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

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

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

Overview Grover’s algorithm 1 Entanglement evaluation 2 Properties 3 Results 4 Future work 5 H. de Boutray Evolution of entanglement in Grover algorithm 3 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Grover algorithm in a nutshell ◮ Search an item x 0 in an unsorted database Ω of N = 2 n objects ◮ Just by applications of the Boolean function f : Ω → { 0 , 1 } such that f ( z ) = 1 ⇔ z = x 0 √ ◮ O ( N ) complexity: quadratic improvement over classical search ◮ Oracle U f defined by U f | x , y � = | x , y ⊕ f ( x ) � ◮ Amplitude amplification / n · · · | 0 � D H ⊗ n +1 U f · · · | 1 � � �� � √ � � Repeated π N / 4 times H. de Boutray Evolution of entanglement in Grover algorithm 4 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Grover’s amplitude amplification / n · · · · · · | 0 � D H ⊗ n +1 U f | 1 � · · · · · · x 0 State before U f H. de Boutray Evolution of entanglement in Grover algorithm 5 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Grover’s amplitude amplification / n · · · · · · | 0 � D H ⊗ n +1 U f | 1 � · · · · · · x 0 x 0 State before U f State after U f H. de Boutray Evolution of entanglement in Grover algorithm 5 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Grover’s amplitude amplification / n · · · · · · | 0 � D H ⊗ n +1 U f | 1 � · · · · · · x 0 x 0 State before U f State after U f x 0 Effect of D H. de Boutray Evolution of entanglement in Grover algorithm 5 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Grover’s amplitude amplification / n · · · · · · | 0 � D H ⊗ n +1 U f | 1 � · · · · · · x 0 x 0 State before U f State after U f x 0 x 0 Effect of D State after D H. de Boutray Evolution of entanglement in Grover algorithm 5 / 17

