One Complexity Theorist

1
One Complexity Theorists View of Quantum
Computing

• Lance FortnowNEC Research Institute

2
Comp.Theory FAQ

• 8. Complexity Theory
• (a) Lower Bounds
• (b) YACC (Yet Another Complexity Class)
• Our ability to understand and handle new models
  of computation comes from our experience studying
  previous notions.
• Case in Point Quantum Computing

3
BQP Yet AnotherComplexity Class

• Lance Fortnow
• NEC Research Institute

4
Quantum Computation

• A computation model based on quantum principles
  of physics.
• Ability to enter many parallel states and use
  interference to recover important information.
• Transformations must be unitary.

5
Dephysicfying Quantum

• To understand the computational powers of quantum
  computing, we should ignore the underlying
  physical model.
• Nondeterministic computation has no known
  underlying physical model yet we have a good
  understanding of its computational power.

6
The Quantum Class BQP

• The set of languages L such that there is a
  Polynomial-time Quantum Turing machine M such
  that for all strings x,
• If x is in L then the measured probability of
  acceptance of M on input x is at least 2/3.
• If x is not in L then the measured probability of
  acceptance of M on input x is at most 1/3.

7
Oddities of Quantum Computing

• Many Parallel States
• Similar to Probabilistic Computation.
• Interference
• Similar ideas in Counting Complexity.
• Unitary Transformations
• New and what makes quantum computing so hard to
  classify precisely.

8
A Product Machine

• Traditional nondeterministic Turing machine has a
  transition function
• Consider a generalized machine with transition
  function

9
The Computation Matrix

• The function d imposes a linear function mapping
  configurations to themselves.
• Consider the matrix Md capturing this linear
  function. The value of the computation after t
  steps is

10
NP as Matrix Multiplication
11
P as Matrix Multiplication
12
GapP as Matrix Multiplication
13
BPP as Matrix Multiplication
14
Small Changes
15
Small Changes
16
Small Changes
17
BQP as Matrix Multiplication
18
Questions

• Wheres the Physics?
• Wheres the ltbra and ketgts?
• Wheres the real/complex numbers?
• Dont we need reversibility?
• What if there is more than one accepting
  configuration?
• Wheres the measurements?

19
Wheres the Physics?

• Car makers have given us a model from which we
  can drive a car. Details of how the car works are
  not necessary.

20
Wheres ltbra ketgts?

• Fancy way that physicists specify row and column
  vectors.
• Dont need to deal with them when studying
  quantum complexity.
• Computer scientists like balance.
• Whats wrong with braT and ket?
• Scares away newcomers.

21
Wheres the complex numbers?

• For BQP one can assume the transitions come from
  -1,-4/5,-3/5,0, 3/5,4/5,1 instead of computable
  complex numbers.
• Noncomputable numbers allow encoding of
  noncomputable functions. Similar problem in
  classical model.

22
Dont we need reversibility?

• The set of matrices M that preserve the L2 norm
  are unitary M(M)T is the identity.
• In particular M is invertible so the computation
  could be reversed.
• Reversibility is not a requirement of quantum
  computing but a consequence.

23
One accepting configuration?

• In most models, can assume one accepting
  configuration by having machine erase work tape
  and moving to single accept state.
• Not reversible process.
• Can be simulated in quantum with negligible
  additional error by writing answer and reversing
  the rest of the computation.

24
Wheres the measurements?

• Squaring value simulates process of measurement
  at end.
• Taking measurements during computation does not
  give additional power.

25
BQP - A good definition

• Simple and Robust.
• Based on a physical model.
• Contains interesting problems.
• Other Quantum Classes not as robust
• EQP - Differences in set of allowable amplitudes
  may affect class.
• BQL - When measurements are made may affect class.

26
BQP as Matrix Multiplication
27
(No Transcript)
28
The Class AWPP
29
The Class AWPP

• Almost-Wide Probabilistic Polynomial Time
• Previously Studied
• Fenner-Fortnow-Kurtz-Li - 1993
• Lide Lis Thesis - 1993
• AWPP contains BQP

30
Properties of AWPP

• BQP Í AWPP Í PP Í PSPACE
• AWPP is low for PP
• PPAWPP PP
• For any L in AWPP, PPL PP.
• There exists a relativized world where AWPP P
  and the polynomial-time hierarchy is infinite.

