Characterizations of symmetric partial Boolean functions with exact quantum query complexity

——Deutsch-Jozsa Problem

1

Daowen Qiu School of Data and Computer Science

Sun Yat-sen University, China

Joint work with Shenggen Zheng

arXiv:1603.06505(Qiu and Zheng)

2017AQIS

September 4-8, 2017, NUS, Singapore

2017AQIS

Abstract In this talk, I would like to report a recent work regarding: 1. An optimal exact quantum query algorithm for generalized Deutsch-Jozsa problem 2. The characterization of all symmetric partial Boolean functions with exact quantum 1-query complexity

2

2017AQIS

Basic background Discover more problems to show that

quantum computing is more powerful than classical computing These problems have the potential of

applications in other areas such as cryptography.

3

2017AQIS

1. Motivations, Problems, Results 2. Preliminaries 3. Main Results 4. Methods of Proofs 5. Conclusions & Further Problems

4

2017AQIS

1. Motivations, Problems, Results

(1) Generalized Deutsch-Jozsa (2) Problems with exact quantum 1-query

complexity

5 2017AQIS

Motivation I

6

The Deutsch-Jozsa promise problem [DJ’92]: 𝑥 ∈ 0,1 𝑛, |x| is the Hamming weight of 𝑥,

𝐷𝐷 𝑥 = �01 𝑖𝑖 x = 0 𝑜𝑜 |x| = 𝑛𝑖𝑖 |x| = 𝑛/2

𝑄𝐸 𝐷𝐷 = 1,𝐷 𝐷𝐷 =𝑛2

+ 1

𝐷𝐷𝑛1 = �01 𝑖𝑖|x| ≤ 1 𝑜𝑜|x| ≥ 𝑛 − 1

𝑖𝑖|x| = 𝑛/2 [MJM’15]

𝑄𝐸 𝐷𝐷𝑛1 ≤ 2 [DJ’92] D. Deutsch, R. Jozsa, Rapid solution of problems by quantum computation, In Proceedings of the Royal Society of London, 439A (1992): 553—558. [MJM’15] A. Montanaro, R. Jozsa, G. Mitchison, On exact quantum query complexity, Algorithmica419 (2015) 775--796.

2017AQIS

Problem I

7

2017AQIS

Motivation II Deutsch-Jozsa problem that is a symmetric partial Boolean function can be solved by DJ algorithm (exact quantum 1-query algorithm). Then how to characterize the other symmetric partial Boolean functions with exact quantum 1-query complexity? Can such functions be solved by DJ algorithm?

8

Problem and Result II

9

2017AQIS

Preliminaries Symmetric partial Boolean functions Classical query complexity Quantum query complexity Multilinear polynomials

10

2017AQIS

Symmetrical partial Boolean functions

11

Representation of symmetric partial Boolean functions

12

Isomorphism of symmetric partial Boolean functions

13

Classical query complexity

14

An exact classical (deterministic) query algorithm to compute a Boolean function 𝑖: {0,1}𝑛 → {0,1} can be described by a decision tree.

If the output of a decision tree is 𝑖(𝑥), for all 𝑥 ∈ {0,1}𝑛, the decision tree is said to "compute" 𝑖. The depthof a tree is the maximum number of queries that can happen before a leaf is reached and a result obtained.

𝐷(𝑖), the deterministic decision tree complexity of 𝑖 is the smallest depth among all deterministic decision trees that compute 𝑖.

2017AQIS

Example

15

Deterministic query complexity (how many times we need to query the input bits)

Example: 𝑖 𝑥1, 𝑥2 = 𝑥1 ⊕ 𝑥2

Decision treeThe minimal depth over all decision trees computing 𝑖 is the exact classical query complexity (deterministic query complexity, decision tree complexity) 𝐷(𝑖).

2017/1/2 1

𝑥1

𝑥2

0 1

0 1

𝑥2

1 01 1

2017AQIS

Quantum query algorithms

16

U0 Q Q U1 UT …

2017AQIS

Black box

17

𝑄𝑥

𝑄𝑥

2017AQIS

Quantum query complexity

18

2017AQIS

Multilinear polynomials

19

2017AQIS

Multilinear polynomials representing symmetric partial Boolean functions

20

Methods of Proofs We would like to outline the basic ideas and methods for the proofs of main results.

21

2017AQIS

𝑫𝑱𝒏𝒌 = �𝟎𝟏 𝒊𝒊 𝒙 ≤ 𝒌 𝒐𝒐 |𝐱| ≥ 𝒏 − 𝒌

𝒊𝒊|𝐱| = 𝒏/𝟐

22

𝑄𝐸 𝐷𝐷𝑛𝑘 ≤ 𝑘 + 1

23

2017AQIS

24

deg 𝐷𝐷𝑛𝑘 ≥ 2𝑘 + 2

25

2017AQIS

Theorem:𝑸𝑬(𝒊) = 𝟏 𝒊𝒊 𝒂𝒏𝒂 𝒐𝒏𝒐𝒐 𝒊𝒊 𝒊 can be computed by DJ algorithm

26

where 𝑛 − 1 ≥ 𝑘 ≥ ⌊𝑛/2⌋, and 𝑛/2 ≥ 𝑙 ≥ ⌊𝑛/2⌋.

Two Lemmas

27

2017AQIS

Equivalence transformation

28

2017AQIS

𝐷𝐷𝑛𝑘 = �01 𝑖𝑖 x ≤ 𝑘 𝑜𝑜 |x| ≥ 𝑛 − 𝑘

𝑖𝑖|x| = 𝑛/2

Theorem. 𝑄𝐸 𝐷𝐷𝑛𝑘 = 𝑘 + 1 and 𝐷 𝐷𝐷𝑛𝑘 = 𝑛/2 + 𝑘 + 1. Theorem. Any symmetric partial Boolean function 𝑖 has 𝑄𝐸(𝑖) = 1 if and only if 𝑖 can be computed by the Deutsch-Jozsa algorithm. Theorem.Any classical deterministic algorithm that solves Simon's problem requires Ω( 𝟐𝐧) queries.

2017/9/14 29

Conclusions

2017AQIS

Let 𝑖: {0,1}𝑛 → {0,1} be an 𝑛-bit symmetric partial Boolean function with domain of definition 𝐷, and let 0 ≤ 𝑘 < ⌊𝑛

2⌋ . Then, for 2𝑘 + 1 ≤ deg 𝑖 ≤ 2(𝑘 + 1),

how to characterize 𝑖by giving all functions with degrees from 2𝑘 + 1 to 2𝑘 + 2?

For the function 𝐷𝑊𝑛𝑘,𝑙defined as:

can we give optimal exact quantum query algorithms for any 𝑘 and 𝑙?

2017/9/14 30

Problems

2017AQIS

Thank you for your attention！

31