New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers

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New Evidence That Quantum Mechanics Is Hard to
Simulate on Classical Computers
?

• Scott Aaronson
• Parts based on joint work with Alex Arkhipov

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In 1994, something big happened in the
foundations of computer science, whose meaning is
still debated today
Why exactly was Shors algorithm
important? Boosters Because it means well build
QCs! Skeptics Because it means we wont build
QCs! Me For reasons having nothing to do with
building QCs!
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Shors algorithm was a hardness result for one of
the central computational problems of modern
science Quantum Simulation
Shors Theorem Quantum Simulation is not in
probabilistic polynomial time, unless Factoring
is also
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Today New kinds of hardness results for
simulating quantum mechanics
Advantages of the new results Based on more
generic complexity assumptions than classical
hardness of Factoring Give evidence that QCs have
capabilities outside the entire polynomial
hierarchy Use only extremely weak kinds of QC
(e.g. nonadaptive linear optics)
Disadvantages Most apply to sampling problems
(or problems with many possible valid outputs),
rather than decision problems Harder to convince
a skeptic that your QC is indeed solving the
relevant hard problem Problems not useful (?)
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What Is The Polynomial Hierarchy?
Example of a PH problem For all n-bit strings
x, does there exist an n-bit string y such that
for all n-bit strings z, ?(x,y,z) holds?
such-and-such is true ? PH collapses to a finite
level is complexity-ese for such-and-such
would be almost as insane as PNP
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BQP vs. PH A Timeline
1990
1995
2000
2005
2010
Bernstein and Vazirani define BQP They construct
an oracle problem, Recursive Fourier Sampling,
that has quantum query complexity n but classical
query complexity n?(log n) First example where
quantum is superpolynomially better! A simple
extension yields RFS?MA Natural conjecture
RFS?PH Alas, we cant even prove RFS?AM!
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Results In The Oracle WorldFrom arXiv0910.4698
There exist oracle sampling and relational
problems in BQP that are not in BPPPH
Assuming the Generalized Linial-Nisan
Conjecture, there exists an oracle decision
problem in BQP but not in PH Original
Linial-Nisan Conjecture was recently proved by
Braverman, after being open for 20 years
Unconditionally, there exists an oracle decision
problem that requires ?(N1/4) queries classically
(even using postselection), but only 1 query
quantumly
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Results In The Real WorldFrom not-yet-arXived
joint work with Alex Arkhipov
Suppose the output distribution of any
linear-optics circuit can be efficiently sampled
classically (e.g., by Monte Carlo). Then
PPBPPNP, and hence PH collapses. Indeed, even
if such a distribution can be sampled in BPPPH,
still PH collapses. Suppose the output
distribution of any linear-optics circuit can
even be approximately sampled in BPP. Then a
BPPNP machine can additively approximate Per(X),
with high probability over a matrix X of i.i.d.
N(0,1) Gaussians. Permanent-of-Gaussians
Conjecture The above problem is P-complete.
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Fourier Fishing Problem
Given oracle access to a random Boolean function
The Task Output strings z1,,zn, at least 75 of
which satisfy and at least 25 of which satisfy
where
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Fourier Fishing Is In BQP
Repeat n times output whatever you see
Algorithm
Distribution over Fourier coefficients
Distribution over Fourier coefficients output by
quantum algorithm
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Fourier Fishing Is Not In PH
Basic Strategy Suppose an oracle problem is in
PH. Then by reinterpreting every ? quantifier as
an OR gate, and every ? quantifier as an AND
gate, we can get an AC0 (constant-depth,
unbounded-fanin, quasipolynomial-size) circuit
for an exponentially-scaled down version of the
problem And AC0 circuits are one of the few
things in complexity theory that we can actually
lower-bound! In particular, it was proved in the
1980s that any AC0 circuit for Majority (or for
computing a Fourier coefficient) must have
exponential size Problem In our case, the AC0
circuit C doesnt have to compute the Fourier
coefficientsit just has to sample from some
probability distribution defined in terms of
them! To deal with that, we use a
nondeterministic reduction (which adds more
layers to the circuit), to show that C would
nevertheless lead to an AC0 circuit for Majority
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Decision Version Fourier Checking
Given oracle access to two Boolean functions

