Optimal control of the quantum gate operations for quantum computing

Hsi-Sheng Goan 管 希 聖

Department of Physics and Center for Theoretical Sciences,

National Taiwan University, Taipei, Taiwan

with Dung-Bang Tsai and Po-Wen Chen

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

• 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:

• Simulation of quantum systems: up to exponential speed-up.

( )O n

Best classical algorithm:

1024 steps

Shor’s quantum algorithm:

1010 steps

On classical THz computer:

150,000 years

On quantum THz computer:

<1 second

Peter Shor

Quantum bits

• Classical bit: 0 or 1; voltage high or low• Quantum bit (QM two-state system): • Spin states; • 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 …

| 0 and |1

and

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 consideredfor 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

– 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)]

• Donor electron-nuclear spin pairs– Digital Approach [Skinner et al., PRL (2003)]

• Donor electron charges– 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

• 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

B. Kane, Nature (1998)

Phosphorus Donor in Si

P donor behaves effectively like a hydrogen-like atom embedded in Si

P shallow donor energy levels in Si

*

**

2 ,

1 HnB n

e

eB

ma a

m

mE E

m

Silicon-based quantum computingTwo 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

– 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)]

• Donor electron-nuclear spin pairs– Digital Approach [Skinner et al., PRL (2003)]

• Donor electron charges– P/P+ charge qubit [Hollenberg et al., (2004)]

• Electron spins in silicon-based quantum dots [Friesen et al., PRB (2002)]

R. Vrijen et al, Concept device: spin-resonance transistor, Phys. Rev. A 62, 012306 (2000)

Donor electron spin in Si-Ge structure

Silicon-based electron-mediated nuclear spin quantum computer

• 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

B. Kane, Nature (1998)

Single-qubit system

2where (2 / 3) | (0) |e e n nA g g

(2)

0

2(2)

1

(2)

1

2(2)

0

/ 2 / 2

2/ 2 / 2

( / 2 / 2)

/ 2 / 2

2/ 2 / 2

( / 2 / 2)

e B n n

e B n ne B n n

e B n n

e B n ne B n n

E g B g B A

AE g B g B A

g B g B

E g B g B A

AE g B g B A

g B g B

Effective low-energy low-temperature Hamiltonian:

2 2e n

AB e e n n e n

Z ZH g B gH AB σ σ

Notations: ; ; .x y zX Y Z

geBEnergy separation:

Single-qubit system

geB

(2) (2)

0 0

222

( / 2 / 2)e Be B n n

E E E

Ag B A

g B g B

Qubit energy separation (if nuclear spins is initialized in spin-up state):

24(

(

2)

2

)

)

(

eeff z

e Be B n n

AA g B A

B

H

g g

A

Effective single-qubit Hamiltonian:

[ cos( ) sin( )]ac e e ac e ac e acH g B X t Y t

Hamiltonian in a Bac field:

Single-qubit controlHaving 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)

Laboratory frame Reference framez

x

y

ac

zB

acB

z

x

acz

B

Bg

acB

effB

Single qubit rotations

22 20

2 2

0

0

with an initial state;

( / )( ) sin ( ) ( )

2( ) ( )2

/

B ac B ac ac

B

a

z

B ac c

g

g B g BP t t

g

B

B

0ac

P(ac

4 B acg B

/ 2

[ cos( ) sin( )] / 2e ee e ac x

e

y

e

ac

B

ac

zH

g B

g

t t

B

0( ) / 2

/ 2e e ac

eac zH

g B

Rx() rotation in the global control donor e-spin QC

wher

1,

2 2e ( ) ,

e eeff z e B a

a

x

c

cH g B

A

• Set a detuned target qubit to perform a 2 rotation, and then every other spectator qubit will undergo a rotation around x-axis with an angle

2 22 2

( ) ( )e B ac

e B ac

g B

g B

• Perform an on-resonance Rx() rotation on every qubit to correct the spectator qubits’ rotations.

