When Exactly Do Quantum Computers Provide A Speedup?

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When Exactly Do Quantum Computers Provide A
Speedup?

• Scott Aaronson (MIT)Papers slides at
  www.scottaaronson.com

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We all hear about the experimental progress
toward building quantum computers but in the
meantime, what about the applications? Its been
20 years since Peter Shor discovered his famous
factoring algorithm. Where are all the amazing
new applications we were promised?
Genesis of This Talk
Who promised you more quantum algorithms? Not me!
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The Parallelism Fallacy
Whats the source of the popular belief that
countless more quantum algorithms should exist?
To me, it seems tied to the idea that a quantum
computer could just try every possible answer in
parallel
But thats not how quantum computing works! You
need to choreograph an interference pattern,
where the unwanted paths cancel
The miracle, Id say, is that this trick yields a
speedup for any classical problems, not that it
doesnt work for more of them
Underappreciated challenge of quantum algorithms
research beating 60 years of classical
algorithms research
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An Inconvenient Truth
A problem has to be special even to be a
plausible candidate for an exponential quantum
speedup
NP-hard
3SAT
NP-complete
P?BQP, NP?BQP Plausible conjectures, which we
have no hope of proving given the current state
of complexity theory
Graph Iso
NP
Lattice Problems
BQP(Quantum P)
Factoring
Quantum Sim
P
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Rest of the Talk

1. Survey of the main families of quantum algorithms
   that have been discovered (and their
   limitations)
2. Results in the black-box model, which aim toward
   a general theory of when quantum speedups are
   possible
3. Lemons into lemonade implications for physics of
   the limitations of quantum computers

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Quantum SimulationWhat a QC does in its sleep
The original application of QCs!
My personal view still the most important one
Major applications (high-Tc superconductivity,
protein folding, nanofabrication, photovoltaics)
High confidence in possibility of a quantum
speedup
Can plausibly realize even before universal QCs
are available
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Shor-like Algorithms
The magic of the Fourier transform
In BQP Pretty much anything you can think of
that reduces to finding hidden structure in
abelian groupsFactoring, discrete log, elliptic
curve problems, Pells equation, unit groups,
class groups, Simons problem
Breaks almost all public-key cryptosystems used
todayBut theoretical public-key systems exist
that are unaffected
Can we go further? Hidden Subgroup
ProblemGeneralizes Shor to nonabelian groups.
Captures e.g. Graph Isomorphism
Alas, nonabelian HSP has been the Afghanistan of
quantum algorithms!
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Grover-like Algorithms
Quadratic speedup for any problem involving
searching an unordered list, provided the list
elements can be queried in superposition Implies
subquadratic speedups for many other basic
problems
Bennett et al. 1997 For black-box searching, the
square-root speedup of Grovers algorithm is the
best possible
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Quantum Walk Algorithms
Childs et al. 2003 Quantum walks can achieve
provable exponential speedups over classical
walks, but for extremely fine-tuned graphs
THE GLUED TREES
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Quantum Adiabatic Algorithm(Farhi et al. 2000)
Hi
Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete
problem
Problem Eigenvalue gap can be exponentially
small
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Landscapeology
Adiabatic algorithm can find global minimum
exponentially faster than simulated annealing
(though maybe other classical algorithms do
better)
Simulated annealing can find global minimum
exponentially faster than adiabatic algorithm (!)
Simulated annealing and adiabatic algorithm both
need exponential time to find global minimum
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Quantum Machine Learning Algorithms
Exponential quantum speedups for solving linear
systems, support vector machines, Google
PageRank, computing Betti numbers, EM scattering
problems

• THE FINE PRINT
• Dont get solution vector explicitly, but only as
  vector of amplitudes. Need to measure to learn
  anything!
• Dependence on condition number could kill
  exponential speedup
• Need a way of loading huge amounts of data into
  quantum state (which, again, could kill
  exponential speedup)
• Not ruled out that there are fast randomized
  algorithms for the same problems

