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Optimal control of the quantum gate
operations for quantum computing
Hsi-Sheng Goan
管
希
聖
with Dung-Bang Tsai and Po-Wen Chen
Department of Physics
and Center for Theoretical Sciences,
National Taiwan University, Taipei, Taiwan
Ref: Phys. Rev. A 79, 060306
(Rapid Communications) (2009).
Quantum Computation and Quantum
Information
• The study of the information processing and
computing tasks that can be accomplished using
quantum mechanical systems.
• To exploit quantum effects, based on the principles
of quantum mechanics to compute and process
information in ways that are faster or more efficient
than or even impossible on conventional computers
or information processing devices.
RSA cryptography
• The difficulty of factorizing large numbers forms the basis of RSA
encryption system: standard industrial strength encryption on the
Internet
– Example: 4633 = 41 x 113
• RSA systems offers each prizes to people who factor number like
(US $200K for this one):
Example: factor a 300-digit number; Best algorithm: takes 1024 steps;
On computer at THz speed: 150,000 years
編碼保密傳輸
編碼金鑰
解碼金鑰
網路銀行 (internet banking): N = p q
•Public key: 公開的編碼金鑰 (N,e)
•Private key:不公開的解碼金鑰 (N, p,q)
Quantum algorithms and
computational speed-ups
• Algorithm: a detailed step-by-step method for solving a problem
• Computer: a universal machine that can implement any algorithm
• Quantum factoring algorithm : exponential speed-up (Shor’s Algorithm)
Example: factor a 300-digit number
Best classical algorithm:
1024 steps
Shor’s quantum algorithm:
1010 steps
On classical THz computer:
150,000 years
On quantum THz computer:
<1 second
• Quantum search of an unsorted database: quadratic speed-up
(Grover’s Algorithm)
– Example: name  phone number (easy)
– phone number  name (hard)
– Classical: O(n), Grover’s:
O( n )
• Simulation of quantum systems: up to exponential speed-up.
Peter Shor
Quantum bits
•
•
•
•
•
•
•
•
•
Classical bit: 0 or 1; voltage high or low
| 0 and |1
Quantum bit (QM two-state system):
Spin states;  and 
Charge states; left or right
Flux states; L or R
Energy states, ground or excited states
Photon polarizations; H or V; L or R
Photon number (Fock) states;
More …
Requirements for physical
implementation
of quantum computation
• A scalable physical system with well characterized
qubits
• The ability to initialize the state of the qubits to a
simple fiducial state, such as |000……〉.
• Long relevant decoherence times, much longer
than the gate operation time
• A universal set of quantum gates
• A qubit-specific measurement capability
Physical systems actively considered
for quantum computer implementation
• Liquid-state NMR
• NMR spin lattices
• Linear ion-trap
spectroscopy
• Neutral-atom optical
lattices
• Cavity QED + atoms
• Linear optics with
single photons
• Nitrogen vacancies in
diamond
• Electrons on liquid He
• Small Josephson junctions
– “charge” qubits
– “phase” qubits
– “flux” qubits
• Impurity spins in
semiconductors
• Coupled quantum dots
– Qubits: spin,charge,
excitons
– Exchange coupled,
cavity coupled
Electron spins in quantum dots
•
•
•
•
•
Top electrical gates define quantum dots in 2DEG.
Coulomb blockade confines excessive electron number at one per
dot.
Spins of electrons are qubits.
Qubits can be addressed individually:
 Back gates can move electrons into magnetized or high-g layer
to produce locally different Zeeman splitting.
 Or a current wire can produce magnetic field gradient.
Exchange coupling is controlled by electrically lowing the tunnel
barrier between dots
Silicon-based quantum bits
•
•
Donor nuclear spins [Kane, Nature (1998)]
Donor electron spins
–
–
–
•
Donor electron-nuclear spin pairs
–
•
Digital Approach [Skinner et al., PRL (2003)]
Donor electron charges
–
•
Si-Ge hetero-structures [Vrijen et al., PRA (2000)]
Dipolar coupling [de Sousa et al., PRA (2004)]
Surface gate and global control [Hill et al., (2005)]
P/P+ charge qubit [Hollenberg et al., (2004)]
Electron spins in silicon-based quantum dots
[Friesen et al., PRB (2002)]
Silicon-based electron-mediated
nuclear spin quantum computer
B. Kane, Nature (1998)
• Exploiting the existing
strength of Si technology
• Qubits are nuclear spins of P
donors in a regular array in
pure silicon
• Low temperature:
– Effective Hamiltonian
involves only spins
– Long spin coherence and
relaxation times
• Magnetic field B to polarized
