Ground State Preparation by Master Equation
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(2) Outline. 1. Universal Quantum Simulation. 2. Rydberg Quantum Simulation. 3. Ground State Preparation by Master Equation.
(3) Universal Quantum Simulation. Outline. 1. Universal Quantum Simulation Motivation Definition by Feynman and Lloyd. 2. Rydberg Quantum Simulation. 3. Ground State Preparation by Master Equation. i. |1i j.
(4) Universal Quantum Simulation. Motivation. Strongly Correlated Electronic Systems Hubbard Model The Hamiltonian in second quantization reads X † X H = −t ci,σ cj,σ + U ni,↑ ni,↓ hi,ji,σ. i. Problems in condensed matter physics High-Tc superconductors, Magnets, etc. Not analytically solvable i. j. Numerically impossible for many particles Problem Exponential growth of the Hilbert space with the particle number. 1/21.
(5) Universal Quantum Simulation. Motivation. Strongly Correlated Electronic Systems Hubbard Model The Hamiltonian in second quantization reads X † X H = −t ci,σ cj,σ + U ni,↑ ni,↓ hi,ji,σ. i. Problems in condensed matter physics High-Tc superconductors, Magnets, etc. Not analytically solvable i. j. Numerically impossible for many particles Problem Exponential growth of the Hilbert space with the particle number. 1/21.
(6) Universal Quantum Simulation. Feynman’s Answer. Definition by Feynman and Lloyd. [Fey82; Llo96]. Current state of the art: 40 particles, 240 variables 300 particles one would require 2300 variables, which is the number of particles in the universe. Simulating Physics with Computers “Let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws.” Further elaborated by Lloyd: A Universal Quantum Simulator could simulate the dynamics of other systems with short-range interactions. i i i i exp Ht ≈ exp H1 t exp H2 t · · · exp Hn t ~ ~ ~ ~ Digital Quantum Simulator: A Universal Quantum Simulator which advances in discrete time steps. 2/21.
(7) Universal Quantum Simulation. Feynman’s Answer. Definition by Feynman and Lloyd. [Fey82; Llo96]. Current state of the art: 40 particles, 240 variables 300 particles one would require 2300 variables, which is the number of particles in the universe. Simulating Physics with Computers “Let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws.” Further elaborated by Lloyd: A Universal Quantum Simulator could simulate the dynamics of other systems with short-range interactions. i i i i exp Ht ≈ exp H1 t exp H2 t · · · exp Hn t ~ ~ ~ ~ Digital Quantum Simulator: A Universal Quantum Simulator which advances in discrete time steps. 2/21.
(8) Universal Quantum Simulation. Feynman’s Answer. Definition by Feynman and Lloyd. [Fey82; Llo96]. Current state of the art: 40 particles, 240 variables 300 particles one would require 2300 variables, which is the number of particles in the universe. Simulating Physics with Computers “Let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws.” Further elaborated by Lloyd: A Universal Quantum Simulator could simulate the dynamics of other systems with short-range interactions. i i i i exp Ht ≈ exp H1 t exp H2 t · · · exp Hn t ~ ~ ~ ~ Digital Quantum Simulator: A Universal Quantum Simulator which advances in discrete time steps. 2/21.
(9) Rydberg Quantum Simulation. Outline. 1. Universal Quantum Simulation. 2. Rydberg Quantum Simulation Why Rydberg Atoms? Mesoscopic CNOT Gate. 3. |2i Ensemble qubit |Ai|Bi. Ground State Preparation by Master |0i|1i Equation Control qubit. Rydberg interaction.
(10) Rydberg Quantum Simulation. Rydberg Gates Revisited. Why Rydberg Atoms? [Urb+09]. Large dipole moment gives rise to strong Rydberg-Rydberg interaction The van der Waals coefficient of the repulsion scales like C6 ∼ n11 One atom can be excited into a Rydberg state, but a second one in the vicinity cannot The Ryd-Ryd interaction shifts the Rydberg level of the second atom out of resonance. Control Target |ri. |1i |0i. 3/21.
