Rich dynamics of qubits Tams Kiss Wigner Research Center for - PowerPoint PPT Presentation

Rich dynamics of qubits Tamás Kiss Wigner Research Center for Physics Collaboration: I. Jex, S. Vymˇ etal, A. Gábris, M. Malachov (Prague) G. Alber, M. Torres, Zs. Bernád (Darmstadt) O. Kálmán, A. Gilyén, D. L. Tóth April 2020, Bolyai seminar 1 / 21

Quantum theory: linear or nonlinear? 1., Closed systems - unitary operators - linear evolution 2., Quantum channels - completely positive maps - linear evolution If quantum states evolved nonlinearly ◮ then hard problems (NP complete) would be easily solved (in polynomial time) D. S. Abrams and S. Lloyd, PRL 81, 3992 (1998) ◮ e.g. search using the Gross-Pitaevskii equation D. A. Meyer and T. G. Wong, New J. Phys. 15, 063014 (2013) ◮ quick discrimination of nonorthogonal states - generic feature A. M. Childs and J. Young, Phys. Rev. A, 93, 022314 (2016) 2 / 21

Nonlinear transformations by selective evolution 3., Measurements � projection (von Neumann) � probabilistic (Born) � information gained � information feed-back � post-selection � breaking linearity � 3 / 21

Transformation of a qubit | ψ 1 � | ψ 0 � A A 0 1 U CNOT | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 0 � | 0 � � B B H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 4 / 21

Transformation of a qubit | ψ 1 � | ψ 0 � A A 0 1 U CNOT | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 0 � | 0 � � B B H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 1 | 00 � + z | 01 � + z | 10 � + z 2 | 11 � � � Ψ in � � � AB = | ψ 0 � A ⊗ | ψ 0 � B = � 1 + | z | 2 4 / 21

Transformation of a qubit | ψ 1 � | ψ 0 � A A 0 1 U CNOT | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 0 � | 0 � � B B H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 1 | 00 � + z | 01 � + z | 10 � + z 2 | 11 � � � Ψ in � � � AB = | ψ 0 � A ⊗ | ψ 0 � B = � 1 + | z | 2 1 | 00 � + z | 01 � + z | 11 � + z 2 | 10 � � � Ψ in � � � U CNOT AB = � 1 + | z | 2 4 / 21

Transformation of a qubit | ψ 1 � | ψ 0 � A A 0 1 U CNOT | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 0 � | 0 � � B B H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 1 | 00 � + z | 01 � + z | 10 � + z 2 | 11 � � � Ψ in � � � AB = | ψ 0 � A ⊗ | ψ 0 � B = � 1 + | z | 2 1 | 00 � + z | 01 � + z | 11 � + z 2 | 10 � � � Ψ in � � � U CNOT AB = � 1 + | z | 2 ◮ after projecting qubit B to | 0 � : 1 | 0 � + z 2 | 1 � � � | ψ 1 � A = � 1 + | z | 2 4 / 21

Transformation of a qubit | ψ 1 � | ψ 0 � A A 0 1 U CNOT | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 0 � | 0 � � B B H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 1 | 00 � + z | 01 � + z | 10 � + z 2 | 11 � � � Ψ in � � � AB = | ψ 0 � A ⊗ | ψ 0 � B = � 1 + | z | 2 1 | 00 � + z | 01 � + z | 11 � + z 2 | 10 � � Ψ in � � � � U CNOT AB = � 1 + | z | 2 ◮ after projecting qubit B to | 0 � : 1 | 0 � + z 2 | 1 � � � f ( z ) = z 2 | ψ 1 � A = −→ � 1 + | z | 2 4 / 21

Transformation of a qubit 1 | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 1 � | ψ 0 � A � A 0 U CNOT ↓ | ψ 0 � | 0 � B B 1 | 0 � + z 2 | 1 � � � | ψ 1 � = � 1 + | z | 4 H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). 2 Iteration of f ( z ) = z 2 1 (complex plane) Im( z ) ◮ | z | < 1 → 0 (stable fixed point) 0 ◮ | z | > 1 → ∞ (stable fixed point) − 1 ◮ | z | = 1 → no convergence − 2 − 2 − 1 0 1 2 Re( z ) 4 / 21

