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
Threshold theorem for quantum supremacy
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
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
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
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
DQC1(one-clean qubit model)
I/2n U}|0i HH
Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF et al, ‘16
NMR spin ensemble
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
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
DQC1(one-clean qubit model)
I/2n U}|0i HH
Knill-Laflamme ‘98Morimae-KF-Fitzsimons ’14KF et al, ‘16
NMR spin ensemble
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
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)
Noisy quantum circuits approaching fault-tolerance threshold
Delft (QuTech): Riste et al., Nat. Comm. 6 6983 (2015)
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)
Noisy quantum circuits approaching fault-tolerance threshold
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%
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)
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
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.
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)
universal QC(standard) noise threshold
noisyclean
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.
classically simulatable?
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)
universal QC(standard) noise threshold
noisyclean
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.
classically simulatable?
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)
universal QC(standard) noise threshold
classically simulatableby GK theorem
phenomenological noise 14.6% magic
state
convex mixture ofPauli basis states x
y
distillability of of magic state
noisyclean
Outline• Motivations
• Hardness proof by postselection
• Threshold theorem for quantum supremacy
• Applications: 3D topological cluster computation & 2D surface code
• 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!
Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).
classical+ postselection quantum+ postselection
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!
Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).
postBPP
classical+ postselection quantum+ postselection
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!
PP=postBQP[Aaronson05]
Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).
postBPP
classical+ postselection quantum+ postselection
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!
PP=postBQP[Aaronson05]
Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).
postBPP
classical+ postselection quantum+ postselection
≠
unless the PH collapses to the 3rd level.
P
PNP
PNP NP …
PHPPP
PpostBPP
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!
PP=postBQP[Aaronson05]
Aaronson, Proc. of the Royal Society A: Math., Phys. and Eng. Sci. 461, 3473 (2005).
postBPP
classical+ postselection quantum+ postselection
≠
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]
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]
・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]
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)
Small amount of noise can easily break these conditions.
[Bremner-Jozsa-Shepherd, ’11]
・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]
Outline• Motivations
• Hardness proof by postselection
• Threshold theorem for quantum supremacy
• Applications: 3D topological cluster computation & 2D surface code
• 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
Main idea: simulation of fault-tolerant quantum computation under postselection
Cw
… …
xy
|0i⌦n
p(x, y)
logical output
…
xy
fault-tolerantversionwith noisycircuits
z
p(x, y, z)
classical processing
arXiv:1610.03632
Main idea: simulation of fault-tolerant quantum computation under postselection
Cw
… …
xy
|0i⌦n
p(x, y)
logical output
…
xy
fault-tolerantversionwith noisycircuits
z
p(x, y, z)
classical processing
error syndrome
arXiv:1610.03632
Main idea: simulation of fault-tolerant quantum computation under postselection
Cw
… …
xy
|0i⌦n
p(x, y)
logical output
…
xy
fault-tolerantversionwith noisycircuits
z
p(x, y, z)
classical processing
error syndrome
p(x, y, |z = 0)under the condition ofnull syndrome
arXiv:1610.03632
Threshold theorem for quantum supremacy
・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
Threshold theorem for quantum supremacy
・Part2: The exponentially small additive error
is achievable by quantum error correction under postselection.
|p(x, y)� p(x, y|z = 0)| < e
�
・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
・Part1: An exponentially small additive error is enough.
・Part2: The exponentially small additive error
is achievable by quantum error correction under postselection.
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
Threshold theorem for quantum supremacy
p(x, y)
|p(x, y)� p(x, y|z = 0)| < e
�
|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
・Part1: An exponentially small additive error is enough.
・Part2: The exponentially small additive error
is achievable by quantum error correction under postselection.
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
Threshold theorem for quantum supremacy
p(x, y)
|p(x, y)� p(x, y|z = 0)| < e
�
|p(x, y)� p(x, y|z = 0)| < e
�
arXiv:1610.03632
|0i|0i
|0i
……
initial state
P
x,y
= |x, yihx, y|⌦ I
⌦n�2
z Qz = |zihz|error syndrome
… …
…
U1
U2 Uk
N1
N2 Nk
Unoisy =Y
k
(NkUk)
fault-tolerant circuitincluding classical processing
arXiv:1610.03632
Part2: error reduction under postselection (sketch)
|0i|0i
|0i
……
initial state
P
x,y
= |x, yihx, y|⌦ I
⌦n�2
z Qz = |zihz|error syndrome
… …
…
U1
U2 Uk
N1
N2 Nk
Unoisy =Y
k
(NkUk)
fault-tolerant circuitincluding classical processing
arXiv:1610.03632
Part2: error reduction under postselection (sketch)
|0i|0i
|0i
……
initial state
P
x,y
= |x, yihx, y|⌦ I
⌦n�2
z Qz = |zihz|error syndrome
… …
…
U1
U2 Uk
N1
N2 Nk
ideal operationstochastic Nk = (1� ✏k)I + Eknoise CP map
Unoisy =Y
k
(NkUk)
fault-tolerant circuitincluding classical processing
arXiv:1610.03632
Part2: error reduction under postselection (sketch)
|0i|0i
|0i
……
initial state
P
x,y
= |x, yihx, y|⌦ I
⌦n�2
z Qz = |zihz|error syndrome
… …
…
U1
U2 Uk
N1
N2 Nk
ideal operationstochastic Nk = (1� ✏k)I + Eknoise CP map
Unoisy =Y
k
(NkUk)
fault-tolerant circuitincluding classical processing
arXiv:1610.03632
p(x, y, z)
= Tr[Px,y
Q
z
Unoisy(⇢ini
)]
Part2: error reduction under postselection (sketch)
Part2: error reduction under postselection (sketch)
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
Part2: error reduction under postselection (sketch)
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]
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
Part2: error reduction under postselection (sketch)
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]
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
Outline• Motivations
• Hardness proof by postselection
• Threshold theorem for quantum supremacy
• Applications: 3D topological cluster computation & 2D surface code
• 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
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)
✏
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
• Faulty part comes from either clifford operations or magic state distillation ⇢faulty = ⇢cl + ⇢magic
• Noise: independent single-qubit dephasing w prob. (phenomenological noise model)
✏
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
• 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)
• Noise: independent single-qubit dephasing w prob. (phenomenological noise model)
✏
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
• 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)
• Noise: independent single-qubit dephasing w prob. (phenomenological noise model)
✏
magic state distillation ✏magic = 0.146(Bravyi-Kitaev ’05; Reichardt ‘06)
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 done with an arbitrary accuracy poly(logarithmic) overhead.
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)
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
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.
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
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.
ν: independent error rateµ: correlated error rate⌫ = 54p/15, µ = 6p/5
• 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
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.
ν: independent error rateµ: correlated error rate⌫ = 54p/15, µ = 6p/5
• 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