UNIVERSITY OF INNSBRUCK Quantum Reservoir Engineering (Discussion in the Quantum Coherent Control Program) IQOQI AUSTRIAN ACADEMY OF SCIENCES

€U AQUTE €U COHERENCE Peter Zoller

UQUAM

AFOSR

Tuesday, March 12, 13 Quantum reservoir engineering?

Environment E dissipation as System a resource S

fundamentally quantum information unevitable coupling diss. preparation of multi-qubit quantum states dissipative quantum computation

non-equilibrium dynamics t ⇢mixed eL result in open many-body quantum systems t !1

bosons spins

Tuesday, March 12, 13 Entanglement by Dissipation

“Optical Pumping” theoretical concepts:

Review:

M. Müller, S. Diehl, G. Pupillo, P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, arXiv:1203.6595; published in Advances of Atomic, Molecular and Optical Physics 2012

F. Verstraete, M.M. Wolf, J.I.Cirac, Nature Physics (2009) first experiments:

J. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller & R. Blatt Nature 470, 486 (2011) H. Krauter, E. Polzik, I. Cirac et al. PRL 2011. 3 Tuesday, March 12, 13 Entanglement via Unitary Evolution

• quantum logic network model

qubits quantum gates read out

time √ U √ | ⇥ t | ⇥ coherent Hamiltonian evolution - quantum gates - deterministic

4 Tuesday, March 12, 13 Entanglement via Unitary Evolution

• quantum logic network model • atomic physics: trapped ions

qubits quantum gates read out

time √ Ut √ | ⇥ | ⇥ R. Blatt coherent Hamiltonian evolution - quantum gates - deterministic

4 Tuesday, March 12, 13 Entanglement via Unitary Evolution

• quantum logic network model • atomic physics: trapped ions

qubits quantum gates read out

time √ Ut √ | ⇥ | ⇥ R. Blatt coherent Hamiltonian evolution - quantum gates - deterministic • decoherence ☹ spontaneous dissipation by coupling to environment - spontaneous emission etc. emission ☹

4 Tuesday, March 12, 13 Entanglement via Unitary Evolution

• quantum logic network model • atomic physics: trapped ions

qubits quantum gates read out

time √ Ut √ | ⇥ | ⇥ R. Blatt coherent Hamiltonian evolution - quantum gates - deterministic • decoherence ☹ spontaneous dissipation by coupling to environment - spontaneous emission etc. emission ☹

Q.: Dissipation as an engineering tool? For entanglement? 4 Tuesday, March 12, 13 Open System Dynamics [& Decoherence ☹]

• open system dynamics

system

environ- not ment observed

5 Tuesday, March 12, 13 Open System Dynamics [& Decoherence ☹]

• open system dynamics

system

environ- not ment observed

completely positive maps:

† Ω E (Ω) Ek ΩEk = k X

Kraus operator

5 Tuesday, March 12, 13 Entanglement from (Engineered) Dissipation

• open system dynamics

system √ √ | ⇥ |

environ- not ment observed

6 Tuesday, March 12, 13 Entanglement from (Engineered) Dissipation

• open system dynamics

system √ √ | ⇥ |

environ- not ment observed

engineering Kraus operators:

† Ω E (Ω) Ek ΩEk = k ! X √ √ =| ⇥ |

desired (pure) quantum state “cooling” into a pure state - non-unitary - deterministic

6 Tuesday, March 12, 13 Entanglement from (Engineered) Dissipation

• open system dynamics • atomic physics: single particle

optical pumping system √ √ | ⇥ |

environ- not ment observed

engineering Kraus operators:

† Ω E (Ω) Ek ΩE t = k (t) ⇥ D D k ⇥ | ⌅ ⇤ | ! X √ √ =| ⇥ | pumping into a pure “dark state”

desired (pure) quantum state “cooling” into a pure state - non-unitary - deterministic

6 Tuesday, March 12, 13 Entanglement from (Engineered) Dissipation

• open system dynamics • atomic physics: single particle

optical pumping system √ √ | ⇥ |

environ- not ment observed

engineering Kraus operators:

† Ω E (Ω) Ek ΩE t = k (t) ⇥ D D k ⇥ | ⌅ ⇤ | ! X √ √ =| ⇥ | pumping into a pure “dark state”

desired (pure) quantum state “cooling” into a pure state - non-unitary Q.: generalize to entangled states? - deterministic

6 Tuesday, March 12, 13 Entanglement from (Engineered) Dissipation

• open system dynamics • atomic physics: single particle

optical pumping system √ √ | ⇥ |

environ- not ment observed

engineering Kraus operators:

† Ω E (Ω) Ek ΩE t = k (t) ⇥ D D k ⇥ | ⌅ ⇤ | ! X √ √ =| ⇥ | pumping into a pure “dark state”

desired (pure) quantum state “cooling” into a pure state - non-unitary Q.: generalize to entangled states? - deterministic see also: D Bacon et al. PRA 2001; S. Lloyd & L. Viola, PRA 2001; D. Lidar, et al. PRL 1998; G. Baggio, et al. arxiv:1209.5568 (2012). 6 Tuesday, March 12, 13 Dark States: Many Particle qubits or particles on a lattice

c

quasi-local Lindblad operators

• master equation quantum jump operator (nonhermitian) Ω˙ i[H,Ω] = † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ω c cÆ + Æ 2 Æ 2 Æ Æ µ ∂ X

7 Tuesday, March 12, 13 Dark States: Many Particle qubits or particles on a lattice

c

quasi-local Lindblad operators

• master equation quantum jump operator (nonhermitian) Ω˙ i[H,Ω] = † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ω c cÆ + Æ 2 Æ 2 Æ Æ µ ∂ X • desired state as “dark state”

t H D E D Ω(t) ⇥ D D | = | ⇧ | ⌅⇤ | Æ cÆ D 0 | ⇥ = construct a parent Liouvillian desired state 7 Tuesday, March 12, 13 Dark States: Many Particle qubits or particles on a lattice

c Questions: ✓ given resources → states (?) ✓ uniqueness quasi-local ✓ implementation Lindblad operators

• master equation quantum jump operator (nonhermitian) Ω˙ i[H,Ω] = † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ω c cÆ + Æ 2 Æ 2 Æ Æ µ ∂ X • desired state as “dark state”

t H D E D Ω(t) ⇥ D D | = | ⇧ | ⌅⇤ | Æ cÆ D 0 | ⇥ = construct a parent Liouvillian desired state 7 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits 1 = ( 01 + 10 ) | ⇥2 | | 1 + = ( 00 11 ) | ⇥ ⇤2 | ⇥| ⇥ 1 = ( 01 10 ) | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) | ⇥2 | |

8 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

8 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + z1z2

9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + z1z2

quantum jump c1 X1(1 Z1Z2) operators = + 9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + + +

2 x 1

z1z2 x

quantum jump c1 X1(1 Z1Z2) operators = + 9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + + +

2 x 1

z1z2 x

quantum jump c1 X1(1 Z1Z2) c2 Z1(1 X1X2) operators = + = + 9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + + +

2 x 1

z1z2 x

quantum jump c1 X1(1 Z1Z2) c2 Z1(1 X1X2) operators = + = + 9 Tuesday, March 12, 13 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + + +

2 x 1

z1z2 x

quantum jump c1 X1(1 Z1Z2) c2 Z1(1 X1X2) 3-particle operators ☹ operators = + = + 9 Tuesday, March 12, 13 40 + P Ca Ions confined in 1/2

a string by a Paul trap D5/2

Detecon Quantum 0 bit | Manipulaon

S 1 1/2 |

Innsbruck ion trap

Tuesday, March 12, 13 Quantum operations on Innsbruck ion-trap quantum computer

Individual light-shift gates (0) (1) (2) z , z , z

Collective spin flips S , S x y + + ... +

Mølmer-Sørensen gate 2 (0) (1) (1) (2) (0) (2) Sx = x x + x x + x x

(0)(2) (0) (1) x x x x

(1) (2) x x

... 14-qubit entanglement, T. Monz et al. PRL (2011) talk U6.4

Tuesday, March 12, 13 Quantum operations on Innsbruck ion-trap quantum computer

Individual light-shift gates (0) (1) (2) z , z , z

Collective spin flips Env. System S , S 0 1 n x y + + ... +

Mølmer-Sørensen gate Physical 2 (0) (1) (1) (2) (0) (2) environment Sx = x x + x x + x x

(0)(2) (0) (1) x x x x Coupling of environment with physical environment

(1) (2) x x Optical pumping of ... 14-qubit entanglement, T. Monz et al. PRL (2011) talk U6.4 „environmental“ ion

Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

12 Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

(i) (ii) (iii) one(iv) pumping (i)cycle(ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

12 Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

(i) (ii) (iii) one(iv) pumping (i)cycle(ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

12 Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

(i) (ii) (iii) one(iv) pumping (i)cycle(ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

mapping eigenvalue to ancilla

12 Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

(i) (ii) (iii) one(iv) pumping (i)cycle(ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

mapping eigenvalue controlled to ancilla “correction”

12 Tuesday, March 12, 13 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

(i) (ii) (iii) one(iv) pumping (i)cycle(ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

controlled mapping eigenvalue dissipation to ancilla “correction”

12 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + z1z2

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + z1z2

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + z1z2

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011 13 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p 1 p = 0.5 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p 1 p = 0.5 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p 1 p = 0.5 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

14 Tuesday, March 12, 13 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) 1〉 0 ) ⎥1〉 1

2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle

pumping probability p 1 p = 0.5 0.8 0.6 0.4 + Ω˙ i[H,Ω] Populations 0.2 = + 2 0 † 1 † 1 † ∞Æ cÆΩcÆ cÆcÆΩ ΩmixedcÆcÆ 1 2 3 + 2 state2 Pumping cycles Æ 1 µ ∂ X= 14 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 0 0001 4 2 0010 0011 3 0100 z3z4 z2z3 0101 0110 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

z2z3

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z3z4 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

z2z3

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z3z4 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 1111

z2z3

z2z3

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z3z4 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000

Populations 1001 1010 1011 1100 1101 1110 0000 + 1111 1111 | | F = 55.8(4)% z2z3

z2z3 x1x2x3x4

15 Tuesday, March 12, 13 Stabilizer pumping: 1+4 ions

Ion: 1234 x x x x 1 2 3 4 z1z2 1 0000 z3z4 z2z3 0 0001 4 2 0010 z1z2 0011 3 0100 z3z4 z2z3 0101 0110 z1z2 0111 1000 16 five-ion entangling gates

Populations 1001 1010 20 collective rotations 1011 34 single-ion gates 1100 + hiding and optical pumping 1101 + state preparation 1110 + readout pulses 0000 + 1111 1111 | | F = 55.8(4)% z2z3

z2z3 x1x2x3x4

15 Tuesday, March 12, 13 Environment E System S

Non-Equilibrium Condensed Matter Physics with Engineered Reservoirs

• open-system many-body dynamics & quantum phase transitions Review: M. Müller, S. Diehl, G. Pupillo, P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, arXiv:1203.6595; published in Advances of Atomic, Molecular and Optical Physics 2012

Exp.:

P. Schindler, M. Müller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, R. Blatt, Quantum simulation of open-system dynamical maps with trapped ions, arXiv:1212.2418 • topology via (engineered) dissipation Review:

C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, S. Diehl, Topology by dissipation [with cold atoms], arXiv:1302.5135

16 Tuesday, March 12, 13 S. Diehl et al., Nat Phys 2008 Non-equilibrium cond mat S. Diehl et al., PRL 2010 • thermodynamic equilibrium • non-equilibrium

O(N) Rotor Model Phase Diagram interaction temperature CONTINUUM HIGH T phase diagram T or QUANTUM CRITICAL interesting (quantum) phases

LOW T phase transition MAGNETIC Quantum LONG RANGE paramagnet ORDER ? 0 0 g 0 gc g 0 c interaction vs. dissipation

quantum phase transition dynamical quantum phase transition Superfluid “thermal” H Eg = Eg Eg | | (pure state) d (mixed state) = i [H, ] + H = H0 + gH1 dt L Hamiltonian (many body) 17 Tuesday, March 12, 13 A. Griessner et al., Driven Dissipative Hubbard Models PRL 2006

• BEC as a “phonon reservoir” - quantum reservoir engineering

BEC

18 Tuesday, March 12, 13 “think quantum optics”

• driven two-level atom + spontaneous trapped atom in a BEC reservoir emission •

e energy scale! | ! optical 1 Ω Γ photon | ! 0 g BEC | ! | ! “phonon” (cycling transition) laser atom photon laser assisted atom + BEC collision

• reservoir: vacuum modes of the radiation field (T=0)

19 Tuesday, March 12, 13 “think quantum optics”

• driven two-level atom + spontaneous trapped atom in a BEC reservoir emission •

e energy scale! | ! optical 1 Ω Γ photon | ! 0 g BEC | ! | ! “phonon” (cycling transition) laser atom photon laser assisted atom + BEC collision

• reservoir: vacuum modes of the • reservoir: Bogoliubov excitations of the radiation field (T=0) BEC (@ temperature T)

19 Tuesday, March 12, 13 “think quantum optics”

• Λ-system

three electronic levels dark state bright state • N atom on M sites local dissipation

cond mat: ~ array of dissipative Josephson junctions quantum optics: spatial chain of Λ-systems

Tuesday, March 12, 13 “think quantum optics”

• Λ-system

three electronic levels dark state bright state • N atom on M sites local dissipation

t (t) ⇥ BEC BEC ⇥ | ⌅ ⇤ | driven dissipative BEC

cond mat: ~ array of dissipative Josephson junctions quantum optics: spatial chain of Λ-systems

Tuesday, March 12, 13 “think quantum optics”

• Λ-system

dark state bright state • 1 atom on 2 sites

1 2

J

~ dissipative Josephson junction

Tuesday, March 12, 13 “think quantum optics”

• Λ-system

dark state bright state • 1 atom on 2 sites

1 2 (a† + a†) vac (a† a†) vac 1 2 | 1 2 | ⇥ symmetric anti-symmetric J “in-phase” “out-of-phase”

~ dissipative Josephson junction pumping into symmetric state “phase locking”

Tuesday, March 12, 13 cij Driven Dissipative Bose Hubbard Model

• master equation

d ⇥ = i [H, ⇥]+ 2c ⇥c† c† c ⇥ ⇥c† c dt ij ij ij ij ij ij ij ⇥ ⇥ ⇤ adjacent sites

1. Bose Hubbard Hamiltonian 2. Liouvillian

1 2 2 H = J a†a + U a† a cij = a† + a† a aj i j 2 i i i j i i jump operator (quasi-local)⇥ ⇥

SF - Mott insulator physics “cooling” to a BEC local “driving/dissipation” causes long range order

~ ~ ~ ... ~

Tuesday, March 12, 13 cij Driven Dissipative Bose Hubbard Model

• master equation

d ⇥ = i [H, ⇥]+ 2c ⇥c† c† c ⇥ ⇥c† c dt ij ij ij ij ij ij ij ⇥ ⇥ ⇤ adjacent sites

1. Bose Hubbard Hamiltonian 2. Liouvillian

1 2 2 H = J a†a + U a† a cij = a† + a† a aj i j 2 i i i j i i jump operator (quasi-local)⇥ ⇥

