Holonomic quantum computation in decoherence-free subspaces

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Holonomic quantum computation in
decoherence-free subspaces
Center for Quantum Information and Quantum
Control

• Lian-Ao Wu

In collaboration with Polao Zanardi and Daniel
Lidar
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• Background
• Decoherence-free subspace (DFS) symmetry-aided
  protection of quantum information, against such
  as collective noise
• Holonomic (adiabatic) quantum computation (HQC)
  all-geometrical Quantum Information Processing
  strategy, robust against operational errors
• Question is
• Can we combine the advantages of the two?
• Bringing together the best of two worlds....!!

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First recall Univerisal Quantum Computation
(requires)
Have 2-level product space (qubits), prepare
initial state
Have universal set of gates
(gates can take one state to arbitrary state)
e.g. 1-qubit X, Z gates for each qubit plus
CPHASE gates. Gates usually are evolution
operators given by
Measure the final state
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Introduce DFS Decoherence-Free Subspace (DFS) I
Decoherence? a quantum information processor
(system) cannot be isolated from its environment
(Bath),
due to the interaction of the system with its
bath, i.e.
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Introduce DFS Decoherence-Free Subspace (DFS) II
Invariant subspaces if there is symmetry in
H e.g. collective dephasing
For example, subspace spanned by 01gt and 10gt
will be invariant. (0001gt,0010gt,0100gt
1000gt)
B-S interaction not harmful for the system
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Introduce HQC Holonomic Quantum Computation (HQC)
I
Control time-dependent periodical Hamiltonian
through M parameters
where
T- period, evolution operator
HQC is based on the adiabatic theorem, which shows
If start with an eigenstate of H(t), the system
will stick on it but 2 phases

if H has non-degenerate eigenvalues
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Introduce HQC Holonomic Quantum Computation (HQC)
II
One is interested in the case at time T, if start
with
The Berry phase is all geometrical, independent
of speed of parameters
Example,
The Berry phase is the solid angle swept out by
the vector
Allow operational error, as long as solid angle
same, the Berry phase same.

Geometrical Phase Gate if
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Introduce HQC Holonomic Quantum Computation (HQC)
III
Dark Eigen State
Using dark state to generate all-geometrical
phase gate
We need 4 states 0gt, 1gt,2gt 3gt for 1
particle, a controllable Hamiltonian in terms of
parameters q(t) and f(t)
The dark state
The Hamiltonian does nothing on 0gt and add a
Berry phase on state 1gt after evolution from 0
to T if q(0)0 If we define our qubit by 0gt
1gt

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Introduce HQC Holonomic Quantum Computation (HQC)
V
Using dark state to generate all-geometrical X
gate eigX/2
In the 4-state space by 0gt, 1gt,2gt 3gt for
one particle , we need a controllable Hamiltonian
The dark state
The Hamiltonian will do nothing on gt and add a
Berry phase for state -gt after evolution from 0
to T.

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Introduce HQC Holonomic Quantum Computation (HQC)
V
Using dark state to generate all-geometrical
2-qubit gate
We have 16 states for 2 particles, 00gt, 01gt,
02gt,. Chose a controllable Hamiltonian,
The dark state
The Hamiltonian will do nothing on 00gt,01gt
10gt and add a Berry phase for state 11gt after
evolution from 0 to T.

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A brief Sum-Up of HQC Holonomic Quantum
Computation (HQC) VI
Use dark states to generate all-geometrical
universal set of gates, 1 qubit Z, X gate
2-qubit CPHASE gate (by controlling Hz, Hx and
H4)
For a dark state, the wave function at T
Using this relation to perform phase gates.
In above cases, g is half of solid angle swept
out by the vector (q,f)

We have to use ancillas 2gt and 3gt for each
qubit when make gates. We pay more price. We need
to Have 4-dimensional working space to support 1
qubit.
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• Come to our work
• A Decoherence-Free Subspace as working space

4 qubit DFS Cspan 1000gt, 0100gt, 0010gt,
0001gt
If interaction between system and bath is
C is a DFS against collective dephasing, will be
our working space to support encoded logical qubit
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Time-dependent controllable Hamiltonian Set
Assume that the system dynamics is generated by
the Hamiltonian
where parameters are dynamically controllable.

Every eigenspace of Z is invariant under action
of
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Dark-states in the DFS
Turn on the parameters in such a way to get
In the basis 100gt,010gt,001gt for qubits l, m
and n, the above Hamiltonian has a dark state
Satisfying
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One-qubit geometrical gates
Turn on the parameters in such a way to get
Acting only on 2, 3 and 4 qubit states1000gt4321,
0100gt4321 and 0010gt4321 nothing on 0001gt4321 .
Define the logic qubit supported by state
0gtL0001gt and 1gtL0010gt. Dark state in
qubits 2,3 and 4
Adiabatically changing parameters a Berry Phase
for 1gtL
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sx gate Turn on the parameters in such a way to
get
where
Easy to prove
Dark state is
Adiabatically changing parameters a Berry Phase
for -gtL
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Two-qubit geometrical gates
Suppose that one can engineer the four-body
interaction
Dark state is
Adiabatically changing parameters a Berry Phase
for 11gtL
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Compare General HQC with HQC in DFS
General HQC
HQC in DFS
4D working space 0gt, 1gt, 2gt, 3gt
4D working space 0001gt, 0010gt,0100gt,1000gt
Logic qubit by 0gtL0001gt and 1gtL 0010gt
Qubit by 0gt and 1gt
Hx, Hz and H4 have same matrix representations
but acting different spaces
Controllable Hamiltonian Hx, Hz and H4
Robust against operational error
Robust operational error and collective
dephasing
Experimental implementation depend on the system
Interesting to note X, Z need only 2-body
interaction
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Implementations
Spin-based quantum dot proposals
One qubit Hamiltonians achievable
confining potential, pulse shaping (Stepanenko
etal al 2003,2004)
Ion Traps
Sorensen-Moelmer scheme (two lasers
control) (Kielpinski et al 2002)
Realizable as well! SM over two pairs of trapped
ions geometrical gates already realized
(Leibfried et al 2003)
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• Summary
• We have discussed how to merge together
  universal HQC DFS by using dark-states in
  decoherence-free subspace against collective
  dephasing. The scheme can be extended to the
  cases against general collective noise.

Thank you for the attention!
LA Wu, PZ, DA Lidar, PRL 95, 130501 (2005)
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geometrical phase factor is precisely the
holonomy in a Hermitian line bundle since the
adiabatic theorem naturally defines a connection
in such a bundle.