Digital quantum simulation with trapped ions

Tobias Thiele, Damian Berger presented papers

2008:

2011: Overview

• Description of the Ion Trap: -Atomic Energy Levels and Motional States -Measurement • Operations and Gates -Single Qubit Rotation, Cirac Gate, Mølmer–Sørensen Gate • Quantum Simulation • Digital Quantum Simulation with Ions pictures from „Entangled states of trapped atomic ions“ R.Blatt & D.Wineland, Nature, 2008

• The qubit is represented by two internal states of the ion. • Motional States mediate qubit-qubit interactions Internal States of 40Ca+

• The qubits are encoded in the S1/2 and the D3/2 level. • The probability of the qubit being in the S1/2 state is determined by measuring the scattered light of the S1/2 - P3/2 transition. Single Qubit Operations

• Laser Light can do all single qubit operations • Laser frequency w0 and transition frequency w have to be equal • Internal Hamiltonian of the two level system: iφ -iφ H = ħΩ(�+e + �-e ) (Assumptions: w0 = w, classical described laser field) • By choosing φ = 0 or φ = π/2 this allows rotation arount x and y axis of the bloch sphere • Rotations around z-axis can be done by composition of x and y rotations or by detuning the laser light The Cirac Zoller CNOT Gate

• narrow Laser light to adress only one ion • tuning the laser light to ,sideband‘ frequencies wion ± wm drives transitions

|g,n⟩ ⟷ |e,n±1⟩ The Cirac Zoller CNOT Gate

picture from „Entangled states of trapped atomic ions“ R.Blatt & D.Wineland, Nature, 2008 Mølmer–Sørensen Gate

from „Towards fault-tolerant quan...“ by R.Blatt et al. , Nat. Phys. 4, 463 (2008)

-both qubits in bichromatic laser field with frequency ω0 ± δ -vibrational degrees of freedom only enter virtually from „Towards fault-tolerant quan...“ by R.Blatt et al. , Nat. Phys. 4, 463 (2008) from „Towards fault-tolerant quan...“ by R.Blatt et al. , Nat. Phys. 4, 463 (2008) Red Line: Fidelity of the state |SS⟩+i|DD⟩ after m gate operations Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results Quantum Simulation

• Problem: – Quantum System with Hamiltonian H • Ground state? • Phase transitions? • Time evolution? • Correlations? – Not solvable with classical methods/computers Quantum Simulation

• Problem: – Quantum System with Hamiltonian H • Ground state? • Phase transitions? • Time evolution? • Correlations? – Not solvable with classical methods/computers

• Solution: – Use quantum system to simulate H • „measure result“ Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results Analog Quantum Simulation

• Analog Quantum Simulation – Build a system that implements only H Analog Quantum Simulation

• Analog Quantum Simulation – Build a system that implements only H • Famous system: Bose-Hubbard Hamiltonian 1 – ˆt ˆ ˆ ˆ ˆ H  J ai a j  Uni (ni 1)  ini i, j 2 i i Analog Quantum Simulation

• Analog Quantum Simulation – Build a system that implements only H • Famous system: Bose-Hubbard Hamiltonian 1 – ˆt ˆ ˆ ˆ ˆ H  J ai a j  Uni (ni 1)  ini i, j 2 i i – Neutral atoms in optical lattice – Changing detuning of laser  change J U Analog Quantum Simulation

• Analog Quantum Simulation – Build a system that implements only H • Famous system: Bose-Hubbard Hamiltonian 1 – ˆt ˆ ˆ ˆ ˆ H  J ai a j  Uni (ni 1)  ini i, j 2 i i – Neutral atoms in optical lattice – Changing detuning of laser  change J U – Superfluid to Mott Insulator Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations Implement any unitary operation Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations Implement any unitary operation Implement any (local) Hamiltonian H Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations String of Ions + toolbox! Implement any unitary operation Implement any (local) Hamiltonian H Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations String of Ions + toolbox! Implement any unitary operation Implement any (local) Hamiltonian H – i.e. 2 Spin Ising model:

i i j 1 2 1 2 H X  B z  J  x x  B( z  z )  J x x i i, j Digital Quantum Simulation

• Digital Quantum Simulation – System of qubits with universal set of qubit operations String of Ions + toolbox! Implement any unitary operation Implement any (local) Hamiltonian H – i.e. 2 Spin Ising model:

i i j 1 2 1 2 H X  B z  J  x x  B( z  z )  J x x i i, j – Goal: Determine time evolution of system Digital Quantum Simulation

1 2 1 2 H X  B( z  z )  J x x Digital Quantum Simulation

1 2 1 2 H X  B( z  z )  J x x • Time evolution of  :  i  – t Ut0  exp Ht0    Digital Quantum Simulation

1 2 1 2 H X  B( z  z )  J x x • Time evolution of  :  i  – t Ut0  exp Ht0   

• Strategy: – Divide Hamiltonian into operations that can be applied at once

1 2 1 2 H X  B( z  z )  J x x  H1  H2

Single Qubit Operations Multi Qubit Operations Digital Quantum Simulation

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Unitary Time evolution:  i  – Ut  exp H1  H2 t     Digital Quantum Simulation

