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When Exactly Do Quantum
Computers Provide A Speedup?
Scott Aaronson (MIT)
Papers & slides at www.scottaaronson.com
Genesis of This Talk
“We all hear about the experimental progress
toward building quantum computers … but in
the meantime, what about the applications?
It’s been 20 years since Peter Shor discovered
his famous factoring algorithm. Where are all
the amazing new applications we were
promised?”
Who promised you more quantum algorithms? Not me!
The Parallelism Fallacy
What’s 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 that’s not how quantum computing works!
You need to choreograph an interference
pattern, where the unwanted paths cancel
The miracle, I’d say, is that this trick yields a
speedup for any classical problems, not that it
doesn’t work for more of them
Underappreciated challenge of quantum algorithms
research: beating 60 years of classical algorithms research
An Inconvenient Truth
A problem has to be special even to be a plausible
candidate for an exponential quantum speedup
3SAT
NP-hard
NP-complete
NP
BQP
(Quantum P)
P
P≠BQP, NPBQP:
Plausible conjectures,
which we have no
hope of proving given
the current state of
complexity theory
Rest of the Talk
I.
Survey of the main families of quantum algorithms
that have been discovered (and their limitations)
II. Results in the black-box model, which aim toward
a general theory of when quantum speedups are
possible
III. Lemons into lemonade: implications for physics of
the limitations of quantum computers
Quantum Simulation
“What 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
Shor-like Algorithms
“The magic of the Interesting
Fourier transform”
In BQP: Pretty much anything you can think of that
reduces to finding hidden structure in abelian groups
Factoring, discrete log, elliptic curve problems, Pell’s
equation, unit groups, class groups, Simon’s problem…
Breaks almost all public-key cryptosystems used today
But theoretical public-key systems exist that are unaffected
Can we go further? Hidden Subgroup Problem

Generalizes Shor to nonabelian groups. Captures e.g. Graph Isomorphism
Alas, nonabelian HSP has been the Afghanistan of quantum algorithms!
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 squareroot speedup of Grover’s algorithm is the best possible
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
Quantum Adiabatic Algorithm
(Farhi et al. 2000)
Hi
Hamiltonian with easilyprepared ground state
Hf
Ground state encodes solution
to NP-complete problem
Problem: “Eigenvalue gap”
can be exponentially small
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
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:
1. Don’t get solution vector explicitly, but only as vector of
amplitudes. Need to measure to learn anything!
2. Dependence on condition number could kill exponential speedup
3. Need a way of loading huge amounts of data into quantum state
(which, again, could kill exponential speedup)
4. Not ruled out that there are fast randomized algorithms for the
same problems
BosonSampling
Suppose we just want a quantum system for which there’s
good evidence that it’s hard to simulate classically—we don’t
care what it’s useful for
A.-Arkhipov 2011, Bremner-Jozsa-Shepherd 2011: In that
case, we can plausibly improve both the hardware
We showed: if a fast, classical
requirements and the evidence for classicalExperimental
hardness,
exact
of algorithm
demonstrations with 3-4
compared
to simulation
Shor’s factoring
BosonSampling is possible,
photons achieved (by
Ourhierarchy
proposal:
then the polynomial
groups in Oxford,
Identical
single Brisbane, Rome, Vienna)
collapses to the third
level.
photons sent
through network of
interferometers,
For more: My complex
quantum systems seminar tomorrow
then measured at
output modes
“But you just listed a bunch of examples
where you know a quantum speedup, and
other examples where you don’t! What
you guys need is a theory, which would tell
you from first principles when quantum
speedups are possible.”
The Quantum Black-Box Model
The setting for much of what we know about the power of
quantum algorithms
i
X=x1…xN
xi
X
“Query complexity” of f: The minimum
number of queries used by any
i,a,w i, athat
, w outputs
 if(X),
, a high
xi , w
algorithm
,a, w iwith
probability,
every a=“answer
X of interest
to us
(i=“queryfor
register,”
register,”
w=“workspace”)
An algorithm can make query transformations, which map


as well as arbitrary unitary transformations that don’t depend
on X (we won’t worry about their computational cost).
Its goal is to learn some property f(X) (for example: is X 1-to-1?)
Total Boolean Functions
f : 0,1  0,1
N
D(f): Deterministic query complexity of F
R(f): Randomized query complexity
Q(f): Quantum query complexity
Example: DORN   RORN   N ,
QORN  ~ N
Theorem (Beals et al. 1998): For all Boolean functions f,

