Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase...
Transcript of Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase...
![Page 1: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/1.jpg)
Phase transitions in the complexity of simulating random
shallow quantum circuits
John Napp, Rolando La Placa, Alex Dalzell, Fernando Brandão, Aram HarrowSimons workshop
May 6, 2020
![Page 2: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/2.jpg)
Complexity from entanglementOriginal motivation for quantum computing [Feynman ‘82]
Nature isn't classical, dammit, and if you want to make a simulation of
Nature, you'd better make it quantum mechanical, and by golly
it's a wonderful problem, because it doesn't look so easy.
This talk: do typical quantum dynamics achieve this?
N systems in product state à O(N) degrees of freedomN entangled systems à exp(N) degrees of freedomDescribes cost of simulating dynamics or even describing a state.
![Page 3: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/3.jpg)
easier quantum simulations
1. solve trivial special case(e.g. non-interacting theory)
2. treat corrections to theoryas perturbations
![Page 4: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/4.jpg)
easier quantum simulationLightly entangling dynamicsproduct states + non-interacting gates are easy.Cost grows exponentially with # of entangling gates.
Stabilizer circuitsPoly-time simulation of stabilizer circuits,growing exponentially with # of non-stabilizer gates.
Likewise for matchgates / non-interacting fermions.
Ground states of 1-D systemsEffort grows exponentially with correlation length.
![Page 5: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/5.jpg)
quantum circuits
+ Complexity interpolates between linear and exponential.- Treating all gates as “potentially entangling” is too pessimistic.
T gates
n qu
bits
Classical simulation possible in time O(T)⋅exp(k), where• k = treewidth [Markov-Shi ‘05]• k = max # of gates crossing any single qubit
[Yoran-Short ’06, Jozsa ‘06]
![Page 6: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/6.jpg)
noisy dynamics?Time evolution of quantum systems
d⇢
dt= �i(H⇢� ⇢H) + noise terms that are linear in ⇢
noise per gate
0 1idealQC
10-2-ishFTQCpossible
≈0.3 classicalsimulationpossible? ?
conjectured to exhibit phase transition(possibly with intermediate phases)
![Page 7: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/7.jpg)
phase transitions?
Complexity smoothly increases with-entanglement-correlation length-# of non-stabilizer gates
Complexity jumps discontinuously with-noise rate
Today: what about circuit depth?
![Page 8: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/8.jpg)
task chosen to favor quantum computers and clear comparison
![Page 9: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/9.jpg)
quantum circuits
U9U7
U6
U5
U4
U2
U1
U8 U3
depth T=7
N=5
qubit
s
|0⟩|0⟩|0⟩|0⟩|0⟩
p(z) = p(z1, z2, z3, z4, z5) =|⟨z1z2z3z4z5| U9U8U7U6U5U4U3U2U1 |00000⟩|2
Other parameters: connectivity, # of gates, fidelity.
![Page 10: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/10.jpg)
random circuit sampling
Photo credit: Google Quantum AI Lab
Conjecture:Output distributionp(z) is hard tosample from onclassical computer.
Google used N=53 qubits in 2D geometry with T=20.
Conjecture: T≥ ! à classical simulation time exp(N).[Aaronson, Bremner, Jozsa, Montanaro, Shepherd, …]
![Page 11: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/11.jpg)
low-depth circuits
Photo credit:Google Quantum AI Lab
Google proposal is ! × ! grid for depth #~ !.
[Terhal-DiVincenzo ’04] showed worst-casehardness of simulation as soon as T≥3. (T=2 is easy.)
