**Multilinear Formulas and Skepticism of Quantum Computing** Scott Aaronson UC Berkeley IAS

**“We have never seen a physical law valid to over a dozen** decimals”(Leonid Levin) Even if QM is right, maybe our complexity model is wrong(Oded Goldreich) Is Quantum Computing Fundamentally Impossible? It violates the extended Church-Turing thesis Analogy to analog computing and unit-cost arithmetic

**DIVIDING LINE** “Sure/Shor Separator” Question: If quantum computing is impossible, then what criterion separates the quantum states that suffice for factoring from the states we’ve already seen?

**YAWN** 10000 polarized photons Buckyballs, superconducting coils, … The simplest candidates for such a criterion are nonstarters… Exponentially small amplitudes? Thousands of coherent qubits? Goal of Paper: Provide a formal framework for studying possible Sure/Shor separators

**AmpP** Circuit Tree P TSH OTree Vidal MOTree 2 2 1 1 Strict containmentContainmentNon-containment Classical Key Idea:Complexity classification of pure quantum states

**Intuitively, once we admit | and | into our set** of possible states, we’re almost forced to admit || and |+| as well! But what if we take the closure under a polynomial number of those operations?

**+** + + |11 |12 |01 |11 |02 |12 Tree size of |: Minimum number of vertices in such a tree representing | Tree states: Infinite families of states {|n}n1 for which TS(|n)=nO(1)

** If then TS(|)** equals the multilinear formula size of f to within a constant factor Minimum size of an arithmetic formula for f(x1,…,xn) involving +, , and complex constants, every node of which computes a multilinear polynomial in x1,…,xn Basic Facts About Tree Size • Many interesting states have polynomial TS1D spin chain, superposition over multiples of 3, … • A random state can’t even be approximated by states with subexponential TS

**Want to know the proof technique? Go to my talk** RAN RAZ Main Result: Tree Size Lower Bound Let C = {x | Axb (mod 2)}, where A is chosen uniformly at random from Then w.h.p. over A, the coset state has tree size n(log n)

**Significance: Coset states underlie quantum error** correction. Our result gives a second reason to prepare them: refuting the hypothesis that “all states in Nature are tree states” Purely classical corollary: First superpolynomial gap between general and multilinear formula size of functions Extensions:Tree size n(log n) is needed even to approximate coset statesCan derandomize, to get an n(log n) lower bound for an explicit coset state

**But what about states arising in Shor’s algorithm?** Conjecture: Let A consist of 5+log(n1/3) uniform random elements of {20,…,2n-1}. Let S consist of all 32n1/3 sums of subsets of A. If a prime p is drawn uniformly from [n1/3,1.1n1.3], then |Sp| 3n1/3/4with probability at least ¾, where Sp= {x mod p : xS} Theorem: Assuming the conjecture, Shor states have tree size n(log n)

**What’s been done in liquid NMR:** (Knill et al. 2001) Non-tree states might already have been observed in solid state! 2D or 3D spin lattices with nearest-neighbor Hamiltonians—similar to “cluster states” 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 Experimental Situation To interpret experiments, would be nice to have explicit tree size lower bounds for small n Key question: What kind of evidence is needed to prove a state’s existence?

**If a quantum computer is in a tree state at every time step,** can it then be simulated classically?Current result: Can be simulated in Open Problems Can we show exponential tree size lower bounds? Do all codeword states have superpolynomial tree size? Relationship between tree size and persistence of entanglement?