Contextuality and Wigner-function negativity in qubit quantum computation

We describe schemes of quantum computation with magic states on qubits for which contextuality and negativity of the Wigner function are necessary resources possessed by the magic states. These schemes satisfy a constraint. Namely, the non-negativity of Wigner functions must be preserved under all available measurement operations. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.

[1]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[2]  T. Fritz,et al.  A Combinatorial Approach to Nonlocality and Contextuality , 2012, Communications in Mathematical Physics.

[3]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[4]  Daniel Gottesman,et al.  Classicality in discrete Wigner functions , 2005, quant-ph/0506222.

[5]  Carlton M. Caves,et al.  Sufficient Conditions for Efficient Classical Simulation of Quantum Optics , 2015, 1511.06526.

[6]  V. Scarani,et al.  Nonlocality of cluster states of qubits , 2004, quant-ph/0405119.

[7]  H. Bombin,et al.  Topological subsystem codes , 2009, 0908.4246.

[8]  Matty J Hoban,et al.  Measurement-based classical computation. , 2013, Physical review letters.

[9]  Joel J. Wallman,et al.  Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities. , 2015, Physical review letters.

[10]  J. Emerson,et al.  Corrigendum: Negative quasi-probability as a resource for quantum computation , 2012, 1201.1256.

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

[12]  Stephen D. Bartlett,et al.  Non-negative subtheories and quasiprobability representations of qubits , 2012, 1203.2652.

[13]  D. Bacon.  Operator quantum error-correcting subsystems for self-correcting quantum memories , 2005, quant-ph/0506023.

[14]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[15]  N. Mermin.  Hidden variables and the two theorems of John Bell , 1993, 1802.10119.

[16]  R. Hudson.  When is the wigner quasi-probability density non-negative? , 1974 .

[17]  D. Poulin.  Stabilizer formalism for operator quantum error correction. , 2005, Physical review letters.

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

[19]  Keisuke Fujii,et al.  On the hardness of classically simulating the one clean qubit model , 2013, Physical review letters.

[20]  G. Tóth,et al.  Bell inequalities for graph states. , 2004, Physical review letters.

[21]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[22]  Lane A. Hemaspaandra,et al.  Threshold Computation and Cryptographic Security , 1993, ISAAC.

[23]  Sergey Bravyi.  Universal quantum computation with the v=5/2 fractional quantum Hall state , 2006 .

[24]  Jirí Vala,et al.  Quantum Contextuality with Stabilizer States , 2013, Entropy.

[25]  M. V. D. Nest,et al.  A monomial matrix formalism to describe quantum many-body states , 2011, 1108.0531.

[26]  J Eisert,et al.  Positive Wigner functions render classical simulation of quantum computation efficient. , 2012, Physical review letters.

[27]  David Poulin,et al.  Operator quantum error correction , 2006, Quantum Inf. Comput..

[28]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[29]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[30]  Huangjun Zhu.  Permutation Symmetry Determines the Discrete Wigner Function. , 2015, Physical review letters.

[31]  Todd A. Brun,et al.  Quantum Computing , 2011, Computer Science, The Hardware, Software and Heart of It.

[32]  G. Vidal,et al.  Classical simulation versus universality in measurement-based quantum computation , 2006, quant-ph/0608060.

[33]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[34]  Discrete phase space based on finite fields , 2004, quant-ph/0401155.

[35]  Samson Abramsky,et al.  The sheaf-theoretic structure of non-locality and contextuality , 2011, 1102.0264.

[36]  Mark Howard,et al.  Application of a Resource Theory for Magic States to Fault-Tolerant Quantum Computing. , 2016, Physical review letters.

[37]  J. Smolin,et al.  Trading Classical and Quantum Computational Resources , 2015, 1506.01396.

[38]  Robert W Spekkens,et al.  Negativity and contextuality are equivalent notions of nonclassicality. , 2006, Physical review letters.

[39]  Victor Veitch,et al.  Contextuality supplies the ‘magic’ for quantum computation , 2014, Nature.

[40]  A. Winter,et al.  Graph-theoretic approach to quantum correlations. , 2014, Physical review letters.

[41]  R. Jozsa,et al.  Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[42]  A. Peres.  Incompatible results of quantum measurements , 1990 .

[43]  E. Galvão.  Discrete Wigner functions and quantum computational speedup , 2004, quant-ph/0405070.

[44]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[45]  R. Raussendorf,et al.  Wigner Function Negativity and Contextuality in Quantum Computation on Rebits , 2014, 1409.5170.

[46]  Scott Aaronson,et al.  The computational complexity of linear optics , 2010, STOC '11.

[47]  David Gross,et al.  Computational power of quantum many-body states and some results on discrete phase spaces , 2008 .