threshold theorem for quantum supremacy - microsoft.com · 1/20/2017 · noisy quantum circuits...

Threshold theorem for quantum supremacy

2017.1.16-20 QIP2017 Seattle, USA

Keisuke FujiiPhoton Science Center, The University of Tokyo

/PRESTO, JST

arXiv:1610.03632


2017.1.16-20 QIP2017 Seattle, USA

Keisuke FujiiPhoton Science Center, The University of Tokyo

/PRESTO, JST

(ascendancy)arXiv:1610.03632

Outline• Motivations

• Hardness proof by postselection

• Threshold theorem for quantum supremacy

• Applications: 3D topological cluster computation & 2D surface code

• Summary

How can we best achieve quantum supremacy with the relatively small systems that may be experimentally accessible fairly soon, systems with of order 100 qubits?

The 25th Solvay Conference on Physics19-22 October 2011; arXiv:1203.5813

“QUANTUM COMPUTING AND THE ENTANGLEMENT FRONTIER” by J Preskill

and Talk by S. Boixo et al

Quantum supremacy with near-term quantum devices

https://www.technologyreview.com/s/601668/google-reports-progress-on-a-shortcut-to-quantum-supremacy/

Intermediate models for non-universal quantum computation


Boson Sampling

Experimental demonstrationsJ. B. Spring et al. Science 339, 798 (2013)M. A. Broome, Science 339, 794 (2013)M. Tillmann et al., Nature Photo. 7, 540 (2013)A. Crespi et al., Nature Photo. 7, 545 (2013)N. Spagnolo et al., Nature Photo. 8, 615 (2014)J. Carolan et al., Science 349, 711 (2015)

Universal linear opticsScience (2015)

Linear optical quantum computation

Aaronson-Arkhipov ‘13


Boson Sampling





IQP(commuting circuits)

Bremner-Jozsa-Shepherd ‘11

Ising type interaction

|+i|+i|+i|+i

T

T

|+i T

… …

Bremner-Montanaro-Shepherd ‘15Gao-Wang-Duan ‘15

KF-Morimae ‘13

Farhi-Harrow ‘16


Boson Sampling





DQC1(one-clean qubit model)

I/2n U}|0i HH

Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF et al, ‘16

NMR spin ensemble




|+i|+i|+i|+i

T

T

|+i T

… …


KF-Morimae ‘13

Farhi-Harrow ‘16


Boson Sampling





DQC1(one-clean qubit model)

I/2n U}|0i HH

Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF et al, ‘16

NMR spin ensemble




|+i|+i|+i|+i

T

T

|+i T

… …


KF-Morimae ‘13

Farhi-Harrow ‘16

The purpose of this study:→ universal but (very) noisy quantum circuits

Noisy quantum circuits approaching fault-tolerance threshold


IBM: Chow et al., Nat. Comm. 5 4015 (2015)C orcoles et al., Nat. Comm. 6, 6979 (2015)Gambetta et al., npj quant. info. 3, 2 (2017)


Delft (QuTech): Riste et al., Nat. Comm. 6 6983 (2015)



Delft (QuTech): Riste et al., Nat. Comm. 6 6983 (2015)

UCSB(Martinis)+Google: Kelly et al., Nature 519, 66 (2015)Barends et al., Nature 508, 500 (2014)[fidelities] single-qubit gate: 99.92% two-qubit gate: 99.4% measurement: 99%


Noisy quantum circuits above standard noise threshold

Threshold theorem: if the noise strength is smaller than a certain constant threshold value, quantum computation can be performed with an arbitrary accuracy.

universal QC(standard) noise threshold

noisyclean



phenomenological noise 2.9-3.3%circuit-based noise 0.75%

R Raussendorf, J Harrington and K GoyalNew Journal of Physics 9 (2007) 199Ann. Phys. 321 2242 (2006)


noisyclean



classically simulatable?




noisyclean



classically simulatable?




classically simulatableby GK theorem

phenomenological noise 14.6% magic

state

convex mixture ofPauli basis states x

y

distillability of of magic state

noisyclean





• Summary

Postselected computation

not zero but can be exponentially small

x

y=1or

p(x|y = 1) = p(x, y)/p(y = 1)

yes:

no: p(x = 0|y = 1) � 2/3

p(x = 1|y = 1) � 2/3

while(y==1)

= solving a decision problem by using conditional probability distribution.

Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).

