Encoded Universality an Overview

1
Encoded Universality an Overview

• Julia Kempe
• University of California, Berkeley
• Department of Chemistry Computer Science
  Division

Sponsors
2
Encoded Universality
Whaley group people Dave Bacon (now
Caltech) Mike Hsieh (undergraduate) Julia
Kempe (postdoc) Simon Myrgren (graduate
student) Prof. Birgitta Whaley Jiri Vala
(postdoc) Jerry Vinokurov (undergraduate)
3
Overview

• Universal quantum computation - a bit of history
• Change of paradigm
• Example Heisenberg interaction
• Lie algebra formalism for encoded universal
  computation
• Results Heisenberg interaction, symmetric XY
  interaction, asymmetric XY with crossterms

4
Quantum circuits
Barenco et al. 95


U

Mantra
Single-qubit gates and CNOT generate every
unitary transformation!
5
The problem
Easy and hard interactions (system-dependent)
Easy intrinsic interactions natural to the
system, easy to tune, rapid Hard slower,
require higher device complexity, high decoherence
Can we avoid hard interactions?
quantum dots, donor-atom nuclear spins, electron
spins 1 Ralph, Munroe and Milburn, 2001
6
Almost every interaction is universal!

• Deutsch et al.(95), Lloyd (95)
• Almost any interaction on two qubits is
  universal.
• In the generic sense.
• Does not include the most frequentes
  interactions.
• Nature is not generic!

qubit i
qubit j
qubit i
qubit j
Hij
Hji
7
Change of paradigm

• Traditionally
• manipulate the physical system
• to produce

H1,H2,...



Independent of systems natural talents (fast,
robust interactions) often difficult, certain
gates can only be implemented with noise high
decoherence ...
8
Change of paradigm

• Traditionally
• manipulate the physical system
• to produce

H1,H2,...


Universal encoded computation interactions
given by the physical system
find a way to make them universal
H
H
H
Encoding?
Independent of systems natural talents (fast,
robust interactions) often difficult, certain
gates can only be implemented with noise high
decoherence ...
9
Classical  Analogy 
1
1

• Two coins
• can only flip the two coins together
•  encode 
•  0 -
•  1 -

flip
0
0
1
0
0
1
1
1
0
0
00 11 01 10
0
Encoded  coin 
0
0
1
1
0
10
Language of Hamiltonians

• U(t) exp(iHt) Which interactions are
  universal?
• Given ??H1, H2,, Hn? can one generate any
  unitary transformation (exactly or
  approximatively)?
• H has to generate the Lie algebra of su(N) of the
  unitary group SU(N)!
• 1) scalar
  multiple
• 2)
• linear combination
• 3)
• Lie bracket

11
Heisenberg interaction
(Pauli matrices)

• omnipresent in solid state physics ( Easy )
• is not universal preserves the total spin of the
  qubits
• Lie algebra of E
• On three qubits

E12
E23
E13
su(2)
12
The algebra L(E) of E (3 qubits)

• the algebra L3(E) splits as
• L3(E) ? L3(E) ? S1?I4 ? S2?I2
• su(2)?S2
• Encoded qubit ?

?su(2)
2
?su(2)
2
Simulation of all operations of one qubit (su(2))
with L3(E) on the encoded qubit !
13
The algebra Ln(E) of E (n qubits)

• the algebra Ln(E) splits as
• Ln(E) ?

...
...
...
Commutant L of Ln(E)
L is generated by
( spin  algebra su(2))
As a Lie algebra L splits into irreducible
representations of su(2).
14
Useful theorem

• Let S be a -closed algebra closed under
  multiplication and linear combination. Then the
  underlying space H is isomorphic to
• such that S and its commutant S split as
• where M(Cd) (M(Cn)) is the algebra of all
  matrices on Cd (Cn).

...
...
Universal computation for free?
...
15
Useful theorem

• Let S be a -closed algebra closed under
  multiplication and linear combination. Then the
  underlying space H is isomorphic to
• such that S and its commutant S split as
• where M(Cd) (M(Cn)) is the algebra of all
  matrices on Cd (Cn).