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

Grover’s algorithm Entanglement evaluation Properties Results Future work Mermin polynomials Definition (Mermin polynomials) Let ( a n ) n ≥ 1 and ( a ′ n ) n ≥ 1 be two families of observables, let’s also generalize ( · ) ′ as such: A ′′ = A , ( λ A + γ B ) ′ = λ A ′ + γ B ′ and ( A ⊗ B ) ′ = A ′ ⊗ B ′ . The Mermin polynomial M n is defined by: � M 1 = a 1 and M n = 1 2 M n − 1 ⊗ ( a n + a ′ n ) + 1 2 M ′ n − 1 ⊗ ( a n − a ′ n ) for n ≥ 2 H. de Boutray Evolution of entanglement in Grover algorithm 7 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Mermin polynomials Definition (Mermin polynomials) Let ( a n ) n ≥ 1 and ( a ′ n ) n ≥ 1 be two families of observables, let’s also generalize ( · ) ′ as such: A ′′ = A , ( λ A + γ B ) ′ = λ A ′ + γ B ′ and ( A ⊗ B ) ′ = A ′ ⊗ B ′ . The Mermin polynomial M n is defined by: � M 1 = a 1 and M n = 1 2 M n − 1 ⊗ ( a n + a ′ n ) + 1 2 M ′ n − 1 ⊗ ( a n − a ′ n ) for n ≥ 2 Example: For two qubits, M 2 = 1 2 ( a 1 ⊗ a 2 + a 1 ⊗ a ′ 2 + a ′ 1 ⊗ a 2 − a ′ 1 ⊗ a ′ 2 ) 2 , a ′ 1 = Z and a ′ Remark: When a 1 = X , a 2 = Z + X 2 = Z − X 2 , M 2 is the Bell √ √ operator H. de Boutray Evolution of entanglement in Grover algorithm 7 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Mermin polynomials Definition (Mermin polynomials) Let ( a n ) n ≥ 1 and ( a ′ n ) n ≥ 1 be two families of observables, let’s also generalize ( · ) ′ as such: A ′′ = A , ( λ A + γ B ) ′ = λ A ′ + γ B ′ and ( A ⊗ B ) ′ = A ′ ⊗ B ′ . The Mermin polynomial M n is defined by: � M 1 = a 1 and M n = 1 2 M n − 1 ⊗ ( a n + a ′ n ) + 1 2 M ′ n − 1 ⊗ ( a n − a ′ n ) for n ≥ 2 Example: For two qubits, M 2 = 1 2 ( a 1 ⊗ a 2 + a 1 ⊗ a ′ 2 + a ′ 1 ⊗ a 2 − a ′ 1 ⊗ a ′ 2 ) 2 , a ′ 1 = Z and a ′ Remark: When a 1 = X , a 2 = Z + X 2 = Z − X 2 , M 2 is the Bell √ √ operator To detect entanglement of a given state, we instantiate those Mermin polynomials M n with specific values of a n and a ′ n . H. de Boutray Evolution of entanglement in Grover algorithm 7 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Mermin evaluation and classical limit ◮ Mermin evaluation: f M n : | ϕ � �→ � ϕ | M n | ϕ � ◮ | ϕ � classical = ⇒ f M n ( | ϕ � ) ≤ 1 ◮ Mermin evaluation is an entanglement witness H. de Boutray Evolution of entanglement in Grover algorithm 8 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Mermin operator optimization ◮ | ϕ � non-local? Find an M n such that f M n ( | ϕ � ) > 1 ◮ M n is a function of ( a i ) 1 ≤ i ≤ n ∀ i , a i = α X + β Y + δ Z Find ( α, β, δ ) such that f M n ( | ϕ � ) > 1 H. de Boutray Evolution of entanglement in Grover algorithm 9 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work States enumeration in the Grover algorithm | 0 � | 0 � D D D H ⊗ n +1 U f U f U f | 0 � | 0 � | 1 � | ϕ 0 � | ϕ 1 � | ϕ 2 � | ϕ 3 � H. de Boutray Evolution of entanglement in Grover algorithm 10 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Preamble | ϕ k � = α k | x 0 � + β k | + � ⊗ n | x 0 � X � � � ϕ ⌊ k opt / 2 ⌋ | + � ⊗ n the middle point is | ϕ ent � = | x 0 � + | + � ⊗ n K � � � ϕ k opt / 2 ≈ | ϕ ent � H. de Boutray Evolution of entanglement in Grover algorithm 11 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Graph trends If M n is chosen to optimize f M n ( | ϕ ent � ), then we expect f M n to behave like a distance measure from | ϕ ent � . Thus we anticipate that: ◮ f M n ( | ϕ k � ) reaches maximum around k opt / 2 ◮ f M n ( | ϕ k � ) grows for k in [0 , ⌊ k opt / 2 ⌋ ] ◮ f M n ( | ϕ k � ) decreases for k in [ ⌊ k opt / 2 ⌋ + 1 , k opt ] H. de Boutray Evolution of entanglement in Grover algorithm 12 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Non locality Assumption: some states are non local: ∃ k , f M n ( | ϕ k � ) > 1 � � { Maximum reached around k opt / 2 } = ⇒ f M n ( � ϕ ⌊ k opt / 2 ⌋ ) > 1 (in fact probably for more k ’s than just ⌊ k opt / 2 ⌋ ) H. de Boutray Evolution of entanglement in Grover algorithm 13 / 17

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. 4 qubits 5 qubits 3 6 qubits 7 qubits Mermin evaluation 2 8 qubits 1 1 0 − 1 0 5 10 Number of iterations 4 5 6 7 8 n k opt 2 3 5 8 12 H. de Boutray Evolution of entanglement in Grover algorithm 14 / 17

Grover’s algorithm Entanglement evaluation Properties Results Future work Results, 9 to 12 On the Mesocenter: 9 qubits 8 10 qubits 11 qubits 6 12 qubits Mermin evaluation 4 2 1 0 0 5 10 15 20 25 30 35 40 45 50 55 Number of iterations n 9 10 11 12 17 25 36 50 k opt H. de Boutray Evolution of entanglement in Grover algorithm 15 / 17