(11) Rydberg Quantum Simulation. Why Rydberg Atoms? [Urb+09]. Rydberg Gates Revisited. Large dipole moment gives rise to strong Rydberg-Rydberg interaction The van der Waals coefficient of the repulsion scales like C6 ∼ n11 One atom can be excited into a Rydberg state, but a second one in the vicinity cannot The Ryd-Ryd interaction shifts the Rydberg level of the second atom out of resonance. |ci |ti. |ri 2π. 4π. 32π. = |1i |0i. 1π/2. 5π/2. control target ,→ talk by Niklas. 3/21.
(12) Rydberg Quantum Simulation. Rydberg Gates. Why Rydberg Atoms?. [Wei10; Mül+09]. Common setup: Atoms trapped in deep optical lattice Rydberg atoms possess long-range interactions Allows for large spacing and gives rise to better single-site addressability Mesoscopic Gate Coupling to many atoms in the vicinity allows to change the state of N atoms CNOT → CNOTN. |0i|1i. Control qubit. |Ai|Bi. Rydberg interaction. Ensemble qubits. 4/21.
(13) Rydberg Quantum Simulation. Rydberg Gates. Why Rydberg Atoms?. [Wei10; Mül+09]. Common setup: Atoms trapped in deep optical lattice Rydberg atoms possess long-range interactions Allows for large spacing and gives rise to better single-site addressability Mesoscopic Gate Coupling to many atoms in the vicinity allows to change the state of N atoms CNOT → CNOTN. |0i|1i. Control qubit. |Ai|Bi. Rydberg interaction. Ensemble qubits. 4/21.
(14) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Principles of the CNOT Gate The CNOT gate flips the target qubit depending on the state of the control qubit CNOT Mapping Rule Let |α, βi be a product of control and target qubit, where α ∈ {0, 1} denotes the control and β ∈ {A, B} the target qubit CNOT |0, Ai = |0, Ai. CNOT |1, Ai = |1, Bi |0, Ai → |0, Ai ,. |1, Ai → |1, Bi. |0, Bi → |0, Bi , |1, Bi → |1, Ai. 5/21.
(15) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. [Mül+09; Wei10]. To implement a mesoscopic CNOT Q gate we need to find a way to flip N qubits at once. Suppose |AN i = i |Aii |0, AN i → |0, AN i ,. |1, AN i → |1, B N i. |0, B N i → |0, B N i , |1, B N i → |1, AN i. Independent of the actual number and position of the particles Properly pulsed laser light drives required transitions. CNOT Gate Operator The gate operation is unitary, so it can be easily reversed N Y U = |0ih0|c ⊗ 11 + |1ih1|c ⊗ σx(i) i=1. Ensemble qubit |Ai|Bi |0i|1i. Rydberg interaction. Control qubit. 6/21.
(16) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. [Mül+09; Wei10]. To implement a mesoscopic CNOT Q gate we need to find a way to flip N qubits at once. Suppose |AN i = i |Aii |0, AN i → |0, AN i ,. |1, AN i → |1, B N i. |0, B N i → |0, B N i , |1, B N i → |1, AN i. Independent of the actual number and position of the particles Properly pulsed laser light drives required transitions. CNOT Gate Operator The gate operation is unitary, so it can be easily reversed N Y U = |0ih0|c ⊗ 11 + |1ih1|c ⊗ σx(i) i=1. Ensemble qubit |Ai|Bi |0i|1i. Rydberg interaction. Control qubit. 6/21.
(17) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. Control Atom in |0i:. π. Ωr. Ωp. π. Ωc Ωr t. CNOT |0, AN i = |0, AN i. |ri. EIT condition fulfilled (target is transparent for Ωp ) Raman transfer is blocked. |Ri Ωc. Control Atom in |1i:. Ωr. CNOT |1, AN i = |1, B N i EIT condition violated (Rydberg level shifted off resonance) Raman transfer is feasible. π. [Mül+09; Wei10]. ∆ Ωp. |0i. |1i. |Ai. |P i. Ωp. |Bi 7/21.
(18) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. Control Atom in |0i:. π. Ωr. Ωp. π. Ωc Ωr |Ri. t. CNOT |0, AN i = |0, AN i. |ri. EIT condition fulfilled (target is transparent for Ωp ) Raman transfer is blocked. Ωc. V. Control Atom in |1i:. Ωr. CNOT |1, AN i = |1, B N i EIT condition violated (Rydberg level shifted off resonance) Raman transfer is feasible. π. [Mül+09; Wei10]. ∆ Ωp. |0i. |1i. |Ai. |P i. Ωp. |Bi 7/21.