electron spins
• Control with surface gates and
NMR pulses
• Donor separation ~ 20nm
• Gate width < 10nm
Phosphorus Donor in Si
P donor behaves effectively like a
hydrogen-like atom embedded in Si
me
1 m* H
a   * a B , En  2
En
m
 me
*
B
P shallow donor energy levels in Si
Silicon-based quantum computing
Two interactions: hyperfine
and exchange interactions .
Determining the strength of
these two interactions as
function of donor depth, donor
separation and surface gate
configuration and voltage.
• L.M. Kettle, H.-S. Goan, S.C. Smith, C.J. Wellard, L.C.L. Hollenberg and C.I. Pakes, “A numerical study of
hydrogenic effective mass theory for an impurity P donor in Si in the presence of an electric field and
interfaces'', Physical Review B 68, 075317 (2003).
• C.J. Wellard, L.C.L. Hollenberg, F. Parisoli, L.M. Kettle, H.-S. Goan, J.A.L. McIntosh and D.N. Jamieson,
“Electron exchange coupling for single donor solid-state spin qubits”, Physical Review B 68, 195209
(2003).
• L.M. Kettle, H.-S. Goan, S.C. Smith, L.C.L. Hollenberg and C.J. Wellard, ”Effect of J-gate potential and
interfaces on donor exchange coupling in the Kane quantum computer architecture'', Journal of Physics:
Condensed Matter 16, 1011 (2004).
• C.J. Wellard, L.C.L. Hollenberg, L.M. Kettle and H.-S. Goan, “Voltage control of exchange coupling in
phosphorus doped silicon”, Journal of Physics: Condensed Matter 16, 5697 (2004).
• L. M. Kettle, H.-S. Goan, and S. C. Smith, “Molecular orbital calculations of two-electron states for P
donor solid-state spin qubits”, Physical review B 73, 115205 (2006).
Quantum gate operation, and quantum
algorithm modelling
CNOT
•
•
•
•
•
C. D. Hill and H.-S. Goan, “Fast non-adiabatic two-qubit gates for the Kane
quantum computer”, Physical Review A 68, 012321 (2003).
C.D. Hill and H.-S. Goan, “Comment on Grover search with pairs of trapped ions“,
Physical Review A 69, 056301 (2004).
C.D. Hill and H.-S. Goan, “Gates for the Kane quantum computer in the presence of
dephasing”, Physical Review A 70, 022310 (2004).
C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree, and H.-S.
Goan, “Global control and fast solid-state donor electron spin quantum computing”,
Physical Review B 72, 045350 (2005).
C.D. Hill and H.-S. Goan, “Fast non-adiabatic gates and quantum algorithms on the
Kane quantum computer in the presence of dephasing”, AIP Conference
Proceedings Vol. 734, pp167-170 (2004).
The CNOT gate
• After using some single qubit identities to simplify, this circuit becomes:
• Under typical expected
conditions, numerical
simulation shows that the
CNOT gate has a systematic
error of 4.0 x 10-5 and takes a
total time of 16.0 s.
• Similar circuits can be found
for any two qubit gate,
including swap and square
root of swap gates.
Silicon-based quantum bits
•
•
Donor nuclear spins [Kane, Nature (1998)]
Donor electron spins
–
–
–
•
Donor electron-nuclear spin pairs
–
•
Digital Approach [Skinner et al., PRL (2003)]
Donor electron charges
–
•
Si-Ge hetero-structures [Vrijen et al., PRA (2000)]
Dipolar coupling [de Sousa et al., PRA (2004)]
Surface gate and global control [Hill et al., (2005)]
P/P+ charge qubit [Hollenberg et al., (2004)]
Electron spins in silicon-based quantum dots
[Friesen et al., PRB (2002)]
Donor electron spin in Si-Ge structure
R. Vrijen et al, Concept
device: spin-resonance
transistor, Phys. Rev. A 62,
012306 (2000)
Silicon-based electron-mediated
nuclear spin quantum computer
B. Kane, Nature (1998)
• Exploiting the existing
strength of Si technology
• Qubits are nuclear spins of P
donors in a regular array in
pure silicon
• Low temperature:
– Effective Hamiltonian
involves only spins
– Long spin coherence and
relaxation times
• Magnetic field B to polarized
electron spins
• Control with surface gates and
NMR pulses
• Donor separation ~ 20nm
• Gate width < 10nm
Single-qubit system
Effective low-energy low-temperature Hamiltonian:
Ze
Zn
H B  H A  g e e B  g n  n B
 Aσ e  σ n
2
2
where A  (2 / 3) g e e g n n |  (0) |2
Notations:  x  X ;  y  Y ;  z  Z .
geBB
Energy separation:
E (2)
 g e  B B / 2  g n n B / 2  A
0
E (2)
1
2 A2
 g e  B B / 2  g n n B / 2  A 
( g e  B B / 2  g n n B / 2)
E (2)
  g e  B B / 2  g n n B / 2  A
1
E
(2)
0
2 A2
  g e  B B / 2  g n n B / 2  A 
( g e  B B / 2  g n n B / 2)
Single-qubit system
Qubit energy separation (if nuclear spins is
initialized in spin-up state):
(2)
E  E (2)

