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Digital Quantum Simulation with Trapped Ions

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Digital Quantum Simulation with Trapped Ions 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 Uni (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 Uni (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 Uni (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 Ut0 exp Ht0 Digital Quantum Simulation 1 2 1 2 H X B( z z ) J x x • Time evolution of : i – t Ut0 exp Ht0 • 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 Single Qubit Operations Multi Qubit Operations • Unitary Time evolution: i – Ut exp H1 H2 t 1 2 1 2 H X B( z z ) J x x H1 H2 Digital Quantum Simulation Single Qubit Operations Multi Qubit Operations • Unitary Time evolution: i – Ut exp H1 H2 t • In experiment only possible to apply: i – Ui t exp Hit 1 2 1 2 H X B( z z ) J x x H1 H2 Digital Quantum Simulation Single Qubit Operations Multi Qubit Operations • Unitary Time evolution: i – Ut exp H1 H2 t • In experiment only possible to apply: i – Ui t exp Hit i i i exp H H t exp H t exp H t • And: 1 2 1 21 2 1 2 H X B( z z ) J xx H1 H2 – since H1, H2 0 Digital Quantum Simulation Single Qubit Operations Multi Qubit Operations • Trotter formula n i i t i t exp H1 H2 t limexp H1 exp H2 n n n 1 2 1 2 H X B( z z ) J x x H1 H2 Digital Quantum Simulation Single Qubit Operations Multi Qubit Operations • Trotter formula n i i t i t exp H1 H2 t limexp H1 exp H2 n n n • Therefore: planned sequence t 1 2 1 2 H n B( ) J H H f i X z z x x 1 2 . 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 Single Qubit Operations Multi Qubit Operations • We have: 1 2 1 2 H X B( z z ) J x x H1 H2 DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits (Ca+) 1 2 1 2 H X B( z z ) J x x H1 H2 DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits (Ca+) with measurement e P1/ 2 D5/ 2 H B( 1 2 ) J 1 2 H H X z z x x 1 2 S1/ 2 DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits – Measurement e P1/ 2 0 D5/ 2 – A universal set of operations 1 S1/ 2 1 2 1 2 H X B( z z ) J x x H1 H2 DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits e P1/ 2 0 D5/ 2 – A universal set of operations 1 S1/ 2 j O1(, j) expi z 1 2 1 2 H X B( z z ) J x x H1 H2 DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits e P1/ 2 0 D5/ 2 – A universal set of operations 1 S1/ 2 j O1(, j) expi z OH () B(exp 1 i2 ) Jj1 2 H H 2X z z z x x 1 2 j DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits e P1/ 2 0 D5/ 2 – A universal set of operations 1 S1/ 2 O (, j) exp i j O (,) expi j 1 z 3 j OH () B(exp 1 i2 ) Jj1 2 H H 2X z z z x x 1 2 j DQS with Ions Single Qubit Operations Multi Qubit Operations • We have: – A string of ions/qubits e P1/ 2 0 D5/ 2 – A universal set of operations 1 S1/ 2 O (, j) exp i j O (,) expi j 1 z 3 j OH () B(exp 1 i2 ) Jj1 2 OH (,H) expi i j 2X z z z x x 41 2 j i j Ising Model Single Qubit Operations Multi Qubit Operations • Easy now to implement! O (, j) exp i j O (,) expi j 1 z 3 j OH () B(exp 1 i2 ) Jj1 2 OH (,H) expi i j 2X z z z x x 41 2 j i j Ising Model Single Qubit Operations Multi Qubit Operations • Easy now to implement! • Prepare qubits in state OH () B(exp 1 i2 ) Jj1 2 OH (,H) expi i j 2X z z z x x 41 2 j i j Ising Model Single Qubit Operations Multi Qubit Operations • Easy now to implement! • Prepare qubits in state • Choose t and n and apply: t n t 1 2 j1 2 i j OH (n) B(exp i) J OH (,H) expi 2X z z z x x 41 2 .
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