Quantum Complexity Classes
http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif
By:
Larisse D. Voufo
On:
November 28
th, 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
Key quantum property
for quantum complexity studies:
Randomness of quantum measurement process
Algorithm performed by a quantum computer
is probabilistic.
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.
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
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
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.
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:
… 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):
… 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))
… to
Quantum Complexicity Classes:
•
BQNP ( = QMA)
•
QMA-complete
•
QIP
BPP
Interactive Proof System: IP
Polynomial Number of
Messages
?, r, …
Deterministic
Polynomial-time TM
Merlin-Arthur Protocol: NP
Constant Number of
Messages
Merlin-Arthur Protocol: MA
BPP
Constant Number of
Messages
Merlin-Arthur Protocol: QMA(C)
•
QMA-Completeness:
ground state energy problem: (5-local hamiltonian).
BQP
Constant Number of
Messages
Merlin-Arthur Protocol: QIP
Q-Polynomial Number of
Messages
BQP
?, r, …
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: |x
i>
|y
i>
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.
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
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:
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…
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
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