SF - Mott insulator physics “cooling” to a BEC local “driving/dissipation” causes long range order

Rem.: we have a microscopic justification for this model ~ ~ ~ ... ~

Tuesday, March 12, 13 Steady State Quantum Phases & Phase Transition

What happens when we switch on interactions?

superfluid order parameter thermal superfluid state

0 interaction U/κ ~ interaction + ”temperature”

for NO interactions Blochband

• perfect (driven dissipative) BEC 2J

N N BEC ( a†) vac (a† ) vac | ⇤⇥ i | ⇤ q=0 | ⇤ i q t (t) ⇥ BEC BEC ⇥ | ⌅⇤ | 23 Tuesday, March 12, 13 Steady State Quantum Phases & Phase Transition

superfluid order parameter thermal superfluid state

0 interaction U/κ ~ interaction + ”temperature”

for WEAK interactions Blochband 3D: depleted condensate (Bogoliubov) • 2J • 1D/2D critical Kosterlitz-Thouless / Luttinger

q

24 Tuesday, March 12, 13 Steady State Quantum Phases & Phase Transition

superfluid order parameter thermal superfluid state

0 interaction U/κ ~ interaction + ”temperature”

for STRONG interactions Blochband quantum phase transition • 2J (here: in mean field for 3D)

n¯n q = thermal state: n,n (¯n + 1)n+1 “fixed temperature”

25 Tuesday, March 12, 13 1 | 0 | bosons spins ion trap experiment

✓in the present form not scalable From bosons … to spins, and ... ✓a very small system

Quantum Simulation of Dynamical Maps with Trapped Ions

1 2 3, 4 3, 4 3, 4 P. Schindler⇤, M. M¨uller⇤, D. Nigg, J. T. Barreiro†, E. A. Martinez, M. Hennrich,3, 4 T. Monz,3, 4 S. Diehl,3, 4 P. Zoller,4 and R. Blatt4 1Institut f¨urExperimentalphysik, Universit¨atInnsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria 2Departamento de F´ısica Te´orica I, Universidad Complutense, Avenida Complutense s/n, 28040 Madrid, Spain 3Institut f¨urTheoretische Physik,http://arxiv.org/abs/1212.2418 Universit¨atInnsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria 4Institut f¨ur Quantenoptik und Quanteninformation, Osterreichische¨ Akademie der Wissenschaften,Technikerstrasse 21A, 6020 Innsbruck, Austria ⇤ These authors contributed equally to this work. Present address: Fakult¨atf¨ur Physik, Ludwig-Maximilians-Universit¨at † M¨unchen & Max-Planck Institute of Quantum Optics, Germany 26 Tuesday, March 12, 13

Dynamical maps describe general transforma- its resulting dissipative dynamics – introduces new sce- tions of the state of a physical system, and their narios of dissipative quantum state preparation [21–25], iteration can be interpreted as generating a dis- dissipative variants of quantum computing and memo- crete time evolution. Prime examples include ries [26–28] and non-equilibrium many-body physics [29– classical nonlinear systems undergoing transitions 31]. The experimental combination of both coherent and to chaos. Quantum mechanical counterparts show dissipative evolution allows us to explore the dynamics intriguing phenomena such as dynamical localiza- of novel classes of non-equilibrium many-body quantum tion on the single particle level. Here we extend systems. the concept of dynamical maps to a many-particle The dynamics of these systems is often considered as context, where the time evolution involves both continuous in time, described by many-body Lindblad coherent and dissipative elements: We experi- master equations, cf. e.g. [32]. This may be conceived as mentally explore the stroboscopic dynamics of a special instance of a more general setting, where a dis- a complex many-body spin model by means of crete time evolution of a system’s reduced density matrix a universal quantum simulator using up to five is generated by concatenated dynamical maps. So far, ions. In particular, we generate long-range phase the concept of dynamical maps has proven useful for the coherence of spin by an iteration of purely dis- description of periodically driven classical nonlinear sys- sipative quantum maps. We also demonstrate tems [33], and their quantum mechanical counterparts, the characteristics of competition between com- such as the kicked rotor, providing one of the paradig- bined coherent and dissipative non-equilibrium matic models of quantum chaos [34–36]. Remarkable ex- evolution which represents the hallmark feature periments have been performed with periodically driven of a novel dynamical phase transition. In order systems of cold atoms, which have demonstrated some of to do so, we employ a new spectroscopic decou- the basic phenomena of quantum chaos such as dynami- pling technique that facilitates the simulation of cal localization [37–40]. At present all these studies are complex many-body systems in an ion trap quan- on the level of single particle physics. tum information processing architecture. We fur- Below we present a first experimental study of many- ther assess the influence of experimental errors in particle open-system dynamical maps for complex spin quantum simulations and tackle this problem by models (representing hardcore bosons), implemented in developing an ecient error detection and reduc- a linear ion-trap quantum computing architecture using tion toolbox based on quantum feedback. up to five ions. The dynamical maps are realized by a Obtaining full control of the dynamics of many-particle digital simulation strategy, contrasting analog Hamilto- quantum systems represents a fundamental scientific nian quantum simulation with trapped ions [41–45]. Our and technological challenge. Impressive experimental study of new physical phenomena, involving the com- progress on various physical platforms has been made [1– petition between coherent and dissipative multi-particle 12], complemented with the development of a detailed dynamics, is enabled by recent progress in performing quantum control theory [13–16]. Controlling the coher- high-fidelity quantum operations in systems of trapped ent dynamics of systems well-isolated from the environ- ions. This allows us not only to engineer (program) com- ment enables, for example, quantum computation in the plex individual dynamical maps, but provides the high circuit model [17]. But this also allows for digital coher- fidelities required to iterate dynamical maps in a mean- ent quantum simulation with time evolution realized by ingful way to follow the time evolution for multiple it- sequences of small Trotter steps [18], as demonstrated in erated maps, and thus to observe for the first time the recent experiments [19, 20]. On the other hand, engineer- novel physics associated with the competition between ing the coupling of a system to its environment – and thus coherent and dissipative dynamics. Non-equilibrium dynamics for bosons and spins - a sketch

• dissipation: delocalizes particles over the lattice • Hamiltonian: „asks“ for particle number, tends to localize particles

0 =! @ ⇢ = i [H,⇢]+ [⇢] t L spins / translation table bosons (cold atoms) hardcore bosons (trapped ions)

jump operators in

+

dissipative target dark state

competing Hamiltonian with with

boson scenario: S. Diehl S. Diehl, A. Tomadin, A. Micheli, R. Fazio, P. Zoller, PRL 2010 Tuesday, March 12, 13 Non-equilibrium dynamics for bosons and spins - a sketch

• dissipation: delocalizes particles over the lattice Hamiltonian: „asks“ for particle number, • a spin analog tends to localize particles • (hard-core bosons) 0 =! @ ⇢ = i [H,⇢]+ [⇢] = t L | ⇥⇥⇤ spins / translation table bosons (cold atoms) hardcore bosons (trapped ions)

jump operators in

+

dissipative target dark state

competing Hamiltonian with with

boson scenario: S. Diehl S. Diehl, A. Tomadin, A. Micheli, R. Fazio, P. Zoller, PRL 2010 Tuesday, March 12, 13 2

