Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Caltech - PowerPoint PPT Presentation

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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Caltech. Starting September 1st, 2005: Dept of Combinatorics & Optimization Institute of Quantum Computing University of Waterloo wcleung@iqc.ca wcleung@math.uwaterloo.ca

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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, Caltech


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Starting September 1st, 2005:

Dept of Combinatorics & OptimizationInstitute of Quantum Computing University of Waterloo

wcleung@iqc.ca

wcleung@math.uwaterloo.ca

wcleung@cs.caltech.edu

wcleung@caltech.edu

*


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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, Caltech


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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, CaltechGSQC-FT: 0503130 (AL)

Prior & related results:

- Raussendorf PhD thesis (2003)

- Nielsen & Dawson, 0405134

- Dawson, Haselgrove, & Nielsen (in-prep)

Tools GSQC: 0404132 (Childs, L, Nielsen)FT: 0504218 (A, Gottesman, Preskill)


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RBB 0301052

N 0402005

Graph State Quantum Computation (GSQC)

To simulate a circuit C :

- prepare graph state |gCi (e.g. using |+i, CP)

- apply single qubit measurements

(with feedforward –

meas bases can depend on prior meas outcomes)

- simulation is “element-wise”

FT Qn 1: Can we achieve FT by simulating a FT circuit ?

Physical noise ?! noise in simulated circuit

FT Qn 2: Will errors propagate via feedforward ?

If 1 error corrupts a meas outcome, can it affect

the simulation of a subsequent op ... ?


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Graph State Quantum Computation (GSQC)

FT Qn 1: Can we achieve FT by simulating a FT circuit ?

Physical noise ?! noise in simulated circuit

FT Qn 2: Will errors propagate via feedforward ?

If 1 error corrupts a meas outcome, will it affect

the simulation of a subsequent op ... ?

Here: answer the above using different tools than prior works

(by “robbing” pieces of recent results)

- Use the notion of composable simulation (CLN04)

(that shows how&why GSQC works) to show why errors

don’t propagate and to relate to circuit noise models

- FT & threshold then follows by AGP05

R03, NC04, DHN

What else do we learn?


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x1

x2

x3

General quantum circuit C

0/1

|0i

|0i

|0i

U3

U5

U1

0/1

U4

U2

0/1

p(x1,x2,x3)

Abstract representation of state transform of the logical space.

Need not correspond to physical implementation.


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x1

x2

x3

General quantum circuit C

0/1

|0i

|0i

|0i

U3

U5

U1

0/1

U4

U2

0/1

p(x1,x2,x3)

Abstract representation of state transform of the logical space.

Need not correspond to physical implementation.

e.g. fault-tolerant encoded logical operations.


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x2

x3

x1

General quantum circuit C

k1

k7

0/1

|0i

|0i

|0i

k5

U3

U5

U1

k2

k3

0/1

k8

U4

k6

U2

0/1

k4

p(x1,x2,x3)

Abstract representation of state transform of the logical space.

Need not correspond to physical implementation.

e.g. in GSQC, the simulations use measurements & teleportation,

with extra, uncontrolled, known Pauli operations.


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x1

x2

x3

Composable simulation

S(U3)

S(|0i)

S(|0i)

S(|0i)

0/1

S(U5)

S(U1)

0/1

S(U4)

S(U2)

0/1

p(x1,x2,x3)

It is lazy (but understandly) to hope that simulating circuit

elements one by one automatically simulates the entire circuit.

The simulation O ! S(O) is composable

if S(O2) ± S(O1) = S(O2± O1)


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in

U

U(in)

ein

eout

S(U)

eout

ein

in

x »p(x)

|ai

S( )

Pein(in)

(Peout± U)(in)

S( )

in

x »p(x)

Peout(|ai)

Composable simulation in GSQC

  • The simulation O ! S(O) is composable

  • if S(O2) ± S(O1) = S(O2± O1)

  • For GSQC :

  • If, 8indemand: 8in 8ein 9 eout s.t.

  • s(u)

  • i.e. ein­ Pein(in)!eoutpr(eout) eout­(Peout±U)(in)

  • If, 8indemand: 8in8 ein

  • For |ai  demand:

Explain symbols

& interpretation

Pein( )

9 eout s.t.

eout pr(eout) eout­Peout (|ai)


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x1

x2

x3

Composable simulation in GSQC

S(U3)

S(|0i)

S(|0i)

S(|0i)

S(U5)

S(U1)

S(U4)

S(U2)

1 2 6

State at various stages of the simulation:

1: eout11,eout12,eout13 Pr(eout11 eout12 eout13)

eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i)

2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13) Pr(eout21eout22|eout11eout12)

eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i)

p(x1,x2,x3)


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x1

x2

x3

Composable simulation in GSQC

S(U3)

S(|0i)

S(|0i)

S(|0i)

S(U5)

S(U1)

S(U4)

S(U2)

1 2 6

State at various stages of the simulation:

1: eout11,eout12,eout13 Pr(eout11 eout12 eout13)

eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i)

2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22)

eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i)

....

