Quantum Computing: What

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Quantum ComputingWhats It Good For?

• Scott Aaronson
• Computer Science Department, UC Berkeley
• January 10, 2002
• www.cs.berkeley.edu/aaronson

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Overview

1. History and background
2. The quantum computation model
3. Example Simons algorithm
4. Other algorithms (Shors, Grovers)
5. Limits of quantum computing, including recent
   work
6. The future

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Richard Feynman (1981) ...trying to find a
computer simulation of physics, seems to me to be
an excellent program to follow out...and I'm not
happy with all the analyses that go with just the
classical theory, because nature isnt classical,
dammit, and if you want to make a simulation of
nature, you'd better make it quantum mechanical,
and by golly it's a wonderful problem because it
doesn't look so easy.
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David Deutsch (1985) Computing machines
resembling the universal quantum computer could,
in principle, be built and would have many
remarkable properties not reproducible by any
Turing machine Complexity theory for such
machines deserves further investigation.
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What Is Quantum Mechanics?
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What Is Quantum Mechanics?
Traditional Physics View Quantum Computing View
Framework for atomic-scale physical theories Computational model with amplitudes instead of probabilities
Complicated (lots of integral signs) Simple
Pessimistic (i.e. Heisenberg uncertainty relation) Optimistic (i.e. Shors factoring algorithm)
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The Model

• Computer has n bits of memory

• Classical case if n2, possible states are 00,
  01, 10, 11

• Randomized case States are vectors of 2n
  probabilities in 0,1
• i.e. Pr000.2 Pr010.2 Pr100.1
  Pr110.5

• Quantum case States are vectors of 2n complex
  numbers called amplitudes

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The Model (cont)

• Dirac ket notation We write state as, i.e.,
• 0.5 00? - 0.5 01? 0.5i 10? - 0.5i 11?
• Superposition over basis states

• Normalization If state is ?i?ii?, then ?i?i2
  1
• (Why complex numbers? Why ?i2 and not ?i2?)

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Measurement

• When we measure state, see basis state i? with
  probability ?i2

• Furthermore, state collapses to i?

• Can also make partial measurements

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Time Evolution

• Matrix U is unitary iff UUI, conjugate
  transpose
• Equivalently U preserves norm

• Can multiply amplitude vector by some unitary U
  (i.e. replace state ?? by U??)

• Quantum analogue of Markov transitions

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Example Square Root of NOT

• Hadamard matrix

H0? (0?1?)/?2 H1? (0?-1?)/?2
H(0?1?)/?2 0? H(0?-1?)/?2 1?
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Quantum Circuits

• Unitary operation is local if it applies to only
  a constant number of bits (qubits)

• Given a yes/no problem of size n
• Apply order nk local unitaries for constant k
• Measure first bit, return yes iff its 1

• BQP class of problems solvable by such a circuit
  with error probability at most 1/3
• ( technical requirement uniformity)

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The Power of Quantum Computing

• Bernstein-Vazirani 1993
• BPP ? BQP ? PSPACE
• BPP solvable classically with order nk time
• PSPACE solvable with order nk memory

• Apparent power of quantum computing comes from
  interference
• Probabilities always nonnegative
• But amplitudes can be negative (or complex), so
  paths leading to wrong answers can cancel each
  other out

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Simons Problem
Given a black box
f(x)
x
Promise There exists a secret string s such that
f(x)f(y) ? yx?s for all x,y (? bitwise
XOR) Problem Find s with as few queries as
possible
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Example
Input x Output f(x)
000 4
001 2
010 3
011 1
100 2
101 4
110 1
111 3
Secret string s 101 f(x)f(x?s)
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Simons Algorithm

• Classically, order 2n/2 queries needed to find s
• - Even with randomness

• Simon (1993) gave quantum algorithm using only
  order n queries

• Assumption given x?, can compute x?f(x)?
  efficiently

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Simons Algorithm (cont)
1. Prepare uniform superposition
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Simons Algorithm (cont)
4. Apply to each bit of
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Simons Algorithm (cont)
5. Measure. Obtain a random y such that
7. Solve for s. Can show solution is unique with
high probability.
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Schematic Diagram
O
0?
b
s
0?
e
r
0?
v
e
O
0?
b
0?
s
f(x)
e
r
0?
v
e
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Period Finding

• Given Function f from 12n to 12n
• Promise There exists a secret integer r such
  that f(x)f(y) ? r x-y for all x
• Problem Find r with as few queries as possible

• Classically, order 2n/3 queries to f needed

• Inspired by Simon, Shor (1994) gave quantum
  algorithm using order poly(n) queries

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Example r5
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Factoring and Discrete Log

• Using period-finding, can factor integers in
  polynomial time (Miller 1976)

• Also discrete log given a,b,N, find r such that
  ar?b(mod N)

• Breaks widely-used public-key cryptosystems
  RSA, Diffie-Hellman, ElGamal, elliptic curve
  systems

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Grovers Algorithm

Unsorted database of n items
Goal Find one marked item

• Classically, order n queries to database needed

• Grover 1996 Quantum algorithm using order ?n
  queries

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Limits of Quantum Computing

• Bennett et al. 1996 Grovers algorithm is
  optimal
• (Quantum search requires order ?n queries)

• Beals et al. 1998 For all total Boolean
  functions f 0,1n?0,1,
• if quantum algorithm to evaluate f uses T
  queries,
• exists classical algorithm using order T6
  queries.

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Collision Problem

• Given a function f 1,,n?1,,N, n even
• Promise f is either 1-1 (i.e. 3,7,9,2) or 2-1
  (5,2,2,5)
• Problem Decide which

• Models graph isomorphism, breaking cryptographic
  hash functions

• Classical algorithm needs order ?n queries to f

• Brassard et al. 1997 Quantum algorithm using
  n1/3 queries

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Collision Lower Bound

• Can a quantum algorithm do better than n1/3?
  Previously couldnt even rule out constant
  number of queries!

• A 2001 Any quantum algorithm for collision
  needs order n1/5 queries

• Shi 2001 Improved to order n1/3

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The Future
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The Future

• When processor components reach atomic scale,
  Moores Law breaks down
• Quantum effects become important whether we
  want them or not
• But huge obstacles to building a practical
  quantum computer!

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Implementation
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Implementation

• Key technical challenge prevent decoherence,
  or unwanted interaction with environment

• Approaches NMR, ion trap, quantum dot,
  Josephson junction, optical

• Recent achievement 153?5 (Chuang et al. 2001)

• Larger computations will require quantum
  error- correcting codes

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Quantum Computing Whats It Good For?

• Potential (benign) applications
• Faster combinatorial search
• Simulating quantum systems

• Spinoff in quantum optics, chemistry, etc.

• Makes QM accessible to non-physicists

• Surprising connections between physics and CS

• New insight into mysteries of the quantum