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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Mar 26, 2015

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Page 1: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

/ 2

Multilinear Formulas and Skepticism of Quantum

Computing

Scott Aaronson, UC Berkeley

http://www.cs.berkeley.edu/~aaronson

Page 2: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Outline(1) Four objections to quantum computing

(2) Sure/Shor separators

(3) Tree states

(4) Result: QECC states require n(log n) additions and tensor products

(5) Experimental (!) proposal

(6) Conclusions and open problems

Page 3: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Four Objections

Theoretical Practical

Physical (A): QC’s can’t be built for fundamental reason

(B): QC’s can’t be built for engineering reasons

Algorithmic (C): Speedup is of limited theoretical interest

(D): Speedup is of limited practical value

Page 4: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

(A): QC’s can’t be built for fundamental reason—Levin’s 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 couldn’t measure its state to prove a breakdown of QM—so 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”

Page 5: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Responses (1) Continuity in amplitudes more benign than in

measurable quantities—should we dismiss classical probabilities of order 10-1000?

(2) How do we know QM’s successor won’t lift us to PSPACE, rather than knock us down to BPP?

(3) 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 today’s experiments to where QM hasn’t been tested?

Page 6: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

How Good Is The Evidence for QM?(1) Interference: Stability of e- orbits, double-slit, etc.

(2) Entanglement: Bell inequality, GHZ experiments

(3) Schrödinger cats: C60 double-slit experiment, superconductivity, quantum Hall effect, etc.

C60

Arndt et al., Nature 401:680-682 (1999)

Page 7: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Alternatives to QM

Roger Penrose Gerard ‘t Hooft(+ King of Sweden)

Stephen Wolfram

Page 8: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Exactly what property separates the Sure States we know we can create, from the Shor States that

suffice for factoring?

DIV

IDIN

G L

INE

Page 9: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

My View: Any good argument for why quantum computing is impossible must answer this question—but I haven’t seen any that do

What I’ll 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

Page 10: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Classes of Pure States

2nH

Classical

Vidal

Circuit

AmpP

MOTree

OTree

TSH

Tree

P

1

2

1

2

Page 11: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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: 00 2 01 10 11 / 7

+

|01 |12

++

|01 |11 |02 |12

1

2

1

2

1

2

1

2

2

7

3

7

TS 11

Page 12: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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?

Tree States: Families such that TS(|n)p(n) for some polynomial p

Will abuse and refer to individual states

2 1

nn n

H

Page 13: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Example Tree State= equal superposition over n-bit strings of parity i

inP

2TS inP O n

0 0 0 1 1/ 2 / 2 / 2 / 2

1 0 1 1 0/ 2 / 2 / 2 / 2

1

21

2

n n n n n

n n n n n

P P P P P

P P P P P

0 1 0 111 TS

2 2 2

n nii i

n nP P O n

Page 14: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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

Multilinear Formulas

+

-3i x1

x1 x2

: 0,1 ,n

f

Theorem: If

0,1

,n

x

f x x

TS MFS f

then

Page 15: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Theorem: Any tree state has a tree of polynomial size and logarithmic depth

Proof Idea: Follows Brent’s Theorem (1974), that any function with a poly-size arithmetic formula has a formula of polynomial size and logarithmic depth

Depth Reduction

Page 16: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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 U|0n=|, V|0n=|. Then

1 10 0 1 0 0 0 0 0

0

n n n nU V U V

2nH

Add OR of 2nd register to 1st register

Page 17: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Why It’s Not Obvious:

Theorem: If is chosen uniformly under the Haar measure, then with 1-o(1) probability, no state | with TS(|)=2o(n) satisfies |||215/16

2nH

2nH

: TS 2o n

Proof Idea: Use Warren’s Theorem from real algebraic geometry

Page 18: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Theorem:

Proof Idea: Guess and verify trees; use Goldwasser-Sipser approximate counting

Evidence that TreeBQP BQP?

