Hybrid quantum error prevention, Hybrid quantum error prevention, reduction, and correction reduction, and correction methods methods Daniel Lidar Daniel Lidar University of Toronto University of Toronto Quantum Information & Quantum Control Conference Toronto, July 23, 2004 $: D-Wave Systems DFS- encoded no encoding Science Science 291 291, 1013 (2001) , 1013 (2001) Isot ope- subs titu ted glyc ine: Phys. Rev. Lett. 91, 217904 (2003)
27
Embed
Hybrid quantum error prevention, reduction, and correction methods Daniel Lidar University of Toronto Quantum Information & Quantum Control Conference.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
2. Enforce decoherence model by “bang-bang” decoupling pulses generated from naturally available interactions/controls
3. Offer decoherence protection by encoding into decoherence-free subspace (against enforced decoherence model)
4. Quantum compute universally over DFS using only the naturally available interactions
5. Combine with - Composite pulse method to deal with systematic gate errors- FT-QECC to deal with random gate errors
Why don’t you just do QECC?Why don’t you just do QECC?
In non-Markovian regime FT-QECC and BB are subject to same In non-Markovian regime FT-QECC and BB are subject to same strength/speed conditions; BB more economicalstrength/speed conditions; BB more economical
Much lower encoding overhead (fewer qubits), fewer gates?
12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU i
Rabi freq. Laser phase on ions 1,2
Trapped IonsTrapped Ions
Naturally Available Interactions:Naturally Available Interactions: E.g., Sorensen-Molmer gates E.g., Sorensen-Molmer gates (work with hot ions)(work with hot ions)
XY Hamiltonian generating SM gates provides commutant structure
' '
The "collective dephasing" algebra
J J
J J
n dJ
n dJ
A M I
A I M Have option to encode into
collective dephasing DFS
Naturally compatible decoherence model is “collective dephasing”
Collective DephasingCollective Dephasing
Often (e.g., spin boson model at low temperatures) errors on different Often (e.g., spin boson model at low temperatures) errors on different qubits are qubits are correlatedcorrelated
Long-wavelength magnetic field B (environment) couples to spins
1
ˆ( )B t z
2
Effect: Random " ":
0 1 0 1j j j j jj j j j
ia b a e b
Collective Dephasing
random j-independent phase (continuously distributed)
1 2
1 2
0 0 1
1 1 0
L
L
DFS encoding
““A Decoherence-Free Quantum Memory Using Trapped Ions”A Decoherence-Free Quantum Memory Using Trapped Ions”D. Kielpinski et al., Science D. Kielpinski et al., Science 291291, 1013 (2001), 1013 (2001)
DFS-encoded
Bare qubits
Bare qubit: two hyperfine states of trapped 9Be+ ion
Chief decoherence sources: (i) fluctuating long-wavelength ambient magnetic fields; (ii) heating of ion CM motion during computation
DFS encoding: into pair of ions
1 2
1 2
0 0 1
1 1 0
L
L
Other sources of decoherence necessarily appear…Can we enforce the symmetry?
Classification of Classification of allall decoherence processes on two qubits: decoherence processes on two qubits:
Enforce DFS conditions by “bang-bang” pulses
““Bang-Bang” DecouplingBang-Bang” DecouplingViola & Lloyd PRA Viola & Lloyd PRA 5858, 2733 (1998), inspired by NMR, 2733 (1998), inspired by NMR
system bath
SBSystem-bath Hamiltonian: S BH
z
x
y
SBH
z
x
y
SBH
SBH
Apply rapid pulsesflipping sign of S
SB
"time reversal",
averaged to zero.
XZX Z
H
SBH Z B
SBH SBH
X X Unlike spin-echo, BB relies in essential way on non-Markovian bath; information is retrieved before it’s lost to bath.
