Guest Column: NP-complete problems and physical reality

@article{Aaronson2005GuestCN,
  title={Guest Column: NP-complete problems and physical reality},
  author={Scott Aaronson},
  journal={SIGACT News},
  year={2005},
  volume={36},
  pages={30-52}
}
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, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing." The section on soap bubbles even includes some "experimental" results. While I do not believe that any of… 

Figures from this paper

Grover Search and the No-Signaling Principle.

It is shown that in four classes of deviations from quantum mechanics, the physical resources required to send a superluminal signal scale polynomially with the resources needed to speed up Grover's algorithm are equivalent to the inability to solve NP-hard problems efficiently by brute force within the classes of theories analyzed.

How quantum is the speedup in adiabatic unstructured search?

  • I. Hen
  • Physics, Computer Science
    Quantum Inf. Process.
  • 2019
It is shown that similarly to its quantum counterpart, the classical construction may also provide a quadratic speedup over standard digital unstructured search and the meaning and the possible implications are discussed in the context of adiabatic quantum computing.

Can Biological Quantum Networks Solve NP‐Hard Problems?

  • G. Wendin
  • Biology
    Advanced Quantum Technologies
  • 2019
The conclusion is that biological quantum networks can only approximately solve small instances of nonpolynomial (NP)‐hard problems, and artificial intelligence and machine learning implemented in complex dynamical systems based on genuine quantum Networks can certainly be expected to show enhanced performance and quantum advantage compared with classical networks.

Quantum Approximate Optimization for Hard Problems in Linear Algebra

Simulated annealing can outperform QAOA for BLLS at a QAoa depth of p\leq3p≤3 for the probability of sampling the ground state, and some of the challenges involved in current-day experimental implementations of this technique on cloud-based quantum computers are pointed out.

Probing the limits of quantum theory with quantum information at subnuclear scales

Modern quantum engineering techniques enabled successful foundational tests of quantum mechanics. Yet, the universal validity of quantum postulates is an open question. Here we propose a new

What the foundations of quantum computer science teach us about chemistry.

This Perspective takes the position that direct chemical simulation is best understood as a digital experiment and builds to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution, known in computer science as undecidable problems.

Revisiting integer factorization using closed timelike curves

A flaw is found in Brun’s algorithm and a modified algorithm is proposed to circumvent the flaw and solve problems like factoring and quantified satisfiability problem.

Linear and nonlinear quantum algorithms made explicit

A thorough breakdown of common quantum algorithms into their component parts, and the explicit cost of each component in terms of fundamental quantum gates is given, and a new state-of-the-art algorithm for producing a superposition of all permutations is determined.

The Info-Computation Turn in Physics

The common Informational basis of computation and communication brings about a foundational shift in scientific reasoning with deep – potentially problematic as well as intriguing – philosophical ramifications.

Analog nature of quantum adiabatic unstructured search

It is found that the unstructured search algorithm’s quadratic speedup is generally not robust to the presence of any one of the above non-idealities, and in some cases it imposes unrealistic conditions on how the strength of these noise sources must scale to maintain the quadratics speedup.
...

References

SHOWING 1-10 OF 86 REFERENCES

Limits on Efficient Computation in the Physical World

This thesis attacks the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known, and studies the relationship of the quantum computing model to physical reality.

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

NONLINEAR QUANTUM MECHANICS IMPLIES POLYNOMIAL-TIME SOLUTION FOR NP-COMPLETE AND P PROBLEMS

This work provides algorithms that solve NP-complete and #P oracle problems by exploiting nonlinear quantum logic gates, and it is argued that virtually any deterministic non linear quantum theory will include such gates.

How powerful is adiabatic quantum computation?

It is argued that the adiabatic approach may be thought of as a kind of 'quantum local search', and a family of minimization problems that is hard for such local search heuristics are designed, and an exponential lower bound is established for the ad iabatic algorithm for these problems.

Quantum computing, postselection, and probabilistic polynomial-time

  • S. Aaronson
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2005
It is shown that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently, or probabilistic polynomial-time, and implies, as an easy corollary, a celebrated theorem of Beigel, Reingold and Spielman that PP is closed under intersection.

A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem

For the small examples that the authors could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.

Quantum Algorithm for Hilbert's Tenth Problem

It is argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.

Simulation of Topological Field Theories¶by Quantum Computers

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a

One complexity theorist's view of quantum computing

  • L. Fortnow
  • Physics
    Electron. Notes Theor. Comput. Sci.
  • 2000

Subquantum information and computation

It is argued that immense physical resources — for nonlocal communication, espionage, and exponentially-fast computation — are hidden from us by quantum noise, and that this noise is not fundamental
...