(19) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. [Mül+09; Wei10]. Many-Body Gate We exploited long-range many-body Rydberg-Rydberg interactions to realise a many-body quantum gate Many-Body Quantum Simulation Can we reverse the process and simulate many-body interactions using a many-body quantum gate?. Ensemble qubit |Ai|Bi |0i|1i. ⇐. i. j. Rydberg interaction. Control qubit. 8/21.
(20) Rydberg Quantum Simulation. Mesoscopic CNOT Gate. Mesoscopic Rydberg Gate Based on EIT. [Mül+09; Wei10]. Many-Body Gate We exploited long-range many-body Rydberg-Rydberg interactions to realise a many-body quantum gate Many-Body Quantum Simulation Can we reverse the process and simulate many-body interactions using a many-body quantum gate?. Ensemble qubit |Ai|Bi |0i|1i. ?. ⇒. i. j. Rydberg interaction. Control qubit. 8/21.
(21) Ground State Preparation by Master Equation. Outline. 1. Universal Quantum Simulation. Ap =. Q. (i). σx. i. 2. Rydberg Quantum Simulation. 3. Ground State Preparation by Master Equation Simple Lattice Model Dissipative State Preparation Cooling into the Ground State Rydberg Setup Implementation of a Single Step More Interesting Systems. |3i Bs =. Q j. (j). σz.
(22) Ground State Preparation by Master Equation. The Toric Code. Simple Lattice Model. [Wei10; Wei+11]. Spins are located on the edges of a two-dimensional lattice Two types of four-body interaction Q (i) Plaquette terms Ap = i σx Q (j) Star terms Bs = j σz. Ap =. Q. (i). σx. i. Q. (j). σz. j. Toric Code Hamiltonian Linear superposition of local interactions X X A(i) Bs(j) H=− p − i. Bs =. j. Global ground state |ψi is eigenstate of both stabilisers Ap |ψi = |ψi ,. Bs |ψi = |ψi. 9/21.
(23) Ground State Preparation by Master Equation. The Toric Code. Simple Lattice Model. [Wei10; Wei+11]. Spins are located on the edges of a two-dimensional lattice Two types of four-body interaction Q (i) Plaquette terms Ap = i σx Q (j) Star terms Bs = j σz. Ap =. Q. (i). σx. i. Q. (j). σz. j. Toric Code Hamiltonian Linear superposition of local interactions X X A(i) Bs(j) H=− p − i. Bs =. j. Global ground state |ψi is eigenstate of both stabilisers Ap |ψi = |ψi ,. Bs |ψi = |ψi. 9/21.
(24) Ground State Preparation by Master Equation. Simple Lattice Model. Excitations of the Toric Code. [Wei10; Wei+11]. Violations of the stabiliser constraints are called excitations “Magnetic” excitation Ap |mi = − |mi “Charge” excitation Bs |ei = |ei a). b). c). 10/21.
(25) Ground State Preparation by Master Equation. Dissipative State Preparation. Intermezzo: Dissipative State Preparation. [BP06; Sei14]. Dissipation is described by a coupling V (t) to a heat bath Markovian evolution of the system %(t) = V (t)%(0) = eLt %(0) with the superoperator L Lindblad Master Equation The evolution of the density matrix %(t) is given by a generalised Liouville-von-Neumann equation X d i 1 † † % = − [H, %] + γi ci %ci − {ci ci , %} dt ~ 2 i with jump operators ci and decay rates γi . 11/21.
(26) Ground State Preparation by Master Equation. Dissipative State Preparation. Intermezzo: Dissipative State Preparation. [BP06; Sei14]. Dissipation is described by a coupling V (t) to a heat bath Markovian evolution of the system %(t) = V (t)%(0) = eLt %(0) with the superoperator L Lindblad Master Equation The evolution of the density matrix %(t) is given by a generalised Liouville-von-Neumann equation X d i 1 † † % = − [H, %] + γi ci %ci − {ci ci , %} dt ~ 2 i with jump operators ci and decay rates γi . 11/21.
