Quantum Complexity Classes
By: Larisse
D. Voufo
On: November 28th, 2006
lvoufo@cs.indiana.edu
http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif
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  P  {} => 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  NP  AWPP  PP  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 = PSPACE
 NP  MA
 BPP  MA  IP
 BQP  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
Interactive Proof System: IP
?, r, …
Polynomial
Number of
Messages
Proof (x  L)
BPP
Merlin-Arthur Protocol: NP
?, r, …
Constant
Number of
Messages
Deterministic
Polynomialtime TM
Merlin-Arthur Protocol: MA
?, r, …
Constant
Number of
Messages
BPP
Merlin-Arthur Protocol: QMA(C)
?, r, …
BQP
Constant
Number of
Messages
• QMA-Completeness:
ground state energy problem: (5-local hamiltonian).
Merlin-Arthur Protocol: QIP
Q- ?, r, …
Polynomial
Number of
Messages
Q-
Proof (x  L)
BQP
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
•
•
2.
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.
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  BPP  BQP  AWPP  PP  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.
2.
3.
Non-exponential speedup:
Eg: Grover’s Quantum Speed-up of the Search of an unsorted
database.
“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.
Exponential Speed-up for “apparently” hard problems
Eg: Shor’s factoring algorithm.
References:
•
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•
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•
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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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