Two-qubit Hamiltonian

• Full Hamiltonian in the Lab. frame

• Effective e-spin Hamiltonian in the rotating frame

1 2 1 2 1 2

0

1

2

1

where ( ) ,

4 ( ) 2

(

) ,2

)

1(

2 2e e e e e e

eff z z e B a

i i ac

e Bn

c x

B n

x

e

H g

A

AA

B J

g B Ag g B

σ σ

Exchange interaction J

Strain

See also:B. Koiller, X. Hu and S. Das Sarma, PRL 88, 072903 (2002).

• L. M. Kettle, H.-S. Goan, and S. C. Smith, PRB 73, 115205 (2006).

Two-qubit control2

2 ac1

q B A Ji

H H H H H

• Two qubit Hamiltonian:

1 2J e eH J

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

|

|

1

00

0

| 00

| 01 | 0

|11

|11 |10

1

1

0 0 0

0 0 0

0 0 0

0

CNOT

0

1

1

0 1

• CNOT + single qubit rotations are universal for quantum computation.

• Any gate can be constructed using CNOT and single qubit rotations.

• Task is to demonstrate that the CNOT gate and single qubit rotations may be constructed.

• What is the CNOT (Controlled-Not) gate:

( ), CNO + ( ), ( )T X Y ZR R R

Constructing CNOT gate from the controlled Z Gate

1

0 0 0

0 0 0

0 0 0

0 10

1

1

0

1

Z

1 ( )C ( )NO T H I HZI

( ) ( )1 11

, 1 12

( )2 2 2Z X ZH R R RH

Hadamard gate:

Controlled-Z gate,

( )2

4 4

2

4

1

( ) ( ) i

I Z I Z

ZZ

i

iZ i Ze I e

Z e

I e

Controlled-Not gate:

Construction of two-qubit gates

1 2 3 4( ) ( ) X Y Zi X X i Y Y i Z ZW W We WV

( )( ) i X X Y Y Z ZU e

4i Z Z

T e

• Any two-qubit gate may be expressed in the following way:

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:

• What we want:

( ) ( )8

4

8 ( ) 8

( ) ( ) ( )8

i Z Z i X X Y Y Z ZY

i Z Z

X X YU eZ I U I e

e

Z

• Isolate the Z-Z term:

Canonical decomposition of CNOT gate for global control e-spin QC

1 2exp8 8

e eU i

σ σ

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).

CNOT gate operation time: 297ns

• 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 fault-tolerant 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 (GRAPE)

• Propagator during time step j (t=T/N)

01

( ) exp kj

m

j kk

iU t t H u H

• Propagator at final time T

1F NU U U • Performance function (fidelity)

2† where : dTr , esired op.D F DU U U

• 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.

• N. Khaneja et al., J. Magn. Reson. 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)

Nielsen et al.,Science; PRA(2006)

Trace Fidelity versus gate time

Stopping criteria of the error threshold : 10-9

30 piecewise constant steps is sufficient

†1Tr

2tr D FnF U U

Optimizer:spectral projected gradient method

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

• If x 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,

e B acx

g Bt

2

e B ac

nt

g B

4 For 100ns and 1, 56 .3. 10 Tact n B

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

1 2exp8 8

e eU i

σ σ

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).

CNOT gate operation time: 297ns

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 time-optimal CNOT gate with input states |

00> and |01>

Simulations performed using the full Hamiltonian

Time evolution of the near time-optimal 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

1 1 2 2 1 21 3

2J

2

2

2

( ) 0; ( ) ( ') ( ')

and are the

( ) ( )

spectral density of the noise signal,

which have the dimension of (en

( )

ergy) /H .

,

,

z

i

e n e n e eN A

j j

J

A

i

A

i

H t t

t t t t t

t

σ σ σ σ σ σ

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 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).

2/1 ,t TrF e

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

• 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

B. Kane, Nature (1998)

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 detection50% confidence in each 2-donor device

Bottom-up approach for large-scale qubit arrays

Using scanning tunnelling microscope lithography and epitaxial silicon overgrowth to construct devices at an atomic scale precision.

350 C

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.