On The Power Of Coherently Controlled Quantum Adiabatic Evolutions

Transcription

1 On The Power Of Coherently Controlled Quantum Adiabatic Evolutions Maria Kieferova (RCQI, Slovak Academy of Sciences, Bratislava, Slovakia) Nathan Wiebe (QuArC, Microsoft Research, Redmond, WA, USA) [arxiv: ]

2 Ground State H = real eigenvalues = energies ground state = the lowest energy

3 Ground State H = real eigenvalues = energies ground state = the lowest energy Example: H = N Z j g = j=0

4 Evolution With A Hamiltonian The Schrödinger equation Eigenstates Evolution d ψ(t) = ih ψ(t) dt H n = E n n n m = δ m,n n(t) = e ient n(0)

5 Evolution With A Time Dependent Hamiltonian The Schrödinger equation Instantaneous eigenstates d ψ(t) = ih(t) ψ(t) dt H(t) n(t) = E n (t) n(t) Evolution n(t) m(t) = δ m,n n(t) e ient n(0)

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7 goal: stay in the ground state

8 Probability Of Error probability of error grover search 1e+2 linear interpolation 1e+0 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 1e-14 1e-16 1e T

9 Probability Of Error probability of error 1e+2 1e+0 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 1e-14 1e-16 small errors grover search 1e T linear interpolation The adiabatic regime

10 Adiabatic Computation H 0 has a ground state which is easy to prepare H 1 has an interesting ground state Complexity [Farhi, E. et al., (2000)] Universal [Aharonov, D. et al.,(2004)] Quantum simulation, state preparation

11 The Adiabatic Theorem The error is small when T 1 min γ 2 e1,g (s) for bounded H and its derivatives

12 The Adiabatic Theorem Let H : [0, 1] C N N be a Hamiltonian that is differentiable three times and has a minimum eigenvalue gap between any two non-degenerate eigenvectors of γ min > 0. If we take g(s) to be a non-degenerate instantaneous eigenstate of H(s) and 1 o(γ min T 2 ), then the error in the adiabatic approximation, (1 g(1) g(1) ) U(1, 0) g(0), is n g e iφn s ṅ(s) g(s) e i γ g,n(s) T 0 γg,n(ξ)dξt 1 s=0 ( ) 1 n (0) + O γ min T 2 where Φ n = 1 0 E N (χ)dχt, γ g,n = E g(s) E n(s), 1 = γ 1 H Ḧ 3 Ḣ 6 + min γ min γ min 3 + γ min 6 and the phase of each instantaneous eigenstate is chosen such that ṅ(s) n(s) = 0 for every s [0, 1] and every n. [Cheung, Hoyer, Wiebe (2011)]

13 Error minimization methods 1e+2 probability of getting an excited state 1e+0 1e-2 1e-4 1e-6 1e-8 1e-10 local adiabatic evolution gets fast to the adiabatic regime make use of dips 1e-12 boundary cancellation 1e-14 minimizes local adiabatic evolution boundary cancelation the adiabatic error 1e total evolution time T

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15 Coherently Controlled Adiabatic Evolution U k : H(f k (s)) = f k (s)h 1 + (1 f k (s))h 0 ψ k U k ψ k measurement and post-selection allow us to implement a wider class of operations [I. Hen, (2014)], [P. Zanardi and M. Rasetti, (1999)]

16 Linear Combinations ψ 0 ψ (cos θ 0 + sin θ 1 ) cos θ ψ 0 + sin θu A ψ 1 cos θu B ψ 0 + sin θu A ψ 1 ( ) cos 2 θu B + sin 2 θu A ψ 0 + sin θ cos θ (U A U B ) ψ 1. [Wiebe, N. and Childs, A.M., 2012.] apply U A, U B in parallel

17 Success Probability U A : H(f (s)) = f (s)h 1 + (1 f (s))h 0 U B : H(g(s)) = g(s)h 1 + (1 g(s))h 0 Success probability= 1 (U A U B ) ψ 2 Ground states must pick similar phases U e +i 1 0 E 0(s)dsT U

18 Error 0 = cos 2 θ n g + sin 2 θ n g [ ṅ(1) g(1) γ g,n(1) T [ ṅ (1) g(1) γ g,n(1) T 1 ṅ(0) g(0) ei γ g,n(0) T ṅ (0) g(0) e i 1 γ g,n(0) T 0 γg,n(ξ)dξt 0 γ g,n (ξ)dξt ] n (1) ] n (1)

19 One excited state H(f (s)) = f (s)h 1 + (1 f (s))h 0 H(g(s)) = g(s)h 1 + (1 g(s))h 0 f, g generate evolutions with opposite errors derivatives of the Hamiltonians at the end must be opposite freedom for the initial boundary 0 = cos 2 θ + sin 2 θ [ ṅ(1) g(1) γ g,n(1) T [ ṅ (1) g(1) γ g,n(1) T 1 ṅ(0) g(0) ei γ g,n(0) T 0 γg,n(ξ)dξt ṅ (0) g(0) e i 1 0 γ g,n(ξ)dξt γ g,n(0) T ] n (1) ] n (1)

20 One excited state 1e+0 1e+0 probability of getting an excited state 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 LAE 1e-14 zero derivatives average of almost LAE 1e cost probability of getting an excited state 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 LAE 1e-14 zero derivatives linear combination 1e cost gets fast to the adiabatic regime favorable error scaling in the AR

21 More transitions 1. Linear combinations from more evolutions 3 levels and 4 evolutions equations for higher dimensional system

22 More transitions 1. Linear combinations from more evolutions 3 levels and 4 evolutions equations for higher dimensional system 2. Exploit symmetry

23 Symmetric evolutions H 0 and H 1 have the same spectrum H(0.5 s) = H (0.5 + s) ḟ (s) = ḟ (s) s=1 s=

24 0 = cos 2 θ n g + sin 2 θ n g Symmetric evolutions [ ṅ(1) g(1) γ g,n(1) T [ ṅ (1) g(1) γ g,n(1) T 1 ṅ(0) g(0) ei γ g,n(0) T ṅ (0) g(0) e i 1 γ g,n(0) T 0 γg,n(ξ)dξt 0 γ g,n (ξ)dξt ] n (1) ] n (1)

25 probability of getting an excited state 1e-2 1e-3 1e-4 1e-5 original linear combination 1e cost

26 1e-16 1e+0 probability of getting an excited state 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 1e-14 LAE zero derivatives average of almost LAE cost

27 1e-16 1e+0 probability of getting an excited state 1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 1e-14 LAE zero derivatives average of almost LAE cost Thanks for your attention!