1. Dynamical Error Correction for Encoded Quantum Computation Kaveh Khodjasteh and Daniel Lidar University of Southern California December, 2007 QEC07

2. Outline Ideal Evolution and Errors Hamiltonian Description Error Inequality Dynamical Decoupling Seamless Decoupling of Operations Not so Seamless Example Encoded Adiabatic Quantum Computation

3. Ideal Evolution and Errors The goal is to perform a desired unitary operation U on a quantum system. neither unitary nor desired … because of errors. always-on undesired terms Qubits Coupling to the Environment Coupling terms among qubits in the system “In the fight between you and the world, back the world.” F. Kafka

4. Hamiltonian Description Take a control Hamiltonian H ctrl (t) that ideally generates a logical rotation Trace out to obtain the state of the system U i d ea l = T + " exp à Z T 0 H c t r l ( t ) d t !# = e ¡ i µ R H ( t ) = H c t r l ( t ) ­ I B + H err + I S ­ H B U b are = T + " exp à Z T 0 H ( t ) d t !# acts on bath acts on system perfectl y acts on system AND bath Secular Hamiltonian H sec

5. Hamiltonian Description of Errors Interaction picture of secular Hamiltonian “error phase” from Magnus expansion Minimize error phase to minimize errors. J = ||H err || is a measure of initial error rate  = ||H sec || is a measure of the bath’s mixing power U err ( T ) = exp ( ¡ i © err ) © err = Z T 0 H err ( s ) d s + i 2 Z s 1 0 Z T 0 [ H err ( s 1 ) ; H err ( s 2 )] d s 2 d s 1 + ¢¢¢ U b are ( t ) = U sec ( t ) U err ( t ) H err ( t ) = U sec ( t ) H err U sec ( t ) y “This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it! ” P. Dirac

6. Magnus Expansion Absolutely converges if [Casas arXiv:0711.2381 ] No discretization unless you want it Always unitary Truncates nicely Is hard to calculate to higher orders: The number of commutator integrals that need to be calculated grows exponentially. Iserles, Amer. Math. Soc. April 2002 Carinena et al, math/0701010 k H err k T < ¼

7. Error Inequalities No matter what control you exercise on your system the error phase cannot increase Proof sketch [Thompson’s theorem] e iA e iB = e iC then C = UAU † +VBV † Use Thompson’s theorem to show that Then use the triangle inequality. Certain restrictions apply to interpretations. No purchase neessary. k © err k · k H err k T © err = 1 X k = 0 V k H err V y k

8. Comparing Error Rates Our focus will be on the error phase. F Q [ ½ S ( T ) ; ½ i d ea l S ( T )] ¸ 1 ¡ D [ ½ 0 S ( T ) ; ½ i d ea l S ( T )] ¡ 1 2 ( e 2 jj © E ( T ) jj 1 ¡ 1 ) Control Error Error due to the environment

9. Dynamical Decoupling Dynamical decoupling (DD) control sequences reduce error phase up to the first order Magnus in the basic form Variations [ Randomized dynamical decoupling ] [ Concatenated dynamical decoupling ] [ Uhrig dynamical decoupling ] [ Multi-qubit decoupling and recoupling ] Generic DD is designed for quantum memory (NOOPeration) Not suitable for correcting quantum operations (but is used in designing them)

10. Undecoupled Terms U err is equivalent to 1 st order Magnus 2 nd (and higher) order Magnus H ( t ) = D i H err ( t ) D y i f or t 2 [ i ¿ ; ( i + 1 ) ¿ ] © ( 1 ) err = Z D i H err ( s ) D y i d s = ¿ X i D i H err D y i + O ( ¿ 2 ¯ J ) © ( 2 ) err = O ( ¿ 2 J 2 ) + O ( ¿ 3 J 2 ¯ ) will be zero will NOT be zero but will be similar to H err ok for higher order decoupling will NOT be zero parts that look like H sec ok for NOOP higher order decoupling

11. Comparing Sequences Constrain duration of the experiment T long minimum pulse width  minimum pulse interval  system-bath coupling strength J secular Hamiltonian strength  let the sequence be chosen based on the above AND Compare It is a resource to quickly vary system Hamiltonian per gate errors consider pulse shaping Source of Errors Who wants a computer without a lifetime warranty.

12. Combining DD wih Quantum Operations Encoding with logical operations that commute with DD H DD generates DD operations and H ctrl generates logical operations Seamlessly blends [ quantum operations that do the job ] & [ decoupling operations that reduce errors ] Top it with measurements if you like [ H DD ( t ) ; H c t r l ( t 0 )] = 0 8 t ; t 0

13. Seamless is just a word Apply control Hamiltonian of strength ||H ctrl ||= for a time T long Apply and spread a DD sequence over this time Arbitrary high fidelities are harder than quantum memory Errors in encoded operation: O ( J 2 T long ) presently uncorrectable with higher order sequences scale like per gate errors

14. Timeline Carr & Purcell 1954 Zanardi 1998 Viola & Lloyd 1999,2000 Haeberlen:book KKh & Lidar 2005,2007 Ührig 2007 Viola & Knill 2005 Santos & Viola 2005 Viola 2000 Lidar 2007 KKh & Lidar in prep

15. Cat Farm Code Encodes n physical qubit into n -1 logical qubits Logical Zero |0…0  L = |0…0  + |1…1  Logical Pauli Operators X j =X 1 X j+1 Z j =Z j+1 Z n Error Hamiltonian Decoupling Sequence X. . Z. . X. . Z.  where X=X 1 X 2 … X n, Z=Z 1 Z 2... Z n H err = X S ® i ­ B ® i

16. Simulate Encoded Adiabatic Deutsch-Jozsa {side result: get a bigger and better computer for your simulations} 2 qubit Deutsch-Jozsa with varying non-physical many-body Hamiltonians ( or someone teach me how to use the gadgets in Biamonte & Love 2007 ) encoded into 4 physical qubits bath: 1 spin interacting via Heisenberg T long =100, J=  =0.01, ||H ctrl ||=0.1

17. Skipped Pulse width issues Composite Pulses, Eulerian Decoupling, Self-correcting Operations Interval Synchronization Lamb shift on the bath Does it heat up the bath? Decoupling/Recoupling multiple spins among themselves Higher order generic decoupling Number combinatorics or tree algebra mess? Coupling of QECC and DD Applying Magnus Expansion to QECC