Grover Search and the No-Signaling Principle
Abstract
Two of the prime properties of quantum physics are the no-signaling principle and the Grover search lower bound. That is, despite admitting stronger-than-classical correlations, quantum mechanics does not imply superluminal signaling, and despite a form of exponential parallelism, quantum mechanics does not imply polynomial-time brute force solution of NP-complete problems. Herein, we investigate the degree to which these two properties are connected. We explore four classes of deviations from quantum mechanics, for which we draw inspiration from the literature on the black hole information paradox. We show that in these models, the physical resources required to send a superluminal signal scale polynomially with the resources needed to speed up Grover’s algorithm. Hence the no-signaling principle is equivalent to the inability to solve NP-hard problems efficiently by brute force within the classes of theories analyzed.
- Authors:
-
- California Institute of Technology (CalTech), Pasadena, CA (United States)
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
- National Inst. of Standards and Technology (NIST), Gaithersburg, MD (United States); Univ. of Maryland, College Park, MD (United States)
- Publication Date:
- Research Org.:
- California Institute of Technology (CalTech), Pasadena, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); Gordon and Betty Moore Foundation
- OSTI Identifier:
- 1600495
- Alternate Identifier(s):
- OSTI ID: 1324491
- Grant/Contract Number:
- SC0011632; PHY-1125565; GBMF-12500028
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review Letters
- Additional Journal Information:
- Journal Volume: 117; Journal Issue: 12; Journal ID: ISSN 0031-9007
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 79 ASTRONOMY AND ASTROPHYSICS
Citation Formats
Bao, Ning, Bouland, Adam, and Jordan, Stephen P. Grover Search and the No-Signaling Principle. United States: N. p., 2016.
Web. doi:10.1103/PhysRevLett.117.120501.
Bao, Ning, Bouland, Adam, & Jordan, Stephen P. Grover Search and the No-Signaling Principle. United States. https://doi.org/10.1103/PhysRevLett.117.120501
Bao, Ning, Bouland, Adam, and Jordan, Stephen P. Wed .
"Grover Search and the No-Signaling Principle". United States. https://doi.org/10.1103/PhysRevLett.117.120501. https://www.osti.gov/servlets/purl/1600495.
@article{osti_1600495,
title = {Grover Search and the No-Signaling Principle},
author = {Bao, Ning and Bouland, Adam and Jordan, Stephen P.},
abstractNote = {Two of the prime properties of quantum physics are the no-signaling principle and the Grover search lower bound. That is, despite admitting stronger-than-classical correlations, quantum mechanics does not imply superluminal signaling, and despite a form of exponential parallelism, quantum mechanics does not imply polynomial-time brute force solution of NP-complete problems. Herein, we investigate the degree to which these two properties are connected. We explore four classes of deviations from quantum mechanics, for which we draw inspiration from the literature on the black hole information paradox. We show that in these models, the physical resources required to send a superluminal signal scale polynomially with the resources needed to speed up Grover’s algorithm. Hence the no-signaling principle is equivalent to the inability to solve NP-hard problems efficiently by brute force within the classes of theories analyzed.},
doi = {10.1103/PhysRevLett.117.120501},
journal = {Physical Review Letters},
number = 12,
volume = 117,
place = {United States},
year = {Wed Sep 14 00:00:00 EDT 2016},
month = {Wed Sep 14 00:00:00 EDT 2016}
}
Web of Science
Works referenced in this record:
Optimal state discrimination and unstructured search in nonlinear quantum mechanics
journal, February 2016
- Childs, Andrew M.; Young, Joshua
- Physical Review A, Vol. 93, Issue 2
Strengths and Weaknesses of Quantum Computing
journal, October 1997
- Bennett, Charles H.; Bernstein, Ethan; Brassard, Gilles
- SIAM Journal on Computing, Vol. 26, Issue 5
NP is as easy as detecting unique solutions
journal, January 1986
- Valiant, L. G.; Vazirani, V. V.
- Theoretical Computer Science, Vol. 47
Quantum search with general nonlinearities
journal, January 2014
- Meyer, David A.; Wong, Thomas G.
