Dynamical Error Correction for Encoded Quantum Computation Kaveh

Dynamical Error Correction for Encoded Quantum Computation Kaveh Khodjasteh and Daniel Lidar University of Southern California December, 2007 QEC 07 Encoded Decoupling of Operations

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

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

Hamiltonian Description " ÃZ !# Take a control Hamiltonian Hctrl(t) that ideally generates a logical rotation ¡ T = e iµR Uideal = T+ exp Hctrl (t)dt 0 acts on system perfectl H(t) y Secular Hamiltonian =H Hsec ctrl acts on bath (t) IB + Herr + IS HB " ÃZ !# T Ubare = T+ exp 0 H(t)dt acts on system AND bath Trace out to obtain the state of the system

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

Magnus Expansion k k Absolutely converges if [Casas ar. Xiv: 0711. 2381] Herr T < ¼ 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

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

Comparing Error Rates ¡ ¸ ¡ jj jj ¡ FQ [½S (T ); ½ideal (T )] 1 D[½ 0 (T ); ½ideal (T )] 1 (e 2 ©E (T ) 1 1) S S S 2 Control Error due to the environment Our focus will be on the error phase.

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)

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

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

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

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

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

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

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 Tlong=100, J= =0. 01, ||Hctrl||=0. 1

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