Multilinear Formulas and Skepticism of Quantum Computing

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Multilinear Formulas and Skepticism of Quantum
Computing

• Scott Aaronson, UC Berkeley
• http//www.cs.berkeley.edu/aaronson

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Outline

• Four objections to quantum computing
• Sure/Shor separators
• Tree states
• Result QECC states require n?(log n) additions
  and tensor products
• Experimental (!) proposal
• Conclusions and open problems

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Four Objections
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(A) QCs cant be built for fundamental
reasonLevins arguments
(1) Analogy to unit-cost arithmetic model (2)
Error-correction and fault-tolerance address only
relative error in amplitudes, not absolute (3)
We have never seen a physical law valid to over
a dozen decimals (4) If a quantum computer
failed, we couldnt measure its state to prove a
breakdown of QMso no Nobel prize
The present attitude is analogous to, say,
Maxwell selling the Daemon of his famous thought
experiment as a path to cheaper electricity from
heat
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Responses

• Continuity in amplitudes more benign than in
  measurable quantitiesshould we dismiss classical
  probabilities of order 10-1000?
• How do we know QMs successor wont lift us to
  PSPACE, rather than knock us down to BPP?
• To falsify QM, would suffice to show QC is in
  some state far from eiHt??. E.g. Fitch Cronin
  won 1980 Physics Nobel merely for showing CP
  symmetry is violated

Real Question How far should we extrapolate from
todays experiments to where QM hasnt been
tested?
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How Good Is The Evidence for QM?

• Interference Stability of e- orbits,
  double-slit, etc.
• Entanglement Bell inequality, GHZ experiments
• Schrödinger cats C60 double-slit experiment,
  superconductivity, quantum Hall effect, etc.

C60
Arndt et al., Nature 401680-682 (1999)
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Alternatives to QM
Roger Penrose
Gerard t Hooft( King of Sweden)
Stephen Wolfram
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(No Transcript)
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My View Any good argument for why quantum
computing is impossible must answer this
questionbut I havent seen any that do

• What Ill Do
• Initiate a complexity theory of (pure) quantum
  states, that studies possible Sure/Shor
  separators
• Prove a superpolynomial lower bound on tree
  size of states arising in quantum error
  correction
• Propose an NMR experiment to create states with
  large tree size

