1. Quantum Complexity Classes
By: Larisse D. Voufo
On: November 28th, 2006
We are not concerned with the feasibility, nor the origination of Quantum computing.
 Ignore fundamental questions of quantum physics: (eg. Wave and particle Duality)
 Ignore the different interpretations of Q-mechanics: EPR paradox, Shrodinger’s Cat, etc…
 Imagine a quantum world!

2. 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.

3. Key quantum property for quantum complexity studies:
Randomness of quantum measurement process  Algorithm performed by a quantum computer
is probabilistic. (== multiple runs, different results)

4. 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
 Rules of Quantum Probability.

5. 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

6. 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.

7. 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.
AWPP: (remove unitary restriction from BQP)

8. 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.

9. … 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

10. … 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 ?

11. … to Quantum Complexicity Classes:
BQNP ( = QMA)
QMA-complete
QIP  
EQP  BQP  QMA  QIP

12. Interactive Proof System: IP
BPP
?, r, …
Polynomial Number of Messages
Proof (x  L)

13. Merlin-Arthur Protocol: NP
Deterministic Polynomial-time TM
?, r, …
Constant Number of Messages

14. Merlin-Arthur Protocol: MA
BPP
?, r, …
Constant Number of Messages

15. Merlin-Arthur Protocol: QMA(C)
BQP
?, r, …
Constant Number of Messages
ground state energy problem is BQNP complete:
Does a "local" Hamiltonian (a sum of Hermitian operators, each involving a constant number of qubits) have an eigenvalue smaller than a specified energy E.
QMA-Completeness:
ground state energy problem: (5-local hamiltonian).

16. Merlin-Arthur Protocol: QIP
BQP Q-
?, r, …
Polynomial Number of Messages Q-
Proof (x  L)

17. 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
Classical Reversible gate <==> Unitary Transformation x(n)  y(n) = f(x(n)) <==> U: |xi>  |yi> Perm. Of n-bit strings <==> computational basis { |xi> }.
Unitary because: all 2n strings |yi> are mutually orthogonal
A quantum computation constructed from such classical gates takes |00…0> to one of the computational basis states, so that the final measurement is deterministic.

18. 3 Issues with this model:
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.
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.
A Classical computer can still simulate a Quantum Computer,
even if limited to an amount of memory space that is polynomial in n.

19. 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).
 A polynomial size circuit family that solves a hard problem would not be of much interest if the gates in the circuit were required to have an exponential accuracy.  An idealized T-gate q-circuit can be simulated with acceptable accuracy by noisy gates, provided that the error probability per gate scales like 1/T.  If we have designed a quantum algorithm that solves a decision problem correctly with probability greater than ½ +  (in the ideal case), then we can perform the gates with an accuracy T < O().  A quantum circuit family in the BQP class can really solve hard problems, as long as, we can improve the accuracy of the gates linearly with the computation size T.

20. 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

21. 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)

22. Analyzing Quantum Algorithm Performances Over Classical Ones:
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.
Although we still do not have a proof for BPP  BQP, there are 3 approaches to comparing the efficiency of quantum algorithms over classical ones.

23. References:
Tarsem S. Purewal Jr. “Revisiting a Limit on Efficient Quantum Computation”. ACM Southeastern Conference 2006, Melbourne, FL. March 10, Slides at
John Preskill. “Lecture Notes on Physics 229: Quantum Information and Computation”. Sept 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.
Artur Ekert, Patrick Hayden, Hitoshi Inamori. “Basic concepts in quantum computation”. converted to wiki format by Burgarth 22:37, 8 Jun 2005 (BST).
Qbit.com. “Introduction to Quantum Theory”.
Elham Kashefi. “Complexity Analysis and Semantics for Quantum Computation”. November 26,
Tarsem S. Purewal Jr.
Lance Fortnow. “One Complexity Theorist's View of Quantum Computing”.

24. -- Thank You!