(27) Ground State Preparation by Master Equation. Dark States. Cooling into the Ground State. [BP06]. Definition: Dark State Here we define a dark state to be a state for which all coupling to the reservoir vanishes ci |Di = 0 The dark state is now a stationary state of the system and a trivial solution to the master equation is % = |DihD| d i % = − H |DihD| − |DihD| H dt ~ X 1 † † † c ci |DihD| + |DihD| ci ci + γi ci |DihD| ci − 2 i i. Contrive a jump operator with the properties The dark state is the ground state The system cools itself into the ground state 12/21.
(28) Ground State Preparation by Master Equation. Cooling into the Ground State. State Preparation of the Toric Code Review: Toric Code Hamiltonian X X H=− A(i) Bs(j) p − i. j. Jump operator for the magnetic excitations 1 cp = σz(i) (1 − Ap ) 2. [Wei10; Wei+11]. a). b). The ground state is a dark state, i.e. cp |ψi = 0 The jump operator cools any density matrix into the unique ground state by diffusion of excitations annihilation of identical excitations. c). 13/21.
(29) Ground State Preparation by Master Equation. Cooling into the Ground State. State Preparation of the Toric Code Review: Toric Code Hamiltonian X X H=− A(i) Bs(j) p − i. j. Jump operator for the magnetic excitations 1 cp = σz(i) (1 − Ap ) 2. [Wei10; Wei+11]. a). b). The ground state is a dark state, i.e. cp |ψi = 0 The jump operator cools any density matrix into the unique ground state by diffusion of excitations annihilation of identical excitations. c). 13/21.
(30) Ground State Preparation by Master Equation. Rydberg Setup [Wei10; Wei+11]. The Toric Code with Rydberg Atoms Rydberg atoms in a large-spacing optical lattice. |Ri. Control atoms placed in the middle of plaquettes Recent developments: Rydberg blockade between two atoms Group of M. Saffman: E. Urban et al. Nature Physics 5, 2 (2009), pp. 110–114. Selective excitation based on the Rydberg Blockade. |ri. Ωc ∆ Ωp |P i Ωp |Ai. |Bi. Ωr. |0i |1i. Group of P. Grangier: A. Gaëtan et al. Nature Physics 5, 2 (2009), pp. 115–118. 14/21.
(31) Ground State Preparation by Master Equation. Rydberg Setup. Quantum Simulation with Ultra Cold Ions. [Bar+11]. Quantum simulation with five trapped ions J. T. Barreiro et al. Nature 470, 7335 (2011), pp. 486–491. Minimal instance of Toric Code stabiliser Implements dissipative dynamics through optical pumping. Proof of concept, the experiment is not scalable For efficient quantum simulation we need n ∼ 100 ions. 15/21.
(32) Ground State Preparation by Master Equation. Rydberg Setup. Quantum Simulation with Ultra Cold Ions. [Bar+11]. Quantum simulation with five trapped ions J. T. Barreiro et al. Nature 470, 7335 (2011), pp. 486–491. Minimal instance of Toric Code stabiliser Implements dissipative dynamics through optical pumping. Proof of concept, the experiment is not scalable For efficient quantum simulation we need n ∼ 100 ions. 15/21.
(33) Ground State Preparation by Master Equation. Single Time Step. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. Because interactions are local we can focus on single plaquette H = Ap = σx(1) σx(2) σx(3) σx(4) Gate sequence for the simulation consists of four steps: G entangles the control and the target atom e−iφσz is the coherent evolution of the control atom, U (θ) is a controlled spin flip on one ensemble atom G−1 reverses the entanglement of control and target atom Optical pumping of the control atom back to |0ic introduces dissipation |0ic. |0ic e. −iφAp. cp. |0ic. =. e G. optical |0ic. −iφσz. U (θ). pumping −1. G. 16/21.
(34) Ground State Preparation by Master Equation. Single Time Step. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. Because interactions are local we can focus on single plaquette H = Ap = σx(1) σx(2) σx(3) σx(4) Gate sequence for the simulation consists of four steps: G entangles the control and the target atom e−iφσz is the coherent evolution of the control atom, U (θ) is a controlled spin flip on one ensemble atom G−1 reverses the entanglement of control and target atom Optical pumping of the control atom back to |0ic introduces dissipation |0ic. |0ic e. −iφAp. cp. |0ic. =. e G. optical |0ic. −iφσz. U (θ). pumping −1. G. 16/21.