NO!
The multiplicative algebra is not at our
disposition! However the Lie algebra splits into
irreducible components in the same basis
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Problem of Encoded Universality

• Given an ensemble of generators H with Lie
  algebra L(H) which splits as
• can one find a component s.t.
  contains su(nj )?
• Encode the quantum information into
• the corresponding sub-space.
• dimension nj

...
Yes
...
...
D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar,
K.B. Whaley, Encoded Universality in Physical
Implementations of a Quantum Computer,
Proceedings of IQC 01, Australia
17
Previous Results - Heisenberg interaction

• E is universal with encoding
• introduce tensor structure, ex. blocks with 3
  qubits

efficient implementation of encoded
gates numerical search
serial coupling - 19 operations for CNOT, 4
operations for 1-qubit parallel coupling - 7
operations for CNOT, 3 operations for 1-qubit
Kempe, Bacon, Lidar, Whaley, Phys. Rev. A
63042307 (2001) DiVincenzo, Bacon,
Kempe,Whaley, NATURE 408 (2000)
18
Exchange-only CNOT
Nearest neighbor exchange coupling
exchange gate
i
j
DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature
408, 339 (2000)
Tradeoffs factor of 3 in space (encoding) factor
of 10 in time
19
Conjoining a new tensor structure
Introduce a cutoff that defines a single qudit.
In principle
...
For larger n one could find larger component
with better encoding ratio ?
...
...
Need to guarantee uniformity of quantum
circuits! (Form of the circuit should not
depend on size of problem.) Introduce cutoff -gt
tensor product structure.
Conjoining subsystems
20
Anisotropic Exchange
XY-interaction
i.e., 1-qutrit operations

• HXY generates su(9) on this subspace
• Truncated qubit use and
  only
• effectively with an ancillary qubit for
  gate-applications

J. Kempe, D. Bacon, D.P. DiVincenzo, K.B.
Whaley, Encoded universality from a single
physical interaction, in Quantum Information
and Computation Special Issue, Vol. 1, 2001
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The Gates
Gate sequences 7 operations for single qubit
operations (serial) 5 operations for Sqrt (-ZZ)
(equiv. to controlled phase) P3-gate
?/4
-?/4
?/2
P3(?)

?/2
-?/2
Truncated qubit
Two-qubit operation
Single qubit operations
P3(-?/2)

?

P3(-?)

(Euler angles)
J. Kempe and K.B.Whaley, Exact gate-sequences
for universal quantum computation using the
XY-interaction alone , quant-ph/0112014, to
appear in Phys. Rev. A
22
Layout Anisotropic Exchange
a) triangular array (qutrit) b)
truncated qubit
or
23
Results Asymmetric Anisotropic Exchange
Poster No. 21 by Jiri Vala
Asymmetric exchange
Asymmetric exchange with crossterm
Universal Encodings and Gate-Sequences
J. Vala and K.B. Whaley, Encoded Universality
with Generalized Anisotropic Exchange
Interactions, in preparation 2002
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Results Asymmetric Anisotropic Exchange
The total exchange Hamiltonian consists of two
components 1) symmetric, which couples the
physical qubit states 01gt and 10gt Hij J (
sx,i sx,j sy,i sy,j ) K ( sx,i sy,j - sy,i
sx,j ) 2) and antisymmetric , coupling the
states 00gt and 11gt hij j ( sx,i sx,j - sy,i
sy,j ) k ( sx,i sy,j sy,i sx,j ) which both
simultaneously transform pairs of code words in
two code-subspaces.
code space II
code space I
This allows to apply similar techniques as in the
symmetric XY-case!
Vala and Whaley, in preparation
Poster No. 21 by Jiri Vala
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Is encoded universality always possible?
NO!

• non-interacting fermions (Valiant,
  TerhalDiVincenzo, Knill 01)
• nearest-neighbor XY-interaction
• linear optics quantum computation

Criterion
If a set of Hamiltonians (over n qubits) allows
for (encoded) universal computation then the Lie
algebra L(H) contains exponentially many linearly
independent elements.
Some component has to contain where
is a polynomial function of n.
...
...
...
ex is not universal with any encoding.

