One Complexity
Theorist’s View of
Quantum Computing
Lance Fortnow
NEC Research Institute
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
BQP: Yet Another
Complexity Class
Lance Fortnow
NEC Research Institute
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.
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.
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.

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.
A Product Machine
Traditional nondeterministic Turing
machine has a transition function
 : Qx  2
Qxx { L , R}
Consider a generalized machine with
transition function
 : Qx  
Qxx { L , R}
The Computation Matrix
 : Qx  
Qxx { L , R}
The function  imposes a linear function
mapping configurations to themselves.
Consider the matrix M capturing this
linear function. The value of the
computation after t steps is:
M  Initial , Accepting
t
NP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
  {TRUE , FALSE}
Multiplication  AND
Addition  OR
M  Initial , Accepting
t
#P as Matrix
Multiplication
 : Qx  
Z
Qxx { L , R}
0
M  Initial , Accepting
t
GapP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
Z
M  Initial , Accepting
t
BPP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
0
Q
L1 ( M (v))  L1 (v)
2
x  L  M  Initial , Accepting  
3
1
t
x  L  M  Initial , Accepting  
3
t
Small Changes
 : Qx  
Qxx { L , R}
Q
L1 ( M (v))  L1 (v)
2
x  L  M  Initial , Accepting  
3
1
t
x  L  M  Initial , Accepting  
3
t
Small Changes
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)
2
x  L  M  Initial , Accepting  
3
1
t
x  L  M  Initial , Accepting  
3
t
Small Changes
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
BQP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
Questions
Where’s the Physics?
Where’s the <bra| and |ket>‘s?
Where’s the real/complex numbers?
Don’t we need reversibility?
What if there is more than one
accepting configuration?
Where’s the measurements?
Where’s 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.
Where’s <bra| & |ket>’s?
Fancy way that physicists specify row
and column vectors.
Don’t need to deal with them when
studying quantum complexity.
Computer scientists like balance.
What’s wrong with braT and ket?
Scares away newcomers.
Where’s 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.
Don’t 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.
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.
Where’s the
measurements?
2
2
t
x  L   M  Initial , Accepting  
3

x  L  M  Initial , Accepting 
t

2
1

3
Squaring value simulates process of
measurement at end.
Taking measurements during computation
does not give additional power.
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.

BQP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
The Class AWPP
 : Qx  
Qxx { L , R}
Q

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
The Class AWPP
“Almost-Wide Probabilistic Polynomial
Time”
Previously Studied
Fenner-Fortnow-Kurtz-Li - 1993
 Lide Li’s Thesis - 1993

AWPP contains BQP
Properties of AWPP
BQP  AWPP  PP 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.
Properties of BQP
BQP  PP 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-time
hierarchy is infinite.
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
P
BPP
BQP
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
P
BPP
BQP
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
P
BPP
BQP
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
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.
Diagram of Classes
PSPACE
PH
PP
NP
AWPP
PP-Low
P
BPP
BQP
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.
Black Box Model
Black Box Model
I
N
P
U
T
Black Box Model
Black Box Model
N
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.
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).
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?
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.
BQP as Matrix
Multiplication
 : Qx  
Qxx { L , R}
Q
L2 ( M (v))  L2 (v)

t

t
x  L  M  Initial , Accepting 
x  L  M  Initial , Accepting 


2
2
2

3
1

3
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).
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.
General Result
Any function f:{0,1}n  {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-BuhrmanCleve-Mosca-de Wolf.
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.
Isn’t quantum better?
What about Shor’s 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.

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.
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.
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.