(35) Ground State Preparation by Master Equation. Single Time Step. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. G is a three step process Uc = exp(−iπσy /4) is the standard π/2 qubit rotation. Ug maps the eigenstate of the ensemble atoms onto the control atom Ug = |0ih0|c ⊗ 11 + |1ih1|c ⊗. N Y. σx(i). i=1. Uc−1 = exp(iπσy /4) reverses the rotation |0ic. |1ic G. |0ic. Uc. |0ic + |1ic. = |λ, −i. |0ic − |1ic. Uc−1. |1ic. Ug |λ, −i. |λ, −i. |λ, −i 17/21.
(36) Ground State Preparation by Master Equation. Single Time Step |0ic. |0ic e. −iφAp. cp. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. |0ic. =. e G. G maps the internal state of the ensemble atoms on the control atom G |0ic ⊗ |λ, +i → |0ic ⊗ |λ, +i. G |0ic ⊗ |λ, −i → |1ic ⊗ |λ, −i. |λ, ±i is eigenstate of the Q (i) interaction Ap = i σx with eigenvalue ±1. optical |0ic. −iφσz. U (θ). pumping −1. G. Phase rotation on the control atom and applying G−1 is equivalent to the many-body interaction Ap exp(−iφAp ) = G−1 exp(−iφσz(c) ) G Controlled spin flip onto one of the ensemble atoms Ui (θ) = |0ih0|c ⊗ 11 + |1ih1|c ⊗ exp(iθσz(i) ) Leaves |λ, +i invariant. 18/21.
(37) Ground State Preparation by Master Equation. Single Time Step |0ic. |0ic e. −iφAp. cp. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. |0ic. =. e G. G maps the internal state of the ensemble atoms on the control atom G |0ic ⊗ |λ, +i → |0ic ⊗ |λ, +i. G |0ic ⊗ |λ, −i → |1ic ⊗ |λ, −i. |λ, ±i is eigenstate of the Q (i) interaction Ap = i σx with eigenvalue ±1. optical |0ic. −iφσz. U (θ). pumping −1. G. Phase rotation on the control atom and applying G−1 is equivalent to the many-body interaction Ap exp(−iφAp ) = G−1 exp(−iφσz(c) ) G Controlled spin flip onto one of the ensemble atoms Ui (θ) = |0ih0|c ⊗ 11 + |1ih1|c ⊗ exp(iθσz(i) ) Leaves |λ, +i invariant. 18/21.
(38) Ground State Preparation by Master Equation. Single Time Step |0ic. |0ic e. −iφAp. cp. Implementation of a Single Step. [Wei10; Wei+11; Wei+10]. |0ic. =. e G. G maps the internal state of the ensemble atoms on the control atom G |0ic ⊗ |λ, +i → |0ic ⊗ |λ, +i. G |0ic ⊗ |λ, −i → |1ic ⊗ |λ, −i. |λ, ±i is eigenstate of the Q (i) interaction Ap = i σx with eigenvalue ±1. optical |0ic. −iφσz. U (θ). pumping −1. G. Phase rotation on the control atom and applying G−1 is equivalent to the many-body interaction Ap exp(−iφAp ) = G−1 exp(−iφσz(c) ) G Controlled spin flip onto one of the ensemble atoms Ui (θ) = |0ih0|c ⊗ 11 + |1ih1|c ⊗ exp(iθσz(i) ) Leaves |λ, +i invariant. 18/21.
(39) Ground State Preparation by Master Equation. Implementation of a Single Step. Cooling to the Ground State. [Wei10; Wei+11; Wei+10]. Controlled spin flip onto one of the ensemble atoms Ui (θ) = |0ih0|c ⊗ 11 + |1ih1|c ⊗ exp(iθσz(i) ). If a flip occurs the control atom is not mapped back to |0ic. Entanglement is not reversed and whole system evolves according to 1 † † ∂t % = γ ci %ci − {ci ci , %} + O(θ3 ) 2. Picture: Numerical simulation with 32 particles. θ = π/4 θ = π/2 θ=π. −16 E[E0 ]. Each spin flip moves excitation to adjacent plaquette. For θ = π move takes place with unity probability, i.e. fastest cooling. −20 −24 −28 −32 0. 10. 20 t[τ ]. 30. 40 ,→ [Wei+11]. 19/21.
