/ 2Multilinear Formulas and Skepticism

of Quantum Computing

Scott Aaronson, UC Berkeley

Trailers for Future Talks

“The Proving Of” Documentary Spanish Version

Announcements Start Talk

/ 2Multilinear Formulas and Skepticism

of Quantum Computing

Scott Aaronson, UC Berkeley

Trailers for Future Talks

“The Proving Of” Documentary Spanish Version

Announcements Start Talk

/ 2Multilinear Formulas and Skepticism

of Quantum Computing

Scott Aaronson, UC Berkeley

Trailers for Future Talks

“The Proving Of” Documentary Spanish Version

Announcements Start Talk

/ 2Multilinear Formulas and Skepticism

of Quantum Computing

Scott Aaronson, UC Berkeley

Trailers for Future Talks

“The Proving Of” Documentary Spanish Version

Announcements Start Talk

Live Coverage of QIP’2004

http://fortnow.com/lance/complog

/ 2Multilinear Formulas and Skepticism

of Quantum Computing

Scott Aaronson, UC Berkeley

Trailers for Future Talks

“The Proving Of” Documentary Spanish Version

Announcements Start Talk

Four Objections to Quantum Computing

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

(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”

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?

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)

Alternatives to QM

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

Stephen Wolfram

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

I hereby propose a complexity theory of

pure quantum

states

one of whose goals is to

study possible

Sure/Shor separators.

2nH

Classical

Vidal

Circuit

AmpP

MOTree

OTree

TSH

Tree

P

1

2

1

2

Strict containmentContainmentNon-containment

Boring Bonus Feature: Relations Between Computational and Quantum

State Complexity Questions

BQP = P#P implies AmpP P

AmpP P implies NP BQP/poly

P = P#P implies P AmpP

P AmpP implies BQP P/poly

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.

Tree states are states with polynomially-bounded TS

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

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

Theorem 1: Any tree state has a tree of polynomial size and logarithmic depth

Theorem 2: Any orthogonal tree state (where all additions are of orthogonal states) can be prepared by a polynomial-size quantum circuit

Theorem 3: Most quantum states can’t even be approximated by a state with subexponential tree size

Theorem 4: A quantum computer whose state is always a tree state can be simulated in the 3rd level of the classical polynomial-time hierarchy. Yields weak evidence that TreeBQP BQP

Grab Bag of Theorems

Coset StatesLet C be a coset in then

Codewords of stabilizer codes (Gottesman, Calderbank-Shor-Steane)

Take the following distribution over cosets: choose uniformly at random (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

Lower Bound To Be Proven:

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)

Cartoon 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

Theorem:

fR(y,z)MR =

y{0,1}k

z{0,1}k

logPr rank 2 1 MFS nkRM c f n

ALL QUESTIONS WILL BE

ANSWERED BY THE NEXT TALK

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

Non-Quantum Corollary: First superpolynomial gap between general and multilinear formula size of functions

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

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

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

Theorem: Assuming a number-theoretic conjecture, there exist p,a for which TS(|pZ+a)=n(log n)

Shor States

0

1 w

i

a p a piw

Bonus Feature: My original conjecture has been falsified by Carl Pomerance

Revised Conjecture (Not Yet Falsified & Obviously True)

Let A consist of 5+log(n1/3) subsets of {20,…,2n-1} chosen uniformly at random. For all 32n1/3 subsets B of A, let S contain the sum of the elements of B. Let S mod p = {x mod p : xS}. If p is chosen uniformly at random from [n1/3,1.1n1.3], then

Prp [|S mod p| 3n1/3/4] 3/4

Theorem: Assuming this conjecture, quantum states that arise in Shor’s algorithm have tree size n(log n)

Partial results toward proving the revised conjecture by Don Coppersmith

Bonus Feature: Cluster StatesEqual superposition over all settings of qubits in a nn lattice, with phase=(-1)m where m is the number of pairs of neighboring ‘1’ qubits

Conjecture: Cluster states have superpolynomial tree size

0 0 1 0 1

0 1 1 1 0

0 0 1 1 1

1 0 1 0 0

1 0 0 1 1

Given an nn unitary matrix U and string x1…xn with Hamming weight k, let Ux be the kk submatrix of U formed by the first k rows and the columns corresponding to xi=1. Then a Terhal state is

(Amazingly, these are always normalized)

Conjecture: Terhal states have superpolynomial tree size

Bonus Feature: Terhal States

0,1 :

detn

x

x x k

U x

Challenge for 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

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)

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

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?

Important for experiments