6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63)

eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i)

p(x1,x2,x3)

S(meas)


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Composable simulation in GSQC

The state is a mixture over possible

each will give the correct output distribution

“Pauli-frame history” of the simulation

State at various stages of the simulation:

1: eout11,eout12,eout13 Pr(eout11 eout12 eout13)

eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i)

2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22)

eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i)

....

6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63)

eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i)

p(x1,x2,x3)


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Composable simulation in GSQC

“Pauli-frame history”

Info of the Pauli-frame at any time:

“classical part” of the simulation.

State at various stages of the simulation:

1: eout11,eout12,eout13 Pr(eout11 eout12 eout13)

eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i)

2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22)

eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i)

....

6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63)

eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i)

p(x1,x2,x3)


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Composable simulation in GSQC

“Pauli-frame history”

“quantum part”

“classical part”

State at various stages of the simulation:

1: eout11,eout12,eout13 Pr(eout11 eout12 eout13)

eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i)

2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22)

eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i)

....

6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63)

eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i)

p(x1,x2,x3)


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S(|0i):

S(|+i):

a

a

0

0

0

b

Mz

b

Mx

0

XaZ0|0i

X0Zb|+i

S( Mx ):

S( Mz ):

a

b

a

b

c

XaZb|i

Mx

c

a©c

XaZb|i

Mz

a©c

Composable simulation

for state prep & meas


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S(H):

£

a

b

b©d

a

£

d

XaZb|i

Mx

|+i

Xb©d Za H|i

a1

b1

S(CP):

a1

b1© a2

S(Z):

a2

b2

a

b

a2

b2© a1

a

b©c

c

XaZb|i

Z(-1)a

Mx

Xa1Zb1 ­

Xa2Zb2 |i

Xa1Z b1©a2­

Xa2Z b2©a1(CP|i)

|+i

Xa Zb©c Z|i

H

Composable simulation

for a universal gate set

Aside:

1. Recipe for GSQC

Quantum part:

- |+i

- CP

- matching in/out

- meas

2. Also for 1-way QC via

- deletion principle

- optional CP


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a’

a

b

a©a’

b©b’

b’

XaZb|i

Xa©a’ Zb©b’ |i

Composable simulation

for Pauli’s

S(Xa’ Zb’) (or S(I)):

Note:

Known Pauli operations : shifts in the classical parts.

Unknown Pauli errors : errors in the classical parts.


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~

O

~

O

UF

O

»

»

UF

O

env

env

What about noise?

0. Add each |+i to the graph state only slightly before it’s

being measured

~

1. Model noisy elementary operations O

 storage, gate, or state prep  meas

where UF = I ­ A0 + i Pi­ Ai on sys ­ env

Pi = nontrivial Pauli’s


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S(Z):

a

b

a

b©c

c

Z(-1)a

XaZb|i

Mx

|+i

Xa Zb©c Z|i

H

How physical errors affect each simulation?

What about noise?

2. Noisy simulation: interperse UFs between ideal operations

e.g.

: where UFs act

  • Expand each UF as Isys­ A0 env + i Pi sys­ Ai env

  • Commute each summand towards end of simulation

  • - “Joint I term” : ideal simulation

  • - Else:(1) classical part may flip

  • (2) quantum part may suffer unknown Pauli error

  • But they’re equivalent !!

  • i.e., classical part is always correct, & fault do not propagate

  • – localization


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S(Z):

a

b

a

b©c

c

Z(-1)a

XaZb|i

Mx

|+i

Xa Zb©c Z|i

H

How physical errors affect each simulation?

What about noise?

2. Noisy simulation: interperse UFs between ideal operations

e.g.

Simulation output:

Noiseless term:

eineout pr(eineout) eout­ (Peout ± U)(in)

Noisy term:

eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in)

= eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPe’i out PiPeout± U(in)


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S(Z):

a

b

a

b©c

c

Z(-1)a

XaZb|i

Mx

|+i

Xa Zb©c Z|i

H

How physical errors affect each simulation?

What about noise?

2. Noisy simulation: interperse UFs between ideal operations

e.g.

Simulation output:

Noiseless term:

eineout pr(eineout) eout­ (Peout ± U)(in)

Noisy term:

eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in)

= eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPe’i out PiPeout± U(in)


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S(Z):

a

b

a

b©c

c

Z(-1)a

XaZb|i

Mx

|+i

Xa Zb©c Z|i

H

How physical errors affect each simulation?

