Efficient Simulation of Quantum Mechanics Collapses the Polynomial Hierarchy

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Efficient Simulation of Quantum Mechanics
Collapses the Polynomial Hierarchy
(yes, really)

• Scott Aaronson Alex Arkhipov
• MIT

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In 1994, something big happened in our field,
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
BPP, unless Factoring is also
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Today A completely different kind of hardness
result for simulating quantum mechanics
Advantages of our result Based on PP?BPPNP
rather than Factoring?BPP Applies to an extremely
weak subset of QC(Non-interacting bosons, or
linear optics with a single nonadaptive
measurement at the end) Even gives evidence that
QCs have capabilities outside PH
Disadvantages Applies to distributional and
relation problems, not to decision
problems Harder to convince a skeptic that your
QC is really solving the relevant hard problem
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Let C be a quantum circuit, which acts on n
qubits initialized to the all-0 state
C defines a distribution DC over n-bit output
strings
QSampling? Given C as input, sample a string x
from any probability distribution D such that
Certainly this problem is BQP-hard
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More generallySuppose QSampling0.01 is in
probabilistic polytime with A oracle. Then
PP?BPPNP So QSampling cant even be in BPPPH
without collapsing PH!
A
Extension to relational problemsSuppose
FBQPFBPP. Then PPBPPNP
QSampling is P-hard under BPPNP-reductions (Pro
vided the BPPNP machine gets to pick the random
bits used by the QSampling oracle)
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Warmup Why Exact 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) Related to my result that
PostBQPPP Proof If we can estimate p, then we
can also compute ?xf(x) using binary search and
padding
Claim 2 Suppose QSampling was classically easy.
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 QSampling0 is easy. Then
PPBPPNP
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So Why Arent We Done?
Ultimately, our goal is to show that Nature can
actually perform computations that are hard to
simulate classically, thereby overthrowing the
Extended Church-Turing Thesis
But any real quantum system is subject to
noisemeaning we cant actually sample from DC,
but only from some distribution D such that
Could that be easy, even if sampling from DC
itself was hard?
To rule that out, we need to show that even a
fast classical algorithm for QSampling? would
imply PPBPPNP
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The Problem
Suppose M knew that all we cared about was the
final amplitude of 0?0? (i.e., thats where we
shoehorned a hard P-complete instance) Then it
could adversarially choose to be wrong about that
one, exponentially-small amplitude and still be a
good sampler So we need a quantum computation
that more robustly encodes a P-complete problem
Indeed. But to bring the permanent into quantum
computing, we need a brief detour into particle
physics (!) Well have to work harder but as a
bonus, well not only rule out approximate
samplers, but approximate samplers for an
extremely weak kind of QC
Hmm robust P-complete problem you mean like
the Permanent?
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Particle Physics In One Slide
There are two types of particles in Nature
BOSONS Force-carriers photons, gluons Swap two
identical bosons ? quantum state ?? is
unchanged Bosons can pile on top of each other
(and do lasers, Bose-Einstein condensates)
FERMIONS Matter quarks, electrons Swap two
identical fermions ? quantum state picks up -1
phase Pauli exclusion principle no two fermions
can occupy same state
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Consider a system of n identical, non-interacting
particles
Let aij?C be the amplitude for transitioning from
initial state i to final state j
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1
2
2
All I can say is, the bosons got the harder job
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Let
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tinitial
tfinal
Then whats the total amplitude for the above
process?
if the particles are bosons
if theyre fermions
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The BosonSampling Problem
Input An m?n complex matrix A, whose n columns
are orthonormal vectors in Cm (here m?n2) Let a
configuration be a list S(s1,,sm) of
nonnegative integers with s1smn Task Sample
each configuration S with probability
where AS is an n?n matrix containing si copies of
the ith row of A
Neat Fact The pSs sum to 1
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Physical Interpretation Were simulating a
unitary evolution of n identical bosons, each of
which can be in mpoly(n) modes. Initially,
modes 1 to n have one boson each and modes n1 to