In particular, we demonstrate the purely dissipative Fig. 1a, with time steps t` t`+1 represented by creation of quantum mechanical long-range phase order. ! K Furthermore, we implement a competition between co- (l) (l) (l) ⇢(t ) ⇢(t )= [⇢(t )] = E ⇢(t )E †. (1) herent and dissipative many-particle dynamics by alter- ` 7! `+1 E ` k ` k nating sequences of unitary and non-unitary maps out- kX=1 lined in Fig. 1a, and observe the destruction of phase co- (l) The set of Kraus operators Ek satisfies herence as a result. This reflects the hallmark feature of K (l) (l) { } a strong coupling non-equilibrium phase transition pre- k=1 Ek †Ek = 1 [17]. While the familiar se- quences of unitary maps are obtained for a single Kraus dicted in a closely related driven-dissipative model of P bosons [46]. In the actual implementation of the sim- operator K = 1, dissipative dynamics corresponds to ulation, the engineered dissipative and coherent dynam- multiple Kraus operators K>1. In particular, the ics compete with undesired dissipative processes mainly continuous time evolution of a Lindblad master equation caused by imperfect gate operations. We address this is recovered in the limit of infinitesimal time steps, cf. generic, though so far widely disregarded aspect, by care- Methods. The dissipative dynamics studied in our spin fully assessing the experimental errors. As a step towards model is governed by dynamical maps according to solving this problem, we develop a novel and ecient spe- two-body Kraus operators acting on pairs of neighboring cial purpose error reduction technique. spins i, i + 1: Competing dissipative and unitary dynamics in a com- Ei,1 = ci,Ei,2 =1 c†ci. (2) plex spin model – Two competing, non-commuting con- i tributions to a Hamiltonian give rise to a quantum phase The elementary operators generating the dynamics2 are transition, if the respective ground states of each con- given by tribution separately favor states with di↵erent symme- In particular, we demonstrate the purely dissipative Fig. 1a, with time steps t+` t`++1 represented by ci =( + )( ), (3) creationtries of [47]. quantum The mechanical transition takes long-range place at phase a critical order. value i ! i+1 i i+1 2 of the dimensionless ratio g of the two competing energy x y K Furthermore, we implement a competition between co- where i± =(i (l) ii )/2 are spin-1/2(l) raising(l) and low- scales. A non-equilibrium analog can be achieved in open ⇢(t`) ⇢(t`+1)= ± [⇢(t`)] = E ⇢(t`)E †. (1) herentIn and particular, dissipative we demonstrate many-particle the dynamics purely dissipative by alter- Fig. 1eringa,7! with operators time stepsE actingt` t on`+1 spinrepresentedik. In by thek continuous many-body quantum systems, where coherent Hamilto- ! k=1 natingcreation sequences of quantum of unitary mechanical and non-unitary long-range phase maps order. out- time limit, the operators ci correspondX precisely to Lind- nian and dissipative dynamics compete with each other: K linedFurthermore, in Fig. 1a, weand implement observe the a destruction competition of between phase co- co- blad quantum jump(l) operators and(l) generate(l) (l) a dissipative The role of the ground state is played by the stationaryThe⇢(t`) set⇢(t of`+1)= Kraus[⇢(t operators`)] = E ⇢E(t`)E †satisfies. (1) herenceherent as and a result. dissipative This many-particle reflects the hallmark dynamics feature by alter- of evolution7! describedE by a quantumk { masterk }k equation, cf. state of the combined evolution, and the dimensionless K (l) (l) k=1 a strongnating coupling sequences non-equilibrium of unitary and non-unitary phase transition maps pre- out- k=1Methods.Ek †Ek The= operators 1 [17]. act WhileX bi-locally the on familiar pairs of se- spins, ratio g is provided by a Hamiltonian energy scale vs. a dictedlined in in aFig. closely1a, and related observe driven-dissipative the destruction of model phase co- of quencesas visualized of unitary in maps Fig. are1b: obtained Physically, for(l a) they single map Kraus any an- dissipative rate. Such a situation has been addressed pre-PThe set of Kraus operators Ek satisfies bosonsherence [46]. as a In result. the actual This reflects implementation the hallmark of feature the sim- of operatorK tisymmetric(l)K (=l) 1, component dissipative in dynamics the wave{ corresponds function} on to a pair viously theoretically in the context of driven-dissipative E †E = 1 [17]. While the familiar se- ulation,a strong the coupling engineered non-equilibrium dissipative and phase coherent transition dynam- pre- multiplek=1ofk sites Krausk into operators the symmetricK>1. one, In or particular, – in the language the dynamics of atomic bosons on a lattice [46]: A dissipa-quences of unitary maps are obtained for a single Kraus icsdicted compete in with a closely undesired related dissipative driven-dissipativeDynamical processes model mainly Maps of continuousP asof Iteration hardcore time evolution bosons of Kraus – of2 symmetrically a Lindblad Maps master delocalize equation particles tive dynamics can be devised to drive the system from anoperator K = 1, dissipative dynamics corresponds to causedbosons by [46]. imperfect In the gate actual operations. implementation We address of the this sim- is recoveredover pairs in ofthe neighboring limit of infinitesimal sites. Since time this steps, process cf. takes ulation,arbitrary the engineered initial state dissipative with density and matrix coherent⇢in dynam-into a Bose-multiple Kraus operators K>1. In particular, the In particular, we demonstrate the purely dissipative Fig. 1a, with time steps t`Methods.t`+1placerepresented The on each dissipative by pair of dynamics neighboring studied sites, in eventually our spin only generic,Einstein though condensate so far widely with disregarded long-rangeKraus aspect, phase map: by coherence care- ascontinuous! time evolution of a Lindblad master equation creation of quantum mechanicalics compete long-range with undesired phase order. dissipative processes mainly modelthe is symmetric governed superposition by dynamical of spinmaps excitations according over to the fully assessing the experimental errors. As a step towards is recoveredK in the limit of infinitesimal time steps, cf. Furthermore, we implementcausedthe a competition unique, by imperfect pure between “dark” gate operations. co- state D of We the address dissipative this evo- bosons spins solving this problem, we develop a novel| i and ecient spe- (l) two-bodywhole Kraus array(l) operators persists(l) as acting the stationary on pairs of state neighboring of the evolu- lution, i.e. ⇢ ⇢ = for⇢(t` long) ⇢ enough(t`+1)= waiting[⇢(Methods.t`)] = E The⇢( dissipativet`)E †. (1) dynamics studied in our spin herent and dissipative many-particlegeneric, though dynamics soin far widely byD alter- disregardedD aspect,7! by care-E spinstion:i, i + Iteration 1:k k of the dissipative dynamical map attracts cial purpose error reduction! technique.| i h | modelk=1 is governed by dynamical maps according to nating sequences of unitaryfullytime. and assessing non-unitary Supplementing the experimental maps this out- dynamics errors. As with a step a Hamiltoniantowards X Competing dissipative and unitary dynamics in a com- two-bodythe system Kraus operators towards a acting unique on dynamical pairs of neighboring fixed point, or lined in Fig. 1a, and observesolvingrepresenting the thisdestruction problem, local of we phase interactions, develop co- a novel beingDissipative and incompatible ecient spe-maps with E(i,l)1 = ci,Ei,2 =1 ci†ci. (2) The• set of Kraus operatorsdark state,E characterizedsatisfies by ⇢(t`+1)=⇢(t`) ⇢D,re- plex spinthe model dissipative– Two tendency competing, to delocalize non-commuting the bosons, con- givesspins i, i + 1: k herence as a result. Thiscial reflects purpose the hallmark error reduction feature technique. of K (l) (l) { } ⌘ tributions to a Hamiltonian give rise to a quantumE †E phase= 1 [17].The elementarysulting While the from familiaroperators the property se- generatingci⇢D = the 0 for dynamics all i separately. are a strong coupling non-equilibriumCompetingrise to phase a strong dissipative transition coupling and pre- unitary dynamicalk dynamics=1 two-body phasek ink transition. Kraus a com- operators It acting More specifically,Ei,1 = ci,E for m i,spin2 =1 excitationsci†ci. initially(2) present dicted in a closely relatedtransition,plexshares driven-dissipative spin if model features the respective– Two of model a competing, quantum ground of phasequences statesnon-commuting transitionon of pairs unitary each of neighboring con- in con- maps that are itgiven spins obtained by for a single Kraus tribution separately favor states withoperatorP di↵erentK = symme- 1, dissipative dynamicsin the array corresponds of N spins, to this pure dark state is given by bosons [46]. In the actualtributions implementationis driven to by a Hamiltonian the of competition the sim- give rise of non-commuting to a quantum phase quantumThe elementary operators+ + generating the dynamics are the Dickec state=( + )( ), (3) ulation, the engineeredtries dissipativetransition, [47].mechanical The and if transition the coherent operators, respective dynam- takes and ground place a classicalmultiple statesat a one critical Krausof in each that value operators con- the or-givenK> by1. In particular,i i thei+1 i i+1 continuous time evolution of a Lindblad master equation ics compete with undesiredof thetribution dissipative dimensionless separately processes ratio favor mainlyg of states the two with competing di↵erent symme- energy x y N dered phase terminates in a strongly mixed state. where ± =( i+ )/2+ are spin-1/2 raisingm and low- is recovered in the limit of infinitesimali ci timei=( steps,i+ cf. )( ),+ N(3) caused by imperfect gatescales.tries operations. A [47].In non-equilibrium our The experiment, We transition address analog we takes this consider can place be analogous at achieveddark a critical state in open-system open value D±=i D(m,i+1 N) i i+1 ⌦ (4) Methods. The dissipativeering dynamics operators studied| i acting in our| spinon spini⇠i. In thei continuous|#i generic, though so far widelymany-bodyof thedynamics disregarded dimensionless quantum of aspect, a quantum systems, ratio by care-g of spin-1/2 where the two coherent – competing or hardcore Hamilto- energy boson – x y i=1 model is governed by dynamicaltimewhere limit,± maps=( the operators accordingi )/2ci toarecorrespond spin-1/2⇣ X raisingprecisely⌘ and to low- Lind- fully assessing the experimentalnianscales. andmodel, errors. dissipativeA non-equilibrium realized As a step dynamics with towards trapped analog compete can ions. be with A achieved schematic each in other: open overview i i ± i two-body Kraus operatorsbladering acting quantumwith operators on pairsm collective jump of acting neighboring operatorsspin on spin excitations. andi. generate In theThe a continuous delocalization dissipative of solving this problem, weThe developmany-body roleof the of a novel the relation quantum ground and e ofcient state systems, the ionicspe- is played where spin- by coherent andUnitary the the stationary atomic Hamilto- boson spins• i, i + 1: evolutiontimethe limit, spin described the excitations operators by ac overi quantumcorrespond the whole master precisely array equation, gives to Lind- rise cf. to the cial purpose error reductionstatenian ofmodel technique. and the dissipative combined is given in dynamics evolution, Fig. 1b, compete whereas and the with a dimensionless more each detailed other: de- Competing dissipative and unitary dynamics in a com- Methods.bladcreation quantum The of jump operators entanglement operators act bi-locally and and generate quantum on pairs a dissipative mechanical of spins, o↵- ratioTheg roleis provided of the ground by a Hamiltonian state is played energy by the scale stationaryE vs.