Download Computation, Quantum Theory, and You

The Complexity of Sampling
Histories
Scott Aaronson, UC Berkeley
http://www.cs.berkeley.edu/~aaronson
August 5, 2003
Words You
Should Stop
Me If I Use
Words I’ll
Stop You If
You Use
Words To
Be Careful
With
polysize
oracle
relativizing
zero-knowledge
#P-complete
nonuniform
holonomy
gauge
SU(2)
intertwinor
kinematical
Lagrangian
loop
string
Outline
•
•
•
•
Why you should stay up at night
worrying about quantum mechanics
Dynamical quantum theories
Solving Graph Isomorphism by
sampling histories
Search in N1/3 queries (but not fewer)
Quantum
mechanics
What we
experience
Assumption
 4
1
1  2
 4
 3
3
 2
 2
1 1   2
1
1
1  2
1 1   2
Time
1
 4
2  3
3
2  3
 2
2  3
1
2  3
3  4
3  4
3
3  4
3  4
 4
 2
1
Quantum state of the universe
4
4
4
4
A Puzzle
• Let |OR = you seeing a red dot
|OB = you seeing a blue dot
t2 :  R OR   B OB
( H )
t1 :  R OR   B OB
• What is the probability that you see the
dot change color?
The Goal
 1   u11
Quantum 

state
  
  N  u1N
uN 1   1 
   Quantum
Unitary
state
matrix
 
uNN   N 
2
  2  s


s


11
N1
1
Probability  1 
Probability


 Stochastic 
distribution 
distribution

 



matrix


2
2


s
s
NN    N 
  N   1N


Why Look for This?
• Quantum theory says nothing about
multiple-time or transition probabilities
• Reply:
“But we have no direct knowledge of
the past anyway, just records”
• Then what is a “prediction,” or the “output
of a computation,” or the “utility of a
decision”?
Bohm’s Theory
• Gives a deterministic evolution rule for
particle positions and momenta
• But doesn’t make sense for discrete
observables:
1 
 1   1
 2  2
 
 1   1
 2   2
2  1 
 
1  0
2 
• Mathematicianly approach: Study the set
of all discrete dynamical rules, without
presupposing one of them is “true”
Our Results
• We define dynamical theories for obtaining
classical histories, and investigate what axioms they
can satisfy
• We give evidence that by examining a history, one
could solve problems that are intractable even for a
quantum computer
- Graph Isomorphism and Approximate Shortest Lattice
Vector in polynomial time
- Unordered search in N1/3 steps instead of N1/2
• We obtain the first model of computation “slightly”
more powerful than quantum computing
Dynamical Theory
• Fix an N-dimensional Hilbert space (N finite)
and orthogonal basis
• Given an NN unitary U and state  acted on,
returns a stochastic matrix S  D ,U 
• Must marginalize to single-time probabilities of
quantum mechanics: diagonal entries of  and
UU-1
Axiom: Symmetry
D is invariant under relabeling of basis states:

D PP , QUP
1
1
  QD ,U P
1
Axiom: Indifference
If U acts on H1  H 2 and is the identity on H2,
then S should also be the identity on H2
Can formalize without tensor products: partition U
into minimal blocks of nonzero entries
Not the same as commutativity:
D U A  ABU A1 ,U B  D   AB ,U A   D U B  ABU B1 ,U A  D   AB ,U B 
Theorem: No dynamical theory satisfies both
indifference and commutativity
Proof: Suppose A and B share an EPR pair
   00  11  / 2.
UA applies /8 rotation to first
qubit, UB applies -/8 to second qubit. Consider
probability p of being at |00 initially and |10 at the end
If UA applied first:
If UB applied first:
1

cos 2 00
2
8
1 2
sin
01
2
8
1

cos 2 00
2
8
1 2
sin
01
2
8
1 2
sin
10
2
8
1

cos 2 11
2
8
1 2
sin
10
2
8
1
2 
cos
11
2
8
1 2
p  sin
2
8
1 1 2 1 2
p   sin
 sin
4 2
8 2
8
Axiom: Robustness
Small (1/poly(N)) change to  or U

Small (1/poly(N)) change to joint
probabilities matrix, S·diag()
Arguably that’s needed for any physical
theory or model of computation
Example 1: Product Dynamics
Take probabilities at any two times to be
independent of each other
  2   2
1
1

 




2
2
  N    N
2


1
1 




2
2
 N    N 
2
Symmetric, robust, commutative, but not indifferent
Example 2: Dieks Dynamics
Partition U into minimal blocks, then apply product
dynamics separately to each
 4 / 5 0 1   3/ 5 
 3/ 5   1 0  4 / 5

 


 4 / 52  0 1    3/ 52 





2
2
  3/ 5  1 0  4 / 5 
Symmetric, indifferent, but not commutative or robust
Theorem: Suppose
 1   u11  u N 1   1 
   
