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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



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.

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):

… 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.

• 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:

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

• Lance Fortnow. “One Complexity Theorist's View of Quantum Computing”. http://people.cs.uchicago.edu/~fortnow/papers/quantview.pdf

-- Thank You!

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:

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

• Lance Fortnow. “One Complexity Theorist's View of Quantum Computing”. http://people.cs.uchicago.edu/~fortnow/papers/quantview.pdf

-- Thank You!