• Decide whether
• ?f,g? are drawn from the uniform distribution U,
  or
• ?f,g? are drawn from the following forrelated
  distribution F pick a random unit vector

then let
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Fourier Checking Is In BQP
H
H
0?
H
H
H
f
0?
g
H
H
H
0?
H
Probability of observing 0??n
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Evidence That Fourier Checking ? PH
We can prove that, even after you condition on
any setting for any polynomial number of f(x)s
and g(y)s, you still have almost no
information about whether f and g are independent
or forrelated We conjecture that this property,
by itself, is enough to imply an oracle problem
is not in PH. We call this the Generalized
Linial-Nisan Conjecture The original Linial-Nisan
Conjecturethe same statement, but without the
almostwas proved last year by Braverman, in a
major breakthrough in complexity theory
(indirectly inspired by this work ?)
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Coming back to the first result, whats
surprising is that we showed hardness of a BQP
sampling problem, by using a nondeterministic
reduction from Majoritya P problem! This
raises a question is something similar possible
in the unrelativized (non-black-box) world?
Indeed it is. Consider the following
problem QSampling Given a quantum circuit C,
which acts on n qubits initialized to the all-0
state. Sample from Cs output distribution.
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Why QSampling Is Hard
Let f0,1n?-1,1 be any efficiently computable
function. Suppose we apply the following quantum
circuit
Then the probability of observing the all-0
string is
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Claim 1 p is P-hard to estimate (up to a
constant factor) Proof If we can estimate p,
then we can also compute ?xf(x) using binary
search and padding
Claim 2 Suppose QSampling?BPP. Then we could
estimate p in BPPNP Proof Let M be a classical
algorithm for QSampling, and let r be its
randomness. Use approximate counting to estimate
Conclusion Suppose QSampling?BPP. Then PPBPPNP
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Related Results
A. 2004 PostBQPPP Bremner, Jozsa, Shepherd
(poster 1) PostIQPPP, hence efficient
simulation of IQP collapses PH Fenner, Green,
Homer, Pruim 1999 Determining whether a quantum
circuit accepts with nonzero probability is hard
for PH
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Ideally, we want a simple, explicit quantum
system Q, such that any classical algorithm that
even approximately simulates Q would have
dramatic consequences for classical complexity
theory
We believe this is possible, using
non-interacting bosons
There are two basic types of particle in the
universe
All I can say is, the bosons got the harder job
BOSONS
FERMIONS
Their transition amplitudes are given
respectively by
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Linear Optics for Dummies (or computer scientists)
Computational basis states have the form
S?s1,,sm?, where s1,,sm are nonnegative
integers such that s1smn n of photons
m of modes (boxes) that each photon can
be in
Theorem (Lloyd 1996 et al.) BosonP ? BQP Proof
Idea Decompose U into a product of O(m2)
elementary linear-optics gates (beamsplitters
and phase-shifters), then simulate each gate
using standard qubit gates
Starting from a fixed basis state (like
??1,,1,0,0?), you get to choose an arbitrary
m?m unitary U to apply U induces an
unitary V on
n-photon states, defined by
Theorem (Knill, Laflamme, Milburn 2001) Linear
optics with adaptive measurements can do all of
BQP By contrast, well use just a single
(nonadaptive) measurement of the photon numbers
at the end
where US,T is an n?n submatrix of U indexed by
S,T (containing an si?tj block of uijs for each
i,j)
Then you get to measure V?? in the computational
basis
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Our Result Take a system of n identical photons
with mO(n2) modes. Put each photon in a known
mode, then apply a Haar-random m?m unitary
transformation U
U
Let D be the distribution that results from
measuring the photons. Suppose theres a BPP
algorithm that takes U as input, and samples any
distribution even 1/poly(n)-close to D in
variation distance. Then in BPPNP, one can
estimate the permanent of a matrix X of i.i.d.
N(0,1) complex Gaussians, to additive error
with high probability over X.
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PGC?Hardness of BosonSampling
Idea Given a Gaussian random matrix X, well
smuggle ?X into the unitary transition matrix U
for mO(n2) photonsin such a way that
?SV??Per(?X), for some basis state S? Useful
fact we rely on given a Haar-random m?m unitary
matrix, an n?n submatrix looks approximately
Gaussian
Then the sampler has no way of knowing which
submatrix of U we care aboutso even if it has
1/poly(n) error, with high probability it will
return S? with probability ?Per(?X)2
Then, just like before, we can use approximate
counting to estimate PrS??Per(?X)2 in BPPNP,
and thereby solve P
Difficulty The bosonic birthday paradox!
Identical bosons like to pile on top of each
other, and thats bad for us
SO WE DEAL WITH IT
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Experimental Prospects

• What would it take to implement the requisite
  experiment?
• Reliable phase-shifters and beamsplitters, to
  implement an arbitrary unitary on m photon modes
• Reliable single-photon sources Fock states, not
  coherent states
• Reliable photodetector arrays
• But crucially, no nonlinear optics or
  postselected measurements!

Our Proposal Concentrate on (say) n30 photons,
so that classical simulation is difficult but not
impossible
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Prize Problems
Prove the Generalized Linial-Nisan
Conjecture!Yields an oracle A such that
BQPA?PHA Prove the Permanent of Gaussians
Conjecture!Would imply that even approximate
classical simulation of linear-optics circuits
would collapse PH
Do a linear optics experiment that overthrows the
Polynomial-Time Church-Turing Thesis
?