Computation, Quantum Theory, and You

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Computation, Quantum Theory, and You

• Scott Aaronson, UC Berkeley
• Qualifying Exam
• May 13, 2002

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Talk Outline

• Sermon
• 2. Quantum Computing Overview
• Collision Lower Bound
• Dynamical Models
• 5. Current and Future Work

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1. Sermon
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The Computer Scientists Idea of Physics
details
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What Does Our World Have That Conways Doesnt?

• 3 or more spatial dimensions

• Continuity?

• Relativistic covariance

Quantum theory

• Quantum theory

• And more?

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My Own View
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Research Goal Prove complexity results, focusing
on quantum computing, that are motivated by this
gap between physics and what we experience.
(Disclaimer I will not bridge the gap in my
thesis.)
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2. Quantum Computing
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Some Milestones
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
1992 1993 1994
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The Quantum Model

• State of computer superposition over binary
  strings
• To each string Y, associate complex amplitude ?Y
• ?Y ?Y2 1
• On measuring, see Y with probability ?Y2
• Dirac ket notation State written
• ?? ?Y ?Y Y?
• Each Y? is called a basis state

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Unitary Evolution

• Quantum state changes by multiplying amplitude
  vector with unitary matrix ?(t1)? U?(t)?
• U is unitary iff U-1U, conjugate transpose
• (Linear transformation that preserves norm1)
• Example
• Circuit model U must be efficiently computable
• Black-box model No such restriction

1/?2 -1/?2
1/?2 1/?2
(0? 1?)/?2 1?
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Quantum Query Model

• State after
• t queries
• ? workbits i index to query z output

• Query ?,i,z? ? ??xi,i,z?

• Arbitrary unitaries that dont depend on X

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3. Collision Lower Bound
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Collision Problem

• Given

• Promised
• (1) X is one-to-one (permutation) or
• (2) X is two-to-one

• Problem Decide which w.h.p., using few queries
  to the xi

• Randomized alg ?(?n)

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Result

• Any quantum algorithm for the collision problem
  uses ?(n1/5) queries (A, STOC2002)

• Shi improved to ?(n1/4)
• ?(n1/3) when range gtgt n

• Previously no lower bound better than ?(1). Open
  since 1997

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Implications

• Oracle A for which SZKA ? BQPA
• SZK Statistical Zero Knowledge
• No trivial polytime quantum algorithms for
• graph isomorphism
• nonabelian hidden subgroup
• breaking cryptographic hash functions

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Brassard-Høyer-Tapp (1997)

• ?(n1/3) quantum alg for collision problem

Grovers algorithm over n2/3 xis

Do I collide with any of the pink xis?
n1/3 xis, queried classically, sorted for fast
lookup

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Previous Lower Bound Techniques

• Block sensitivity (Beals et al. 1998)
• Q2(f) ?(?bs(f))

• Quantum adversary method (Ambainis 2000)

• Problem Every 1-1 input differs in at least n/2
  places from every 2-1 input

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• Lemma (follows Beals et al. 1998) Let ?(xi,h)1
  if xih, 0 otherwise. Then P(X) is poly of deg ?
  2T over the ?(xi,h).

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Input Distribution

• D(g) Uniform distribution over g-1 inputs

• Technicality g might not divide n
• But assume for simplicity that it does

• Exercise Show that, if TO(?n), then P(g) is a
  polynomial of degree ? 2T in g for integers
  1?g??n.

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Monomials of P(X)

• I(X) product of r variables ?(xi,h)

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Calculating ?(I,g) 1

• Range of I Y. wY.

• ?(I,g) 0 unless Y?S (range of X)

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Calculating ?(I,g) 2

• Given an S containing Y,
• of g-1 inputs of size n n!/(g!)n/g

• Let y1,,yw be distinct values in Y
• ri of times yi appears in Y
• r1 rw r

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Becomes polynomial(g)
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Markovs Inequality

• Let P(x) be a poly with b1?P(x)?b2 for all
  a1?x?a2 and dP(x)/dx?c for some a1?x?a2.
  Then

Large derivative
Short
Long
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Lower Bound

• 0 ? P(g) ? 1 for all 0 ? g ? ?n

• P(1) ? 1/10 and P(2) ? 9/10
• So dP/dg ? 4/5 somewhere

• ?(n1/4) lower bound would follow if g always
  divided n

• Can fix to obtain an ?(n1/5) bound
• Shi found a better way to fix

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4. Dynamical Models
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A Puzzle

• Let OR? you seeing a red dot
• OB? you seeing a blue dot

• What is the probability that you see the dot
  change color?

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Why Is This An Issue?

• Quantum theory says nothing about multiple-time
  or transition probabilities

• Reply
• But we have no direct knowledge of the past
  anyway, just records

• But then what is a prediction, or the output
  of a computation, or the utility of a decision?

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Results
(submitted to PRL, quant-ph/0205059)

• What if you could examine an observers entire
  history? Defined class DQP

• Showed SZK ? DQP. Combined with collision
  bound, implies oracle A for which BQPA ? DQPA

• Can search an N-element list in order N1/3
  steps, though not fewer

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DQP
BQP
SZK
BPP
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5. Current and Future Work
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BQP versus PH

• Almost-complete (?!) joint work with Umesh
• Conjecture BQPA ? PHA for an oracle A
• (Best known BQPA ? (?2)A)
• Use Recursive Fourier Sampling
• Have reduced problem to generalizing the
  Razborov-Smolensky circuit lower bound
• Need to show replacer gates dont help us
  compute sum modulo 3

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BPPA vs. BQPA for random A

• Conjecture If BPPBQP, then BPPABQPA with
  probability 1
• What I can show If BPPBQP then
  BPTimepolylogBQTimepolylog
• Whats missing Extend the result of Beals et
  al. (1998) that D(f)O(Q2(f)6) for all total f to
  almost-total f
• Does the same hold for BPP vs. SZK, or even P
  vs. NP?coNP? (cf. Rudichs thesis)

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Limitations of Shor-like algorithms

• Defined a class BPP?BQPshor?BQP
• Subclass of quantum algorithms that prepare a
  state ?xx?f(x)?, then ignore f(x)? and do
  something simple to x?
• Conjecture 1 BQPshor?AM. Implies that if
  NP?BQPshor then PH?2
• Conjecture 2 Shor-like query algorithms yield
  no asymptotic speedup for any total function

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Physics Modulo Complexity Assumptions

• Can some version of M-theory decide SAT? (cf.
  Preskills talk)
• If so, move on to the next version!

• Anthropic computer for solving NP-complete
  problems efficiently

• Stupid question Why cant I just will myself
  to solve NP-complete problems? (Or generate
  truly random sequences?)

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Postulate No matter who you are, someone can
give you a 3SAT instance that you cant decide
with probability ½?.
What constraints does that impose?