Separating quantum communication and approximate rank Anurag Anshu a - - PowerPoint PPT Presentation

Separating quantum communication and approximate rank

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc, Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f

a CQT, National University of Singapore b Massachusetts Institute of Technology c Microsoft Research, New England d Dept. of CS, National University of Singapore e MajuLab, UMI 3654, Singapore f SPMS, Nanyang Technological University

July 8, 2017

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 1 / 22

Roadmap

1

Some background

2

Separating quantum communication and approximate rank

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 2 / 22

Models of query complexity

For a function F, Randomized (two-sided error of ε) query complexity Rdt

ε (F), Quantum (two sided error of ε) query complexity Qdt ε (F).

Quadratic separation: using Grover’s search algorithm [Grov95] and its variant proved in [BBHT96]. OR: {0, 1}n → {0, 1} outputs 1 if the input contains at least one 1. Rdt

1/3

Qdt

1/3

2 [BBHT96]

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 3 / 22

Lower bounds on quantum query complexity

For a function F, approximate polynomial degree degε(F) is the minimum among the degrees of all polynomials p(x) satisfying |p(x) − F(x)| ≤ ε, for all x. It lower bounds quantum query complexity [Beals, Buhrman, Cleve, Mosca, de Wolf 1998]: Qdt

ε (F) ≥ 1 2degε(F).

Example: deg1/3(OR) = Θ(√n). Other well known bounds: Adversary bound [Ambainis 2000], Negative weights adversary bound [Hoyer, Lee, Spalek 2005].

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 4 / 22

Degree not a tight lower bound

It is known that Qdt

1/3(F) = O(deg1/3(F))6.

Moreover, there exists a function F, such that Qdt

1/3(F) = Θ(deg1/3(F))1.3219 [Ambainis 2003].

Is this the best possible separation?

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 5 / 22

Cheat sheets

Aaronson, Ben-David and Kothari [2016] introduced the technique of cheat sheet. Follow up to the works G¨

• ¨
• s, Pitassi and Watson [2015] and

Ambainis, Balodis, Belovs, Lee, Santha and Smotrovs [2015]. A transformation from F → Fcs. Rdt

1/3

Qdt

1/3

2.5 [ABK16]

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 6 / 22

Cheat sheets

Aaronson, Ben-David and Kothari [2015] introduced the technique of cheat sheet. Follow up to the works G¨

• ¨
• s, Pitassi and Watson [2015] and

Ambainis, Balodis, Belovs, Lee, Santha and Smotrovs [2015]. A transformation from F → Fcs. Qdt

1/3

deg1/3 4 [ABK16]

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 7 / 22

Cheat sheet review

Fcs has two components: ‘c’ copies of a parent function F and a cheat sheet cs. Compute based on inputs to functions and content at ‘decimal(b)’. b = F1, . . . Fc F1 Fc 1 2 2c

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 8 / 22

Communication complexity

F x y Randomized communication complexity R1/3(F): number of bits communicated in a randomized protocol. Quantum communication complexity Q1/3(F): number of qubits communicated in an entanglement assisted quantum protocol.

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 9 / 22

Lower bound on quantum communication complexity

Approximate rank for F, rkε(F) = minF ′{rk(F ′) : |F ′(x, y) − F(x, y)| ≤ ε}. Lower bound on quantum communication complexity [Buhrman and de Wolf 2001, Lee and Shraibman 2008]: For F : {0, 1}n × {0, 1}n → {0, 1}, Q1/3(F) ≥ Ω(log rk1/3(F) − log n). Quantum log-rank conjecture: are Q1/3(F) and log rk1/3(MF) polynomially related?

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 10 / 22

Lower bound on quantum communication complexity

Approximate rank for F, rkε(F) = minF ′{rk(F ′) : |F ′(x, y) − F(x, y)| ≤ ε}. Lower bound on quantum communication complexity [Buhrman and de Wolf 2001, Lee and Shraibman 2008]: For F : {0, 1}n × {0, 1}n → {0, 1}, Q1/3(F) ≥ Ω(log rk1/3(F) − log n). Quantum log-rank conjecture: are Q1/3(F) and log rk1/3(F) polynomially related? Other lower bound: quantum information complexity ([Touchette 2015]).

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 11 / 22

Cheat sheets in communication complexity

Notion of cheat sheet extended to communication complexity in A., Belovs, Ben-David, G¨

• ¨
• s, Jain, Kothari, Lee and Santha [2016].

A similar transformation: F → FG, called look-up function. Super-quadratic separation between R1/3(F) and Q1/3(F).

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 12 / 22

Look-up function FG

F : X ⊗ Y → {0, 1} F1, F2 . . . Fc ≡ F G : X ⊗c ⊗ Y⊗c ⊗ W → {0, 1} W is set of strings u0, v0, u1, v1 . . . u2c, v2c ∈ W F1 x1 y1 Fc xc yc u0 v0 u1 v1 u2c v2c

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 13 / 22

Look-up function FG

F1 x1 y1 Fc xc yc compute b = (F1, F2, . . . Fc) u0 v0 u1 v1 u2c v2c

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 14 / 22

Look-up function FG

F1 x1 y1 Fc xc yc goto block number decimal(b) u0 v0 u1 v1 u2c v2c

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 15 / 22

Look-up function FG

F1 x1 y1 Fc xc yc u0 v0 u1 v1 u2c v2c Iff G(ub ⊕ vb, x1, y1 . . . xc, yc) = 1 FG = 1

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 16 / 22

Quantum communication complexity of look-up function

For reasonably non-trivial function G, we show the following.

Theorem

Q1/3(FG) = Ω(log 1 disc(F)). disc(F) is the discrepancy of F.

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 17 / 22

An outline of proof

We show that for any r-round protocol Π for FG that makes an error

• f 1

3, there exists a protocol Π′ for F that makes an error of 1 2 − 1 r2

and communicates the same as in Π. So, Q1/3(FG) = Ω(Q 1

2 − 1 r2 (F)) = Ω(log

1 disc(F) − log r2).

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 18 / 22

An outline of proof

Key idea: Quantum cut and paste theorem [Jain, Radhakrishnan and Sen 2003, Nayak and Touchette 2016]. In a protocol where each player has low information about content of the correct location of other player’s ‘look up part’, output cannot be correct.

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 19 / 22

Choice of G

Recall: in cheat sheet of Aaronson, Ben-David and Kothari, correct cheat sheet location must certify the evaluation of F1, F2, . . . Fc on their inputs. Fix a circuit C for F, with number of gates size(F). We require that ub ⊕ vb certifies the evaluation of inputs (to F1, F2, . . . Fc) on C.

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 20 / 22

Choice of G

Theorem

For G as defined above, log rk1/3(FG) = O(

• size(F)).

Now choose F to be inner product function IPn(x, y) =

i xiyi mod 2. We have size(IPn) = O(n) and

log

1 disc(IPn) = Θ(n).

Theorem

There exists a total function F such that Q(F) = ˜ Ω(log rk1/3(F))2.

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 21 / 22

Open questions

Can the round dependence in our main result be removed or weakened? Is there a general lifting theorem from quantum query complexity to quantum communication complexity?

Recently, a lifting theorem shown from randomized query complexity to randomized communication complexity [GPW17].

Quantum log-rank conjecture?

Anurag Anshua, Shalev Ben-Davidb, Ankit Gargc , Rahul Jaina,d,e, Robin Kotharib, Troy Leea,e,f (CQT) Separations in communication complexity July 8, 2017 22 / 22