quantum complexity classes
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By: Larisse D. Voufo On: November 28 th , 2006 [email protected]. Quantum Complexity Classes. http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif. Introduction. 1982 (Trend toward miniaturization and microcircuitry) , Paul Benioff & Richard Feynman : - PowerPoint PPT PresentationTRANSCRIPT
Quantum Complexity Classes
http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif
By: Larisse D. VoufoOn: November 28th, 2006
Introduction
• 1982 (Trend toward miniaturization and microcircuitry),
Paul Benioff & Richard Feynman:
Quantum Systems could perform computation.
• 1985, David Deutch.
Quantum Computer Turing Machine
possibility of new Complexity of algorithms
• Later On,
Universality of Quantum Circuits
Machine independent notion of quantum complexity.
Key quantum property for quantum complexity studies:
Randomness of quantum measurement process Algorithm performed by a quantum computer
is probabilistic.(== multiple runs, different results)
Probabilistic Computation vs. Quantum Computation.
• Nondeterministic Computation (NC) = tree of configurations of NTM
• Probabilistic Computation
= NC where probabilities
<--> edges and nodes.
Rules of Classical Probability.
• Quantum Computation
= NC where amplitudes
<--> edges and nodes.
Rules of Quantum Probability.
From Classical Complexity classes…
• P – “easy”: languages decided by polynomial-time TMs
• NP: languages decided by polynomial-time NTMs.Guess an answer, verify in polynomial time.
Is answer YES?• NP-hard:
Every hard problem can be polynomially reduced to a problem in this class.
• NP-complete: NPC = NP-hard NP
NP-hard NP-hard P P {} => P = NP {} => P = NP
From Classical Complexity classes…
• NPI: Problems in NP of intermediate difficulty NPI = NP – P – NPC
= NP – P – NP-hard
• Co-NP:
Like NP, but Answer is NO (counter-example based) NP Co-NP
No proof for: P NP.
From Classical Complexity classes…
• AWPP: languages decided by Almost-Wide Probabilistic Polynomial-time NTMs
• PP:languages decided by polynomial-time NTMs where the majority of paths gives the correct answer.
• P#P: functions that count the number of accepting paths through an
NP machine.
P P NP NP AWPP AWPP PP PP P P#P#P..
From Classical Complexity classes…
• IP: Problems solvable by an Interactive Proof System.
• MA:
languages decided by a bounded-error probabilistic Merlin-Arthur protocol.
• BPP:
Bounded-error Probabilistic Polynomial Time.“Problems that admit a probabilistic circuit family of polynomial
size that always gives the right answer with prob > ½ + ”.
• PSPACE:
DPs that can be solved in polynomial-space, but may require exponential time.
… to Quantum Complexity Classes:
• BQP:
Bounded-error Quantum Polynomial Time.
“DPs that can be solved, with high probability, by polynomial-size quantum circuits”.
• EQP (QP):
Exact version of BQP
… to Quantum Complexity Classes:
P BPP BQP PSPACE IP = PSPACENP MABPP MA IPBQP P#P PSPACE No firm proof for: BPP BQP (in general) If P = PSPACE, then P = AWPP “relative to oracle”NP = AWPP “relative to oracle”NP PSPACE (checking if C(x(n), y(n)) = 1 for each y(m))NP BQP ?
… to Quantum Complexicity Classes:
• BQNP ( = QMA)
• QMA-complete
• QIP
EQP BQP QMA QIP
BPP
Interactive Proof System: IP
Polynomial Number of Messages
?, r, …
Proof (x L)
Deterministic Polynomial-time TM
Merlin-Arthur Protocol: NP
Constant Number of Messages
?, r, …
Merlin-Arthur Protocol: MA
BPP
Constant Number of Messages
?, r, …
Merlin-Arthur Protocol: QMA(C)
• QMA-Completeness:
ground state energy problem: (5-local hamiltonian).
BQPConstant
Number of Messages
?, r, …
Merlin-Arthur Protocol: QIP
Q-
Q-
Polynomial Number of Messages
BQP?, r, …
Proof (x L)
A model for quantum circuits:
Facts:
• Quantum gate:
unitary transformation reversible gate.
