Dynamical Error Correction for Encoded Quantum Computation

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Dynamical Error Correction for Encoded Quantum
Computation

• Kaveh Khodjasteh
• and
• Daniel Lidar
• University of Southern California
• December, 2007

QEC07
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Outline

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

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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
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Hamiltonian Description

• Take a control Hamiltonian Hctrl(t) that ideally
  generates a logical rotation
• Trace out to obtain the state of the system

Secular Hamiltonian Hsec
acts on bath
acts on system perfectly
acts on system AND bath
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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 baths mixing
  power

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
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Magnus Expansion

• Absolutely converges if Casas arXiv0711.2381
• 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

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Error Inequalities

• No matter what control you exercise on your
  system
• the error phase cannot increase
• Proof sketch
• Thompsons theorem eiAeiB eiC then C
  UAUVBV
• Use Thompsons theorem to show that
• Then use the triangle inequality.
• Certain restrictions apply to interpretations. No
  purchase neessary.

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Comparing Error Rates

• Our focus will be on the error phase.

Control Error
Error due to the environment
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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)

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Undecoupled Terms

• Uerr is equivalent to
• 1st order Magnus
• 2nd (and higher) order Magnus

will NOT be zero but will be similar to Herr ok
for higher order decoupling
will be zero
will NOT be zero parts that look like Hsec ok for
NOOP higher order decoupling
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Comparing Sequences

• Constrain
• duration 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 above
• AND
• Compare

per gate errors consider pulse shaping
It is a resource to quickly vary system
Hamiltonian
Source of Errors
Who wants a computer without a lifetime warranty.
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Combining DD wih Quantum Operations

• Encoding with logical operations that commute
  with DD
• HDD 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

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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( J2?Tlong )
• presently uncorrectable with higher order
  sequences
• scale like per gate errors

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Timeline

Carr Purcell 1954
Zanardi 1998 Viola Lloyd 1999,2000
Haeberlenbook KKh Lidar 2005,2007 Ührig
2007 Viola Knill 2005 Santos Viola 2005
Viola 2000 Lidar 2007 KKh Lidar in prep
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Cat Farm Code

• Encodes n physical qubit into n -1 logical qubits
• Logical Zero
• 00?L 00? 11?
• Logical Pauli Operators
• XjX1Xj1
• ZjZj1Zn
• Error Hamiltonian
• Decoupling Sequence
• X . ? . Z .? . X .? . Z .?
• where
• XX1X2 Xn, ZZ1Z2... Zn

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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
• Tlong100, J?0.01, Hctrl0.1

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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