A quantum gravity computer is one for which the particular effects of quantum gravity are relevant. In general relativity, causal structure is non-fixed. In quantum theory non-fixed quantities are subject to quantum uncertainty. It is therefore likely that, in a theory of quantum gravity, we will have indefinite causal structure. This means that there will be no matter of fact as to whether a particular interval is time-like or not. We study the implications of this for the theory of computation. Classical and quantum computations consist in evolving the state of the computer through a sequence of time steps. This will, most likely, not be possible for a quantum gravity computer because the notion of a time step makes no sense if we have indefinite causal structure. We show that it is possible to set up a model for computation even in the absence of definite causal structure by using a certain framework (the causaloid formalism) that was developed for the purpose of correlating data taken in this type of situation. Corresponding to a physical theory is a causaloid, Λ (this is a mathematical object containing information about the causal connections between different spacetime regions). A computer is given by the pair {Λ,S} where S is a set of gates. Working within the causaloid formalism, we explore the question of whether universal quantum gravity computers are possible.We also examine whether a quantum gravity computer might be more powerful than a quantum (or classical) computer. In particular, we ask whether indefinite causal structure can be used as a computational resource.
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References
A. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proc. of the London Math. Soc., Series 2, Vol. 42 (1936) 230–265.
D. Deutsch, “Quantum Theory, the Church—Turing Principle and the Universal Quantum Computer”, Proc. of the Roy. Soc. of London, Series A, Math. Phys. Sci., Vol. 400, No. 1818, pp. 97–117 (1985).
L. Hardy, “Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity”, gr-qc/0509120 (2005).
A. Church, “An Unsolvable Problem of Elementary Number Theory”, Am. J. Math., 58, 345– 363 (1936).
C. Soans and A. Stevenson (eds.) Concise Oxford English Dictionary, 11th edition (OUP, 2004) Oxford.
E. Bernstein and U. Vazirani, “Quamtum Complexity Theory”, SIAM J. Comput., 26(5), 1411– 1473 (1997).
R. Penrose, “Gravity and State Vector Reduction”, in R. Penrose and C. J. Isham (eds.), Quantum Concepts in Space and Time, pp 129–146 (Clarendon, Oxford, 1986).
L. Hardy, “Towards Quantum Gravity: A Framework for Probabilistic Theories with Non-Fixed Causal Structure”, J. Phys. A: Math. Theor. 40, 3081–3099 (2007).
L. Hardy, “Causality in Quantum Theory and Beyond: Towards a Theory of Quantum Gravity”, PIRSA#:06070045 (pirsa.org, 2006).
S. Lloyd, “The Computational Universe: Quantum Gravity from Quantum Computation”, quant-ph/0501135 (2005).
M. L. Hogarth, “Does General Relativity Allow an Observer to View an Eternity in a Finite Time?”, Found. Phys. Lett., 5, 173–181 (1992).
M. L. Hogarth, “Non-Turing Computers and Non-Turing Computability”, PSA: Proc. of the Biennial Meeting of the Philos. of Sci. Assoc., Vol. 1994, Volume One: Contributed Papers, pp. 126–138 (1994).
I. Pitowsky, “The Physical Church Thesis and Physical Computational Complexity”, Iyyun 39, 81–99 (1990).
J. Polchinski, String Theory, Vols. 1 and 2 (Cambridge University Press 1998) Cambridge.
C. Rovelli, Quantum Gravity (Cambridge University Press, 2004) Cambridge.
T. Thiemann, “Lectures on Loop Quantum Gravity”, Lecture Notes in Physics 541 (2003).
L. Smolin, “An Invitation to Loop Quantum Gravity”, hep–th/0408048 (2004).
R. D. Sorkin “Causal Sets: Discrete Gravity”, qr-qc/0309009 (2003).
J. Ambjorn, Z. Burda, J. Jurkiewicz, and C. F. Kristjansen, “Quantum gravity represented as dynamical triangulations”, Acta Phys. Polon. B 23, 991 (1992).
S. Aaronson, “NP–complete Problems and Physical Reality”, quant-ph/0502072 (2005).
R. Penrose, Emperors New Mind (OUP, 1989) Oxford.
R. Penrose, Shadows of the Mind (OUP, 1994) Oxford.
M. Tegmark, “Importance of quantum decoherence in brain processes”, Phys. Rev. E 61, 4194– 4206 (2000).
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Hardy, L. (2009). Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure. In: Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. The Western Ontario Series in Philosophy of Science, vol 73. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9107-0_21
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