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Understanding, controlling, and overcoming decoherence and noise in quantum computation

Understanding, controlling, and overcoming decoherence and noise in quantum computation. Kaveh Khodjasteh, D.A.L., PRL 95 , 180501 (2005); PRA 75 , 062310 (2007). NSF September 10, 2007. +. Quantum Computers are Open Systems: Decoherence.

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Understanding, controlling, and overcoming decoherence and noise in quantum computation

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  1. Understanding, controlling, and overcoming decoherence and noise in quantum computation Kaveh Khodjasteh, D.A.L., PRL 95, 180501 (2005); PRA 75, 062310 (2007) NSF September 10, 2007

  2. + Quantum Computers are Open Systems: Decoherence Quantum computers use superposition and entanglement (“massive parallelism”) Every real quantum system interacts with an environment (“bath”). Environment is noisy & uncontrollable. The environment acts as an uncontrollable observer, making random-time measurements, in random basis. Destroys superposition states. Devastating for quantum computation: A sufficiently decohered quantum computer admits efficient simulation on a classical computer.

  3. 2 level system (qubit) Switch time T Rabi flop [sec] Decoherence time T2(upper bound) Quality factorT2/T (no. of ops) Charge of electron in bulk GaAs 10-13 aaaaa 10-9 aaaaa 104 aaaaa Exciton in GaAs quantum dot 10-15 aaaaa 10-12 aaaaa 103 aaaaa Electron spin in GaAs quantum dot 10-10 aaaaa 10-6 aaaaa 104 aaaaa Trapped ion (In) sdddd 10-6 aaaaa 10-1 aaaaa 105 aaaaa Nuclear spin in liquid state NMR 10-3 aaaaa 10+4 aaaaa 107 aaaaa Atom in microwave cavity 10-14 aaaaa 10-5 aaaaa 109 aaaaa Is Decoherence a Problem? How can we overcome decoherence?

  4. l h l ¯ D i i i i t t t ¯ b l d l H J i i i t t t t p o a r n e r a c o n a m o n g e n u c e : y p e r n e n e r a c o n e w e e n e e c r o n s a n n u c e : d f S G A t o m e a a o r a s ( ) J O I § A M H 1 ¼ z H H H H = + + n n = S S B B 2 ( ) ¯ O I § B K H 1 0 ¼ z = X X X < ~ ~ ~ n m n m [ ] ~ ­ Z A I B I I + + ¢ ¾ = i i i i n n n m n m ; i i n m n ; ; Example: electron spins in a semiconductor quantum dot Model: [Merkulov, Efros, Rosen, PRB. 65, 205309, (2002)]

  5. A B 1 1 1 3 ; ; System Bath Robust Hadamard Gate for Spin Bath

  6. ( ) ( ) f l l C H i t t t t o n a n s o u r a u y c o n r o c h l h l b T H i i i i t t t t t e r e e x s s a p u s e s e q u e n c e a e m n a e s a n y a r r a r y S B H I B X B Y B Z B ­ + ­ + ­ + ­ = S X Y Z 0 e ( ) H H I I H H t ­ + ­ + = S B S B S B System-Bath Model of Decoherence and Noise Besides the quantum system S there is always an environment or bath B. Hamiltonian: For a single system qubit Errors come from faulty control and undesired couplings with the environment. Indirectly also from .

  7. ± b l k f l P i 0 t r o ¿ e m : w o r s m p e r e c y ! ; h ± 6 0 w e n ¿ = ; . l E t r r o r s a c c u m u a e a s s e q u e n c e g r o w s . on system Universal Dynamical Decoupling A pulse sequence that eliminates the system-bath interaction for a single qubit: =

  8. Concatenated Universal Dynamical Decoupling To counter error accumulation, correct errors at all timescales: Nest the universal DD pulse sequence into its own free evolution periods f : p(1)= X f Z f X f Z f p(2)= X p(1)Z p(1)X p(1)Z p(1) etc. Length grows exponentially; how about error reduction?