(40) Ground State Preparation by Master Equation. More Interesting Systems. Fermi-Hubbard Model in 2D. [Wei10; Wei+11]. What now about the Hubbard model? Well. . . X X z y y y x x x x σi0 ,j 0 ,σ + t σ2i,j,σ σ2i,j+1,σ + σ2i,j,σ σi,j,σ σi+1,j,σ + σi,j,σ σi+1,j,σ H =−t i,j,σ. +t. X. i,j,σ. +V. X. i,j,σ. y y y x x x (−1)j+1 σ2i σ2i+1,j,σ σ2i+1,j+1,σ + σ2i+1,j,σ σ2i+1,j+1,σ 0 +1,j 0 ,σ σ2i0 +1,j 0. z z x x x x σ2i,2j,σ σ2i+1,2j+1,σ σ2i 0 ,2j 0 ,σ σ2i0 +1,2j 0 ,σ σ2i0 +1,2j 0 ,σ σ2i0 +1,2j 0 +1,σ. i,j,σ. +V. X. z z x x x x σ2i+1,2j+1,σ σ2i,2j+2,σ σ2i 0 ,2j 0 +1,σ σ2i0 +1,2j 0 +1,σ σ2i0 ,2j 0 +2,σ σ2i0 +1,2j 0 +2,σ. i,j,σ. +V. X. y y y y z z σ2i+1,2j,σ σ2i+2,2j+1,σ σ2i 0 +1,2j 0 ,σ σ2i0 +2,2j 0 ,σ σ2i0 +1,2j 0 +1,σ σ2i0 +2,2j 0 +1,σ. i,j,σ. +V. X. y y y y z z σ2i+1,2j+2,σ σ2i+2,2j+1,σ σ2i 0 +1,2j 0 +1,σ σ2i0 +2,2j 0 +1,σ σ2i0 +1,2j 0 +2,σ σ2i0 +2. i,j,σ. H. Weimer et al. Quantum Information Processing 10, 6 (2011), pp. 885–906 20/21.
(41) Ground State Preparation by Master Equation. More Interesting Systems. Fermi-Hubbard Model in 2D. [Wei10; Wei+11]. What now about the Hubbard model? Well. . . X X z y y y x x x x σ2i,j,σ σ2i,j+1,σ + σ2i,j,σ σi0 ,j 0 ,σ + t σi,j,σ σi+1,j,σ + σi,j,σ σi+1,j,σ H =−t i,j,σ. +t. X. i,j,σ. +V. X. i,j,σ. y y y x x x (−1)j+1 σ2i σ2i+1,j,σ σ2i+1,j+1,σ + σ2i+1,j,σ σ2i+1,j+1,σ 0 +1,j 0 ,σ σ2i0 +1,j 0. z z x x x x σ2i,2j,σ σ2i+1,2j+1,σ σ2i 0 ,2j 0 ,σ σ2i0 +1,2j 0 ,σ σ2i0 +1,2j 0 ,σ σ2i0 +1,2j 0 +1,σ. i,j,σ. +V. X. z z x x x x σ2i+1,2j+1,σ σ2i,2j+2,σ σ2i 0 ,2j 0 +1,σ σ2i0 +1,2j 0 +1,σ σ2i0 ,2j 0 +2,σ σ2i0 +1,2j 0 +2,σ. i,j,σ. +V. X. y y y y z z σ2i+1,2j,σ σ2i+2,2j+1,σ σ2i 0 +1,2j 0 ,σ σ2i0 +2,2j 0 ,σ σ2i0 +1,2j 0 +1,σ σ2i0 +2,2j 0 +1,σ. i,j,σ. +V. X. y y y y z z σ2i+1,2j+2,σ σ2i+2,2j+1,σ σ2i 0 +1,2j 0 +1,σ σ2i0 +2,2j 0 +1,σ σ2i0 +1,2j 0 +2,σ σ2i0 +2. i,j,σ. H. Weimer et al. Quantum Information Processing 10, 6 (2011), pp. 885–906 20/21.