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

TreeBQP

3 3P PTreeBQP

Page 19: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

QECC StatesLet C be a coset in then

Codewords of stabilizer codes (Gottesman, CSS) Later we’ll add phases to reduce codeword size

Take the following distribution over cosets: choose u.a.r. (where k=n1/3), then let

2;n 1

x C

C xC

2 2,k n kA Z b Z

2 :nC x Z Ax b

logPr TS 1n

CC n Result:

Page 20: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Raz’s BreakthroughGiven coset C, let

Need to lower-bound multilinear formula size MFS(f)

1 if

0 otherwise

x Cf x

LOOKS HARD

Until June, superpolynomial lower bounds on MFS didn’t exist

Raz: n(log n) MFS lower bounds for Permanent and Determinant of nn matrix

(Exponential bounds conjectured, but n(log n) is the best Raz’s method can show)

Page 21: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Idea of Raz’s MethodGiven 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

: 0,1 ,n

f

, : 0,1 0,1k k

Rf y z

Show MR has large rank with high probability over

choice of fR

fR(y,z)MR =

y{0,1}k

z{0,1}k

Page 22: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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

1 1IP , mod 2n nx w x w x w

1 2 3 4 5 6 7 8 2 1 2

1 2 3 4 5 6 7 8 3 3 1

1 1 1 1 0

0 0 1 0 1

x x x x x x x x y z z

w w w w w w w w z y y

logPr rank 2 1 MFS nkRM c f n

Theorem (Raz):

Page 23: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Lower Bound for Coset States

1

2

3

1

2

3

0

1

11

0

y

y

y

z

z

z

b

xA

If these two kk matrices are invertible (which they are with probability > 0.2882), then MR is a permutation of the identity matrix, so rank(MR)=2k

0 1 1 0 0 0 1 0

1 0 1 1 0 1 1 1

0 0 0 1 0 1 0 0

Page 24: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Corollary

First superpolynomial gap between multilinear and general formula size of functions

• f(x) is trivially NC1—just check whether Ax=b

• Determinant not known to be NC1—best formulas known are nO(log n)

Still open: Is there a polynomial with a poly-size formula but no poly-size multilinear formula?

Page 25: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Inapproximability of Coset States

2rank ij ij

ij

M N m Fact: For an NN 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|20.98

Page 26: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Shor StatesSuperpositions over binary integers in arithmetic progression: letting w = (2n-a-1)/p,

(= 1st register of Shor’s alg after 2nd register is measured)

Conjecture: Let S be a set of integers with |S|=32t and |x|exp((log t)c) for all xS and some c>0. Let Sp={x mod p : xS}. 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(|pZ+a)=n(log n)

0

1 w

i

a p a piw

Page 27: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Challenge for NMR Experimenters• Create a uniform superposition over a “generic” coset of (n9) or even better, Clifford group state• Worthwhile even if you don’t demonstrate error correction• We’ll overlook that it’s really

2n

(1-10-5)I/512 + 10-5|CC|

New test of QM: are all states tree states?

What’s been done: 5-qubit codeword in liquid NMR(Knill, Laflamme, Martinez, Negrevergne, quant-ph/0101034)

00000 10010 01001 10100 01010 11011 00110 110001

4 11101 00011 11110 01111 10001 01100 10111 00101

TS(|) 69

Page 28: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Tree Size Upper Bounds for Coset States

0 1 2 3 4 5 6 7 8 9 10 11 12

1 1 3

2 3 7 7

3 4 9 17 10

4 5 11 21 27 13

5 6 13 25 49 33 16

6 7 15 29 57 77 39 19

7 8 17 33 65 121 89 45 22

8 9 19 37 73 145 185 101 51 25

9 10 21 41 81 161 305 225 113 57 28

10 11 23 45 89 177 353 433 249 125 63 31

11 12 25 49 97 193 385 705 545 273 137 69 34

12 13 27 53 105 209 417 833 993 593 297 149 75 37

log2(# of nonzero amplitudes)

n

#

of qubi ts

“Hardest” cases (to left, use naïve strategy; to right, Fourier strategy)

Page 29: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

For Clifford Group States

0 1 2 3 4 5 6 7 8 9 10 11 12

1 1 3

2 3 7 11

3 4 9 17 25

4 5 11 21 41 53

5 6 13 25 49 89 85

6 7 15 29 57 113 153 133

7 8 17 33 65 129 225 233 189

8 9 19 37 73 145 289 369 345 301

9 10 21 41 81 161 321 545 561 537 413

10 11 23 45 89 177 353 705 865 817 793 541

11 12 25 49 97 193 385 769 1281 1313 1265 1177 733

12 13 27 53 105 209 417 833 1665 1985 1889 1841 1689 957

log2(# of nonzero amplitudes)

n

#

of qubi ts

Page 30: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

Open Problems• Exponential tree-size lower bounds

• Lower bound for Shor states

• Explicit codes (i.e. Reed-Solomon)

• Concrete lower bounds for (say) n=9

• Extension to mixed states

• Separate tree states and orthogonal tree states

• PAC-learn multilinear formulas? TreeBQP=BPP?

• Non-tree states already created in solid state?

Important for experiments

Page 31: Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley aaronson.

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