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Continuing Work
Poster No. 38 by Mike Hsieh

• How find the gate sequences that implement the
  encoded one- and two-qubit gates?
• Numeric search genetic algorithms (Hsieh,
  Kempe)
• Developed numeric tools
• Preliminary results in 4-qubit encoding for
  Heisenberg interaction

27
Summary

• Lie-algebra methods and redefinition of the
  tensor structure of Hilbert space allow for
  universality!
• Use of encoding - the implementation of one-qubit
  gates is obsolete! (change of paradigm)
• Heisenberg interaction is omnipresent (e.g. in
  solid state physics) and easy to implement,
  whereas one-qubit gates are extremely hard to
  obtain ? encoded universality gives attractive qc
  proposals
• Evaluation of the trade-offs in space and time
• Open Questions
• Find better than ad hoc ways to generate the gate
  sequences
• General easy criteria to determine encoded power
  of interaction (e.g. amount of symmetry)
• a general theory allowing for measurements and
  prior entanglement (to incorporate
  Briegel/Rauschendorff and Nielson schemes into
  analysis)

28
References
J. Kempe and K.B.Whaley, Exact gate-sequences
for universal quantum computation using the
XY-interaction alone , quant-ph/0112014 , to
appear in Phys. Rev. A J. Kempe, D. Bacon, D.P.
DiVincenzo, K.B. Whaley, Encoded universality
from a single physical interaction, in Quantum
Information and Computation Special Issue, Vol.
1, 2001, quant-ph/0112013 D. Bacon, J. Kempe,
D.P. DiVincenzo, D.A. Lidar, K.B. Whaley,
Encoded Universality in Physical Implementations
of a Quantum Computer, Proceedings of IQC 01,
Australia, quant-ph/0102140 D.P. DiVincenzo, D.
Bacon, J. Kempe, K.B. Whaley, Universal Quantum
Computation with the Exchange Interaction,
NATURE 408, 339 (2000), quant-ph/0005116 J. Vala
and K.B. Whaley, Encoded Universality with
Generalized Anisotropic Exchange Interactions,
in preparation 2002 Earlier related work on
DFSs J. Kempe, D. Bacon, D. Lidar,K.B. Whaley,
Phys. Rev. A 63042307 (2001) D. Bacon, J. Kempe,
D. Lidar, K.B. Whaley, Phys. Rev. Lett. (2000)

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Conclusions/Open Questions

• Encoding into sub-spaces allows to make certain
  interactions universal
• Representation theory of Lie groups - powerful
  tool
• E and XY alone are universal - important
  simplification of physical implementations
• Which other interactions to investigate?
• General theory when interactions allow for
  encoded universality?
• How find the gate sequences that implement the
  encoded one- and two-qubit gates?

30

• tensor product of encoded qubits
• conjoined codes Bacon et al., PRL 85, 1758
  (2000)
• Bacon et al., quant-ph/0102140
• find entangling operations
• Lie algebraic analysis Kempe et al., PRA 63,
  042307 (2001)
• Kempe et al., JQIC (2001)
• efficient implementation
• numerical search e.g. Heisenberg exchange

serial coupling - 19 operations for CNOT, 4
operations for 1-qubit parallel coupling - 7
operations for CNOT, 3 operations for
1-qubit DiVincenzo, Bacon, Kempe, Burkard,
Whaley, Nature 408, 339 (2000)
31
Results

• tensor product of encoded qubits conjoined
  codes
• (Bacon, Kempe, DiVincenzo, Lidar, Whaley ICQ01)
• universality Lie algebraic analysis
• (Kempe, Bacon, DiVincenzo, Whaley IQC01)

Anisotropic Exchange Interaction
i.e., 1-qutrit operations
HXY generates su(9) on this subspace (Kempe,
Bacon, DiVincenzo, Whaley IQC01)
32
Single-Qubit and Two-Qubit Gates
1) the full su(2) algebra over a single logical
qubit is generated via the commutation relations
between exchange interactions over
physical qubits e.g. H13,H23 i (J2 - j2) s
y,12 and H12 ,s y,12 i 2 J s z,12 2)
entangling two-qubit operation C(Z) results from
application of the encoded s z operation onto the
physical qubits 2-3-4 in the triangular
architecture and single-qubit operations 3) the
commutation relations are applied via selective
recoupling 4) a similar construction is valid
for a general anisotropic interaction containing
the cross-product terms