Nuclear structure (and reactions) with Quantum Computers - II - PowerPoint PPT Presentation

Nuclear structure (and reactions) with Quantum Computers - II Alessandro Roggero figure credit: JLAB collab. figure credit: IBM QC and QIS for NP JLAB – 17 March, 2020

Quantum phase estimation in one slide GOAL: compute eigenvalue φ with error δ using exact eigenvector | φ � Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

Quantum phase estimation in one slide GOAL: compute eigenvalue φ with error δ using exact eigenvector | φ � Hadamard test: one controlled- U operation and O (1 /δ 2 ) experiments | 0 � • H H | φ � U Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

Quantum phase estimation in one slide GOAL: compute eigenvalue φ with error δ using exact eigenvector | φ � Hadamard test: one controlled- U operation and O (1 /δ 2 ) experiments | 0 � • H H | φ � U Quantum Phase Estimation (QPE) uses O (1 /δ ) ∗ controlled- U operations, O (log(1 /δ )) ∗ ancilla qubits and only O (1) ∗ experiments | 0 � • H | 0 � • H V | 0 � • H | φ � U 2 U 4 U Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

Quantum phase estimation in one slide GOAL: compute eigenvalue φ with error δ using exact eigenvector | φ � Hadamard test: one controlled- U operation and O (1 /δ 2 ) experiments | 0 � • H H | φ � U Quantum Phase Estimation (QPE) uses O (1 /δ ) ∗ controlled- U operations, O (log(1 /δ )) ∗ ancilla qubits and only O (1) ∗ experiments | 0 � • H | 0 � • H V | 0 � • H | φ � U 2 U 4 U BONUS: works even if | φ � → α | φ � + β | ξ � with O (1 /α 2 ) ∗ experiments Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

Filling in the details Abrams & Lloyd (1999) · · · | 0 � • H . . . . . . . . . QFT † · · · | 0 � • H | 0 � • · · · H · · · | φ � U 2 U 2 m − 1 U The QPE algortihm has 4 main stages 1 prepare m ancilla in uniform superposition of basis states 2 apply controlled phases using U k with k = 2 0 , 2 1 , . . . , 2 m − 1 3 perform (inverse) Fourier transorm on ancilla register 4 measure the ancilla register Alessandro Roggero JLAB - 17 Mar 2020 2 / 18

Filling in the details: state preparation | 0 � · · · • H . . . . . . . . . QFT † · · · | 0 � • H | 0 � • · · · H | φ � U 2 · · · U 2 m − 1 U 1 prepare m ancilla in uniform superposition of basis states � | 0 � + | 1 � � � | 0 � + | 1 � � � | 0 � + | 1 � � | Φ 1 � = H ⊗ m | 0 � m = √ √ √ ⊗ ⊗ · · · ⊗ 2 2 2 2 m − 1 1 � = √ | k � 2 m k =0 BINARY REPRESENTATION: use | 3 � to indicate | 00011 � see DL lectures Alessandro Roggero JLAB - 17 Mar 2020 3 / 18

Filling in the details: phase kickback | 0 � · · · • H . . . . . . . . . QFT † | 0 � • · · · H · · · | 0 � • H | φ � U 2 · · · U 2 m − 1 U The state | φ � is an eigenstate of U with U | φ � = exp( i 2 πφ ) | φ � 2 each c- U k applies a phase exp( i 2 πkφ ) to the | 1 � state of the ancilla � | 0 � + e i 2 πφ | 1 � ⊗ | 0 � + e i 4 πφ | 1 � ⊗ · · · ⊗ | 0 � + e i 2 m πφ | 1 � � | Φ 2 � = √ √ √ ⊗ | φ � 2 2 2 2 m − 1 1 � √ = exp ( i 2 πφk ) | k � ⊗ | φ � 2 m k =0 Alessandro Roggero JLAB - 17 Mar 2020 4 / 18

Filling in the details: inverse QFT · · · | 0 � • H . . . . . . . . . QFT † | 0 � • · · · H | 0 � • · · · H · · · U 2 m − 1 | φ � U 2 U Recall that: QFT † | k � = � 2 m − 1 � � 1 − i 2 π qk exp | q � see DL lectures √ q =0 2 m 2 m 3 after an inverse QFT the final state is 2 m − 1 2 m − 1 | Φ 3 � = QFT † | Φ 2 � = 1 φ − q � � �� � � exp i 2 πk | q � ⊗ | φ � 2 m 2 m k =0 q =0 Alessandro Roggero JLAB - 17 Mar 2020 5 / 18

Filling in the details: final measurement | 0 � · · · • H . . . . . . . . . QFT † | 0 � • · · · H · · · | 0 � • H | φ � U 2 · · · U 2 m − 1 U 2 m − 1 2 m − 1 � �� 1 � i 2 πk � � 2 m (2 m φ − q ) | Φ 3 � = exp | q � ⊗ | φ � 2 m q =0 k =0 4 if phase φ is a m -bit number we can find 0 ≤ p < 2 m s.t. 2 m φ = p 2 m − 1 � | Φ 3 � = δ q,p | q � ⊗ | φ � = | p � ⊗ | φ � q =0 ⇒ exact solution with only 1 measurement! Alessandro Roggero JLAB - 17 Mar 2020 6 / 18

Final measurement: generic phase . . . � 2 m (2 m φ − q ) � | Φ 3 � = � 2 m − 1 � 2 m − 1 e i 2 πk 1 | 0 � QPE m | q � ⊗ | φ � q =0 2 m k =0 | φ � when 2 m φ is not an integer we can sum the term in parenthesis as 2 m − 1 � sin (2 m x/ 2) e ixk = 1 − e i 2 m x ix � 2(2 m − 1) � = exp 1 − e ix sin ( x/ 2) k =0 Alessandro Roggero JLAB - 17 Mar 2020 7 / 18

Final measurement: generic phase . . . � 2 m (2 m φ − q ) � | Φ 3 � = � 2 m − 1 � 2 m − 1 e i 2 πk 1 | 0 � QPE m | q � ⊗ | φ � q =0 2 m k =0 | φ � when 2 m φ is not an integer we can sum the term in parenthesis as 2 m − 1 � sin (2 m x/ 2) e ixk = 1 − e i 2 m x ix � 2(2 m − 1) � = exp 1 − e ix sin ( x/ 2) k =0 we will measure the ancilla register in | q � with probability sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 where we have defined M = 2 m Alessandro Roggero JLAB - 17 Mar 2020 7 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=32/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=33/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=34/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=35/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=36/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=37/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=38/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=39/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase example example taken from A. Childs lecture notes (2011) sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 EXERCISE: show that if r = ⌈ Mφ ⌋ then P ( r ) ≥ 4 /π 2 ≈ 0 . 4 1 M=32 φ=40/256 0.8 0.6 P(q) 0.4 2 P min = 4/ π 0.2 0 0 4 8 12 16 20 24 28 32 Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

Final measurement: generic phase II sin 2 ( Mπ ( φ − q/M )) 1 P ( q ) = sin 2 ( π ( φ − q/M )) M 2 the best m -bit approximation to φ is p/M with p = ⌈ Mφ ⌋ the probabilty of making an error δ = ( q − p ) /M is 1 Probability of measuring phase with error δ 1 =2 M=2 0.8 0.6 0.4 0.2 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 δ Alessandro Roggero JLAB - 17 Mar 2020 9 / 18