates for holonomic quantum computation , 2005, quant-ph/0510226.

[127]  Exact solutions of holonomic quantum computation , 2003, quant-ph/0312079.

[128]  L. C. Kwek,et al.  Relation between geometric phases of entangled bipartite systems and their subsystems , 2003, quant-ph/0309130.

[129]  Á. Rivas,et al.  Uhlmann phase as a topological measure for one-dimensional fermion systems. , 2013, Physical review letters.

[130]  Holonomic quantum computation with neutral atoms , 2002, quant-ph/0204030.

[131]  M. C. Nemes,et al.  Phases of quantum states in completely positive non-unitary evolution , 2003 .

[132]  Paolo Zanardi,et al.  Holonomic quantum computation , 1999 .

[133]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[134]  Kang Xue,et al.  Braiding transformation, entanglement swapping, and Berry phase in entanglement space , 2007, 0704.0709.

[135]  R. Mosseri,et al.  Geometry of entangled states, Bloch spheres and Hopf fibrations , 2001, quant-ph/0108137.

[136]  Guilu Long,et al.  Experimental realization of nonadiabatic holonomic quantum computation. , 2013, Physical review letters.

[137]  Robustness against parametric noise of nonideal holonomic gates , 2006, quant-ph/0611079.

[138]  W. Demtröder,et al.  Unambiguous Proof for Berry's Phase in the Sodium Trimer: Analysis of the Transition A 2 E ′ ′ ← X 2 E ′ , 1998 .

[139]  J I Cirac,et al.  Geometric Manipulation of Trapped Ions for Quantum Computation , 2001, Science.

[140]  Geometric manipulation of the quantum states of two-level atoms , 2004 .

[141]  D. M. Tong,et al.  Non-adiabatic holonomic quantum computation , 2011, 1107.5127.

[142]  Mikio Nakahara,et al.  Realization of arbitrary gates in holonomic quantum computation , 2003 .

[143]  A symmetry approach to geometric phase for quantum ensembles , 2014, 1411.0635.

[144]  A P Chikkatur,et al.  Direct nondestructive imaging of magnetization in a spin-1 Bose-Einstein gas. , 2005, Physical review letters.

[145]  S. Filipp,et al.  Geometric phase in entangled systems: A single-neutron interferometer experiment , 2009, 0907.4909.

[146]  Frank Wilczek,et al.  Gauge kinematics of deformable bodies , 1989 .

[147]  Paul B. Slater,et al.  Mixed State Holonomies , 2001, math-ph/0111041.

[148]  D. Bohm,et al.  Significance of Electromagnetic Potentials in the Quantum Theory , 1959 .

[149]  A. Sudbery,et al.  Global asymmetry of many-qubit correlations: A lattice-gauge-theory approach , 2011, 1102.5609.

[150]  C. H. Oh,et al.  Kinematic approach to the mixed state geometric phase in nonunitary evolution. , 2004, Physical review letters.

[151]  Rajendra Bhandari Singularities of the mixed state phase. , 2002, Physical review letters.

[152]  Shi-Liang Zhu,et al.  Implementation of universal quantum gates based on nonadiabatic geometric phases. , 2002, Physical review letters.

[153]  P. Kim,et al.  Experimental observation of the quantum Hall effect and Berry's phase in graphene , 2005, Nature.

[154]  E. Sjöqvist Experimentally testable geometric phase of sequences of Everett's relative quantum states , 2009 .

[155]  Shi-Liang Zhu,et al.  Unconventional geometric quantum computation. , 2003, Physical Review Letters.

[156]  W. LiMing,et al.  Representation of the SO(3) group by a maximally entangled state (4 pages) , 2004 .

[157]  Klaus Molmer,et al.  Geometric phase gates based on stimulated Raman adiabatic passage in tripod systems , 2007 .

[158]  N. Buric,et al.  Uniquely defined geometric phase of an open system , 2009 .

[159]  Pérola Milman,et al.  Topological phase for entangled two-qubit states. , 2003, Physical review letters.

[160]  Arun K. Pati,et al.  Generalization of the geometric phase to completely positive maps , 2003 .

[161]  Parallel transport in an entangled ring , 2002, quant-ph/0202048.

[162]  V. Malinovsky,et al.  Adiabatic holonomic quantum gates for a single qubit , 2014 .

[163]  Vlatko Vedral,et al.  Geometric quantum computation , 2000, quant-ph/0004015.

[164]  Experimental measurement of mixed state geometric phase by quantum interferometry using NMR , 2005, quant-ph/0509139.

[165]  V. C. Rakhecha,et al.  Neutron Interferometric Observation of Noncyclic Phase , 1998 .

[166]  Geometric phase for open quantum systems and stochastic unravelings , 2005, quant-ph/0510184.

[167]  Shi-Liang Zhu,et al.  Universal quantum gates based on a pair of orthogonal cyclic states: Application to NMR systems , 2002, quant-ph/0210027.

[168]  Viola,et al.  Theory of quantum error correction for general noise , 2000, Physical review letters.

[169]  H. C. Longuet-Higgins,et al.  Intersection of potential energy surfaces in polyatomic molecules , 1963 .

[170]  Jonathan A. Jones,et al.  Geometric quantum computation using nuclear magnetic resonance , 2000, Nature.

[171]  S. Filipp,et al.  Observation of nonadditive mixed-state phases with polarized neutrons. , 2008, Physical review letters.

[172]  Arun Kumar Pati,et al.  Gauge-invariant reference section and geometric phase , 1995 .

[173]  Daniel A. Lidar,et al.  Scheme for fault-tolerant holonomic computation on stabilizer codes , 2009 .

[174]  Stefan W. Hell,et al.  Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin , 2014, Nature Communications.

[175]  L. C. Kwek,et al.  Geometric phase in open systems: Beyond the Markov approximation and weak coupling limit , 2006 .

[176]  G. Lindblad On the generators of quantum dynamical semigroups , 1976 .

[177]  Paolo Zanardi,et al.  Non-Abelian Berry connections for quantum computation , 1999 .

[178]  T. Brun,et al.  Fault-tolerant holonomic quantum computation in surface codes , 2014, 1411.4248.

[179]  A. Stone,et al.  Spin-orbit coupling and the intersection of potential energy surfaces in polyatomic molecules , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.