What about noise?

2. Noisy simulation: interperse UFs between ideal operations

e.g.

Simulation output:

Noiseless term:

eineout pr(eineout) eout­ (Peout ± U)(in)

Noisy term:

eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in)

= eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPi’± U(in)

subsequent simulations

unaffected (composability)

U’F(S)


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How physical errors affect each simulation?

What about noise?

3. Elementary operations in GSQC:

- prepare |+i

- CP each in a unique simulation

- single qubit meas

- storage} WLOG error acts in later sim

Together with localization -- noise affecting each elementary

operation affects only 1 simulated operation

Simulation output:

Noiseless term:

eineout pr(eineout) eout­ (Peout ± U)(in)

Noisy term:

eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in)

= eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPi’± U(in)

subsequent simulations

unaffected (composability)

U’F(S)


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Thus, in GSQC:

(1) by interpreting errors in the classical part as unknown

Pauli errors + composability, faults of elementary ops in

1 simulation give combined fault the corr simulated op only

(2) the joint no-fault term in 1 simulation gives an ideal

simulated op

E.g. 1.For independent stochastic noise, simulated operation O

has fault with prob at most Tot(S(O)) = sum of prob of

faults of all elementary ops in S(O).

To reliably simulate circuit C , find FT circuit C ’ that

handles error Tot(S(O)) , then simulate C ’ with GSQC .

Threshold for GSQC ¸ circuit / maxO(#ops in S(O)).


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Thus, in GSQC:

(1) by interpreting errors in the classical part as unknown

Pauli errors + composability, faults of elementary ops in

1 simulation give combined fault the corr simulated op only

(2) the joint no-fault term in 1 simulation gives an ideal

simulated op

E.g. 2.For general noise, consider output of 1 simulation:

eineout pr(eineout) iA’i [·]­e’i out­Pe’i outP’i± U(in)

U’F(S)


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Thus, in GSQC:

(1) by interpreting errors in the classical part as unknown

Pauli errors + composability, faults of elementary ops in

1 simulation give combined fault the corr simulated op only

(2) the joint no-fault term in 1 simulation gives an ideal

simulated op

E.g. 2.For general noise, consider output of 1 simulation:

eineout pr(eineout) i A’i [·]­e’i out­Pe’i outP’i± U(in)

.... and compose many simulations:

ein...eout pr(ein ...eout)

ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in)

For each Pauli-frame history, the env-sys state is in the form

treated in circuit model (e.g. AGP05). Sum of amp of fault

terms upper bounds noisy amp of simulated op.

U’F(S)


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Meditating on this ....

ein...eout pr(ein ...eout)

ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in)

GSQC is just QC with constantly changing Pauli-frame.

These eout are like the perfect part of error syndromes.

Qns: NonDeterministic operations?

Insights on using error syndromes or

Pauli-encryption/randomization to tame noise?

e.g., to reduce the extent of non-Markovian-ness ?

Composability of FT analysis?


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Thus, in GSQC:

(1) by interpreting errors in the classical part as unknown

Pauli errors + composability, faults of elementary ops in

1 simulation give combined fault the corr simulated op only

(2) the joint no-fault term in 1 simulation gives an ideal

simulated op

E.g. 2.For general noise, consider output of 1 simulation:

eineout pr(eineout) i A’i [·]­e’i out­Pe’i outP’i± U(in)

.... and compose many simulations:

ein...eout pr(ein ...eout)

ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in)

For each Pauli-frame history, the env-sys state is in the form

treated in circuit model (e.g. AGP05). Sum of amp of fault

terms upper bounds noisy amp of simulated op.

U’F(S)


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Composable simulation for a universal gate set

S(H):

£

a

b

b©d

a

£

d

XaZb|i

Mx

|+i

Xb©d Za H|i

S(Z):

a

b

a

b©c

c

XaZb|i

Z(-1)a

Mx

|+i

Xa Zb©c Z|i

H


Slide33 l.jpg

Composable simulation for a universal gate set

S(Xa’ Zb’):

a’

a

b

a©a’

b©b’

b’

XaZb|i

Xa©a’ Zb©b’ |i

a1

b1

S(CP):

a1

b1© a2

a2

b2

a2

b2© a1

Xa1Zb1 ­

Xa2Zb2 |i

Xa1Z b1©a2­

Xa2Z b2©a1(CP|i)


Slide34 l.jpg

|+i

d1

|+i

d2

|+i

d1

|+i

|+i

Y

Z

|+i

d2

Simulating an optional C-Z, summary:

simulates

To do the C-Z:

To skip the C-Z:

Do:

Skip:

also simulates

up to Z-rotations


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Universal Initial state

3 qubits, 8 cycles

Starting from the cluster state

measurement in Z basis


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