E
0
0
2 A2
 ge B B  2 A 
( g e  B B / 2  g n n B / 2)
geBB
Effective single-qubit Hamiltonian:
H eff   ( A) ze
2
4 A2
 ( A)  ge B B  2 A 
( g e  B  g n n ) B
Hamiltonian in a Bac field:
H ac  ge e Bac [ X e cos(act )  Ye sin(act )]
Single-qubit control
Having control over hyperfine
interaction by applying voltage
to A gate would allow us to:
• Change the resonant
frequency of a particular
qubit.
• Perform X and Y rotations
on a specific qubit using a
resonant magnetic field
• Perform a Z on a specific
qubit (much faster than X
and Y rotations)
These three operations allow
us to do any single qubit
rotation on the nuclear spins.
B. Kane, Nature 393, 133 (1998)
Single qubit rotations
Laboratory frame
z
H  g e  B B ze / 2
Reference frame
z
H  (0  ac ) ze / 2
 ge e Bac / 2
 g e e Bac [ xe cos(ac t )   ye sin(ac t )] / 2
Bz
y
Bac
ac
x
Beff

Bz  ac
g B
x
Bac
P(ac)
 g  B Bac 2 0  ac 2
( g  B Bac / ) 2
P (t ) 
sin  (
) (
)
g  B Bac 2 0  ac 2
2