  9. h l d h l F i i i n 4 t t t t t x e o a s e q u e n c e u r a o n s o a s ¿ p u s e s . l l i t t n c o n c a e n a o n e v e = l l i t ¿ p u s e n e r v a = l d h A i t s s u m e z e r o p u s e w j j a n 4 ( ) ¯ d 1 ¸ ¡ n 2 j j b 4 n Performance of Concatenated Sequences [Khodjasteh & Lidar, PRA 75, 062310 (2007) ]

  10. l f h b h l d h h b f l l b f b h S H i i i i i i t t t t t + m u a o n s o r a s p n c a n s y s e m a c o u p e r o u g e s e n e r g o r a s m a n u m e r o a s p n s , . h l l l h f h f h ¯ l d N T i i i i i t t t t t t t n s e c o n c a e n a o n e v e o r o g e m e a s u r e o e r r o r s e o n e m n u s p u r y o e n a r a c e 4 . h b h h l l b W K i i i 1 t t t t t t t t t t t t t o u s y s e m s a e e s a r e a a a e r m a e q u r u m a a e m p e r a u r e n e a r . . 2 ( ( ) ) j j j j l ¯ d b 1 · ¡ ¡ o g n a n n Dynamical Decoupling for Quantum Memory: Numerically Exact Simulations for Spin Chain Spin bath initially in equilibrium at 1K Theory bound:

  11. 1 j i ( j i j i ) ( j i j i ) 0 0 1 1 0 0 1 1 0 ¡ ¡ = L 2 1 j i ( j i j i j i j i j i j i ) 1 2 0 0 1 1 2 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 + ¡ ¡ ¡ ¡ = L p 2 3 Computation Problem: DD pulses can interfere with logic gates (cancel them too) How can they be reconciled? • Need a commuting structure of pulses and computation. • Use encoded qubits from a DFS. Pick DD pulses to commute with logical gates over DFS, such that DD pulses are still a universal decoupling group. 1 ( ) ( ) =+-+++ 23

  12. ¹ ¹ µ µ X 2 Z 1 i i ¹ ¹ P ( ( ) ) P d b l d d b H X Z E E J X E X Y Y Z Z J E i i i t t t t ¡ ¡ + + + ´ e a n e g e n e r a e a r r a r y s n g e e n c o e q u g a e s = = = H i i j i j i j i j i j i j 1 2 1 3 1 2 i j i j e s p 2 3 ; ; Heisenberg Computation over DFS is Universal • Heisenberg exchange interaction: • Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)] • Over 4-qubit DFS: CNOT involves 14 elementary steps (D. Bacon, Ph.D. thesis)

  13. ( = ) µ N H i ¡ [ ] Z X H Z X X Z 0 t g a e ¢ ¢ ¢ ¢ ¢ ¢ e o r = H M M i 1 1 e s ; Universal Decoupling Group Commutes with Heisenberg Exchange • n levels of concatenation, N=4npulses • Universal decoupling group on M (even) system-spins: p(1)= X U Z U X U Z U p(2)= X p(1)Z p(1)X p(1)Z p(1) … X Z X Z X U U U U t Next: demonstrate discrete set of (encoded) single-qubit gates from universal set

  14. ¯ ¯ M M H H 0 1 0 1 0 z z = = A B 1 1 1 3 : ; ; System Bath 4-Qubit DFS π/8 Gate + CDD: ideal pulses, varying internal bath coupling PDD Internal bath coupling CDD J=10MHz, T=100nsec Concatenation Level

  15. J J M M H H 0 1 0 0 1 z z = = A B 1 1 1 3 : ; ; System Bath 4-Qubit DFS π/8 Gate + CDD: ideal pulses, varying system-bath coupling PDD system-bath coupling CDD β=10MHz, T=100nsec Concatenation Level

  16. A B 1 1 1 3 ; ; System Bath 4-Qubit DFS Logic Gate + CDD, Finite Width Pulses t

  17. Conclusions • Decoherence and noise remain the fundamental obstacle to large scale implementation of quantum computers • A concatenated dynamical decoupling strategy drastically improves fidelity of quantum memory and quantum logic gates • What next? • Consider Hybrid CDD-QEC strategy • What is the fault-tolerance threshold for this hybrid setting? • Optimal Decoupling: Can we do better than CDD?

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