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Unitary Time evolution:  i  – Ut  exp H1  H2 t     • In experiment only possible to apply:  i  – Ui t  exp Hit     Digital Quantum Simulation

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Unitary Time evolution:  i  – Ut  exp H1  H2 t     • In experiment only possible to apply:  i  – Ui t  exp Hit      i   i   i  • And: exp H1  H2 t   exp H1t exp H2t          

– since H1, H2  0 Digital Quantum Simulation

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations

• Trotter formula n  i    i t   i t  exp H1  H2 t   limexp H1 exp H2     n   n    n  Digital Quantum Simulation

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations

• Trotter formula n  i    i t   i t  exp H1  H2 t   limexp H1 exp H2     n   n    n  • Therefore: planned sequence

 t t  i n n . . . f n times Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • We have: DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • We have: – A string of ions/qubits (Ca+) DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • We have: – A string of ions/qubits (Ca+) with measurement

e  P1/ 2

  D5/ 2

  S1/ 2 DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • We have: e  P – A string of ions/qubits 1/ 2 – Measurement 0  D5/ 2

– A universal set of operations 1  S1/ 2 DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations e  P • We have: 1/ 2

– A string of ions/qubits 0  D5/ 2

1  S1/ 2 – A universal set of operations

j O1(, j)  expi z  DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations e  P • We have: 1/ 2

– A string of ions/qubits 0  D5/ 2

1  S1/ 2 – A universal set of operations

j O1(, j)  expi z 

  O ()  expi  j  2   z   j  DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations e  P • We have: 1/ 2

– A string of ions/qubits 0  D5/ 2

1  S1/ 2 – A universal set of operations   O (, j)  exp i j O (,)  expi  j  1  z  3      j    O ()  expi  j  2   z   j  DQS with Ions

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations e  P • We have: 1/ 2

– A string of ions/qubits 0  D5/ 2

1  S1/ 2 – A universal set of operations   O (, j)  exp i j O (,)  expi  j  1  z  3      j      O ()  expi  j  O (,)  expi  i j  2   z  4       j   i j  Ising Model

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Easy now to implement!   O (, j)  exp i j O (,)  expi  j  1  z  3      j      O ()  expi  j  O (,)  expi  i j  2   z  4       j   i j  Ising Model

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Easy now to implement!     O ()  expi  j  O (,)  expi  i j  2   z  4       j   i j  • Prepare qubits in state  Ising Model

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Easy now to implement!     O ()  expi  j  O (,)  expi  i j  2   z  4       j   i j  • Prepare qubits in state  • Choose t and n and apply: t t t  n n . n . . Ising Model

1 2 1 2 Single Qubit Operations H X  B( z  z )  J x x  H1  H2 Multi Qubit Operations • Easy now to implement!     O ()  expi  j  O (,)  expi  i j  2   z  4       j   i j  • Prepare qubits in state  • Choose t and n and apply: t t n n . . . • Measure state at the end Ising Model-Results

1 2 1 2 H X  B( z  z )  J x x Single Qubit Operations Multi Qubit Operationsn 1 n 1

n  2

n  3

n  4 Ising Model-Results

1 2 1 2 H X  B( z  z )  J x x Single Qubit Operations Multi Qubit Operationsn 1 n 1

n  2

n  3

n  4 Ising Model-Results

1 2 1 2 H X  B( z  z )  J x x Single Qubit Operations Multi Qubit Operationsn 1 n 1

n  2

n  3

n  4 Ising Model-Results

1 2 1 2 H X  B( z  z )  J x x Single Qubit Operations Multi Qubit Operationsn 1 n 1

n  2

n  3

n  4 Content

• Quantum Simulation – Analog Quantum Simulation • Digital Quantum Simulation – Ising Model • Digital Quantum Simulation with Ions – More complex system/Results Complex systems       O (, j)  exp i j O ()  expi  j  O (,)  expi  j  O (,)  expi  i j  1  z  2   z  3     4       j   j   i j 

O4 ( / n,0).O2 ( / n)

 O ( / n, ).O ( / n,0).O ( / n) 4 2 4 2

O3 (4 / n,0).O4 ( / n,0).O3 (4 / n,0).

O4 ( / n, ).O4 ( / n,0).O2 ( / n) More complex systems       O (, j)  exp i j O ()  expi  j  O (,)  expi  j  O (,)  expi  i j  1  z  2   z  3     4       j   j   i j 

C  O2 ( / 32)

D  O4 ( /16,0)

E  O1( / 2,1)

F  O1(,1) Even more complex systems       O (, j)  exp i j O ()  expi  j  O (,)  expi  j  O (,)  expi  i j  1  z  2   z  3     4       j   j   i j 

C  O2 ( / 32)

D  O4 ( /16,0)

E  O1( / 2,1)

F  O1(,1) Summary-Take Home Message

• Analog quantum computing – Does well, but only for a single Hamiltonian – Needs a system that implements H • Digital Quantum Computing – Possible to implement any local Hamiltonian – Needs (only) set of qubits and a universal set of operations • The more computational power, the longer and more exact the simulations can be done Ising Model-Results:time-dependence

1 2 Single1 2 Qubit Operations H X  B( z  z )  J(t)Multix x Qubit Operationsn 1