(
) (
)
2
with an initial  state; 0  g  B Bz /

t

4 g  B Bac
0
ac
Rx(q) rotation in the global control
donor e-spin QC
1
H eff    ge B Bac xe ,
2
2
where    ( A)  ac ,
e
z
• Set a detuned target qubit to perform a 2 rotation, and
then every other spectator qubit will undergo a rotation
g e  B Bac
around x-axis with an angle q  2  2
(  ) 2  ( g e  B Bac ) 2
• Perform an on-resonance Rx(q) rotation on every qubit to
correct the spectator qubits’ rotations.
Two-qubit Hamiltonian
• Effective e-spin Hamiltonian in the rotating frame
1  1
2
2
where i   ( Ai )  ac ,
H eff 
1e
z
2e
z
1
 ge  B Bac ( 1xe   x2e )  Jσ1e  σ 2e ,
2
4 A2
 ( A)  ge  B B0  2 A 
( g e  B  g n n ) B
• Full Hamiltonian in the Lab. frame
Exchange interaction J
Strain
• L. M. Kettle, H.-S. Goan, and S. C. Smith, PRB 73, 115205 (2006).
See also:B. Koiller, X. Hu and S. Das Sarma, PRL 88, 072903 (2002).
Two-qubit control
• Two qubit Hamiltonian:
2
H 2 q   H B H A  H ac  H J
i 1
H J  J  e1   e 2
The magnitude of the
exchange interaction, J,
depends on the degrees of
overlap of electronic wave
functions and can be controlled
by the surface J-Gate.
B. Kane, Nature 393, 133 (1998)
Universal and CNOT gate
• CNOT + single qubit rotations are universal for quantum computation.
• Any gate can be constructed using CNOT and single qubit rotations.
CNOT + RX ( ), RY (  ), RZ ( )
•
What is the CNOT (Controlled-Not) gate:
| 00 | 00
| 01 | 01
| 10 |11
|11 |10
1
0
CNOT  
0

0
0
1
0
0
0
0
0
1
0
0

1

0
• Task is to demonstrate that the CNOT gate and single qubit
rotations may be constructed.
Constructing CNOT gate from the
controlled Z Gate
Hadamard gate:
1 1 1 



H
,
H

R
(
)
R
(
)
R
(
)
Z
X
Z
1 1
2
2
2
2

Controlled-Z gate,
1
0
1Z  
0

0
0
1
0
0
0 0
0 0

1 0

0 1
1Z  e
i (
 (e
I Z I Z

)
2
2

i Z
4
Controlled-Not gate:
CNOT  ( I  H ) 1Z (I  H )
 I ) (I  e

i Z
4

)e
i Z Z
4
Construction of two-qubit gates
• Any two-qubit gate may be expressed in the following way:
V  (W1  W2 ) eiq X X  X  iqY Y Y  iqZ Z Z (W3  W4 )
where W1, W2, W3 and W4 are local operations. We can perform
these operations using single-qubit rotations.
• The only challenge is to perform the entangling part of the gate.
• What we have:
U (q )  eiq ( X  X Y Y  Z Z )
• What we want:
i Z Z
4
T e

• Isolate the Z-Z term:


(Z  I ) U ( ) (Z  I ) U ( )  e
8
8

i (  X  X Y Y  Z  Z )
8
e

i Z Z
4

e
i ( X  X Y Y  Z  Z )
8
Canonical decomposition of CNOT
gate for global control e-spin QC
 
  1e 2 e 
U    exp i σ  σ 
8
 8

CNOT gate operation time: 297ns
C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree,
and H.-S. Goan, “Global control and fast solid-state donor electron spin
quantum computing”, Phys. Rev. B 72, 045350 (2005).
• Simulation of electron exchange mediated two-qubit gates in
the Kane donor nuclear spin scheme showed that the gate
fidelity is limited primary by the electron coherence when the
electron dephasing timescale is close to the typical gate
operation time of O(s).
• Experimental indication: P donor electron spin T2 > 60 ms at
4K in purified silicon [Tyryshkin, Lyon et al., PRB (2003)].
• Features of e-spin based QC:
― Fast gate speed,
― Comparatively simpler readout
Optimal control
• One of the important criteria for physical
implementation of a practical quantum computer is to
have a universal set of quantum gates with operation
times much faster than the relevant decoherence time
of the quantum computer.
• High-fidelity quantum gates to meet the error
threshold of about 10-4 (10-3) are also desired for faulttolerant quantum computation (FTQC).
• Thus the goal of optimal control is to find fast and
high-fidelity quantum gates.
Error threshold: P. Aliferis and J. Preskill, Phys. Rev. A 79, 012332 (2009).
GRadient Ascent Pulse Engineering
• N. Khaneja et al.,
(GRAPE)
J. Magn. Reson.
• Propagator during time step j (t=T/N)
m
 i 