  
  N  u1N



    .
 u NN   N 
Then there is a weight-1 “flow” through the network
1
u11
2
t
N
2




uNN
1
2
s
N
2
where flow through an edge can’t exceed the edge’s capacity
Proof Idea: By the Max-Flow-Min-Cut Theorem
(Ford-Fulkerson 1956), it suffices to show that any
set of edges separating s from t (a cut) has total
capacity at least 1. Let A,B be right, left edges
respectively not in cut C. Then the capacity of C is

iA
2
i
 j 

2
jB
iA, jB
uij
so we need to show
1
u
iA, jB
  i    j .
2
2
ij
i A
jB
Fix U and consider maximum of right-hand side.
Equals the max eigenvalue of a positive semidefinite
matrix, which we can analyze using some linear
algebra…
Example 3: Flow Dynamics
Using the previous theorem, we construct a
dynamical theory that satisfies the symmetry,
indifference, and robustness axioms
Not obvious a priori that any such theory exists
Model of Computation
• Polynomial-time classical computation, with
one query to a history oracle
• Oracle takes as input descriptions of quantum
circuits U1,…,UT
• Any dynamical theory D induces a distribution D
over classical histories for
0
n
 U1 0
n

 U T
U1  0
n
• Oracle chooses a symmetric robust indifferent
theory D “adversarially,” then returns a sample from D
• At least as powerful as standard quantum computing
The Graph Isomorphism Problem
• Decide whether two graphs G and H are isomorphic

• The best known algorithm takes about
n = number of vertices
2
n
time
• But we don’t think Graph Isomorphism is NP-complete
• Intuitively, it’s “only” as hard as counting collisions in
 1  G  , ,  n!  G  ,  1  H  , ,  n!  H 
Could be easier than finding a needle in a haystack!
The Collision Problem
361542
vs.
622565
• Given a list of N numbers x1,…,xN, you’re promised
that either every number occurs once, or every
number occurs twice. Decide which.
• Best classical algorithm makes ~ N queries
(“birthday paradox”)
• Brassard, Høyer, Tapp 1997 gave a quantum
algorithm that makes ~N1/3 queries
• Is there a faster quantum algorithm—say, log N
queries? If so, we’d get a polynomial-time quantum
algorithm for Graph Isomorphism!
The Collision Problem (con’t)
• Aaronson 2002: Any quantum algorithm needs at
least ~N1/5 queries
• Improved by Shi to ~N1/3 queries
• Previously, couldn’t even rule out constant number of
queries!
• Proofs use multivariate polynomials
• Implications:
• No “dumb” quantum algorithm for Graph Isomorphism
• “Oracle separation” between the complexity classes
BQP (Bounded-Error Quantum Polynomial-Time) and
DQP (Dynamical Quantum Polynomial-Time)
Conjectured
World Map
NP
Satisfiability, Traveling
Salesman, etc.
DQP
My New
Class
BQP
Quantum
Polynomial
Time
Graph Isomorphism
Approximate
Shortest Vector
Factoring
P
Polynomial
Time
Solving the Collision Problem
by Sampling Histories
Suppose every number occurs twice. Then
1
N
N
i
i 1
xi
“Measurement”
of 2nd register
1
i  j

2
x
1
i  j

2
x
i
Two bitwise Fourier
transforms
i
GOAL: When we inspect the classical history,
see both |i and |j with high probability
Solving the Collision Problem
by Sampling Histories (con’t)
Theorem: Under any dynamical theory satisfying the
symmetry and indifference axioms, the first Fourier
transform makes the hidden variable “forget” whether it
was at |i or |j. So after the second Fourier transform,
it goes to |i half the time and |j half the time; thus with
½ probability we see both |i and |j in the history
Proof Idea: Use symmetry axiom, together with
n
automorphisms of Z 2
Indifference axiom needed to “trace out” second
register
Finding a Marked Item in N1/3 Queries
N1/3
iterations
of
Grover’s
quantum
search
algorithm
Probability of
observing the
marked item after T
iterations is ~T2/N
Hidden variable
N1/3 Search Algorithm Is Optimal
• Bennett, Bernstein, Brassard, Vazirani 1996:
If a quantum computer searches a list of N items
for a single randomly-placed marked item, the
probability of observing the marked item after T
2
steps is at most
T
N
• So probability of observing it in a history of the
first T steps is at most
2
3
T
t

t 1 N
T
N
• Summary: If “your whole life flashed before you
in an instant,” and if you’d prepared for this by
putting your brain in certain superpositions, then
(under reasonable axioms) you could solve
Graph Isomorphism in polynomial time
• But probably still not Satisfiability
• Contrast: Nonlinear quantum mechanics would
put Satisfiability and even harder problems in
polynomial time (Abrams and Lloyd 1998)
• Postulate: NP-complete problems can’t be
efficiently solved in physical reality
• Justification for the postulate: Maybe I’m
wrong, but then I’d be too busy solving NPcomplete problems to care that I was wrong
• What does the postulate imply?
(1) Quantum states evolve linearly
(under plausible
complexity
assumptions)
(2) We can’t make unlimited-precision measurements
(3) The “self-sampling” anthropic principle (Bostrom
2000) is false
(4) Constraints on quantum gravity?
• The postulate does not imply your whole life
couldn’t flash before you in an instant