Superpolynomial speedups from

the quantum Fourier transform on the symmetric group

Sean Hallgren, NECAram Harrow, Bristol

QIP 2007

ANY

(a)

(b): almost any quantum circuit

Guiding principlesOfficial 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.

The plan

1.Review Recursive Fourier Sampling [BV93].

2.Generalize Fourier sampling.

3.Generalize the recursion.

4.Circuits yielding superpolynomial speedups:

1. the quantum Fourier transform over any finite group,

2. a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

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†.

Fourier sampling on

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

[BV93]

The plan

1.Review Recursive Fourier Sampling [BV93].

2.Generalize Fourier sampling.

3.Generalize the recursion.

4.Circuits yielding superpolynomial speedups:

(a) the quantum Fourier transform over any finite group,

1. a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

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.

Generalization:oracle-assisted state

identificationGoal: 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

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,

Dispersing circuitsDefinition: A unitary U on n qubits is (α,β)-dispersing ifthere exists a set A⊆{0,1}n with |A|≥2αn and

for all a∈A.

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.

e.g.: H⊗n and the standard QFT are both (1,1)-dispersing.

The plan

1.Review Recursive Fourier Sampling [BV93].

2.Generalize Fourier sampling.

3.Generalize the recursion.

4.Circuits yielding superpolynomial speedups:

(a) the quantum Fourier transform over any finite group,

1. a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0.

Recursive amplificationIdea: 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 thatO1 (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).

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

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.

The plan

1.Review Recursive Fourier Sampling [BV93].

2.Generalize Fourier sampling.

3.Generalize the recursion.

4.Circuits yielding superpolynomial speedups:

1. a 1-ε fraction of length-Ω(n3) circuits on n qubits for any ε>0,

2. the quantum Fourier transform over any finite group.

Random circuitsDefinition: A random quantum circuit of length T on n qubits is generated by the following process:For t=1,...,TChoose 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.

Expand

where σp are Paulis and γt(p) are coefficients.

Random circuits are usually dispersingProof sketch: based on techniques of [Dahlstein, Oliveira, Plenio;

0605126, 0701125]

Note that γt(p)2 form 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).

After t random 2-qubit unitaries, let the state be |Ψti.

•Let G be a finite group.

•The QFT on G realizes the isomorphism

where 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|.

quantum Fourier transforms

All QFTs are dispersingProof 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.

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 Dam

One shouldn’t read too much into the idea of “using” a particular quantum circuit.

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?