U j (t )  exp   t  H 0   ukj H k 
k 1



• Propagator at final time T
UF  UN
U1
• Performance function (fidelity)
172, 296 (2005).
• A. Sporl et al.,
Phys. Rev. A
75, 012302 (2007)
See also: Montangero et al.,
PRL 99, 170501 (2007) and
Carlini et al., PRL 96,
060503 (2006);PRA 75,
042308 (2007)
  Tr U U F  , where U D : desired op.
†
D
2
Nielsen et al.,
Science; PRA
(2006)
• Optimize the performance function (fidelity) w.r.t. the
control amplitudes ukj in a given time T.
• The minimum time sequence that meets the required
error threshold is the near time-optimal control sequence.
Trace Fidelity versus gate time
1
†
Ftr  n Tr U DU F 
2
Optimizer:
spectral
projected
gradient method
Stopping criteria
of the error
threshold : 10-9
30 piecewise constant steps is sufficient
Choice of the value of Bac
• While the target electron spin qubit will perform a
particular unitary operation within time t, every spectator
qubit will rotate around the x-axis with an angle of
qx 
g e  B Bac
t
• If qx does not equal to 2n, where n is an integer,
another correction step will be required for the spectator
qubits. Therefore, it will be more convenient to choose
the operation time,
2n
t
g e  B Bac
For t  100ns and n  1, Bac  3.56 104 T.
Near time-optimal control sequence
30 steps
in100ns
with an
error of
1.11x10-16
Calculations
performed
using the
effective
e-spin
Hamiltonian
Canonical decomposition of CNOT
gate for global control e-spin QC
 
