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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Caltec

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

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

Dept of Combinatorics & Optimization Institute of Quantum Computing University of Waterloo

wcleung@iqc.ca

wcleung@math.uwaterloo.ca

wcleung@cs.caltech.edu

wcleung@caltech.edu

*

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)

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 ... ?

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?

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.

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.

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.

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)

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, 8in demand: 8in 8ein 9 eout s.t.
- s(u)
- i.e. ein Pein(in)!eoutpr(eout) eout(Peout±U)(in)
- If, 8in demand: 8in8 ein
- For |ai demand:

Explain symbols

& interpretation

Pein( )

9 eout s.t.

eout pr(eout) eoutPeout (|ai)

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)

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)

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)

“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)

“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)

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

£

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

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.

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

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

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)

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)

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)

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)

(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)).

(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)

(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)

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?

(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)

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

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)

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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