Multilinear Formulas and Skepticism of Quantum Computing

Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley “The Proving Of” Documentary Spanish Version Trailers for Future Talks Announcements Start Talk

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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley “The Proving Of” Documentary Spanish Version Trailers for Future Talks Announcements Start Talk

Four Objections to Quantum Computing

(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 • Continuity in amplitudes more benign than in measurable quantities—should we dismiss classical probabilities of order 10-1000? • How do we know QM’s successor won’t 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 today’s experiments to where QM hasn’t been tested?

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 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? DIVIDING LINE

AmpP Circuit  Tree P TSH OTree Vidal  MOTree  2 2 1 1 Classical I hereby propose a complexity theory of pure quantum states one of whose goals is to study possible Sure/Shor separators. 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

+   + + |01 |12 |01 |11 |0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. Tree states are states with polynomially-bounded TS Example:

+ Theorem: If   then x1 x2 -3i x1 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

Grab Bag of Theorems 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

Coset States Let 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 Lower Bound To Be Proven:

Raz’s Breakthrough Given coset C, let Need to lower-bound multilinear formula size MFS(f) LOOKS HARD Until June, superpolynomial lower bounds on MFS didn’t 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 Raz’s method can show)

Cartoon of Raz’s 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 ALL QUESTIONS WILL BE ANSWERED BY THE NEXT TALK MR = fR(y,z) z{0,1}k y{0,1}k Theorem:

Lower Bound for Coset States x A b If these two kk matrices are invertible (which they are with probability > 0.2882), then MR is a permutation of the identity matrix, so rank(MR)=2k Non-Quantum Corollary: First superpolynomial gap between general and multilinear formula size of functions

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(logn) has ||C|20.98

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

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 : xS}. 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 States Equal superposition over all settings of qubits in a nn lattice, with phase=(-1)m where m is the number of pairs of neighboring ‘1’ qubits Conjecture: Cluster states have superpolynomial tree size

Bonus Feature: Terhal States Given an nn unitary matrix U and string x1…xn with Hamming weight k, let Ux be the kk 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

What’s been done: 5-qubit codeword in liquid NMR(Knill, Laflamme, Martinez, Negrevergne, quant-ph/0101034) TS(|)  69 Challenge for Experimenters • Create a uniform superposition over a “generic” coset of (n9) or even better, Clifford group state • Worthwhile even if you don’t demonstrate error correction • We’ll overlook that it’s really (1-10-5)I/512 + 10-5|CC| New test of QM: are all states tree states?

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)

For Clifford Group States 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

Multilinear Formulas and Skepticism of Quantum Computing