Quantum simulations using split-step quantum walks C. M. Chandrashekar Optics and Quantum Information Group The Institute of Mathematical Sciences, Chennai, India 15th February 2016, ISCQI 2016, IOP, Bhubaneswar Quantum simulations using split-step quantum walks
Aim of the talk 1 Discretization of quantum field theories in the era of QIT/QIP 2 Artificial synthesis of topological insulators using discrete-time quantum walks Quantum simulations using split-step quantum walks
Outline 1 Discretization of quantum field theories : need and approaches Quantum Cellular Automaton (QCA) and Dirac Cellular Automaton (DCA) Discrete-time quantum walk and Dirac Hamiltonian (Dirac Equation) Split-step quantum walk and DCA Zitterbewegung oscillations Entanglement spectrum arXiv:1509.08851 (with Arindam Mallick) Artificial synthesis of topological insulators 2 Topological quantum walks and localized states Two split-step Four split-step Entanglement spectrum of topological quantum walks and localized states arXiv:1502.00436 (with H. Obuse & T. Busch) Quantum simulations using split-step quantum walks
Discretization of space and time • Early Proposal to simplify the computation of field theories Divisibility of Space and Time., Yukawa, H. Atomistics and the Prog. Theor. Phys. Suppl. 37 and 38, 512 (1966) Quantum field theory on discrete space-time, Yamamoto, H., Phys. Rev. D 30 1127 (1984) • Discretization of Dirac equation describing the relativistic motion of a spin 1/2 particle (one prominent example) Confinement of quarks, Wilson, K. G., Phys. Rev. D 10, 2445 (1974) Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata, Bialynicki-Birula, I., Phys. Rev. D 49, 6920 (1994) • Lattice guage theories An introduction to lattice gauge theory and spin systems, Kogut, J. B., Rev. Mod. Phys. 51, 659 (1979) • Quantum cellular automaton and quantum lattice gas From quantum cellular automata to quantum lattice gases, Meyer, D. A. J., Stat. Phys. 85, 551 (1996) The Feynman path integral for the Dirac equation, Riazanov, G. V., Sov. Phys. JETP 6 1107-1113 (1958) Quantum simulations using split-step quantum walks
Quantum Cellular automaton and Dirac Cellular Automaton • Lattice gauge theory Evolution is described by the unitary operator which is an exponential of an Hamiltonian involving the whole system at a time • Quantum Cellular Automaton Evolution (update) rule of the system is described by a local unitary operators each involving few subsystems. It can be regarded as a microscopic mechanism for an emergent quantum fields and as a framework to unify a hypothetical Planck scale with the usual Fermi scale of the high-energy physics The QCA which is not derivable by quantizing classical theory can also be used as a framework for quantum theory of gravity • Dirac Cellular Automaton Free field QCA models emerging to Dirac Hamiltonian (DH) for spinor with non-zero mass and massless particles. Quantum simulations using split-step quantum walks
From discrete-time quantum walk to relativistic equations :Klein-Gordon, Dirac Quantum simulations using split-step quantum walks
Discrete-time quantum walk in 1D • Walk is defined on the Hilbert space H = H c ⊗ H p H c (particle) is spanned by | ↑� and | ↓� H p (position) is spanned by | x � , x ∈ Z • Initial state : | Ψ in � = [cos( δ ) | ↑� + e i η sin( δ ) | ↓� ] ⊗ | x = 0 � • Evolution : � 1 � 1 1 Coin operation - Hadamard operation : H = √ 1 − 1 2 Conditional unitary shift operation S : � � S = � | ↑��↑ | ⊗ | x − 1 �� x | + | ↓��↓ | ⊗ | x + 1 �� x | x ∈ Z state | ↑� moves to the left and state | ↓� moves to the right Quantum simulations using split-step quantum walks
Hadamard walk • Each step of QW (Hadamard walk) : W = S ( H ⊗ ✶ ) Quantum walk 0.09 Classical random walk 0.08 0.07 0.06 Probability 0.05 0.04 0.03 0.02 0.01 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 Particle position 100 step of CRW and QW [ S ( H ⊗ ✶ )] 100 on a particle with initial state 1 2 ( | ↑� + i | ↓� ) √ • G. V. Riazanov (1958), R. Feynman (1986) • K.R. Parthasarathy, Journal of applied probability 25, 151-166 (1988) • Y. Aharonov, L. Davidovich and N. Zugury, Phys. Rev. A, 48, 1687 (1993) • Use of word Quantum random walk • Salvador E. Venegas-Andraca, Quantum Information Processing vol. 11(5), pp. 1015-1106 (2012) Quantum simulations using split-step quantum walks