  1e 2 e 
U    exp i σ  σ 
8
 8

CNOT gate operation time: 297ns
C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree,
and H.-S. Goan, “Global control and fast solid-state donor electron spin
quantum computing”, Phys. Rev. B 72, 045350 (2005).
Parallel quantum computing
• Traditional decomposition method that decomposes general
gate operations into several single-qubit and some interaction
(two-qubit) operations in series as the CNOT gate in the
globally controlled electron spin scheme. So the single-qubit
operations and two-qubit (interaction) operations do not act on
the same qubits at the same time.
• The GRAPE optimal control approach is in a sense more like
parallel computing as single-qubit (A1 and A2 both on) and
two-qubit (J on) operations can be performed simultaneously
on the same qubits in parallel.
• As a result, the more complex gate operation it is applied, the
more time one may save, especially for those multiple-qubit
gates that may not be simply decomposed by using the
traditional method.
Time evolution of the near timeoptimal CNOT gate with input states
|00> and |01>
Simulations performed using the full Hamiltonian
Time evolution of the near timeoptimal CNOT gate with input states of
|10> and |11>
Simulations performed using the full Hamiltonian
Summary of the CNOT gate fidelities
• After about 60 (250) times of CNOT operations, the error sums up
to 1.03x10-4 (0.79x10-4) and one has to reinitialize the nuclear spin
state in order to maintain fault-tolerant quantm computation.
• In the paper by A. Sporl et al., Phys. Rev. A 75 012302 (2007),
TCNOT=55ps is about 5 times faster than the pioneering
experiment of coupled superconducting Josepson charge qubits
[canonical decomposition]; with an error of 10-9 using the effective
Hamiltonian, (when including higher charge states, the leakage is
less than 1%).
Control voltage fluctuations (noise)
• Since we apply voltages on the A and J gates to control the
strengths of hyperfine interaction and exchange interaction,
there might be noise induced from the (thermal) fluctuations
in the control circuits, which then cause the uncertainties of
the control parameters and decrease the fidelity of a specific
operation.
• We model the noise on the control parameters A1, A2 and J as
independent white noise with Hamiltonian
H N   A1 (t )σ1e  σ1n   A 2 (t )σ 2 e  σ 2 n   J 3 (t )σ1e  σ 2 e ,
i (t )  0;
i (t ) j (t ')   ij (t  t '),
 2A and  2J are the spectral density of the noise signal,
which have the dimension of (energy) 2 /Hz.
Contour plot of logarithmic errors
We simulate the optimal
control sequence in the
presence of the white
noise through the effective
master equation approach.
• To satisfy the error threshold 10-4(10-3) of FTQC, the spectral densities,
(J/h)2 and (A/h)2 have to be smaller than 6.2Hz and 13Hz (63Hz and
125Hz), respectively.
• This precision of control should be achievable with modern electronic
voltage controller devices as the spectral density of energy fluctuations
of the control parameters for good room temperature devices can be
estimated to be 10-4~10-2Hz..
Effect of decoherence
• The decoherence time T2 for P donor electron spin in
purified Si has been indicated experimentally to be
potentially considerably longer than 60ms at 4K.
• The error with decoherence can be estimated to be
1  Fr e
 t / T2
,
where Fr and t are the trace fidelity and operation
time of the gate, respectively.
• For this simple estimate, the error is about 2.7x10-6,
below the FTQC error threshold of 10-4 (10-3).
Conclusions
• A great advantage of the optimal control gate sequence is that
the maximum exchange interaction is about 500 times smaller
than the typical exchange interaction of J/h=10.2 GHz in the
Kane’s originalproposal and yet the CNOT gate operation time is
still 3 times faster than that in the globally controlled electron
spin scheme.
• This small exchange interaction relaxes significantly the
stringent distance constraint of two neighboring donor atoms of
10-20nm as reported in the original Kane's proposal to about
30nm. To fabricate surface gates within such a distance is within
reach of current fabrication technology.
• Each step of the control sequence is about 3.3ns which may be
achievable with modern electronics.
Conclusions
• The CNOT gate sequence we found is with high fidelity, above the
fidelity threshold required for fault-tolerant quantum computation.
• The fidelity of the gate sequence is shown, by using realistic (device)
parameters, to be robust against control voltage fluctuations,
electron spin decoherence and dipole-dipole interaction.
• The GRAPE time-optimal control approach is in a sense more like
parallel computing. The more complex gate operation it is applied,
the more time one may save, especially for those multiple-qubit
gates that may not be simply decomposed by using the traditional
method.
• The GRAPE technique may be proved useful in implementing
(complex) quantum gate operations.
• Ref: D.-B. Tsai, P.-W. Chen and H.-S. Goan, Phys. Rev. A 79, 060306
(Rapid Communication) (2009).
Silicon-based electron-mediated
nuclear spin quantum computer
B. Kane, Nature (1998)
• Exploiting the existing
strength of Si technology
• Qubits are nuclear spins of P
donors in a regular array in
pure silicon
• Low temperature:
– Effective Hamiltonian
involves only spins
– Long spin coherence and
relaxation times
• Magnetic field B to polarized
electron spins
• Control with surface gates and
NMR pulses
• Donor separation ~ 20nm
• Gate width < 10nm
Top-down approach for few qubit devices
Controlled single-ion implantation
• 14 KeV P ion beam is used to implant
P dopants to an average depth of 15nm
below the Si-SiO2
• Ion-stopping resist defines the array
sites
• Each ion entering the Si substrate
produces e-hole pair that drift in an
applied electric field
• Created single current pulse for each
ion strike is detected by on-chip single
ion detector circuit.
95% confidence in ion detection
50% confidence in each 2-donor device
Bottom-up approach for large-scale qubit arrays
350 C
Using scanning tunnelling
microscope lithography and epitaxial
silicon overgrowth to construct
devices at an atomic scale precision.
Conclusion
• The CNOT gate operation time of 100ns is 3 times faster than the
globally controlled electron spin scheme of 297ns; with an error of
1.11x10-16 using effective Hamiltonian and an error of 1.92x10-6
using the full Hamiltonian.
• One great advantage: the maximum value of the exchange
interaction is J/h=20MHz compared to the typical value of 10.2GHz
in the original Kane's proposal .
• This relaxes significantly the stringent distance constraint of two
neighboring donor atoms (with surface gates on top) from about
10-20nm as reported in the original Kane's proposal to about 30nm
which is within the reach of the current fabrication technology.
• Each step of the control sequence is about 3.3ns which may be
achievable with modern electronics.