31
Properties of BQP

• BQP Í PP Í PSPACE
• BQP is low for PP
• PPBQP PP
• For any L in BQP, PPL PP.
• There exists a relativized world where BQP P
  and the polynomial-timehierarchy is infinite.

32
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
BQP
BPP
P
33
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
BQP
BPP
P
34
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
BQP
BPP
P
35
The Polynomial-Time Hierarchy

• Nondeterministic Computation is a misleading
  title. Really Existential.
• Similarly can have Universal Computation.
• Alternating TM - Switches back and forth between
  Existential and Universal.
• Unbounded Alternations - PSPACE
• Constant Alternations - PH

36
BQP in PH?

• Bernstein-Vazirani relativized language does not
  appear to sit in PH.
• It would if we allowed slightly more than
  polynomial-time or constant alternations.
• Suggestion
• Try to show that BQP can be solved in
  quasipolynomial time and/or polylogarithmic
  alternations.

37
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
BQP
BPP
P
38
NP in BQP?

• Relative to a random oracle NP is in AWPP.
• Two problems
• Random oracles do not give us a good view of the
  world.
• Need unitary transformations to get NP in BQP.
• Make it difficult to obtain bad consequences of
  NP in BQP.

39
Black Box Model
40
Black Box Model
I
N
P
U
T
41
Black Box Model
42
Black Box Model
N
43
Black Box Model
N
T

• Count only number of queries made.
• We do not care about computation time.
• Also known as decision tree or oracle model.
• Hard to define decision trees properly for
  quantum machines.

44
OR Function

• The OR function requires all N queries on some
  input of N bits for a deterministic machine.
• Adversary always answers zero on all queries.
• OR has small nondeterministic black box
  complexity (1 query).

45
Black Box Classes

• P Polylogarithmic in N queries
• NP Nondeterministic polylogarithmic in N
  queries
• The OR functions separates black box P from black
  box NP.
• How about BQP?

46
Black Box BQP

• The probability of acceptance of a black box BQP
  machine using t queries is a polynomial of degree
  at most 2t.
• Easy to see from Matrix Multiplication view of
  BQP.

47
BQP as Matrix Multiplication
48
The OR function

• The OR function has degree n.
• However a BQP black box need only approximate the
  OR function.
• Any polynomial that approximates the OR functions
  has degree ?(?n).

49
Tightness of OR

• Any black box BQP machine must use ?(?n) queries.
• OR function separates NP from BQP.
• Grover shows that O(?n) queries suffice to
  compute OR on a BQP machine.

50
General Result

• Any function f0,1n ? 0,1 that can be
  approximated by a degree d polynomial has a
  deterministic black box algorithm using O(d6)
  queries.
• Due to Nisan-Szegedy, Beals-Buhrman-Cleve-Mosca-de
  Wolf.

51
BQP and P

• Every function computed by a BQP black box
  algorithm using t queries can be computed by a
  deterministic black box algorithm using O(t6)
  queries.
• Black box BQP is the same as black box P.

52
Isnt quantum better?

• What about Shors factoring, discrete logarithm,
  Deutch-Josza, Simon, etc.
• These have black box algorithms with limited
  input space.
• Deutch-Josza only separates all same from same
  number of zeros-ones.
• Factoring problem leads to black box with strong
  algebraic structure.

53
NP and BQP

• If BQP were to contain NP in the traditional
  model it would be because NP problems have a nice
  structure that BQP can take advantage of.
• To me this seems unlikely so I would conjecture
  that BQP cannot solve NP problems.

54
Is quantum computing useful?

• We can factor but
• If the only uses of quantum computation remain
  discrete logarithms and factoring, it will likely
  become a special-purpose technique whose only
  raison d'etre is to thwart public key
  cryptosystems. (Peter Shor)
• Using tools of counting complexity, we have shown
  new bounds on power of quantum machines.

55
Conclusions

• Quantum Complexity very fascinating and worthy of
  future study.
• To study complexity of BQP forget the physics and
  their awful notation.
• Still seeking a definitive answer on usefulness
  of quantum computation.
• So far unable to use unitary property of BQP to
  help in classifying the class.
• Though useful in some oracle worlds.