superpolynomial speedups from the quantum fourier transform on the symmetric group

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?