Superpolynomial speedups from the quantum Fourier transform on the symmetric group

1. (a) ANY Superpolynomial speedups fromthe quantum Fourier transform on the symmetric group • Sean Hallgren, NEC • Aram Harrow, Bristol (b): almost any quantum circuit QIP 2007

2. Official way to find quantum speedups This talk’s approach 1. Find a useful/interesting problem. 1. Start with a (poly-size) quantum circuit U. 2. Prove classical lower bounds for some natural oracle formulation. 2. Cook up an oracle problem which U solves quickly. 3. Find an efficient quantum algorithm. 3. Derive classical lower bounds from information theory. Guiding principles

3. The plan • Review Recursive Fourier Sampling[BV93]. • Generalize Fourier sampling. • Generalize the recursion. • Circuits yielding superpolynomial speedups: • the quantum Fourier transform over any finite group, • a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

4. Fourier sampling on [BV93] Reduce to state identification: 1. For each a, define 2. H⊗n|Ψai = |ai 3. If O |xi|0i= |xi|a∙xi, then we can prepare |Ψai with one call to O and one call to O†. • Goal: Find secret string a ∈ {0,1}n =: A. Classical (randomized) query lower bound of Ω(log |A|) = Ω(n) from information theory. quantum: O(1) queries, poly(n) time. classical: Ω(n) queries

5. The plan • Review Recursive Fourier Sampling [BV93]. • Generalize Fourier sampling. • Generalize the recursion. • Circuits yielding superpolynomial speedups: • the quantum Fourier transform over any finite group, • a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

6. Generalization:oracle-assisted state identification Reduce to state identification: 1. For each a, define 2. If O|xi|0i= |xi|f(a,x)i, then we can prepare |Ψai with one call to O and one call to O†. 3. There exists U s.t. |ha|U|Ψai|2= Ω(1) for all a∈A. • Goal: Find secret string a ∈ A ⊆ {0,1}n . Classical (randomized) query lower bound of Ω(log |A|) from information theory. quantum: O(1) queries, poly(n) time. classical: Ω(log|A|) queries

7. Oracle-assisted state identification:key ingredients • Circuit U of size poly(n) acting on n qubits. • A large set A ⊆ {0,1}n. [i.e. log |A|=Ω(n)] • A function f: A×{0,1}n→{0,1} such that|ha|U|Ψai|=Ω(1), for all a∈A.Recall: • Such an f exists iff, for all a∈A,

8. Dispersing circuits Definition: A unitary U on n qubits is (α,β)-dispersing if there exists a set A⊆{0,1}n with |A|≥2αn andfor all a∈A. e.g.: H⊗n and the standard QFT are both (1,1)-dispersing. Lemma: If U is (α,β)-dispersing and can be constructed in poly(n) time, then we can use it to define an oracle problem solvable using O(1/β2) quantum queries + poly(n/β2) quantum time and requiring Ω(αn) classical queries.

9. The plan • Review Recursive Fourier Sampling [BV93]. • Generalize Fourier sampling. • Generalize the recursion. • Circuits yielding superpolynomial speedups: • the quantum Fourier transform over any finite group, • a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

10. Recursive amplification Idea: Learning f(a,x) requires first solving a subproblem (equivalent to the original problem) depending on x. Define function s:{0,1}n→A and oracle O1 such that O1 (x, s(x)) = f(a,x)O1 (x, s′) = FAIL if s′≠s(x) How do we learn s(x)? A second oracle, O2, on input (x1,x2), outputs f(s(x1),x2).

11. Recursive amplification, cont. • Define l layers of recursion. • s(x1), s(x1, x2), ..., s(x1, ..., xl-1) ∈ A • For 1≤k<l,Ok(x1,...,xk, s(x1,...,xk) = f(s(x1,...,xk-1),xk) [s(Ø)=a]Ok(x1,...,xk, ≠s(x1,...,xk) = FAIL • Ol(x1,...,xl) = f(s(x1,...,xl-1), xl) quantum: Q queries →O((2Q)l) queries (need to uncompute) classical: Ω(log |A|) queries →Ω((log |A|/2) l) queries

12. Superpolynomial speedup • Take l =Θ(log n). • quantum: O(1) queries and poly(n) time becomes poly(n) queries and time. • classical: nΩ(1) queries becomes nΩ(log n) queries. • Corollary: Any (Ω(1),Ω(1))-dispersing circuit gives rise to some superpolynomial speedup. • Note: Unlike [BV93], this construction cannot place BQP outside of PH, or even NP. However, it can handle any Ω(1) probability of success.

13. The plan • Review Recursive Fourier Sampling [BV93]. • Generalize Fourier sampling. • Generalize the recursion. • Circuits yielding superpolynomial speedups: • a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0, • the quantum Fourier transform over any finite group.

14. Random circuits Definition: A random quantum circuit of length T on n qubits is generated by the following process: For t=1,...,T Choose a random pair of qubits (i,j) from 1,...,n. Apply a uniformly random U(4) rotation to qubits i and j.(An efficiently universal discrete gate set would also work.) Theorem: For any α,β>0, a random circuit of length Ω(n3) on n qubits is (α,β)-dispersing with probability Corollary: For any ε>0, a random circuit of length Ω(n3) on n qubits has probability ≥1-ε of yielding a separation between O(n3) quantum time and nΩ(ε log n) classical queries.

15. Random circuits are usually dispersing Proof sketch: based on techniques of [Dahlstein, Oliveira, Plenio; 0605126, 0701125] After t random 2-qubit unitaries, let the state be |Ψti. Expand where σp are Paulis and γt(p) are coefficients. Note that γt(p)2form a probability distribution, and that Eγt(p)2 evolves with t according to a classical Markov chain on {0,1,2,3}n with gap Ω(1/n2). Thus each Eγt(p)2 ≈ 4-n after T=Ω(n3).

16. quantum Fourier transforms • Let G be a finite group. • The QFT on G realizes the isomorphismwhere λ labels irreps of G, Vλ is acted on by left multiplication and Vλ* by right multiplication. • Theorem: The QFT on G is (1/2, 1/√2)-dispersing. • In fact:Can take α=(log Σλ dim Vλ) / log |G|.

17. All QFTs are dispersing • Proof sketch: • Pick an irrep λ and a pure state |Ψλi∈Vλ. Let the state of Vλ* be maximally mixed. • Since this is right-invariant, if we inverse-QFT and measure |gi the answer will be uniformly distributed. • However, we need a pure state with this property. Find it using derandomization and a fourth moment argument. • Note: This is a weaker model of dispersing: “For any a∈A, there exists |φai such that ∑x|ha, φa |U|xi| is large.” However, the speedup results are unchanged.

18. Conclusions • The recursive Fourier sampling speedup appears to be more related to recursion than to Fourier sampling. • Even seemingly worthless quantum circuits are (most of the time) better than classical circuits for at least one task. Intriguingly, these speedups appear to be incomparable. • A skeptical note:“Since H and Toffoli are universal, every quantum speedup can be obtained from the Z2 QFT and reversible classical circuits.” --Wim van DamOne shouldn’t read too much into the idea of “using” a particular quantum circuit.

19. Open problems • Give more candidates for BQPO⊄PHO. • Find tight concentration bounds for the output of random quantum circuits. • Oracle constructions: [see also Aaronson-Kuperberg 06] • Can any n-qubit state be prepared up to error ε using poly(n) time and log(1/ε) oracle calls? • Can any n-qubit unitary be implemented with poly(n, log 1/ε) time and oracle calls? • What can classical circuits do with access to these oracles?