/2 Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley http://www.cs.berkeley.edu/ ~aaronson
Mar 26, 2015
/ 2
Multilinear Formulas and Skepticism of Quantum
Computing
Scott Aaronson, UC Berkeley
http://www.cs.berkeley.edu/~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
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
(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
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
Classes of Pure States
2nH
Classical
Vidal
Circuit
AmpP
MOTree
OTree
TSH
Tree
P
1
2
1
2
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
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
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
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: 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
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
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
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
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:
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)
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
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):
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
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?
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
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
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
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 in solid state?
Important for experiments
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