QW using generalized quantum coin operation • Hadamard walk : Probability | Ψ in � = | ↑� ⊗ | x = 0 � → peak to left 0.12 | Ψ in � = | ↓� ⊗ | x = 0 � → peak to right 0.10 � � 1 | Ψ in � = | ↑� ± i | ↓� ⊗ | x = 0 � → symmetric √ 0.08 2 0.06 0.04 • SU(2) operation : 0.02 Position � e i ξ cos( θ ) � 100 � 50 50 100 e i ζ sin( θ ) � B ξ,θ,ζ ≡ − e − i ζ sin( θ ) e − i ξ cos( θ ) 0.1 θ = 15 ° θ = 15 ° 0.08 Probability 0.06 θ = 45 ° • Each step of generalized QW : 0.04 W ξ,θ,ζ = S ( B ξ,θ,ζ ⊗ ✶ ) 0.02 � t | Ψ in � implements t steps � W ξ,θ,ζ 0 −100 −50 0 50 100 of generalized DQW Position Quantum simulations using split-step quantum walks
Symmetric evolution of DQW and hyperbolic PDE � � � � cos( θ ) sin( θ ) 1 | Ψ in � = | ↑� ± i | ↓� ⊗ | x = 0 � √ B ( θ ) = 2 − sin( θ ) cos( θ ) � � � � 1 cos( θ ) − i sin( θ ) | Ψ in � = | ↑� ± | ↓� ⊗ | x = 0 � √ B ( θ ) = 2 − i sin( θ ) cos( θ ) In the form of left moving and right moving component ψ 0 x , t +1 = cos( θ ) ψ 0 x +1 , t − i sin( θ ) ψ 1 x − 1 , t ψ 1 x , t +1 = cos( θ ) ψ 1 x − 1 , t − i sin( θ ) ψ 0 x +1 , t Differential equation form in continuum limit :Klein-Gordon equation � ∂ 2 ∂ t 2 − cos( θ ) ∂ 2 � ψ 0(1) ∂ x 2 + 2[1 − cos( θ )] x , t = 0 CMC, SB and RS, PRA, 81 062340 (2010) Quantum simulations using split-step quantum walks
Dirac equation from Discrete-time QW Dirac equation � � � � i � ∂ i � ∂ α · ∂ ∂ t − ˆ ∂ x − ˆ β mc 2 Ψ = ∂ t + i � c ˆ Ψ = 0 H D From DTQW when θ = 0, the expression in continuum limit takes the form � i � ∂ ∂ � ∂ t − i � σ 3 Ψ( x , t ) = 0 ∂ x David Mayer (1996) and Fredrick Strauch (2006) For θ � = 0 Giuseppe Molfetta - Fabrice Debbasch (2013) and CMC (2013) Quantum simulations using split-step quantum walks
Dirac Cellular Automaton DH from the QCA by constructing the evolution operator for a system which is (1) unitary, (2) invariant under space translation, (3) covariant under parity transformation, (4) covariant under time reversal and (5) has a minimum of two internal degrees of freedom (spinor). This QCA evolution which recovers DE is named as DCA and is in the form, � � α T − − i β U DA = = α { T − ⊗ |↑� �↑| + T + ⊗ |↓� �↓|} − i β ( I ⊗ σ x ) − i β α T + where α corresponds to the hopping strength, β corresponds to the mass term.Associated Hamiltonian in momentum basis, produces DH, � mc 2 � H ( k ) = a − kc mc 2 kc c τ with the identification β = mac � , k is a eigenvalue of momentum operator. • Derivation of the Dirac equation from principles of information processing, D Ariano, G. M. and Perinotti, P. Phys. Rev. A 90, 062106 (2014) • Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension, Bisio, A., DAriano,G. M., Tosini, A. Annals of Physics 354, 244264 (2015) Quantum simulations using split-step quantum walks
DTQW The general form of C is, C = C ( ξ, θ, φ, δ ) = e i ξ e − i θσ x e − i φσ y e − i δσ z = e i ξ × � e − i δ (cos( θ ) cos( φ ) − i sin( θ ) sin( φ )) − e i δ (cos( θ ) sin( φ ) + i sin( θ ) cos( φ )) � e − i δ (cos( θ ) sin( φ ) − i sin( θ ) cos( φ )) e i δ (cos( θ ) cos( φ ) + i sin( θ ) sin( φ )) � F θ,φ,δ G θ,φ,δ � = e i ξ − G ∗ F ∗ θ,φ,δ θ,φ,δ The general form of the evolution operator � � F θ,φ,δ T − G θ,φ,δ T − U QW = e i ξ − G ∗ F ∗ θ,φ,δ T + θ,φ,δ T + � � � � U QW = F θ T − ⊗ |↑� �↑| + T + ⊗ |↓� �↓| + G θ T − ⊗ |↑� �↓| ) + T + ⊗ |↓� �↑| By taking the value of θ → 0 the off-diagonal terms can be ignored and a massless DH can be recovered. Quantum simulations using split-step quantum walks
Split-step QW � � F θ 1 ,φ 1 ,δ 1 G θ 1 ,φ 1 ,δ 1 C ( θ 1 , φ 1 , δ 1 ) = , − G ∗ F ∗ θ 1 ,φ 1 ,δ 1 θ 1 ,φ 1 ,δ 1 � � F θ 2 ,φ 2 ,δ 2 G θ 2 ,φ 2 ,δ 2 C ( θ 2 , φ 2 , δ 2 ) = − G ∗ F ∗ θ 2 ,φ 2 ,δ 2 θ 2 ,φ 2 ,δ 2 and a two half-shift operators, � T − � I � � 0 0 S − = , S + = 0 0 I T + � � � � U SQW = S + I ⊗ C ( θ 2 , φ 2 , δ 2 ) I ⊗ C ( θ 1 , φ 1 , δ 1 ) S − F θ 2 ,φ 2 ,δ 2 F θ 1 ,φ 1 ,δ 1 T − − G θ 2 ,φ 2 ,δ 2 G ∗ F θ 2 ,φ 2 ,δ 2 G θ 1 ,φ 1 ,δ 1 T − + G θ 2 ,φ 2 ,δ 2 F ∗ θ 1 ,φ 1 ,δ 1 I θ 1 ,φ 1 ,δ 1 I = − G ∗ θ 2 ,φ 2 ,δ 2 F θ 1 ,φ 1 ,δ 1 I − F ∗ θ 2 ,φ 2 ,δ 2 G ∗ − G ∗ θ 2 ,φ 2 ,δ 2 G θ 1 ,φ 1 ,δ 1 I + F ∗ θ 2 ,φ 2 ,δ 2 F ∗ θ 1 ,φ 1 ,δ 1 T + θ 1 ,φ 1 ,δ 1 T + Quantum simulations using split-step quantum walks
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