Quantitative estimation of the evolution of entanglement in Grovers - - PowerPoint PPT Presentation

▶
quantitative estimation of the evolution of entanglement
SMART_READER_LITE
LIVE PREVIEW

Quantitative estimation of the evolution of entanglement in Grovers - - PowerPoint PPT Presentation

Sep 21, 2023 225 likes •454 views

Quantitative estimation of the evolution of entanglement in Grovers algorithm Henri de Boutray Univ. of Bourgogne Franche-Comt e FEMTO-ST Institute, DISC department, VESONTIO team ANR project I-QUINS henri.de_boutray@univ-fcomte.fr

slide-1
SLIDE 1

Quantitative estimation of the evolution of entanglement in Grover’s algorithm

Henri de Boutray

  • Univ. of Bourgogne Franche-Comt´

e FEMTO-ST Institute, DISC department, VESONTIO team ANR project I-QUINS henri.de_boutray@univ-fcomte.fr Joint work with Alain Giorgetti, Fr´ ed´ eric Holweck, Pierre-Alain Masson and Hamza Jaffali

November 28, 2019

  • H. de Boutray

Evolution of entanglement in Grover algorithm 1 / 17

slide-2
SLIDE 2

Estimation of entanglement in Grover’s algorithm

Objectives ◮ Role of entanglement in quantum speed-up? ◮ Establish entanglement-related properties in quantum algorithms Tackled point ◮ Algorithm: Grover’s quantum search ◮ Evaluation method: Mermin polynomials

  • H. de Boutray

Evolution of entanglement in Grover algorithm 2 / 17

slide-3
SLIDE 3

Overview

1

Grover’s algorithm

2

Entanglement evaluation

3

Properties

4

Results

5

Future work

  • H. de Boutray

Evolution of entanglement in Grover algorithm 3 / 17

slide-4
SLIDE 4

Grover’s algorithm Entanglement evaluation Properties Results Future work

Grover algorithm in a nutshell

◮ Search an item x0 in an unsorted database Ω of N = 2n objects ◮ Just by applications of the Boolean function f : Ω → {0, 1} such that f (z) = 1 ⇔ z = x0 ◮ O( √ N) complexity: quadratic improvement over classical search ◮ Oracle Uf defined by Uf |x, y = |x, y ⊕ f (x) ◮ Amplitude amplification |0 /n H⊗n+1 Uf D · · · |1 · · · Repeated

  • π

√ N/4

  • times
  • H. de Boutray

Evolution of entanglement in Grover algorithm 4 / 17

slide-5
SLIDE 5

Grover’s algorithm Entanglement evaluation Properties Results Future work

Grover’s amplitude amplification

|0 /n H⊗n+1 · · · Uf D · · · |1 · · · · · ·

x0

State before Uf

  • H. de Boutray

Evolution of entanglement in Grover algorithm 5 / 17

slide-6
SLIDE 6

Grover’s algorithm Entanglement evaluation Properties Results Future work

Grover’s amplitude amplification

|0 /n H⊗n+1 · · · Uf D · · · |1 · · · · · ·

x0

State before Uf

x0

State after Uf

  • H. de Boutray

Evolution of entanglement in Grover algorithm 5 / 17

slide-7
SLIDE 7

Grover’s algorithm Entanglement evaluation Properties Results Future work

Grover’s amplitude amplification

|0 /n H⊗n+1 · · · Uf D · · · |1 · · · · · ·

x0

State before Uf

x0

State after Uf

x0

Effect of D

  • H. de Boutray

Evolution of entanglement in Grover algorithm 5 / 17

slide-8
SLIDE 8

Grover’s algorithm Entanglement evaluation Properties Results Future work

Grover’s amplitude amplification

|0 /n H⊗n+1 · · · Uf D · · · |1 · · · · · ·

x0

State before Uf

x0

State after Uf

x0

Effect of D

x0

State after D

  • H. de Boutray

Evolution of entanglement in Grover algorithm 5 / 17

slide-9
SLIDE 9

Grover’s algorithm Entanglement evaluation Properties Results Future work

Entanglement evaluations

◮ entanglement quantification: Geometric Mesurement of entanglement [WG03], Bell-Mermin inequalities [Mer90, ACG+16] ◮ entanglement classification: Secant varieties [HJN16]