- Physical Review A, Vol. 89, Issue 1
Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for -Complete and # Problems
journal, November 1998
- Abrams, Daniel S.; Lloyd, Seth
- Physical Review Letters, Vol. 81, Issue 18
Grover’s quantum searching algorithm is optimal
journal, October 1999
- Zalka, Christof
- Physical Review A, Vol. 60, Issue 4
Quantum Algorithms for Quantum Field Theories
journal, May 2012
- Jordan, S. P.; Lee, K. S. M.; Preskill, J.
- Science, Vol. 336, Issue 6085
Black holes: complementarity or firewalls?
journal, February 2013
- Almheiri, Ahmed; Marolf, Donald; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2013, Issue 2
Aspects of the Papadodimas-Raju proposal for the black hole interior
journal, November 2014
- Harlow, Daniel
- Journal of High Energy Physics, Vol. 2014, Issue 11
Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group
journal, March 2006
- Collins, Benoît; Śniady, Piotr
- Communications in Mathematical Physics, Vol. 264, Issue 3
State-dependent bulk-boundary maps and black hole complementarity
journal, April 2014
- Papadodimas, Kyriakos; Raju, Suvrat
- Physical Review D, Vol. 89, Issue 8
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels
journal, March 1993
- Bennett, Charles H.; Brassard, Gilles; Crépeau, Claude
- Physical Review Letters, Vol. 70, Issue 13
Weinberg’s nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox
journal, January 1991
- Polchinski, Joseph
- Physical Review Letters, Vol. 66, Issue 4
Quantum computing, postselection, and probabilistic polynomial-time
journal, August 2005
- Aaronson, Scott
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue 2063
Guest Column: NP-complete problems and physical reality
journal, March 2005
- Aaronson, Scott
- ACM SIGACT News, Vol. 36, Issue 1
Violations of the Born rule in cool state-dependent horizons
journal, January 2016
- Marolf, Donald; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2016, Issue 1
No-Signaling Principle Can Determine Optimal Quantum State Discrimination
journal, October 2011
- Bae, Joonwoo; Hwang, Won-Young; Han, Yeong-Deok
- Physical Review Letters, Vol. 107, Issue 17
Quantum mechanics near closed timelike lines
journal, November 1991
- Deutsch, David
- Physical Review D, Vol. 44, Issue 10
Quantum computation vs. firewalls
journal, June 2013
- Harlow, Daniel; Hayden, Patrick
- Journal of High Energy Physics, Vol. 2013, Issue 6
The black hole final state
journal, February 2004
- Horowitz, Gary T.; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2004, Issue 02
Closed timelike curves make quantum and classical computing equivalent
journal, October 2008
- Aaronson, Scott; Watrous, John
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 465, Issue 2102
Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren
book, January 1931
- Wigner, Eugen
Can Closed Timelike Curves or Nonlinear Quantum Mechanics Improve Quantum State Discrimination or Help Solve Hard Problems?
journal, October 2009
- Bennett, Charles H.; Leung, Debbie; Smith, Graeme
- Physical Review Letters, Vol. 103, Issue 17
Comment on "The black hole final state"
journal, March 2004
- Gottesman, Daniel; Preskill, John
- Journal of High Energy Physics, Vol. 2004, Issue 03
Better Late than Never: Information Retrieval from Black Holes
journal, March 2013
- Braunstein, Samuel L.; Pirandola, Stefano; Życzkowski, Karol
- Physical Review Letters, Vol. 110, Issue 10
Weinberg's non-linear quantum mechanics and supraluminal communications
journal, January 1990
- Gisin, N.
- Physics Letters A, Vol. 143, Issue 1-2
Simulation of Topological Field Theories¶by Quantum Computers
journal, June 2002
- Freedman, Michael H.; Kitaev, Alexei; Wang, Zhenghan
- Communications in Mathematical Physics, Vol. 227, Issue 3
Nonlinear quantum search using the Gross–Pitaevskii equation
journal, June 2013
- Meyer, David A.; Wong, Thomas G.
- New Journal of Physics, Vol. 15, Issue 6
Computers with Closed Timelike Curves Can Solve Hard Problems Efficiently
journal, January 2003
- Brun, Todd A.
- Foundations of Physics Letters, Vol. 16, Issue 3, p. 245-253
Works referencing / citing this record:
Any modification of the Born rule leads to a violation of the purification and local tomography principles
journal, November 2018
- Galley, Thomas D.; Masanes, Lluis
- Quantum, Vol. 2