D f   O Q f 
6

How to reconcile with the exponential
speedup of Shor’s algorithm? Totality of f.
Longstanding Open Problem: Is there any Boolean function
with a quantum quantum/classical gap better than quadratic?
Almost-Total Functions?
Conjecture (A.-Ambainis 2011): Let Q be any quantum
algorithm that makes T queries to an input X{0,1}N.
Then there’s a classical randomized that makes poly(T,1/,1/)
queries to X, and that approximates Pr[Q accepts X] to within
 on a ≥1- fraction of X’s
Theorem (A.-Ambainis): This would follow from an extremely
natural conjecture in discrete Fourier analysis (“every bounded
low-degree polynomial p:{0,1}N[0,1] has a highly influential
variable”)
The Collision Problem
Given a 2-to-1 function f:{1,…,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
functions—a central problem in cryptanalysis
“More structured than Grover search, but less
structured than Shor’s period-finding problem”
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-HøyerTapp’s algorithm is the best possible
Symmetric Problems
A.-Ambainis 2011: Massive generalization of collision lower
bound. If f is any function whatsoever that’s symmetric
under permuting the inputs and outputs, and has sufficiently
many outputs (like collision, element distinctness, etc.), then


R f   O Q f  polylogQ f 
7
New Result (Ben-David 2014): If f:SN{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
What’s the largest possible
quantum speedup?
“Forrelation”: Given two Boolean functions f,g:{0,1}n{-1,1},
estimate how correlated g is with the Fourier transform of f:
1
2
3n / 2
 f x  1
x y
x , y0 ,1n
gy
 1/ 3?
 2 / 3?
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 that’s solvable with k quantum
queries, is also solvable with ~N1-1/2k classical queries
For details: My CS theory seminar on Friday
Can we turn the lemon of QCs’ limitations
into the lemonade of physical insight?
Proposal: Adopt as a principle (conjecture?) that
there’s 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
“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
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,
you’d need the ability to solve the collision problem—
something I showed is generically hard even for a
quantum computer
1
N
x  y
N
 x f x 
x 1
Measure 2nd
register
2
f x 
The Firewall Paradox (AMPS 2012):
Refinement of Hawking’s 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 can’t
we do something to those Hawking
qubits, then dive into the black hole,
and see that we’ve completely
destroyed the spacetime geometry
in the interior?
Entanglement among
Hawking photons detected!
Harlow-Hayden 2013: Striking argument that doing the
AMPS experiment would require solving a problem that’s
exponentially
hard
even
for
a
quantum
computer
A. 2014: Strengthened the Harlow-Hayden
argument,
to showmade
that aa dent
general
ability
to perform
“So, long
before you’ve
in the
problem,
the black
AMPSevaporated
experimentanyway,
would imply
the ability
to to
hole hasthe
already
and there’s
nowhere
invert
any cryptographic one-way function
jump to see
a firewall!”
MODEL SITUATION:

RBH

Is the geometry
x,0ofRspacetime
0 B f x protected
 x,1 Rby1 an
g x 

H
B
1
2narmor
x0,of
1n computational complexity?
1
R: “Old” Hawking photons
B: Hawking photon just now coming out
H: Degrees of freedom still in black hole
H
f,g: Two functions for
which we want to know
whether their ranges
are equal or disjoint
If we could detect entanglement between R and B for any |RBH,
then we could solve a close cousin of the collision problem!

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-blackbox problems of practical interest (e.g., factoring)
The black-box model lets us develop a rich theory of what
kinds of structure do or don’t 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!

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