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BosonSampling
Suppose we just want a quantum system for which
theres good evidence that its hard to simulate
classicallywe dont care what its useful for
A.-Arkhipov 2011, Bremner-Jozsa-Shepherd 2011 In
that case, we can plausibly improve both the
hardware requirements and the evidence for
classical hardness, compared to Shors factoring
algorithm
We showed if a fast, classical exact simulation
of BosonSampling is possible, then the polynomial
hierarchy collapses to the third level.
Experimental demonstrations with 3-4 photons
achieved (by groups in Oxford, Brisbane, Rome,
Vienna)
Our proposal Identical single photons sent
through network of interferometers, then measured
at output modes
For more My complex quantum systems seminar
tomorrow
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But you just listed a bunch of examples where
you know a quantum speedup, and other examples
where you dont! What you guys need is a theory,
which would tell you from first principles when
quantum speedups are possible.
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The Quantum Black-Box ModelThe setting for much
of what we know about the power of quantum
algorithms
Xx1xN
i
xi
X
Query complexity of f The minimum number of
queries used by any algorithm that outputs f(X),
with high probability, for every X of interest to
us
An algorithm can make query transformations,
which map
(iquery register, aanswer register,
wworkspace)
as well as arbitrary unitary transformations that
dont depend on X (we wont worry about their
computational cost).
Its goal is to learn some property f(X) (for
example is X 1-to-1?)
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Total Boolean Functions
D(f) Deterministic query complexity of FR(f)
Randomized query complexityQ(f) Quantum query
complexity
Example
Theorem (Beals et al. 1998) For all Boolean
functions f,
How to reconcile with the exponential speedup of
Shors algorithm? Totality of f.
Longstanding Open Problem Is there any Boolean
function with a quantum quantum/classical gap
better than quadratic?
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Almost-Total Functions?
Conjecture (A.-Ambainis 2011) Let Q be any
quantum algorithm that makes T queries to an
input X?0,1N. Then theres a classical
randomized that makes poly(T,1/?,1/?) queries to
X, and that approximates PrQ accepts X to
within ?? on a 1-? fraction of Xs
Theorem (A.-Ambainis) This would follow from an
extremely natural conjecture in discrete Fourier
analysis (every bounded low-degree polynomial
p0,1N?0,1 has a highly influential variable)
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The Collision Problem
Given a 2-to-1 function f1,,N?1,,N, find a
collision (i.e., two inputs x,y such that
f(x)f(y))
10 4 1 8 7 9 11 5 6 4 2 10 3 2 7 9
11 5 1 6 3 8
Variant Promised that f is either 2-to-1 or
1-to-1, decide which Models the breaking of
collision-resistant hash functionsa central
problem in cryptanalysis More structured than
Grover search, but less structured than Shors
period-finding problem
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Birthday Paradox Classically, ?N queries are
necessary and sufficient to find a collision with
high probability
Brassard-Høyer-Tapp 1997 Quantumly, N1/3
queries suffice
Grover on N2/3 f(x) values

N1/3 f(x) values queried classically

A. 2002 First quantum lower bound for the
collision problem (N1/5 queries are needed no
exponential speedup possible) Shi 2002 Improved
lower bound of N1/3. Brassard-Høyer-Tapps
algorithm is the best possible
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Symmetric Problems
A.-Ambainis 2011 Massive generalization of
collision lower bound. If f is any function
whatsoever thats symmetric under permuting the
inputs and outputs, and has sufficiently many
outputs (like collision, element distinctness,
etc.), then
New Result (Ben-David 2014) If fSN?0,1 is any
Boolean function of permutations, then
D(f)O(Q(f)12)
Upshot Need a structured promise if you want
an exponential quantum speedup
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Whats the largest possible quantum speedup?
Forrelation Given two Boolean functions
f,g0,1n?-1,1, estimate how correlated g is
with the Fourier transform of f
A.-Ambainis 2014 This problem is solvable using
only 1 quantum query, but requires at least
2n/2/n queries classically
Furthermore, this separation is essentially the
largest possible! Any N-bit problem thats
solvable with k quantum queries, is also solvable
with N1-1/2k classical queries
For details My CS theory seminar on Friday
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Can we turn the lemon of QCs limitations into
the lemonade of physical insight?
Proposal Adopt as a principle (conjecture?) that
theres no efficient way to solve NP-complete
problems in the physical world, then investigate
the implications for other issues
Example Implications- No closed timelike curves
(A.-Watrous 2009)- No postselected final state
(probably rules out Horowitz-Maldacena)-
Something like the holographic entropy bound
should hold- Metastable states must be
unavoidable in spin glasses, protein folding,
etc.- Many spectral gaps must decrease
exponentially with number of particles
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Explanation for the linearity of the
Schrödinger equation
Abrams Lloyd 1998 If quantum mechanics were
nonlinear, one could generically exploit that to
solve NP-complete problems in polynomial time
1 solution to NP-complete problem
No solutions
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A complexity-theoretic argument against hidden
variables?
A. 2004 In theories like Bohmian mechanics, in
order to sample the entire trajectory of the
hidden variable, youd need the ability to solve
the collision problemsomething I showed is
generically hard even for a quantum computer
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The Firewall Paradox (AMPS 2012) Refinement of
Hawkings information paradox that challenges
black hole complementarity
If the black hole interior is built out of the
same qubits coming out as Hawking radiation, then
why cant we do something to those Hawking
qubits, then dive into the black hole, and see
that weve completely destroyed the spacetime
geometry in the interior?
Entanglement among Hawking photons detected!
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Harlow-Hayden 2013 Striking argument that doing
the AMPS experiment would require solving a
problem thats exponentially hard even for a
quantum computer
A. 2014 Strengthened the Harlow-Hayden argument,
to show that a general ability to perform the
AMPS experiment would imply the ability to invert
any cryptographic one-way function
So, long before youve made a dent in the
problem, the black hole has already evaporated
anyway, and theres nowhere to jump to see a
firewall!
MODEL SITUATION
Is the geometry of spacetime protected by an
armor of computational complexity?
f,g Two functions for which we want to know
whether their ranges are equal or disjoint
R Old Hawking photonsB Hawking photon just
now coming outH Degrees of freedom still in
black hole
If we could detect entanglement between R and B
for any ??RBH, then we could solve a close
cousin of the collision problem!
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Summary
Exponential quantum speedups depend on
structure For example, abelian group structure,
glued-trees structure, forrelational structure
Sometimes we can even find such structure in
real, non-black-box problems of practical
interest (e.g., factoring)
The black-box model lets us develop a rich theory
of what kinds of structure do or dont suffice
for exponential speedups
Understanding the limitations of quantum
computers has given us new insights about
seemingly-remote issues in physics
Single most important application of QC (in my
opinion) Disproving the people who said QC was
impossible!