Dynamical Error Correction for

Encoded Quantum Computation

Kaveh Khodjastehand

Daniel LidarUniversity of Southern California

December, 2007

QEC07

Outline

Ideal Evolution and ErrorsHamiltonian Description

Error InequalityDynamical Decoupling

Seamless Decoupling of OperationsNot so Seamless

ExampleEncoded Adiabatic Quantum Computation

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

Hamiltonian Description

Take a control Hamiltonian Hctrl(t) that ideally generates a logical rotation

Trace out to obtain the state of the system

Uideal = T+

"

exp

ÃZ T

0Hctrl(t)dt

! #

= e¡ iµR

H(t) = Hctrl(t) I B +Herr + I S HB

Ubare = T+

"

exp

ÃZ T

0H (t)dt

! #

acts on bath

acts on systemperfectl

y

acts on system

AND bath

Secular HamiltonianHsec

Hamiltonian Description of Errors

Interaction picture of secular Hamiltonian

“error phase” from Magnus expansion

Minimize error phase to minimize errors.

J = ||Herr|| is a measure of initial error rate

= ||Hsec|| is a measure of the bath’s mixing power

Uerr(T) = exp(¡ i©err)

©err =Z T

0Herr(s)ds+

i2

Z s1

0

Z T

0[Herr(s1);Herr(s2)]ds2ds1 +¢¢¢

Ubare(t)=Usec(t)Uerr(t)

Herr(t) = Usec(t)HerrUsec(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

Magnus Expansion

Absolutely converges if [Casas arXiv:0711.2381]

No discretization unless you want itAlways 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

kHerrkT < ¼

Error Inequalities

No matter what control you exercise on your system

the error phase cannot increase

Proof sketch

[Thompson’s theorem] eiAeiB = eiC 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©errk · kHerrkT

©err =1X

k=0

VkHerrVyk

Comparing Error Rates

Our focus will be on the error phase.

FQ [½S (T);½idealS (T)] ¸ 1¡ D[½0

S (T);½idealS (T)]¡ 1

2(e2jj©E (T )jj1 ¡ 1)

Control Error

Error due to the environment

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

Uerr is equivalent to

• 1st order Magnus

• 2nd (and higher) order Magnus

H (t) = Di Herr(t)Dyi for t 2 [i¿;(i +1)¿]

©(1)err =

ZDi Herr(s)D

yi ds = ¿

X

i

Di HerrDyi + O(¿2¯J )

©(2)err = O(¿2J 2) + O(¿3J 2¯)

will be zero

will NOT be zero but will

be similar to Herr

ok for higher order decoupling

will NOT be zero

parts that look like Hsec

ok for NOOP higher order decoupling

Comparing Sequences

Constrainduration of the experiment Tlong

minimum pulse width minimum pulse interval

system-bath coupling strength J secular Hamiltonian strength

let the sequence be chosen based on the aboveAND

Compare

It is a resource to quickly vary

system Hamiltonian

per gate errorsconsider pulse

shaping

Source of Errors

Who wants a computer without a lifetime

warranty.

Combining DD wih Quantum Operations

Encoding with logical operations that commute with DDHDD generates DD operations and Hctrl generates logical operations

Seamlessly blends [ quantum operations that do the job ]

&

[ decoupling operations that reduce errors ]Top it with measurements if you like

[HDD(t);Hctrl(t0)]= 0 8t;t0

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( J2Tlong )

presently uncorrectable with higher order sequencesscale like per gate errors

Timeline Carr & Purcell 1954

Zanardi 1998Viola & Lloyd 1999,2000

Haeberlen:bookKKh & Lidar 2005,2007Ührig 2007

Viola & Knill 2005Santos & Viola

2005Viola 2000Lidar 2007

KKh & Lidar in prep

Cat Farm Code

Encodes n physical qubit into n -1 logical qubitsLogical Zero

|0…0L = |0…0 + |1…1Logical Pauli Operators

Xj=X1Xj+1

Zj=Zj+1Zn

Error Hamiltonian

Decoupling SequenceX . . Z . . X . . Z .

where X=X1X2… Xn, Z=Z1Z2... Zn

Herr =X

S®i B®

i

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 qubitsbath: 1 spin interacting via Heisenberg

Tlong=100, J==0.01, ||Hctrl||=0.1

Skipped

Pulse width issuesComposite Pulses, Eulerian Decoupling, Self-correcting Operations

Interval SynchronizationLamb 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 DDApplying Magnus Expansion to QECC