Hardness proof via postBQP = PP

Bremner-Jozsa-Shepherd, Proc. Royal Soc. A: Math. Phys. and Eng. Sic. 467, 2126 (2011)

A (fictitious) ability to postselect a possibly exponentially rare events allows us to distinguish quantum and classical tasks!


classical+ postselection quantum+ postselection





postBPP





PP=postBQP[Aaronson05]


postBPP







postBPP


≠

unless the PH collapses to the 3rd level.

P

PNP

PNP NP …

PHPPP

PpostBPP






postBPP


≠

unless the PH collapses to the 3rd level.

P

PNP

PNP NP …

PHPPP

PpostBPP

If your system is potentially as powerful as postBQP under postselection, then its classical simulation is hard unless the PH collapses!

Difficulty of quantum supremacy with noisy sampling

1

c

p

ideal(x) < p

samp(x) < cp

ideal(x) (c > 1)

・multiplicative error (or exponentially small additive error)

[Bremner-Jozsa-Shepherd, ’11]


1

c

p

ideal(x) < p

samp(x) < cp

ideal(x) (c > 1)



・constant additive error with l1-norm

kpsamp(x)� p

ideal(x)k1 =X

x

|psamp(x)� p

ideal(x)| < c

[Aaronson-Arkhipov, ’11, Bremner-Montanaro-Shepherd ‘16]


1

c

p

ideal(x) < p

samp(x) < cp

ideal(x) (c > 1)


Small amount of noise can easily break these conditions.


・constant additive error with l1-norm

kpsamp(x)� p

ideal(x)k1 =X

x

|psamp(x)� p

ideal(x)| < c

[Aaronson-Arkhipov, ’11, Bremner-Montanaro-Shepherd ‘16]





• Summary

Main idea: simulation of fault-tolerant quantum computation under postselection

Cw

… …

xy

|0i⌦n

p(x, y)

…

xy

fault-tolerantversionwith noisycircuits

z

p(x, y, z)

classical processing

arXiv:1610.03632


Cw

… …

xy

|0i⌦n

p(x, y)

logical output

…

xy


z

p(x, y, z)


arXiv:1610.03632


Cw

… …

xy

|0i⌦n

p(x, y)

logical output

…

xy


z

p(x, y, z)


error syndrome

arXiv:1610.03632


Cw

… …

xy

|0i⌦n

p(x, y)

logical output

…

xy


z

p(x, y, z)


error syndrome

p(x, y, |z = 0)under the condition ofnull syndrome

arXiv:1610.03632


・Part1: An exponentially small additive error is enough.

Cw

… …

xy

error syndrome…

xyfault-

tolerantversion

z

p(x, y, z)

where the overhead is polynomial in . Then, classical simulation of with a multiplicative error is hard.

|0i⌦n

(n,)p(x, y, z) 1 < c <

p2

p(x, y)

|p(x, y)� p(x, y|z = 0)| < e

�

arXiv:1610.03632

Cw

… …

xy

error syndrome…

xyfault-

tolerantversion

z

p(x, y, z)

|0i⌦n

Part1: an exponential small additive error is enough

Solve a PP-complete problem (MAJSAT) using as in [Aaronson05]

→

p(x, y)

p(x|y)p(y = 0) > 2�6n�4probability for postselection:

arXiv:1610.03632

Cw

… …

xy

error syndrome…

xyfault-

tolerantversion

z

p(x, y, z)

|0i⌦n

Part1: an exponential small additive error is enough

Solve a PP-complete problem (MAJSAT) using as in [Aaronson05]

→

p(x, y)

p(x|y)p(y = 0) > 2�6n�4probability for postselection:

Therefore, if with = poly(n)

then we have

p(x, y, z)→　　　　　　can solve the PP-complete problem under postselection.

|p(x, y)� p(x, y|z = 0)| < e

�

|p(x|y = 0)� p(x|y = 0, z = 0)| < 1/2

arXiv:1610.03632

|0i|0i

|0i

……

initial state

P

x,y

= |x, yihx, y|⌦ I

⌦n�2


… …

…

U1

U2 Uk

N1

N2 Nk

ideal operationstochastic Nk = (1� ✏k)I + Eknoise CP map

Unoisy =Y

k

(NkUk)


arXiv:1610.03632

p(x, y, z)

= Tr[Px,y

Q

z

Unoisy(⇢ini

)]



Unoisy(⇢ini

) = ⇢sparse

+ ⇢faulty

Nk = (1� ✏k)I + EkUsing , we decompose into

such that .

Unoisy

p(x, y) / Tr[Px,y

Q

z=0⇢sparse]

arXiv:1610.03632


Unoisy(⇢ini

) = ⇢sparse

+ ⇢faulty


such that .