Transformation of a qubit 1 | ψ 0 � = 1 + | z | 2 ( | 0 � + z | 1 � ) | ψ 1 � | ψ 0 � A � A 0 U CNOT ↓ | ψ 0 � | 0 � B B 1 | 0 � + z 2 | 1 � � � | ψ 1 � = � 1 + | z | 4 H. Bechmann-Pasquinucci et al. Phys. Lett. A 242 , 198 (1998). | 0 � Iteration of f ( z ) = z 2 (Bloch sphere) � | z | < 1 states converge to | 0 � � | z | > 1 states converge to | 1 � � z = 1 weird points: the Julia set | 1 � 4 / 21

Iterative nonlinear quantum protocols ◮ Ensemble of qubits in pure state | ψ 0 � ∼ | 0 � + z | 1 � ( z ∈ C ) 1. Take them pairwise: | Ψ 0 � = | ψ 0 � A ⊗ | ψ 0 � B 2. Apply an entangling two-qubit operation U 3. Measure the state of qubit B — keep A only for result 0 ◮ Smaller ensemble in pure state | ψ 1 � ∼ | 0 � + f ( z ) | 1 � ◮ Quantum magnification bound: exponential downscaling of the ensemble U ↔ f ( z ) = a 0 z 2 + a 1 z + a 2 b 0 z 2 + b 1 z + b 2 A. Gilyén, T. Kiss and I. Jex, Sci. Rep. 6 , 20076 (2016) 5 / 21

Historical remarks on complex dynamics f ◦ n → ? � Iterated rational polynomials: f : ˆ C → ˆ C , � One century of complex chaos: 1871 idea of iterated functions by Ernst Schröder Ueber iterirte Functionen. , Math. Ann. . Fatou: z �→ z 2 / ( z 2 + 2) 1906 first weird example by P 1920ies G. Julia, S. Lattès, & . . . 1970ies Computers help visualize: B. Mandelbrot & . . . � A good book: J.W. Milnor Dynamics in One Complex Variable , (Vieweg, 2000) 6 / 21

Iterative dynamics - examples CNOT gate plus a single qubit gate � sin θ e i ϕ � cos θ U = − sin θ e − i ϕ cos θ Family of maps over ˆ C : z 2 + p p = tan θ e i ϕ z �→ f p ( z ) = 1 − p ∗ z 2 p ∈ C parameter of the gate 7 / 21

Iterative dynamics - Julia sets on the Bloch sphere (a) θ = 0 . 4 , ϕ = π (b) θ = 0 . 55 , ϕ = π (c) θ = 0 . 633 , ϕ = π 2 2 2 (d) θ = 1 . 05 , ϕ = π (f) θ = 0 . 232 , ϕ = 0 (e) θ = 0 . 5 , ϕ = 0 . 5 2 A. Gilyén, T. Kiss and I. Jex, Sci. Rep. 6 , 20076 (2016). 8 / 21

Lattès map: J = ˆ C f ( z ) = z 2 + i iz 2 + 1 , p = i · (1 − i ) · (1 − i ) · (1 − i ) C / Z [ i ] : · · · ℘ ℘ ℘ ℘ Φ Φ Φ Bloch sphere : � � · · · � A commutative diagram: ◮ map on the Bloch sphere ↔ × (1 − i ) n on the torus ◮ all initial states are weird ◮ ergodicity Lattès, S (1918), Les Comptes rendus de l’Académie des sciences, 166: 26-28 A. Gilyén, T. Kiss and I. Jex, Sci. Rep. 6 , 20076 (2016). 9 / 21

Lattès map: ergodic dynamics (a) | z | > 1 (c) | f ◦ 2 ( z ) | > 1 (d) | f ◦ 3 ( z ) | > 1 (b) | f ( z ) | > 1 (h) | f ◦ 7 ( z ) | > 1 (e) | f ◦ 4 ( z ) | > 1 (f) | f ◦ 5 ( z ) | > 1 (g) | f ◦ 6 ( z ) | > 1 (i) | f ◦ 8 ( z ) | > 1 (l) | f ◦ 11 ( z ) | > 1 (j) | f ◦ 9 ( z ) | > 1 (k) | f ◦ 10 ( z ) | > 1 10 / 21