= c a,E=1with Uc†c=exp(. iH(2)) according to the dimensionless spin scription is provided in Methods andHamiltonian Supplementaryi,1 i In-asevolutioni, visualized2 describedi ini Fig. by1b: a quantum Physically, master they equation, map any cf. an- plex spin model – Two competing,state of the non-commuting combined evolution, con- and the dimensionless diagonal long range order, witnessed, e.g., by the single- dissipativeformation rate. Such II. The a situation discrete has time been evolution addressed is generated pre- Hamiltonian + tributions to a Hamiltonian give rise to a quantum phase The elementary operatorstisymmetricMethods. generatingparticle The the component correlations operators dynamics actin are the bi-locally wave= 0 function on for pairsi of onjz spins, a pairz (see viouslyratioby theoreticallyg sequencesis provided of in by dynamical the a Hamiltonian context or Kraus of driven-dissipative energynearest maps scale neighbor(l) vs.acting a interaction on H = i i j i+1 = (1 + i )(1 + i+1)/4. transition, if the respective ground states of each con- given by ofas sites visualized into thein Fig. symmetric|"ih"1b:| Physically,⌦h |"ih" one,i6| or they – in map| the any language| !1 an- dynamicsdissipative of atomic rate. Such bosons a situation on a lattice has been [46]: addressed AE dissipa- pre- Supplementaryi Information III). i tribution separately favor statesthe system’s with di↵erent reduced symme- density matrix ⇢ as illustrated inoftisymmetric hardcore bosons componentX – symmetrically in the wave function delocalizeX on particles a pair tiveviously dynamics theoretically can be devised in the to context drive theof driven-dissipative system from an+ + In our interacting lattice spin system, competing uni- ci =( + of)( sites into the), symmetric(3) one, or – in the language 28 tries [47]. The transitiondynamics takes place of atomicat a critical bosons value on a lattice [46]: A dissipa-i overi+1 pairstaryi dynamics ofi+1 neighboring can be sites. achieved Since by this the process stroboscopic takes re- arbitrary initial state with densityTuesday, matrix March⇢ 12,in into13 a Bose- of the dimensionless ratio g of the two competing energy x y of hardcore bosons – symmetrically delocalize particles tive dynamics can be devised to drivewhere the system± =( fromi an )/2place are spin-1/2alization on each raising of pair coherent of and neighboring low- maps ⇢(t sites,`) eventually⇢(t`+1)=U only⇢(t`)U † scales. A non-equilibriumEinstein analog condensate can be achieved with in long-range open phasei coherencei ± asi over pairs of neighboring sites. Since this7! process takes arbitrary initial state with density matrixering operators⇢in into a Bose- acting onthe spin symmetrici. In the superposition continuous of spin excitations over the many-body quantum systems,the unique, where pure coherent “dark” Hamilto- state D of the dissipative evo- place on each pair of neighboring sites, eventually only Einstein condensate with long-range| i time phase limit, coherence the operators as ciwholecorrespond array precisely persists toas Lind- the stationary state of the evolu- nian and dissipative dynamicslution, i.e. compete⇢in with⇢D = each other:D for long enough waiting the symmetric superposition of spin excitations over the the unique, pure! “dark”| statei h | bladof the quantum dissipative jump evo- operatorstion: and Iteration generate of a the dissipative dissipative dynamical map attracts The role of the groundtime. state Supplementing is played by the this stationary dynamicsD with a Hamiltonian whole array persists as the stationary state of the evolu- lution, i.e. ⇢ ⇢ = | iforevolution long enough described waiting by a quantumthe system master towards equation, a unique cf. dynamical fixed point, or state of the combinedrepresenting evolution, and localin the interactions, dimensionlessD D being incompatible with tion: Iteration of the dissipative dynamical map attracts time. Supplementing! this| i dynamicsh | Methods. with a Hamiltonian The operators actdark bi-locally state, characterizedon pairs of spins, by ⇢(t )=⇢(t ) ⇢ ,re- ratio g is provided bythe a Hamiltonian dissipative energy tendency scale to vs. delocalize a the bosons, gives the system towards a unique dynamical`+1 fixed` point,⌘ D or representing local interactions, beingas visualized incompatible in Fig. with1b:sulting Physically, from they the map property any an-c ⇢ = 0 for all i separately. dissipative rate. Such arise situation to a strong has been coupling addressed dynamical pre- phase transition. It dark state, characterized byi ⇢D(t )=⇢(t ) ⇢ ,re- the dissipative tendency to delocalizetisymmetric the bosons, component gives inMore the wave specifically, function for onm aspin pair excitations`+1 initially` ⌘ D present viously theoretically inshares the context features of ofdriven-dissipative a quantum phase transition in that it sulting from the property ci⇢D = 0 for all i separately. rise to a strong coupling dynamicalof phase sites into transition. the symmetric It one, or – in the language dynamics of atomic bosons on a lattice [46]: A dissipa- inMore the specifically, array of N forspins,m spin this excitations pure dark initially state is present given by is drivenshares by features the competition of a quantum of non-commuting phaseof transition hardcore quantum bosons in that – it symmetrically delocalize particles tive dynamics can be devised to drive the system from an thein the Dicke array state of N spins, this pure dark state is given by mechanicalis driven by operators, the competition and a classical of non-commutingover one pairs in that of quantum neighboring the or- sites. Since this process takes arbitrary initial state with density matrix ⇢in into a Bose- the Dicke state dered phase terminates in a stronglyplace mixed on state. each pair of neighboring sites, eventually only N m Einstein condensate withmechanical long-range operators, phase coherence and a as classical one in that the or- + N In our experiment, we consider analogousthe symmetric open-system superposition of spin excitationsD = D(m, over N the) N i ⌦ (4) the unique, pure “dark” statedered phaseD of the terminates dissipative in evo-a strongly mixed state. | i | i⇠ m |#i dynamics| i of a quantum spin-1/2 –whole or hardcore array persists boson –as the stationary state of the evolu- i=1 + N lution, i.e. ⇢ ⇢ = In ourfor experiment, long enough we waiting consider analogous open-system D = D(m, N) ⇣ Xi ⌘ ⌦ (4) in D D tion: Iteration of the dissipative dynamical| i map| attractsi⇠ |#i time. Supplementing! thismodel,| dynamicsi dynamicsh realized| of with a with quantum a Hamiltonian trapped spin-1/2 ions. A – schematicor hardcore overview boson – i=1 the system towards a uniquewith dynamicalm collective fixedspin point, excitations. or ⇣ X The⌘ delocalization of representing local interactions,ofmodel, the relation being realized incompatible of with the ionic trapped with spin- ions. and A schematic the atomic overview boson dark state, characterizedthe bywith⇢ spin(tm collective excitations)=⇢(t )spin over⇢ excitations.,re- the whole The array delocalization gives rise to of the the dissipative tendencymodelof to the delocalize is relationgiven in the of Fig. bosons, the1 ionicb, giveswhereas spin- and a more the atomic detailed boson de- `+1 ` D sulting from the property creationthec ⇢ spin= 0 excitations of for entanglement all i separately.⌘ over the and whole quantum array gives mechanical rise to the o↵- rise to a strong couplingscriptionmodel dynamical is providedgiven phase in transition. Fig. in Methods1b, whereas It and Supplementarya more detailed In- de- i D More specifically, for m spindiagonalcreation excitations oflong entanglement initially range order, present and witnessed, quantum e.g., mechanical by the single- o↵- shares features of a quantumformationscription phase II. is transition provided The discrete in in Methods that time it evolution and Supplementary is generated In- + in the array(l) of N spins, thisparticlediagonal pure dark correlations long state range is order, giveni byj witnessed, = 0 for e.g.,i byj the single-(see is driven by the competitionbyformation sequences of non-commuting II. of dynamical The discrete quantum or time Kraus evolution maps isacting generated on h + i6 | | !1 the DickeE state Supplementaryparticle correlations Information III).= 0 for i j (see mechanical operators,the andby system’s a sequences classical reduced one of dynamical in that density the or or- matrix Kraus maps⇢ as illustrated(l) acting inon h i j i6 | | !1 E SupplementaryIn our interacting Information lattice III). spin system, competing uni- dered phase terminates inthe a strongly system’s mixed reduced state. density matrix ⇢ as illustrated in N m taryIn dynamics our interacting+ canN be lattice achieved spin system, by the competing stroboscopic uni- re- In our experiment, we consider analogous open-system D = D(m, N) ⌦ (4) | i | i⇠ i |#i alizationtary dynamicsi=1 of coherent can be maps achieved⇢(t`) by the⇢(t stroboscopic`+1)=U⇢(t` re-)U † dynamics of a quantum spin-1/2 – or hardcore boson – ⇣ X ⌘ 7! model, realized with trapped ions. A schematic overview alization of coherent maps ⇢(t`) ⇢(t`+1)=U⇢(t`)U † with m collective spin excitations. The delocalization of 7! of the relation of the ionic spin- and the atomic boson the spin excitations over the whole array gives rise to the model is given in Fig. 1b, whereas a more detailed de- creation of entanglement and quantum mechanical o↵- scription is provided in Methods and Supplementary In- diagonal long range order, witnessed, e.g., by the single- formation II. The discrete time evolution is generated + particle correlations = 0 for i j (see by sequences of dynamical or Kraus maps (l) acting on i j Supplementary Informationh i6 III). | | !1 the system’s reduced density matrix ⇢ asE illustrated in In our interacting lattice spin system, competing uni- tary dynamics can be achieved by the stroboscopic re- alization of coherent maps ⇢(t ) ⇢(t )=U⇢(t )U † ` 7! `+1 ` a c 1 Competition