Eliminating Logical Errors Using “Bang-Bang” Eliminating Logical Errors Using “Bang-Bang” SM GateSM Gate
0 01
1 10
L
L
SBH
SBH12 1 1
( , , ) U12 1 1
( , , ) U
t
2ZI IZ
no differential dephasing
X Z
X Z X Z
Also holds for : Y XY X Y
error also eliminated2
YX XYY
strong & fast
( )n
SBH
SBH12 1 1
( 2 , , ) U12 1 1
( 2 , , ) U
t
z z
Eliminating Leakage Errors Using “Bang-Bang” SM Eliminating Leakage Errors Using “Bang-Bang” SM GateGate
0 01
1 10
L
L
LeakH
no leakage errors
z z zLea zk LeakH H
{ , , , , , , , } Leak
H XI IX YI IY XZ ZX YZ ZY B
For general “leakage For general “leakage elimination via BB” seeelimination via BB” seeWu, Byrd, D.A.L., Wu, Byrd, D.A.L., Phys. Rev. Lett.Phys. Rev. Lett. 8989, 127901 (2002), 127901 (2002)
SM Pulses are Universal on |01>,|10> CodeSM Pulses are Universal on |01>,|10> Code
Can generate a universal set of logic gates by controlling relative laser phase- all single DFS-qubit operations- controlled-phase gate between two DFS qubits [Also: D. Kielpinski et al. Nature 417, 709 (2002), K. Brown et al., PRA 67, 012309 (2003)]
1 2
1 2
0 0 1
1 1 0
L
L
1 2 1 2exp[ ( cos( ) sin( ))]
DFSi X Y
12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU i
Similar conclusions apply to XY & XXZ models of solid-state physics (e.g., q. dots in cavities, electrons on He):
SM and XY/XXZ Pulses are SM and XY/XXZ Pulses are “Super-Universal”“Super-Universal”
For trapped ions can eliminate all dominant errors For trapped ions can eliminate all dominant errors (differential dephasing + leakage) in a 4-pulse sequence(differential dephasing + leakage) in a 4-pulse sequence
To eliminate ALL two-qubit errors (including ) need a To eliminate ALL two-qubit errors (including ) need a 10-pulse sequence.10-pulse sequence.
Scheme entirely compatible with SM or XY/XXZ-based Scheme entirely compatible with SM or XY/XXZ-based gates to perform universal QC inside DFS.gates to perform universal QC inside DFS.
For details, see: D.A.L. and L.-A. Wu, Phys. Rev. A 67, 032313 (2003).
_X
Further applications:Further applications:Quantum DotsQuantum Dots
200nm
Heisenberg SystemsHeisenberg Systems
Same method works, e.g., for Same method works, e.g., for spin-coupled quantum dotsspin-coupled quantum dots QC: QC:
1 2 1 2 1 2HeisBy BB pulsing of
col decoherenclective conditions can be c2
reat :e ed
y yx x z zJH
Details: L.-A. Wu, D.A.L., , 207902
Requires sequence of 6 /2 pulses to create collective decoherence
conditions over blocks of 4 qubits. Leakage elimination requires 7 more pulses.
Phys. Rev. Lett.
88 (2002); L.A. Wu, M.S. Byrd, D.A.L., , 127901 (2002).Phys. Rev. Lett. 89
Earlier DFS work showed universal QC with Heisenberg interaction alone possible [Bacon, Kempe, D.A.L., Whaley, Phys. Rev. Lett. 85, 1758 (2000)]:
SB 1
z
y yx x z zi i i i i i
yx zi i i
n
i
zx x y y
B B BH g g g
S B S B S B
Heisenberg interaction is “super-universal”Heisenberg interaction is “super-universal”
On to fault-tolerance… On to fault-tolerance… (with Kaveh Khodjasteh)(with Kaveh Khodjasteh)
We have neglected so far:We have neglected so far:Control inaccuracy in BB pulse implementation Control inaccuracy in BB pulse implementation (systematic + random)(systematic + random)
HHSBSB, , HHBB on during BB pulse on during BB pulse
Time constraints on BB pulsesTime constraints on BB pulses
Composite pulses (NMR) Concatenated QECC
All of these issues are shared by QECC:All of these issues are shared by QECC:
Related to transition q. Zeno inverse q. Zeno effect; form of bath spectral density plays crucial role K. Shiokawa, D.A.L., Phys. Rev. A 69, 030302(R) (2004); P. Facchi, D.A.L., Pascazio Phys. Rev. A 69, 032314 (2004)