(42) Summary. What You Should Remember. Summary. |ri |Ri Ωc. Whatj You Should Remember Ωr i. ∆ Ωp. |0i. |1i. |Ai. |P i. Ωp. |Bi.
(43) Summary. What You Should Remember. What You Should Remember. Simulating quantum mechanics on a computer is exponentially hard Many-body gates can be used to simulate many-body interactions Rydberg atoms are very suitable, because the interactions are long range and allow for single-site addressability Dissipative preparation of ground states Implementation of complex spin systems Toric code can be set up such that it is self correcting. 21/21.
(44) Acknowledgement ITP3. Przemyslaw Bienias ITP1. Marcel Klett Holger Cartarius PI5. Harald Kübler.
(45) References & Further Reading. [Bar+11] J. T. Barreiro et al. Nature 470, 7335 (2011), pp. 486–491. [BP06] H. P. Breuer and F. Petruccione. The theory of open quantum systems. 1st ed. Oxford University Press, 2006. [Fey82] R. P. Feynman. Int. J. Theo. Phys. 21, 6/7 (1982), pp. 467–488. [Gaë+09] A. Gaëtan et al. Nature Physics 5, 2 (2009), pp. 115–118. [Llo96] S. Lloyd. Science 273 (1996), pp. 1073–1078. [Mül+09] M. Müller et al. Phys. Rev. Lett. 102, 17 (2009), p. 170502. [Sei14] U. Seifert. “Quantenmechanik 2”. Lecture. 2014. [Urb+09] E. Urban et al. Nature Physics 5, 2 (2009), pp. 110–114. [Wei+10] H. Weimer et al. Nature Physics 6, 5 (2010), pp. 382–388. [Wei+11] H. Weimer et al. Quantum Information Processing 10, 6 (2011), pp. 885–906. [Wei10] H. Weimer. “Quantum many-body physics with strongly interacting Rydberg atoms”. PhD thesis. 2010. Important, Experiments.
(46) Appendix. Time Scales [Mül+09]. Appendix: Time Scales of the Gate |ri. |Ri. duration of Raman pulse Ryd-Ryd interaction (c). γ|ri =. 1 (c) τ|ri. . Ω2p. Ω2c. 1 ∼ Vce T ∆ ∆. Numbers for. Ωr. ∆ Ωp. radiative decay of |ri EIT condition 87. Ωc. Rb for a gate fidelity of 99 %:. (c). τ|ri = 66 µs. T = 0.44 µs. ∆ = 2π × 1.2 GHz. |0i. |1i. Ωp = 2π × 70 MHz. |Ai. Vce = 10Ω2c /∆ ≈ 56.3 GHz Ωc = 6Ωp ≈ 2.6 GHz. |P i. Ωp. |Bi.
(47) Appendix. Lindblad Master Equation. Appendix: Decay of a Two-Level System. [BP06; Sei14]. Reminder: Lindblad Master Equation NX −1 i 1 d % = − [H, %] + γi ci %c†i − {c†i ci , %} dt ~ 2 i=1 2. Two-Level System H= c1 = σ+ ,. γ 1 = γ+. c2 = σ− ,. γ2 = γ −. c3 = σz ,. γ 3 = γz. ~ω0 σz 2 σ+ = |eihg| = σx + iσy. σ− = |gihe| = σx − iσy. I.
(48) Appendix. Lindblad Master Equation. Appendix: Decay of a Two-Level System. [BP06; Sei14]. II. Master Equation iω0 1 1 d %=− (σz % − %σz ) + γ+ σ+ %σ− − σ− σ+ % − %σ− σ+ dt 2 2 2 1 1 + γ− σ− %σ+ − σ+ σ− % − %σ+ σ− 2 2 1 1 + γz σz %σz − σz σz % − %σz σz 2 2 Time evolution of matrix elements d %ee = γ+ %gg − γ− %ee he|(∗)|ei : dt d hg|(∗)|gi : %gg = −γ+ %gg + γ− %ee dt. (∗).
(49) Appendix. Lindblad Master Equation. Appendix: Decay of a Two-Level System. [BP06; Sei14]. probability density %. Solution of the differential equations 1. %ee (t) %gg (t). γ− /γ γ+ /γ 0. 1/γ time t. Calculating the coherences he|(∗)|gi and hg|(∗)|ei allows to derive the principle of detailed balance.. III.
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