Lattès map with noisy initial states Dynamics represented by R 3 → R 3 functions: u ′ = u 2 − v 2 2 w w ′ = − 2 uv v ′ = 1 + w 2 , 1 + w 2 , 1 + w 2 No book by Milnor! :-( Asymptotics: all mixed initial states → completely mixed state O. Kálmán, T. Kiss and I. Jex, J Russ Laser Res 39: 382 (2018) 11 / 21

CNOT + Hadamard gate: phase transition Noisy (mixed) initial states: ρ ⊙ ρ M −→ ρ ′ = U H Tr ( ρ ⊙ ρ ) U † ρ H where � 1 � � 1 + w � 1 ρ = 1 1 u − iv U H = √ , 1 − 1 u + iv 1 − w 2 2 P = Tr ( ρ 2 ) = (1 + u 2 + v 2 + w 2 ) / 2 ≤ 1 Purity: 12 / 21

Convergence for different purities P 2 2 2 (b) (c) (d) 1 1 1 y P y P y P 0 0 0 − 1 − 1 − 1 − 2 − 2 − 2 − 2 − 1 0 1 2 − 2 − 1 0 1 2 − 2 − 1 0 1 2 x P x P x P P = 1 P = 0 . 87 P = 0 . 75 Light blue: convergence to | 0 � after an even number of steps Dark blue: convergence to | 0 � after an odd number of steps Red: convergence to the completely mixed state 13 / 21

Fractal dimension D bc as a function of purity P 1.7 1.6 1.5 1.4 D bc 1.3 1.2 1.1 1 0.9 0.75 P c 0.8 0.85 0.9 0.95 1 P M. Malachov, I. Jex, O. Kálmán, and T. Kiss, Chaos 29, 033107 (2019) 14 / 21

LOCC scheme with 2 qubits | ψ � | ψ ′ � | ψ � 0 H ? B A ′ A 0 H ? | ψ � = c 1 | 00 � + c 2 | 01 � + c 3 | 10 � + c 4 | 11 � | ψ ′ � = U H ⊗ U H � N ( c 2 1 | 00 � + c 2 2 | 01 � + c 2 3 | 10 � + c 2 � 4 | 11 � ) 15 / 21

2 qubits: chaotic entanglement Asymptotic states Green: Fully entangled: 1 | ψ ( ∞ ) � = ( | 00 � + | 11 � ) √ 2 Blue: Completely separable, oscillatory: � � | 00 � , 1 | ψ ( ∞ ) � → 2 ( | 0 � + | 1 � ) ⊗ ( | 0 � + | 1 � ) T. Kiss, S. Vymˇ etal, L. D. Tóth, A. Gábris, I. Jex, G. Alber, PRL 107 , 100501 (2011) 16 / 21

An application for state orthogonalization 2 z ◮ nonlinear transformation: f ϕ = 0 = 1 + z 2 J. M. Torres, J. Z. Bernád, G. Alber, O. Kálmán, and T. Kiss, Phys. Rev. A, 95 , 023828 (2017) 17 / 21

An application for state orthogonalization 2 z ◮ nonlinear transformation: f ϕ = 0 = 1 + z 2 2 ◮ two superattractive 1 fixed points: 1 and − 1 Im( z ) 0 ◮ Julia set: imaginary axis − 1 − 2 − 0 . 2 − 0 . 1 0 0 . 1 0 . 2 Re( z ) J. M. Torres, J. Z. Bernád, G. Alber, O. Kálmán, and T. Kiss, Phys. Rev. A, 95 , 023828 (2017) 17 / 21

An application for state orthogonalization 2 z ◮ nonlinear transformation: f ϕ = 0 = 1 + z 2 ◮ Julia set: longitudinal | 0 �−| 1 � √ great circle through y axis 2 ◮ equally separates | 0 � + | 1 � regions of convergence √ 2 J. M. Torres, J. Z. Bernád, G. Alber, O. Kálmán, and T. Kiss, Phys. Rev. A, 95 , 023828 (2017) 17 / 21