i strength 0 10) 0.8 , dissipative map unitary map ⇡ (3

D 16

| 0.6 t i ⇡ t i+1 8 0.4 ⇢mixed ⇡ (i) (ii) 4 0.2 ⇡ 2

(i) Overlap with 0 0 10 20 30 40 50 E Competition strength

i 0.8 strength Competition

10) 0 ,

⇢init ⇢D = D (3 0.6 ⇡

| i h | D 16 | ⇡ 0.4 8 t i t i+1 ⇡ discrete time steps 4 0.2 ⇡ 2 Overlap with 0

b 14 15 16 17 18 D + Uj,j+1 i,i+1 # of composite maps Hardcore i i+1 i i+1 i i+1 j j+1 boson d ⇢mixed model

Di,i+1 Uj,j+1 Uj,j+1 Dj,j+1 ⇢ init ⇢D Spin model Competitionstrength

Di,i+1 + Uj,j+1 t i t i+1 | | | | | discrete time steps

29 Tuesday, March 12, 13 5

a (i) (ii) (iii)

1 D12 2 D23 3 ...... U .... N-1

D Stabilization N N-1N

b c d 42P (i) (ii) (iii) 1/2 2 3 D5/2 mJ -1/2 | -3/2 U Z 12 729 nm -5/2 M U 23 (i) (iii) (ii) ...... mJ U 1/2 N-1N 2 4 S1/2 -1/2 |

Experimental procedure to implement open-system dynamical maps. Figure 2. Experimental procedure to implement open-system dynamical maps. a, Schematic overview of the experimental implementation of a composite dynamical

