dynamical error correction for encoded quantum computation kaveh khodjasteh and daniel lidar...
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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