[WG03] T.-C. Wei and P.M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Physical Review A, 68(4):042307, October 2003. [Mer90] N David Mermin. Extreme quantum entanglement in a superposition of macroscopically distinct states. Physical Review Letters, 65(15):1838–1840, October 1990. [ACG+16] Daniel Alsina, Alba Cervera, Dardo Goyeneche, Jos´ e I. Latorre, and Karol ˙ Zyczkowski. Operational approach to Bell inequalities: Applications to qutrits. Physical Review A, 94(3):032102, September 2016. [HJN16] Fr´ ed´ eric Holweck, Hamza Jaffali, and Isma¨ el Nounouh. Grover’s algorithm and the secant varieties. Quantum Information Processing, 15(11):4391–4413, November 2016.

  • H. de Boutray

Evolution of entanglement in Grover algorithm 6 / 17

slide-10
SLIDE 10

Grover’s algorithm Entanglement evaluation Properties Results Future work

Mermin polynomials

Definition (Mermin polynomials) Let (an)n≥1 and (a′

n)n≥1 be two families of observables, let’s also generalize (·)′

as such: A′′ = A, (λA+γB)′ = λA′ +γB′ and (A⊗B)′ = A′ ⊗B′. The Mermin polynomial Mn is defined by:

  • M1 = a1

and Mn = 1

2Mn−1 ⊗ (an + a′ n) + 1 2M′ n−1 ⊗ (an − a′ n)

for n ≥ 2

  • H. de Boutray

Evolution of entanglement in Grover algorithm 7 / 17

slide-11
SLIDE 11

Grover’s algorithm Entanglement evaluation Properties Results Future work

Mermin polynomials

Definition (Mermin polynomials) Let (an)n≥1 and (a′

n)n≥1 be two families of observables, let’s also generalize (·)′

as such: A′′ = A, (λA+γB)′ = λA′ +γB′ and (A⊗B)′ = A′ ⊗B′. The Mermin polynomial Mn is defined by:

  • M1 = a1

and Mn = 1

2Mn−1 ⊗ (an + a′ n) + 1 2M′ n−1 ⊗ (an − a′ n)

for n ≥ 2 Example: For two qubits, M2 = 1

2(a1 ⊗ a2 + a1 ⊗ a′ 2 + a′ 1 ⊗ a2 − a′ 1 ⊗ a′ 2)

Remark: When a1 = X, a2 = Z+X

√ 2 , a′ 1 = Z and a′ 2 = Z−X √ 2 , M2 is the Bell

  • perator
  • H. de Boutray

Evolution of entanglement in Grover algorithm 7 / 17

slide-12
SLIDE 12

Grover’s algorithm Entanglement evaluation Properties Results Future work

Mermin polynomials

Definition (Mermin polynomials) Let (an)n≥1 and (a′

n)n≥1 be two families of observables, let’s also generalize (·)′

as such: A′′ = A, (λA+γB)′ = λA′ +γB′ and (A⊗B)′ = A′ ⊗B′. The Mermin polynomial Mn is defined by:

  • M1 = a1

and Mn = 1

2Mn−1 ⊗ (an + a′ n) + 1 2M′ n−1 ⊗ (an − a′ n)

for n ≥ 2 Example: For two qubits, M2 = 1

2(a1 ⊗ a2 + a1 ⊗ a′ 2 + a′ 1 ⊗ a2 − a′ 1 ⊗ a′ 2)

Remark: When a1 = X, a2 = Z+X

√ 2 , a′ 1 = Z and a′ 2 = Z−X √ 2 , M2 is the Bell

  • perator

To detect entanglement of a given state, we instantiate those Mermin polynomials Mn with specific values of an and a′

n.

