Closed timelike curves make quantum and classical computing equivalent

  title={Closed timelike curves make quantum and classical computing equivalent},
  author={Scott Aaronson and John Watrous},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  pages={631 - 647}
  • S. Aaronson, J. Watrous
  • Published 19 August 2008
  • Computer Science
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to non-trivial insights into general relativity, quantum information and other areas. In this paper, we show that, if CTCs existed, quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class polynomial space (), consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves… 

Figures from this paper

Quantum state cloning using Deutschian closed timelike curves.
We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the
Quantum computation with indefinite causal structures
It is shown that process matrices correspond to a linear particular case of P-CTCs, and therefore that its computational power is upperbounded by that of PP, and a family of processes that can violate causal inequalities but nevertheless can be simulated by a causally ordered quantum circuit with only a constant overhead, showing that indefinite causality is not necessarily hard to simulate.
Computability Theory of Closed Timelike Curves
The question of what's computable by Turing machines equipped with time travel into the past is asked, and the answer is, closed timelike curves or CTCs (with no bound on their size), which are shown to solve exactly the problems that are Turing-reducible to the halting problem.
Quantum state discrimination circuits inspired by Deutschian closed timelike curves
It is proved that the proposed practical method for discriminating multiple non-orthogonal states, by using a previously known quantum circuit designed to simulate D-CTCs, achieves the multiple Cherno bound when discriminating an arbitrary set of pure qubit states.
Closed timelike curves and the second law of thermodynamics
One out of many emerging implications from solutions of Einstein's general relativity equations are closed timelike curves (CTCs), which are trajectories through spacetime that loop back on
Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves
Bennett and Schumacher’s postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch’s model. We show that even a single
Quantum Entanglement Near Open Timelike Curves
Closed timelike curves are striking predictions of general relativity allowing for time-travel. They are afflicted by notorious causality issues (e.g. grandfather’s paradox). Quantum models where a
Treating Time Travel Quantum Mechanically
The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this
Quantum computation is an island in theoryspace
The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible
The Weakness of CTC Qubits and the Power of Approximate Counting
Results in structural complexity theory concerned with computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting, and computational complexity of finding stationary distributions for quantum channels are presented.


Quantum computational complexity in the presence of closed timelike curves
Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed
Quantum mechanics near closed timelike lines.
  • Deutsch
  • Physics
    Physical review. D, Particles and fields
  • 1991
Several novel and distinctive quantum-mechanical effects occur on and near closed timelike lines, including violations of the correspondence principle and of unitarity, and consideration of these sheds light on the nature of quantum mechanics.
Space-Bounded Quantum Complexity
  • J. Watrous
  • Computer Science
    J. Comput. Syst. Sci.
  • 1999
It is shown that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power and, furthermore, that any QTM running in space s can be simulated deterministically in NC2(2s)?DSPACE(s2)?DTIME(2O(s).
Quantum Information and the PCP Theorem
  • R. Raz
  • Computer Science, Mathematics
    46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
  • 2005
The main result is that the membership x∈SAT can be proved by a logarithmic-size quantum state, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof.
Problem of equilibration and the computation of correlation functions on a quantum computer
The quantum algorithms that are presented could provide an exponential speedup over what can be achieved with a classical device, given a preparation of the equilibrium state.
Fault-tolerant quantum computation with constant error
This paper shows how to perform fault tolerant quantum computation when the error probability, q, is smaller than some constant threshold, q.. the cost is polylogarithmic in time and space, and no measurements are used during the quantum computation.
NP-complete Problems and Physical Reality
  • S. Aaronson
  • Physics
    Electron. Colloquium Comput. Complex.
  • 2005
Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms,
Computers with Closed Timelike Curves Can Solve Hard Problems Efficiently
A computer which has access to a closed timelike curve, and can thereby send the results of calculations into its own past, can exploit this to solve difficult computational problems efficiently. I
QMA/qpoly Is Contained In PSPACE/poly: De-Merlinizing Quantum Protocols
  • S. Aaronson
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2005
A new technique for removing existential quantifiers over quantum states is introduced, and it is shown that there is no way to pack an exponential number of bits into a polynomial-size quantum state, in such a way that the value of any one of those bits can later be proven with the help of a polymouthsize quantum witness.
Guest Column: NP-complete problems and physical reality
This book surveys proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing".