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

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

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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 with

slide-1
SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

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/δ2) experiments |0 H

  • H

|φ U

Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

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SLIDE 4

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/δ2) experiments |0 H

  • H

|φ U Quantum Phase Estimation (QPE) uses O(1/δ)∗ controlled-U

  • perations, O(log(1/δ))∗ ancilla qubits and only O(1)∗ experiments

|0 H

  • V

|0 H

  • |0

H

  • |φ

U U 2 U 4

Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

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SLIDE 5

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/δ2) experiments |0 H

  • H

|φ U Quantum Phase Estimation (QPE) uses O(1/δ)∗ controlled-U

  • perations, O(log(1/δ))∗ ancilla qubits and only O(1)∗ experiments

|0 H

  • V

|0 H

  • |0

H

  • |φ

U U 2 U 4 BONUS: works even if |φ → α |φ + β |ξ with O(1/α2)∗ experiments

Alessandro Roggero JLAB - 17 Mar 2020 1 / 18

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SLIDE 6

Filling in the details

Abrams & Lloyd (1999)

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 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 = 20, 21, . . . , 2m−1 3 perform (inverse) Fourier transorm on ancilla register 4 measure the ancilla register Alessandro Roggero JLAB - 17 Mar 2020 2 / 18

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SLIDE 7

Filling in the details: state preparation

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1

1 prepare m ancilla in uniform superposition of basis states

|Φ1 = H⊗m |0m = |0 + |1 √ 2

  • ⊗

|0 + |1 √ 2

  • ⊗ · · · ⊗

|0 + |1 √ 2

  • =

1 √ 2m

2m−1

  • k=0

|k BINARY REPRESENTATION: use |3 to indicate |00011 see DL lectures

Alessandro Roggero JLAB - 17 Mar 2020 3 / 18

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SLIDE 8

Filling in the details: phase kickback

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 The state |φ is an eigenstate of U with U |φ = exp(i2πφ) |φ

2 each c-U k applies a phase exp(i2πkφ) to the |1 state of the ancilla

|Φ2 = |0 + ei2πφ |1 √ 2 ⊗ |0 + ei4πφ |1 √ 2 ⊗ · · · ⊗ |0 + ei2mπφ |1 √ 2

  • ⊗ |φ

= 1 √ 2m

2m−1

  • k=0

exp (i2πφk) |k ⊗ |φ

Alessandro Roggero JLAB - 17 Mar 2020 4 / 18

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SLIDE 9

Filling in the details: inverse QFT

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 Recall that: QFT † |k =

1 √ 2m

2m−1

q=0

exp

  • −i2π qk

2m

  • |q

see DL lectures

3 after an inverse QFT the final state is

|Φ3 = QFT † |Φ2 = 1 2m

2m−1

  • k=0

2m−1

  • q=0

exp

  • i2πk
  • φ − q

2m

  • |q ⊗ |φ

Alessandro Roggero JLAB - 17 Mar 2020 5 / 18

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SLIDE 10

Filling in the details: final measurement

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 |Φ3 =

2m−1

  • q=0
  • 1

2m

2m−1

  • k=0

exp

  • i2πk

2m (2mφ − q)

  • |q ⊗ |φ

4 if phase φ is a m-bit number we can find 0 ≤ p < 2m s.t. 2mφ = p

|Φ3 =

2m−1

  • q=0

δq,p |q ⊗ |φ = |p ⊗ |φ ⇒ exact solution with only 1 measurement!

Alessandro Roggero JLAB - 17 Mar 2020 6 / 18

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SLIDE 11

Final measurement: generic phase

. . . QPEm |0 | Φ3 = 2m−1

q=0

  • 1

2m

2m−1

k=0

ei 2πk

2m (2mφ−q)

|q ⊗ |φ |φ when 2mφ is not an integer we can sum the term in parenthesis as

2m−1

  • k=0

eixk = 1 − ei2mx 1 − eix = exp

  • ix

2(2m − 1) sin (2mx/2) sin (x/2)

Alessandro Roggero JLAB - 17 Mar 2020 7 / 18

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SLIDE 12

Final measurement: generic phase

. . . QPEm |0 | Φ3 = 2m−1

q=0

  • 1

2m

2m−1

k=0

ei 2πk

2m (2mφ−q)

|q ⊗ |φ |φ when 2mφ is not an integer we can sum the term in parenthesis as

2m−1

  • k=0

eixk = 1 − ei2mx 1 − eix = exp

  • ix

2(2m − 1) sin (2mx/2) sin (x/2) we will measure the ancilla register in |q with probability P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) where we have defined M = 2m

Alessandro Roggero JLAB - 17 Mar 2020 7 / 18

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SLIDE 13

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 14

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=32/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 15

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=33/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 16

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=34/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 17

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=35/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 18

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=36/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 19

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=37/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 20

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=38/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 21

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=39/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 22

Final measurement: generic phase example

example taken from A. Childs lecture notes (2011)

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) EXERCISE: show that if r = ⌈Mφ⌋ then P(r) ≥ 4/π2 ≈ 0.4

4 8 12 16 20 24 28 32

0.2 0.4 0.6 0.8 1

P(q)

M=32 φ=40/256

Pmin = 4/π

2

Alessandro Roggero JLAB - 17 Mar 2020 8 / 18

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SLIDE 23

Final measurement: generic phase II

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) the best m-bit approximation to φ is p/M with p = ⌈Mφ⌋ the probabilty of making an error δ = (q − p)/M is

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 δ 0.2 0.4 0.6 0.8 1 Probability of measuring phase with error δ M=2

1=2

Alessandro Roggero JLAB - 17 Mar 2020 9 / 18

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SLIDE 24

Final measurement: generic phase II

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) the best m-bit approximation to φ is p/M with p = ⌈Mφ⌋ the probabilty of making an error δ = (q − p)/M is

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 δ 0.2 0.4 0.6 0.8 1 Probability of measuring phase with error δ M=2

1=2

M=2

2=4

Alessandro Roggero JLAB - 17 Mar 2020 9 / 18

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SLIDE 25

Final measurement: generic phase II

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) the best m-bit approximation to φ is p/M with p = ⌈Mφ⌋ the probabilty of making an error δ = (q − p)/M is

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 δ 0.2 0.4 0.6 0.8 1 Probability of measuring phase with error δ M=2

1=2

M=2

2=4

M=2

3=8

Alessandro Roggero JLAB - 17 Mar 2020 9 / 18

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SLIDE 26

Final measurement: generic phase II

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) the best m-bit approximation to φ is p/M with p = ⌈Mφ⌋ the probabilty of making an error δ = (q − p)/M is

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 δ 0.2 0.4 0.6 0.8 1 Probability of measuring phase with error δ M=2

1=2

M=2

2=4

M=2

3=8

M=2

4=16

Alessandro Roggero JLAB - 17 Mar 2020 9 / 18

slide-27
SLIDE 27

Final measurement: generic phase II

P(q) = 1 M2 sin2 (Mπ(φ − q/M)) sin2 (π(φ − q/M)) the best m-bit approximation to φ is p/M with p = ⌈Mφ⌋ the probabilty of making an error δ = (q − p)/M is

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 δ 0.2 0.4 0.6 0.8 1 Probability of measuring phase with error δ M=2

1=2

M=2

2=4

M=2

3=8

M=2

4=16

~1/M

Alessandro Roggero JLAB - 17 Mar 2020 9 / 18

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SLIDE 28

Quick recap of QPE for eigenstates

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 given an eigenstate |φ QPE can provide an estimate for the phase φ with precision δ using M ∼ 1/δ with probability P > 4/π2

Alessandro Roggero JLAB - 17 Mar 2020 10 / 18

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SLIDE 29

Quick recap of QPE for eigenstates

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 given an eigenstate |φ QPE can provide an estimate for the phase φ with precision δ using M ∼ 1/δ with probability P > 4/π2 this probability can be amplified to 1 − ǫ using more ancilla qubits* m′ = m +

  • log

1 2ǫ + 2

  • ⇒

M′ ∼ 1 δǫ

*see eg. Nielsen & Chuang

Alessandro Roggero JLAB - 17 Mar 2020 10 / 18

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SLIDE 30

Quick recap of QPE for eigenstates

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|φ U U 2 · · · U 2m−1 given an eigenstate |φ QPE can provide an estimate for the phase φ with precision δ using M ∼ 1/δ with probability P > 4/π2 this probability can be amplified to 1 − ǫ using more ancilla qubits* m′ = m +

  • log

1 2ǫ + 2

  • ⇒

M′ ∼ 1 δǫ

*see eg. Nielsen & Chuang

we can repeat this O(log(1/ǫ)) times and take a majority vote to increase the success probability to 1 − ǫ (see Chernoff bound)

Alessandro Roggero JLAB - 17 Mar 2020 10 / 18

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SLIDE 31

QPE on general states

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|Ψ U U 2 · · · U 2m−1 If we start with a generic state |Ψ =

j cj |φj we find

|Φ3 =

  • j

cj

2m−1

  • q=0
  • 1

2m

2m−1

  • k=0

exp

  • i2πk

2m (2mφj − q)

  • |q ⊗ |φj

The new probability becomes P(q) = 1 M2

  • j

|cj|2 sin2 (Mπ(φj − q/M)) sin2 (π(φj − q/M))

Alessandro Roggero JLAB - 17 Mar 2020 11 / 18

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SLIDE 32

EXAMPLE: Spectrum of 1D Hubbard model

example taken from Ovrum & Horth-Jensen (2007)

HH =

L

  • i
  • σ=↑,↓
  • ǫa†

i,σai,σ − t

  • a†

i+1,σai,σ + a† i,σai+1,σ

  • + U

L

  • i

ni,↑ni,↓ we can estimate the spectrum using QPE with random initial states consider simple case with t = 0, ǫ = U = 1

Ovrum&Horth-Jensen (2007)

2 4 6 8 10 12 14 0.05 0.1 0.15 0.2 Density of states

L=16 -> 256 states 8 qubits

m = 16 ancilla qubits

Alessandro Roggero JLAB - 17 Mar 2020 12 / 18

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SLIDE 33

Fermion to spin mapping: Jordan Wigner transformation

The fermionic Fock space with M modes can be mapped into the Hilbert space of M spins using the following identification see DL lectures ak = k−1

j=0 −Zj

  • Xk+iYk

2

a†

k =

k−1

j=0 −Zj

  • Xk−iYk

2

  • ⇒

{aj, a†

k} = δj,k

  • ccupation encoded into value of spin projection, σ± = Xk±iYk

2

|vac = |↑↑↑↑ a†

2 |vac = |↑↑↓↑

a†

2 |↑↑↓↑ = 0

fermionic phase encoded into the string of Pauli Z operators a†

2a† 1 |vac = −a† 1a† 2 |vac

a†

2a† 0a† 1 |vac = a† 0a† 1a† 2 |vac

Many other mappings available

Bravy-Kitaev, BK-Superfast auxiliary fermions error-correcting codes LDPC codes

Bravyi & Kitaev (2000) , Verstraete & Cirac (2005), Havlicek et al. (2017), Steudtner & Wehner (2017)

Alessandro Roggero JLAB - 17 Mar 2020 13 / 18

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SLIDE 34

Time evolution for Hubbard model

H′

H = L

  • i
  • σ=↑,↓

ni,σ +

L

  • i

ni,↑ni,↓ ni,σ = a†

i,σai,σ

using Jordan-Wigner transformation we can map this into 2L qubits ni,↑ = 1 + Z2i−1 2 ni,↓ = 1 + Z2i 2 HJW = h0 + h1

2L

  • j=1

Zj + h2

2L

  • j<k=1

ZjZk propagator U(τ) can be obtained using

  • ne and two-qubit Z rotatations

EXERCISE

Show that U2 = e−i θ

2 Z1Z2 is

  • Rz(θ)

Alessandro Roggero JLAB - 17 Mar 2020 14 / 18

slide-35
SLIDE 35

QPE as state preparation

|0 H · · ·

  • QFT †

. . . . . . . . . |0 H

  • · · ·

|0 H

  • · · ·

|Ψ U U 2 · · · U 2m−1 |Ψq before the ancilla measurement we have |Φ3 =

  • j

cj

M−1

  • q=0
  • 1

M

M−1

  • k=0

exp

  • i2πk

M (Mφj − q)

  • |q ⊗ |φj

after measuring the integer value q the system qubits are left in |Ψq = 1 M

  • P(q)
  • j

cj sin (Mπ(φj − q/M)) sin (π(φj − q/M)) |φj ≈

  • |φj− q

M | 1 M

cj |φj

Alessandro Roggero JLAB - 17 Mar 2020 15 / 18

slide-36
SLIDE 36

Recap of first day (so far)

given a state |Ψ prepared on a register with n qubits we can compute expectation value Ψ|O|Ψ = Nk

k

  • kΨ|Pk|Ψ with error ǫ

using O(N3

K/ǫ2) repetitions and no additional quantum gate

compute overlap |Ψ|Φ|2 with another state saved in an ancillary register using O(n) gates and O(1/ǫ2) repetitions let |Ψ =

k ck |φk be the decomposition in the eigenbasis of U

we can obtain the eigenvalue spectrum with resolution δ and error ǫ using m = O(log(1/δ)) controlled-U k operations with exponent k = 20, 21, . . . , 2m−1, an additional register of m ancilla qubits and O(1/ǫ2) repetitions project to final state |Ψq ≈ |φq with resolution δ using the same setup if |Ψ is an eigenstate we only need O(1) repetitions

Alessandro Roggero JLAB - 17 Mar 2020 16 / 18

slide-37
SLIDE 37

Recap of first day (so far)

given a state |Ψ prepared on a register with n qubits we can compute expectation value Ψ|O|Ψ = Nk

k

  • kΨ|Pk|Ψ with error ǫ

using O(N3

K/ǫ2) repetitions and no additional quantum gate

compute overlap |Ψ|Φ|2 with another state saved in an ancillary register using O(n) gates and O(1/ǫ2) repetitions let |Ψ =

k ck |φk be the decomposition in the eigenbasis of U

we can obtain the eigenvalue spectrum with resolution δ and error ǫ using m = O(log(1/δ)) controlled-U k operations with exponent k = 20, 21, . . . , 2m−1, an additional register of m ancilla qubits* and O(1/ǫ2) repetitions** project to final state |Ψq ≈ |φq with resolution δ using the same setup if |Ψ is an eigenstate we only need O(1) repetitions**

**Remember there is a non-zero failure probability *Can be relaxed using iterative schemes (see eg. Kitaev (1995), Wiebe&Granade (2015))

Alessandro Roggero JLAB - 17 Mar 2020 16 / 18

slide-38
SLIDE 38

Recap of first day (so far)

given a state |Ψ prepared on a register with n qubits we can compute expectation value Ψ|O|Ψ = Nk

k

  • kΨ|Pk|Ψ with error ǫ

using O(N3

K/ǫ2) repetitions and no additional quantum gate

compute overlap |Ψ|Φ|2 with another state saved in an ancillary register using O(n) gates and O(1/ǫ2) repetitions let |Ψ =

k ck |φk be the decomposition in the eigenbasis of U

we can obtain the eigenvalue spectrum with resolution δ and error ǫ using m = O(log(1/δ)) controlled-U k operations with exponent k = 20, 21, . . . , 2m−1, an additional register of m ancilla qubits* and O(1/ǫ2) repetitions** project to final state |Ψq ≈ |φq with resolution δ using the same setup if |Ψ is an eigenstate we only need O(1) repetitions**

**Remember there is a non-zero failure probability *Can be relaxed using iterative schemes (see eg. Kitaev (1995), Wiebe&Granade (2015))

Alessandro Roggero JLAB - 17 Mar 2020 16 / 18

slide-39
SLIDE 39

Was this worth it?

Can we implement efficiently a controlled-U M operation for M ≫ 1? remember that U(τ)k = U(kτ) = exp(ikτH) for some Hamiltonian

  • S. LLOYD (1996): local hamiltonians can be simulated efficiently

Consider a system of n qubits and a r-local Hamiltonian H = Nj

j

hj where each term hj acts on at most r = O(1) qubits at a time for Nj = O(poly(n)), then using the Trotter-Suzuki decomposition

  • U(τ) −

Nj

  • j

exp (iτhj)

  • ≤ Cτ 2

we can implement U(τ) with error ǫ using O (poly(τ, 1/ǫ, n)4r) gates. PROBLEM: the Jordan-Wigner transformation leads to n-local terms

Alessandro Roggero JLAB - 17 Mar 2020 17 / 18

slide-40
SLIDE 40

Final recap of first day

1 quantum computers can simulate efficiently the time-evolution

  • perator U(τ) = exp(iτH) for r-local Hamiltonians

for target error ǫ this requires O(poly(n, τ, 1/ǫ)4r) gates Jordan Wigner on n qubits leads to n-local terms!

SPOILER: this might not be a problem in practice

tomorrow we’ll generalize this and find better scaling

2 if we can prepare an energy eigenstate |φ we can use this to measure

it’s phase with accuracy ∆ using a total propagation time τ ∼ 1/∆

3 if |Ψ has overlap α = |φ|Ψ|2, we just add O(1/α) repetitions Alessandro Roggero JLAB - 17 Mar 2020 18 / 18