1. Understanding, controlling, and overcoming decoherence and noise in quantum computation NSF September 10, 2007 Kaveh Khodjasteh, D.A.L., PRL 95, 180501 (2005); PRA 75, 062310 (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. Is Decoherence a Problem? 2 level system (qubit) Switch time T Rabi flop [sec] Decoherence time T 2 (upper bound) Quality factor T 2 /T (no. of ops) Charge of electron in bulk GaAs Exciton in GaAs quantum dot Electron spin in GaAs quantum dot Trapped ion (In) sdddd Nuclear spin in liquid state NMR Atom in microwave cavity 10 -13 aaaaa 10 -15 aaaaa 10 -10 aaaaa 10 -6 aaaaa 10 -3 aaaaa 10 -14 aaaaa 10 -9 aaaaa 10 -12 aaaaa 10 -6 aaaaa 10 -1 aaaaa 10 +4 aaaaa 10 -5 aaaaa 10 4 aaaaa 10 3 aaaaa 10 4 aaaaa 10 5 aaaaa 10 7 aaaaa 10 9 aaaaa How can we overcome decoherence?

4. Example: electron spins in a semiconductor quantum dot Model: D i po l ar i n t erac t i onamong t h enuc l e i : ¯ S ome d a t a f or G a A s J = O ( I§ n A n ) ¼ 1 MH z ¯ = O ( I 2 § n < m B nm ) ¼ 10 KH z [Merkulov, Efros, Rosen, PRB. 65, 205309, (2002)] H yper ¯ ne i n t erac t i on b e t weene l ec t ronsan d nuc l e i : J H = H S + H SB + H B = X i ­ i Z i + X i ; n A i ; n ~ ¾ i ¢ ~ I n + X n ; m B nm [ ~ I n ~ I m ]

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

6. 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. C on t a i nsour ( f au l t y ) con t ro l H c ( t ) H = H S ( t ) ­ I B + I S ­ H B + H SB H e = I S ­ B 0 + X ­ B X + Y ­ B Y + Z ­ B Z T h ereex i s t sapu l sesequence t h a t e l i m i na t esanyar b i t rary H SB

7. A pulse sequence that eliminates the system-bath interaction for a single qubit: Universal Dynamical Decoupling = on system ± ; ¿ ! 0 P ro bl em:wor k s i mper f ec t l y w h en ± ; ¿ 6 = 0. E rrorsaccumu l a t eas sequencegrows.

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. LevelConcatenated DD Series after multiplying Pauli matrices 1 XfZfXfZfXfZfXfZf 2 fZfXfZfYfZfXfZffZfXfZfYfZfXfZf 3 XfZfXfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZfXfZf XfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZf Length grows exponentially; how about error reduction?

9. Performance of Concatenated Sequences [Khodjasteh & Lidar, PRA 75, 062310 (2007) ] F i x t h e t o t a l sequence d ura t i onso i t h as 4 n ¿pu l ses. n = conca t ena t i on l eve l ¿ = pu l se i n t erva l A ssumezeropu l sew i d t h ¯d ( n ) ¸ 1 ¡ 4 j a j n 4 j b j n 2

10. Dynamical Decoupling for Quantum Memory: Numerically Exact Simulations for Spin Chain l og ( 1 ¡ ¯d ( n )) · j a j n ¡ j b j n 2 Spin bath initially in equilibrium at 1K Theory bound:

11. 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  j 0 L i = 1 2 ( j 01 i ¡ j 10 i )( j 01 i ¡ j 10 i ) j 1 L i = 1 2 p 3 ( 2 j 0011 i + 2 j 1100 i ¡ j 0110 i ¡ j 1001 i ¡ j 1010 i ¡ j 0101 i )

12. 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) H H e i s = P i ; j J ij ( X i X j + Y i Y j + Z i Z j ) ´ P i ; j J ij E ij ¹ X = ¡ 2 p 3 ( E 13 + 1 2 E 12 ) ¹ Z = ¡ E 12 e i µ ¹ X an d e i µ ¹ Z genera t ear b i t rarys i ng l eenco d e d qu b i t ga t es

13. Universal Decoupling Group Commutes with Heisenberg Exchange n levels of concatenation, N=4 n pulses 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 1 ¢¢¢ X M Z 1 ¢¢¢ Z M [ H H e i s ; X or Z ] = 0 e ¡ i ( µ = N ) H ga t e t XZZXX UUUU Next: demonstrate discrete set of (encoded) single-qubit gates from universal set

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

15. β=10MHz, T=100nsec Concatenation Level CDD PDD System Bath A 1 ; 1 B 1 ; 3 system-bath coupling J = 0 : 01 MH z J = 10 MH z 4-Qubit DFS π/8 Gate + CDD: ideal pulses, varying system-bath coupling

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

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?