quantum complexity classes

Quantum Complexity Classes

http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif

By: Larisse D. VoufoOn: November 28th, 2006

[email protected]

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


• 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.


• 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..


• 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!

http://www.quantiki.org/wiki/images/thumb/d/da/UncertaintyCartoon.jpg/180px-UncertaintyCartoon.jpg

quantum complexity classes

Documents