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Classes of Pure States
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Tree size TS(??) minimum number of
unbounded-fanin and ? gates, 0?s, and 1?s
needed in a tree representing ??. Constants are
free. Permutation order of qubits is
irrelevant. Example
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Tree States Families such that TS(?n?)?p(n)
for some polynomial p Will abuse and refer to
individual states
Motivation If we accept ?? and ??, we almost
have to accept ????? and ??????. But can
only polynomially many tensorings and
summings take place in the multiverse, because
of decoherence?
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Example Tree State
equal superposition over n-bit strings of
parity i
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Multilinear Formulas
Trees involving ,?, x1,,xn, and complex
constants, such that every vertex computes a
multilinear polynomial (no xi multiplied by
itself) Given let MFS(f) be minimum number of
vertices in multilinear formula for f
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Depth Reduction
Theorem Any tree state has a tree of polynomial
size and logarithmic depth Proof Idea Follows
Brents Theorem (1974), that any function with a
poly-size arithmetic formula has a formula of
polynomial size and logarithmic depth
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is an orthogonal tree state if it has a
polynomial-size tree that only adds orthogonal
states Theorem Any orthogonal tree state can be
prepared by a poly-size quantum circuit Proof
Idea If we can prepare ?? and ??, clearly can
prepare ?????. To prepare ?????? where
????0 let U0??n??, V0??n??. Then
Add OR of 2nd register to 1st register
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Theorem If is chosen uniformly under the
Haar measure, then with 1-o(1) probability, no
state ?? with TS(??)2o(n) satisfies
????2?15/16
Why Its Not Obvious
Proof Idea Use Warrens Theorem from real
algebraic geometry
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TreeBQP
Class of problems solvable by a quantum computer
whose state at every time is a tree state.
(1-qubit intermediate measurements are
allowed.) BPP ? TreeBQP ? BQP
Theorem Proof Idea Guess and verify trees use
Goldwasser-Sipser approximate counting Evidence
that TreeBQP ? BQP?
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QECC States
Let C be a coset in then Codewords of
stabilizer codes (Gottesman, CSS) Later well
add phases to reduce codeword size Take the
following distribution over cosets choose
u.a.r. (where kn1/3), then let
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Razs Breakthrough
Given coset C, let Need to lower-bound
multilinear formula size MFS(f)
LOOKS HARD
Until June, superpolynomial lower bounds on MFS
didnt exist Raz n?(log n) MFS lower bounds for
Permanent and Determinant of n?n
matrix (Exponential bounds conjectured, but
n?(log n) is the best Razs method can show)
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Idea of Razs Method
Given choose 2k input bits
u.a.r. Label them y1,,yk, z1,,zk Randomly
restrict remaining bits to 0 or 1 u.a.r. Yields a
new function Let
fR(y,z)
MR
z?0,1k
y?0,1k
Show MR has large rank with high probability over
choice of fR
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Intuition Multilinear formulas can compute
functions with huge rank, i.e. But once we
restrict everything except y1,,yk, z1,,zk, with
high probability rank becomes small
Theorem (Raz)
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Lower Bound for Coset States
x
A
b
If these two k?k matrices are invertible (which
they are with probability gt 0.2882), then MR is a
permutation of the identity matrix, so
rank(MR)2k
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Corollary

• First superpolynomial gap between multilinear and
  general formula size of functions
• f(x) is trivially NC1just check whether Axb
• Determinant not known to be NC1best formulas
  known are nO(log n)
• Still open Is there a polynomial with a
  poly-size formula but no poly-size multilinear
  formula?

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Inapproximability of Coset States
Fact For an N?N complex matrix
M(mij), (Follows from Hoffman-Wielandt
inequality) Corollary With ?(1) probability
over coset C, no state ?? with TS(??)no(log n)
has ??C?2?0.98
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Shor States
Superpositions over binary integers in arithmetic
progression letting w ?(2n-a-1)/p?, ( 1st
register of Shors alg after 2nd register is
measured) Conjecture Let S be a set of integers
with S32t and x?exp((log t)c) for all x?S
and some cgt0. Let Spx mod p x?S.
For sufficiently large t, if we choose a prime p
uniformly at random from t,5t/4, then Sp?3t/4
with probability at least 3/4 Theorem Assuming
the conjecture, there exist p,a for which
TS(pZa?)n?(log n)
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Challenge for NMR Experimenters

• Create a uniform superposition over a generic
  coset of (n?9) or even better, Clifford
  group state
• Worthwhile even if you dont demonstrate error
  correction
• Well overlook that its really

(1-10-5)I/512 10-5C??C
New test of QM are all states tree states?
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Tree Size Upper Bounds for Coset States
log2( of nonzero amplitudes)
n of qubi ts
Hardest cases (to left, use naïve strategy to
right, Fourier strategy)
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For Clifford Group States
log2( of nonzero amplitudes)
n of qubi ts
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Open Problems

• Exponential tree-size lower bounds
• Lower bound for Shor states
• Explicit codes (i.e. Reed-Solomon)
• Concrete lower bounds for (say) n9
• Extension to mixed states
• Separate tree states and orthogonal tree states
• PAC-learn multilinear formulas? TreeBQPBPP?
• Non-tree states already created in solid state?

Important for experiments
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Conclusions

• Complexity theory is relevant for experimental
  QIP
• Complexity of quantum states deserves further
  attention
• QC skeptics can strengthen their case (and help
  us) by proposing Sure/Shor separators
• QC experiments will test quantum mechanics
  itself in a fundamentally new way