map consisting of (i) multiple elementary dissipative maps, 30 Tuesday, March(ii) 12, 13 coherent competition, and (iii) error detection and cor- rection. Decoupled ions are represented as gray bullets and decoupling (re-coupling) operations as gray (blue) squares. b, Scheme for decoupling ions from the interaction with the ma- nipulating light fields: (i) shelve population from 4S1/2(m = 1/2) = to 3D5/2(m = 5/2), (ii) transfer the popu- lation from| "i 3D (m = 1/2) = to 4S (m =+1/2), 5/2 |#i 1/2 and subsequently to (iii) 3D5/2(m = 3/2). c,Asingledis- sipative element is realized using two system spins and one ancilla qubit ( 0 , 1 ) by (i) mapping the informa- tion whether the| i⌘ system|#i | i⌘ is in|"i the symmetric or antisymmet- ric subspace onto the logical states 1 or 0 of the ancilla, | i | i Figure 3. Experimental results of dissipatively in- respectively; (ii) mapping the antisymmetric onto the sym- duced delocalization through composite dynamical metric state using a controlled phase flip conditioned by the maps with 3+1 ions. The results from an ideal model state of the ancilla qubit; and finally (iii) reinitialization of 2 are shown in light-blue bars whereas those from a model in- the ancilla qubit via optical pumping using the 4 P1/2 state cluding depolarization noise are indicated by dark-grey bars. (see Supplementary Information I). d,Schematicviewofthe Blue rectangles indicate the experimentally observed dynam- competing interaction consisting of quasi-local unitary maps ics without any correction scheme whereas red diamonds in- Uj,j+1. clude a post-selective error detection scheme (error bars, 1). a, Dissipative pumping into a three-spin Dicke state: Starting in an initial product state with two localized spin excitations the composite, globally acting unitary map can be re- , the application of the first two elementary dissipative alized by a single unitary sequence (Fig. 2d). The re- maps| "#"i leads to an increase in the delocalization of the two exci- sults displayed in Fig. 4 for experiments with 3+1 and tations over the spin chain, which is reflected by an increasing state overlap fidelity with the three-spin Dicke state D(2, 3) . 4+1 ions show a clear fingerprint of incompatible Hamil- | i tonian dynamics, which competes with the dissipative However, after applying a second and a third composite dissi- pative map, a decrease in the state overlap fidelity sets in and maps driving the spin chains towards the Dicke states. becomes dominant for long sequences of dynamical maps. b, Further measurements with varying excitation number The presence of depolarizing noise results in population leak- and competition strength are discussed (see Supplemen- age out of the initial subspace with m =2spinexcitations. tary Information VI). This e↵ect is evident in the decay of the probability of find- (iii) To reduce the detrimental e↵ect of the experimen- ing the three-spin system in the m =2excitationssubspace tal imperfections and thus to enable the implementa- as a function of the number of applied elementary dissipative tion of longer sequences of dynamical maps, we devel- maps. A single composite dissipative map is indicated by a yellow bar. oped and implemented two counter-strategies (see Fig. 5 for details). In a first approach we applied a quantum non-demolition (QND) measurement of the spin excita- tion number at the end of the sequence of dynamical maps, which allowed us to detect and discard experimen- 5

a a (i) (ii) (iii) 1.0

0.8

0.6 th i b c d 0.4 42P (i) (ii) (iii) 1/2 2 3 D5/2 mJ -1/2 0.2 | -3/2 Overlap w 729 nm -5/2 (i) (iii) (ii) 0.0 mJ 0 1 2 3 4 5 6 7 1/2 2 elementary dissipative maps 4 S1/2 -1/2 | b

1.0 Figure 2. Experimental procedure to implement open-system dynamical maps. a, Schematic overview of the experimental implementation of a composite dynamical 0.8 map consisting of (i) multiple elementary dissipative maps, (ii) coherent competition, and (iii) error detection and cor- 0.6

rection. Decoupled ions are represented as gray bullets and 2 excitations r decoupling (re-coupling) operations as gray (blue) squares. b, 0.4 Scheme for decoupling ions from the interaction with the ma- nipulating light fields: (i) shelve population from 4S1/2(m = abiliy fo

1/2) = to 3D (m = 5/2), (ii) transfer the popu- b 0.2 | "i 5/2 lation from 3D5/2(m = 1/2) = to 4S1/2(m =+1/2), and subsequently to (iii) 3D (m |=#i 3/2). c,Asingledis- Pro 5/2 0.0 sipative element is realized using two system spins and one 0 1 2 3 4 5 6 7 ancilla qubit ( 0 , 1 ) by (i) mapping the informa- elementary dissipative maps tion whether the| i⌘ system|#i | i⌘ is in|"i the symmetric or antisymmet- ric subspace onto the logical states 1 or 0 of the ancilla, | i | i Figure 3. Experimental results of dissipatively in- respectively; (ii) mapping the antisymmetric onto the sym- duced delocalization through composite dynamical metric state using a controlled phase flip conditioned by the maps with 3+1 ions. The results from an ideal model state of the ancilla qubit; and finally (iii) reinitialization of Experimental results of dissipatively in- duced delocalization through 2 are shown in light-blue bars whereas those from a model in- the ancilla qubit via optical pumping using the 4 P1/2 state cluding depolarization noise are indicated by dark-grey bars. (see Supplementary Information I). d,Schematicviewofthe composite dynamical maps with 3+1 ions. Blue rectangles indicate the experimentally observed dynam- 31 competing interaction consisting of quasi-local unitary maps Tuesday, icsMarch without 12, 13 any correction scheme whereas red diamonds in- Uj,j+1. clude a post-selective error detection scheme (error bars, 1). a, Dissipative pumping into a three-spin Dicke state: Starting in an initial product state with two localized spin excitations the composite, globally acting unitary map can be re- , the application of the first two elementary dissipative alized by a single unitary sequence (Fig. 2d). The re- maps| "#"i leads to an increase in the delocalization of the two exci- sults displayed in Fig. 4 for experiments with 3+1 and tations over the spin chain, which is reflected by an increasing state overlap fidelity with the three-spin Dicke state D(2, 3) . 4+1 ions show a clear fingerprint of incompatible Hamil- | i tonian dynamics, which competes with the dissipative However, after applying a second and a third composite dissi- pative map, a decrease in the state overlap fidelity sets in and maps driving the spin chains towards the Dicke states. becomes dominant for long sequences of dynamical maps. b, Further measurements with varying excitation number The presence of depolarizing noise results in population leak- and competition strength are discussed (see Supplemen- age out of the initial subspace with m =2spinexcitations. tary Information VI). This e↵ect is evident in the decay of the probability of find- (iii) To reduce the detrimental e↵ect of the experimen- ing the three-spin system in the m =2excitationssubspace tal imperfections and thus to enable the implementa- as a function of the number of applied elementary dissipative tion of longer sequences of dynamical maps, we devel- maps. A single composite dissipative map is indicated by a yellow bar. oped and implemented two counter-strategies (see Fig. 5 for details). In a first approach we applied a quantum non-demolition (QND) measurement of the spin excita- tion number at the end of the sequence of dynamical maps, which allowed us to detect and discard experimen- 6

a excitation number are strongly suppressed and a reason- able overlap with the ideal evolution can be maintained 0.8 for more simulation time steps. Complementary to this post-selective method, we in- troduced a more powerful, active QND feedback scheme, 0.6 which bears similarities to quantum feedback protocols as realized with photons in a cavity [50]. The key idea is to th i 0.4 actively stabilize the spin system during the sequence of dynamical maps in a subspace with a particular spin ex-

0.2 citation (or hardcore boson) number (see Supplementary Information V). In order to be able to perform this stabi- Overlap w lization with a single ancilla qubit, we break the stabiliza- 0.0 0 1 2 3 4 5 6 7 tion process into two independent parts, where the first elementary dissipative and Hamiltonian maps part removes one excitation if there are too many excita- tions in the system, and the second part adds one excita- b tion if needed. Similarly to the post-selective technique 0.7 presented above, first the information whether there are too many (few) excitations in the system is coherently 0.6 mapped onto the ancilla qubit. Depending on the state 0.5 of the ancilla qubit, a single excitation is removed from (injected into) the system by a quantum feedback proto- 0.4 col. This extraction (injection) is in general an ambigu- th i 0.3 ous process, as the excitation can be removed (injected) on multiple sites. We use a scheme that tries to per- 0.2 form the removal (injection) subsequently on each site and stops once it was successful. Using only a single an-