  • H. de Boutray

Evolution of entanglement in Grover algorithm 7 / 17

slide-13
SLIDE 13

Grover’s algorithm Entanglement evaluation Properties Results Future work

Mermin evaluation and classical limit

◮ Mermin evaluation: fMn : |ϕ → ϕ|Mn|ϕ ◮ |ϕ classical = ⇒ fMn(|ϕ) ≤ 1 ◮ Mermin evaluation is an entanglement witness

  • H. de Boutray

Evolution of entanglement in Grover algorithm 8 / 17

slide-14
SLIDE 14

Grover’s algorithm Entanglement evaluation Properties Results Future work

Mermin operator optimization

◮ |ϕ non-local? Find an Mn such that fMn(|ϕ) > 1 ◮ Mn is a function of (ai)1≤i≤n ∀i, ai = αX + βY + δZ Find (α, β, δ) such that fMn(|ϕ) > 1

  • H. de Boutray

Evolution of entanglement in Grover algorithm 9 / 17

slide-15
SLIDE 15

Grover’s algorithm Entanglement evaluation Properties Results Future work

States enumeration in the Grover algorithm

|0 H⊗n+1 Uf D Uf D Uf D |0 |0 |0 |1 |ϕ0 |ϕ1 |ϕ2 |ϕ3

  • H. de Boutray

Evolution of entanglement in Grover algorithm 10 / 17

slide-16
SLIDE 16

Grover’s algorithm Entanglement evaluation Properties Results Future work

Preamble

|ϕk = αk |x0 + βk |+⊗n

|+⊗n |x0

  • ϕ⌊kopt/2⌋
  • X

the middle point is |ϕent = |x0+|+⊗n

K

  • ϕkopt/2
  • ≈ |ϕent
  • H. de Boutray

Evolution of entanglement in Grover algorithm 11 / 17

slide-17
SLIDE 17

Grover’s algorithm Entanglement evaluation Properties Results Future work

Graph trends

If Mn is chosen to optimize fMn(|ϕent), then we expect fMn to behave like a distance measure from |ϕent. Thus we anticipate that: ◮ fMn(|ϕk) reaches maximum around kopt/2 ◮ fMn(|ϕk) grows for k in [0, ⌊kopt/2⌋] ◮ fMn(|ϕk) decreases for k in [⌊kopt/2⌋ + 1, kopt]

  • H. de Boutray

Evolution of entanglement in Grover algorithm 12 / 17

slide-18
SLIDE 18

Grover’s algorithm Entanglement evaluation Properties Results Future work

Non locality

Assumption: some states are non local: ∃k, fMn(|ϕk) > 1 {Maximum reached around kopt/2} = ⇒ fMn(

  • ϕ⌊kopt/2⌋
  • ) > 1

(in fact probably for more k’s than just ⌊kopt/2⌋)

  • H. de Boutray

Evolution of entanglement in Grover algorithm 13 / 17

slide-19
SLIDE 19

Grover’s algorithm Entanglement evaluation Properties Results Future work

Results, 4 to 8

For 8 qubits, 1 week of computation on personal computer with naive implementation.

5 10 −1 1 2 3 1 Number of iterations Mermin evaluation 4 qubits 5 qubits 6 qubits 7 qubits 8 qubits

n 4 5 6 7 8 kopt 2 3 5 8 12

  • H. de Boutray

Evolution of entanglement in Grover algorithm 14 / 17

slide-20
SLIDE 20

Grover’s algorithm Entanglement evaluation Properties Results Future work

Results, 9 to 12

On the Mesocenter:

5 10 15 20 25 30 35 40 45 50 55 2 4 6 8 1 Number of iterations Mermin evaluation 9 qubits 10 qubits 11 qubits 12 qubits

n 9 10 11 12 kopt 17 25 36 50

  • H. de Boutray

Evolution of entanglement in Grover algorithm 15 / 17

slide-21
SLIDE 21

Grover’s algorithm Entanglement evaluation Properties Results Future work

Future work

◮ Use Qbricks to prove algorithm properties

◮ Grover ◮ Deutsch-Jozsa ◮ Shor

◮ Use the work done with Jessy Colonval [Cd19] to establish more quantum properties to use in quantum program verification (Contextuality)

[Cd19] Jessy Colonval, Henri de Boutray. Formalisation et validation d’une m´ ethode de construction de syst` emes de blocs. AFADL 2019.

  • H. de Boutray

Evolution of entanglement in Grover algorithm 16 / 17

slide-22
SLIDE 22

Grover’s algorithm Entanglement evaluation Properties Results Future work

Thank you for your attention

  • H. de Boutray

Evolution of entanglement in Grover algorithm 17 / 17