Unoisy

p(x, y) / Tr[Px,y

Q

z=0⇢sparse]

Then we can show thatkp(x, y)� p(x, y|z = 0)k1 < 2k⇢faultyk1/qz=0

qz=0

⌘ Tr[Qz=0

Unoisy(⇢ini

)]where .

< 2X

r>d

C(r)

✓✏

1� ✏

◆r

✏ ⌘ max

k✏k( )

(prob. of null syndrome measurement)

�arXiv:1610.03632


Unoisy(⇢ini

) = ⇢sparse

+ ⇢faulty


such that .

Unoisy

p(x, y) / Tr[Px,y

Q

z=0⇢sparse]

Then we can show thatkp(x, y)� p(x, y|z = 0)k1 < 2k⇢faultyk1/qz=0

qz=0

⌘ Tr[Qz=0

Unoisy(⇢ini

)]where .

< 2X

r>d

C(r)

✓✏

1� ✏

◆r

✏ ⌘ max

k✏k( )

(prob. of null syndrome measurement)

�

There is a constant threshold εth below whichthe output from the noisy quantum circuits cannot be simulated efficiently on a classical computer unless the PH collapses to the 3rd level.

p(x, y, z) = Tr[Px,y

Q

z

Unoisy(⇢ini

)]

arXiv:1610.03632





• Summary

Topological MBQC on a 3D cluster state

• MBQC on a graph state of degree log(n)(corresponds to commuting circuits of depth log(n))

see also KF-Tamate ‘16




• Noise: independent single-qubit dephasing w prob. (phenomenological noise model)

✏




• Faulty part comes from either clifford operations or magic state distillation ⇢faulty = ⇢cl + ⇢magic


✏





kp(x, y)� p(x, y|z = 0)k1 <

12

5

poly(n)

✓5✏

1� ✏

◆d

✏cl = 0.167Clifford operations

(counting # of self-avoiding walks: Dennis et al ‘02)


✏





kp(x, y)� p(x, y|z = 0)k1 <

12

5

poly(n)

✓5✏

1� ✏

◆d

✏cl = 0.167Clifford operations

(counting # of self-avoiding walks: Dennis et al ‘02)


✏

magic state distillation ✏magic = 0.146(Bravyi-Kitaev ’05; Reichardt ‘06)


Threshold theorem: if the noise strength is smaller than a certain constant threshold value, quantum computation can be done with an arbitrary accuracy poly(logarithmic) overhead.



universal QCnoise threshold

classically simulatableby GK theorem

phenomenological noise 14.6% magic

state

convex mixture ofPauli basis states x

y

distillability of of magic state

noisyclean classical simulation is hard!

Circuit-based noise modelwith 2D surface code

|+i

|+i

¥H

H

H

H

I

I

I

I

• 2D nearest-neighbor gates on a square grid• circuit-based depolarizing noise model:

prep., meas., 1- and 2-qubit gates with probability p.

(1,3,5,7) 2

3

45

(2,4,6,8)

6

78

9

(1,6)

(5,10)

arXiv:1610.03632


|+i

|+i

¥H

H

H

H

I

I

I

I



ν: independent error rateµ: correlated error rate⌫ = 54p/15, µ = 6p/5

(1,3,5,7) 2

3

45

(2,4,6,8)

6

78

9

(1,6)

(5,10)

arXiv:1610.03632


|+i

|+i

¥H

H

H

H

I

I

I

I




• threshold value: p=2.84% (distillability of magic state)

(1,3,5,7) 2

3

45

(2,4,6,8)

6

78

9

(1,6)

(5,10)

arXiv:1610.03632


|+i

|+i

¥H

H

H

H

I

I

I

I




• threshold value: p=2.84% (distillability of magic state)

• higher than the standard threshold 0.75%

(1,3,5,7) 2

3

45

(2,4,6,8)

6

78

9

(1,6)

(5,10)

arXiv:1610.03632

Summary• Sampling with noisy quantum circuits can exhibitquantum supremacy.

• The threshold for supremacy is much higher than that for universal fault-tolerant quantum computation.

• The threshold is determined purely by distillability of the magic state (in a phenomenological model it sharply separate classically simulatable and not-simulatable regions).

• Can we directly verify or identify quantum supremacy of the near-term noisy quantum devices in a pre-fault-tolerant region?

Thank you for your attention!arXiv:1610.03632

threshold theorem for quantum supremacy - microsoft.com · 1/20/2017 · noisy quantum circuits...

Documents