Overlap w 0.1 cilla qubit, this process cannot be performed eciently 0.0 as a unitary process, therefore we developed a technique 0 1 2 3 4 5 6 elementary dissipative and Hamiltonian maps making use of the resetting and decoupling techniques described above (see Supplementary Information V). Figure 4. Experimental results for competing dissipa- We demonstrate the excitation removal for a chain of tive andExperimental coherent dynamics results with 3+1for competing and 4+1 ions. dissipative3+1 spins, initially and coherent prepared in an equal superposition As in Fig. 3, the results from an ideal model are shown in of all basis states, as shown in Fig. 3c. At the current light-bluedynamics bars whereas with those 3+1 from aand model 4+1 including ions. depo- level of experimental accuracy, the implementation of 32 larization noise are indicated by dark-grey bars. Blue rectan- this stabilization scheme cannot improve the perfor- Tuesday,gles March indicate 12, 13 the experimentally observed dynamics without mance when used in the full simulation sequence (see any correction scheme whereas red diamonds include a post- Supplementary Information V). We emphasize, however, selective error detection scheme (error bars, 1). The appli- cation of dissipative (coherent) maps is indicated by yellow that our approach relies only on a single ancillary qubit, (red) bars. Competing dissipative and coherent dynamics for regardless of the system size. More generally, such m =2excitationsinchainsof a, N =3andb, N =4spins: customized error detection and reduction strategies the spin chains are first driven towards the Dicke-type dark will incur a substantially reduced resource overhead as state by the two and three elementary dissipative maps for a compared with full-fledged quantum error correction system of 3 and 4 spins. The subsequent application of the protocols. We showcased a novel error management non-compatible unitary dynamical maps leads to a strong de- technique consisting of theoretical modeling of the er- crease of the overlap with the respective Dicke states, before subsequent elementary dissipative maps again start to pump rors, as well as designing system specific error reduction the system back towards the Dicke states. techniques. For the simulation of large systems it will be imperative to develop and understand sophisticated error models. Furthermore, we expect our experimental tal runs with a final erroneous excitation number and observations on the interplay of engineered and detri- thereby improve the overall simulation accuracy. This mental dissipation to stimulate theoretical research on global measurement is QND in the sense that only infor- the fundamental issue of robustness of (dynamical) mation about the total number of excitations, but not many-body phenomena in open quantum systems in the on their individual spatial locations along the chain is presence of noise - similar to the persistence of quantum acquired; thus the simulation subspace is not disturbed phases at finite as compared to zero temperature in (see Supplementary Information IV). The results shown Hamiltonian systems. in Fig. 3 and Fig. 4 confirm that the errors in the spin a b QND measurement of the 1 excitation / hardcore boson number Ancilla (i) 1 X 0.8 No Ok No | 1 0.6 m -1 m +1 2 0.4 0 m 0 0 3 m=m0

(iii) ...... 0.2

0 (ii) 0 20 40 60 80 c # of composite dissipative maps Stabilization: extraction protocol (i) No error Ancilla 10 1 1 X 1 60 | | 50 m0-1 m0+1 40 1 5 30 m m 20 0 2 10 0 0 3 800 m>m0 0 20 40 60 80 injection extraction ...... m-1 (ii) Error, no stabilization

10 601

50

40 5 30 d 20 Initial state After excitation removal 10 0 0 0.8 0.8 0 20 40 60 80800 # of excitations 0.6 0.6 (iii) Error, with stabilization

10 601 50 0.4 0.4 40

5 30

20

10 Probability 0.2 Probability 0.2 0 0 0 20 40 60 80800

0 0 0 1 2 3 0 1 2 3 # of excitations # of excitations

Experimental error detection and reduction techniques.

33 Tuesday, March 12, 13 3

Nr. of Nr. of global Nr. of AC- Nr. of Total number Described

Algorithm type MS gates rotations R Stark shifts SZ resets of operations in section Elementary dissipative map 6 7 9 1 23 IIb Hamiltonian competing dynamics with 3+1 ions 2 3 2 0 7 IIc Hamiltonian competing dynamics with 4+1 ions 3 4 4 0 11 IIc QND post-selective error detection 4 8 6 0 18 IV Mapping for the spin excitation removal 2 9 4 0 15 V Mapping for the spin excitation injection 3 12 7 0 22 V Excitation injection / removal step (single site) 4 0 2 0 6 V Spectroscopic decoupling 0 5 4 0 9 Ic Composite dissipative map (3 spins) 12 26 24 2 64 Composite diss. and coh. dyn. map (3 spins) 14 29 26 2 71 Composite diss. and coh. map + QND (3 spins) 18 40 27 3 88 Composite dissipative map (4 spins) 18 33 35 3 89 Composite diss. and coh. dyn. map (4 spins) 21 44 44 3 112

TABLE I Summary of the required resources for the elementary and composite dynamical maps and additional tools used in the quantum simulation. The required operations for the composite maps do not strictly match the sum of the required elementary operations, since in the implementation of composite maps synergy e↵ects in the resources for the spectroscopic decoupling operations can be exploited.

intensity-dependent AC-Stark shift AC . Again, the structed systematically (M¨uller et al., 2011). However, rotation angle ✓ is determined by the pulse length ⌧, as such decompositions are in many cases not optimal ✓ = AC ⌧/⇡, and a ⇡-pulse corresponds to ✓ = 1. Fi- in terms of the number of required gate operations, it is nally, collective entangling operations are implemented convenient to resort to a numerical optimal control al- by a bi-chromatic, globally applied laser field, which ef- gorithm (Nebendahl et al., 2009) to search for optimized 34 Tuesday,fectively March realizes 12, 13 two-body Mølmer-Sørensen (MS) type sequence decompositions involving less gates. Whereas interactions (Mølmer and Sørensen, 1999), the numerical optimization algorithm becomes inecient for general unitary operations acting on a large number of qubits, it is well-suited for the optimization of unitaries i✓⇡ 2 i✓⇡ MS(✓, )=exp S =exp which act only on a small subset of ions (such as 2+1 4 0 2 i j 1 i>j ✓ ◆ X ions in the implementation of an elementary dissipative @ A(3) Kraus map), independently of the total system size. between all pairs i and j of the ion chain (i, j = Numerically optimized pulse sequences may include 0, 1,...N) (first experimental realization in Ref. (Sackett global AC-Stark pulses and MS gates with negative ro- et al., 2000)). Again, the angle allows one to control tation angles ✓ < 0, which are not directly contained in x x y y whether i j (for = 0), i j (for =1/2) or inter- the available gate set discussed in Sec. I.A. However, actions i j corresponding to any other axis in the as any collective rotation around the z-axis of the Bloch x-y-plane are realized. In this notation, the angle ✓ =1/2 sphere can be interpreted as a re-definition of the x- and corresponds to a ”fully-entangling” MS gate, i.e. a uni- y-axes, a global AC-Stark pulse can be omitted if the tary which maps the computational basis states of N +1 phases of the following resonant operations are prop- ions onto multi-particle entangled states, which are (up erly adjusted. Regarding MS gates with ✓ < 0, these can to local rotations) equivalent to N + 1-qubit GHZ states. be implemented by MS gates with positive rotation an- Altogether, operations (1) to (3) form a universal set of gles, as MS( ✓, ) MS(1 ✓, ) up to local rotations ⌘ gates, enabling the implementation of arbitrary unitaries (see Eq. (9) in Ref. (M¨uller et al., 2011)). on any subset of ions (Nebendahl et al., 2009).

C. Spectroscopic decoupling of ions B. Numerical optimization of gate sequences Despite the globally applied beams for the collective Any unitary operation required for the implementation rotations and MS gates, operations on subsets of ions of dissipative and coherent maps, as well the error detec- can be realized by spectroscopically decoupling ions not tion and reduction protocols, needs to be decomposed involved in the realization of a certain Kraus map from into a sequence of available operations. As discussed the dynamics. This is realized by coherently transferring in more detail below, such decompositions can be con- ions to electronic states, where they do not couple to the