m are unoccupied. After applying the unitary, we
measure the number of bosons in each mode.
Example
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Theorem (implicit in Lloyd 1996) BosonSampling ?
QSampling Proof Sketch We need to simulate a
system of n bosons on a conventional quantum
computer The basis states s1,,sm? (s1smn)
just record the occupation number of each
mode Given any scattering matrix U?Cm?m on the
m modes, we can decompose U as a product U1UT,
where TO(m2) and each Ut acts only on
2-dimensional subspaces of the form
for some (i,j)
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Theorem (Valiant 2001, Terhal-DiVincenzo 2002)
FermionSampling?BPP
In stark contrast, we prove the
following Suppose BosonSampling??BPP. Then
given an arbitrary matrix X?Cn?n, one can
approximate Per(X)2 in BPPNP
But I thought we could approximate the permanent
in BPP anyway, by Jerrum-Sinclair-Vigoda!
Yes, for nonnegative matrices. For general
matrices, approximating Per(X)2 is P-complete.
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Outline of Proof
Given a matrix X?Cn?n , with every entry
satisfying xij?1, we want to approximate
Per(X)2 to within ?n! This is already
P-complete (proof standard padding
tricks) Notice that Per(X)2 is a degree-2n
polynomial in the entries of X (as well as their
complex conjugates) As in Lipton/LFKN, we can let
V be some random curve in Cn?n that passes
through X, and let Y1,,Yk?Cn?n be other matrices
on V (where k?n2) If we can estimate Per(Yi)2
for most i, then we can estimate Per(X)2 using
noisy polynomial interpolation
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But Linear Interpolation Doesnt Work!
A random line through X?Cn?n retains too much
information about X
X
We need to redo Lipton/LFKN to work over the
complex numbers rather than finite fields
Solution Choose a matrix Y(t) of random
trigonometric polynomials, such that Y(0)X
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For sufficiently large L and tgtgt0, each yij(t)
will look like an independent Gaussian,
uncorrelated with xij
Furthermore, Per(Y(t)) is a univariate polynomial
in e2?it of degree at most Ln
Questions How do we sample Y(t) and Y1,,Yk
efficiently? How do we do the noisy polynomial
interpolation? Lazy answer Since were a BPPNP
machine, just use rejection sampling!
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The problem reduces to estimating Per(Y)2, for
a matrix Y?Cn?n of (essentially) independent
N(0,1) Gaussians To do this, generate a random
m?n column-orthonormal matrix A that contains Y/m
as an n?n submatrix (i.e., such that ASY/m for
some random configuration S) Let M be our BPP
algorithm for approximate BosonSampling, and let
r be Ms randomness Use approximate counting (in
BPPNP) to estimate
Intuition M has no way to determine which
configuration S we care about. So if its right
about most configurations, then w.h.p. we must
have
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Problem Bosons like to pile on top of each other!
Call a configuration S(s1,,sm) good if every si
is 0 or 1 (i.e., there are no collisions between
bosons), and bad otherwise We assumed for
simplicity that all configurations were good But
suppose bad configurations dominated. Then M
could be wrong on all good configurations, yet
still work Furthermore, the bosonic birthday
paradox is even worse than the classical one!
rather than ½ as with classical particles
Fortunately, we show that with n bosons and m?kn2
boxes, the probability of a collision is still at
most (say) ½
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Experimental Prospects

• What would it take to implement BosonSampling
  with photonics?
• Reliable phase-shifters
• Reliable beamsplitters
• Reliable single-photon sources
• Reliable photodetectors
• But crucially, no nonlinear optics or
  postselected measurements!

Problem The output will be a collection of n?n
matrices B1,,Bk with unusually large
permanentsbut how would a classical skeptic
verify that Per(Bi)2 was large? Our Proposal
Concentrate on (say) n30 photons, so that
classical simulation is difficult but not
impossible
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Open Problems
Does our result relativize? (Conjecture No) Can
we use BosonSampling to do universal QC? Can we
use it to solve any decision problem outside
BPP? Can you convince a skeptic (who isnt a
BPPNP machine) that your QC is indeed doing
BosonSampling? Can we get unlikely complexity
collapses from PBQP or PromisePPromiseBQP? Would
a nonuniform sampling algorithm (one that was
different for each scattering matrix A) have
unlikely complexity consequences? Is Permanent
P-complete for 1/-1 matrices (with no 0s)?
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Conclusion

• I like to say that we have three choices either
• The Extended Church-Turing Thesis is false,
• Textbook quantum mechanics is false, or
• QCs can be efficiently simulated classically.

For all intents and purposes