• Classical Reversible Computer
= special case of Quantum Computer.
• x(n) y(n) = f(x(n)) <==> U: |xi> |yi>
• |00…0> Deterministic final measurement
3 Issues with this model:
1. Universality• Complete Model <==>
There exists no transformation in U(2n) that we cannot reach.
• Simulation of a Q-computer using another Q-computer complexity classes do not depend on the details of the
hardware.
2. Simulating a quantum computer on a classical computer: Better characterize the resources needed.
• A Classical Computer can still simulate a Q-Computer, despite a polynomial limit on memory space available.
3 Issues with this model:
3. Accuracy== growth of error in measurement as the quantum circuit
size increases.• NO Polynomial-size circuit family (hard problems) w/
gates of exponential accuracy. • An idealized T-gate q-circuit (acceptable accuracy):
Error Prob / gate 1/T.• Quantum Algorithm w/ prob > ½ + (in the ideal case)
Gates w/ accuracy T < O().• BQP can really solve hard problems
<==> linear improvement of the accuracy of the gates (computation size T).
More on Relationships between Complexity classes
P P BPP BPP BQP BQP AWPP AWPP PP PP PSPACE. PSPACE.
• Bernstein and Vazirani:
BQP PSPACE
• Adelman, Demarrais and Huang:
BQP PP
• Fortnow and Rogers:
BQP AWPP
Other Complexity Classes
Vary from one literature to another…
• UP, QPSV, NPSV, UPSV, etc…
Elham Kashefi’s PhD thesis (Imperial College London)
• NQP, C=P, coC=P, etc…
Tarsem S. Purewal Jr (University of Georgia)
Analyzing Quantum Algorithm Performances Over Classical Ones:
1. Non-exponential speedup:
Eg: Grover’s Quantum Speed-up of the Search of an unsorted database.
2. “Relativized” Exponential Speed-up
Oracles
BPP BQP “relative to oracle”.
Eg:
Simon’s exponential quantum speedup for finding the period of 2 to 1 function.
Deutch’s algorithm.
3. Exponential Speed-up for “apparently” hard problems
Eg: Shor’s factoring algorithm.
References:• Tarsem S. Purewal Jr. “Revisiting a Limit on Efficient Quantum Computation”. ACM
Southeastern Conference 2006, Melbourne, FL. March 10, 2006. Slides at http://www.cs.uga.edu/~purewal/slides/BQPinPPTalk.pdf
• John Preskill. “Lecture Notes on Physics 229: Quantum Information and Computation”. Sept. 1998. California Institute of Technology.
• Tarsem S. Purewal Jr. “Nondeterministic Quantum Query Complexity”. Combinatorics, Algorithms, and Theory Seminar (CATS), University of Georgia, Athens, GA. May 1, 2006.
• Tarsem S. Purewal Jr. “5-local Hamiltonian is QMA-Complete”. Quantum Computing Journal Club, University of Georgia, Athens, GA. June 6, 2005.
• Prof. Tony Hey. “Quantum Computing: an introduction”. Quantum Technology Center. http://www.qtc.ecs.soton.ac.uk/flecture.html
• Artur Ekert, Patrick Hayden, Hitoshi Inamori. “Basic concepts in quantum computation”. converted to wiki format by Burgarth 22:37, 8 Jun 2005 (BST). http://www.quantiki.org/wiki/index.php/Basic_concepts_in_quantum_computation
• Qbit.com. “Introduction to Quantum Theory”. http://www.quantiki.org/wiki/index.php/Introduction_to_Quantum_Theory
• Elham Kashefi. “Complexity Analysis and Semantics for Quantum Computation”. November 26, 2003. http://web.comlab.ox.ac.uk/oucl/work/elham.kashefi/papers/phdelham.pdf
• Tarsem S. Purewal Jr. http://www.cs.uga.edu/~purewal/vita.html
• Lance Fortnow. “One Complexity Theorist's View of Quantum Computing”. http://people.cs.uchicago.edu/~fortnow/papers/quantview.pdf
-- Thank You!
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