• prepare L x W cluster state in O(1) depth• single-qubit measurements simulate depth-W circuit on line of L qubits• implies classical hardness is ≥ exp(min(L,W)).• tensor contraction achieves this
L
W
How low can we make depth?
measurement-basedquantum computing (MBQC)
![Page 12: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/12.jpg)
tensor contractionU9
U7
U6
U5
U4
U2U1
U8 U3
|0⟩|0⟩|0⟩|0⟩|0⟩
⟨z1z2z3z4z5| U9U8U7U6U5U4U3U2U1 |00000⟩ = ⟨z1|⟨z2|⟨z3|⟨z4|⟨z5|
Ui = tensor with 4 indices, each dim 2
![Page 13: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/13.jpg)
tensor contraction in 1-D|0⟩|0⟩
|0⟩
⟨z1|⟨z2|
⟨zN|
depth T
N qu
bits
N
T
intermediatetensors can be
N
T
run time:T exp(N) or N exp(T)
![Page 14: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/14.jpg)
simulating 2-D circuitsDepth T=O(1) circuiton ! × ! grid
W
L
T
can be simulated in time 2LW or 2LT or 2WT
[Napp, La Placa, Dalzell, Brandão, Harrow, in preparation]
Naively takes time 2$( &)
≈ ! qubits on line for time !.
But 1-D effective evolution is not unitary.
Entanglement has phase transitionfrom area law -> volume law.
T=3 in area law phase àNO(1)-time classical simulation forapproximate sampling of random circuits.Exact or worst-case is #P-hard.
![Page 15: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/15.jpg)
cheaper tensor contractionT=O(1)! × ! grid effective
timeevolution
! qubitsevolving for! time
Ui
sideways gatesgenerically notunitary
≈ U
weakmeasurement
![Page 16: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/16.jpg)
Approximate simulationEn
tang
lemen
t E
acro
ss c
ut
Matrixproductstate
bonddimensionexp(E)
Example:
| i =X
i,j
ci,j |ii ⌦ |ji
=X
k
�k|↵ki|�ki<latexit sha1_base64="xzWMUP4oroKYRyNd67Syxa/1sDY=">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</latexit>
X
k
|ki|↵ki<latexit sha1_base64="cqrXNo0l/0MI6oVF9ceciPqUnh8=">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</latexit>
X
k
|ki|�ki<latexit sha1_base64="PBJu4rfOT0d/LlFezfoK9cLv1io=">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</latexit>
X
k
�khk|hk|<latexit sha1_base64="3ty4OYffKyYgSOeelBqZMnO963U=">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</latexit>
Simulation algorithm:• Do tensor contraction• Truncate bonds to dim
exp(O(E)).Run-time is N2O(E).
![Page 17: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/17.jpg)
Does the algorithm work?1. Yes.
We tested it and simulated 400x400grids on a laptop.
2. Probably.We proved a phase transition in something likethe effective entanglement.
3. Sometimes.The extended brickwork architecture is #P-hardto simulate exactly but our algorithm is provento work on it.
“Beware of bugs in the above code; I have only proved it correct, not tried it.”--Donald Knuth
![Page 18: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/18.jpg)
stat mech model
U
X
�,⌧2Sk<latexit sha1_base64="KcxY3Rk3CYTDJ4wUcv4G59cZCiU=">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</latexit>
σ τWg(σ,τ)
qc = 3.249…
G
G
G
G
Qubits à dim-q particles. E[tr[ρAk]] = partition function
area lawE = O(1)
volume lawE = O( ! )
disordered
ordered
q
k=2 anisotropic Ising
conjecture: decimating yieldsnonnegative weights for all k, q
1+ε ∼ high-T Potts
≫1 ∼ holography
A Ac
Wg-1(σ,τ) = G(σ,τ)
= q-dist(σ, τ)
![Page 19: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/19.jpg)
random tensor networksX
�,⌧2Sk<latexit sha1_base64="KcxY3Rk3CYTDJ4wUcv4G59cZCiU=">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</latexit>
σ τWg(σ,τ) Wg-1(σ,τ)
= G(σ,τ) = q-dist(σ, τ)
G
G
G
G
q = local dim, E[tr[ρk]]
• EU∼Haar[U⊗k ⊗ (U*)⊗k] = ∑σ,τ Wg(σ,τ) |σ⟩⟨τ|• EG∼GUE[G⊗k ⊗ (G*)⊗k] = ∑σ |σ⟩⟨σ|• ⟨σ|τ⟩ = G(σ,τ)• || G – I ||op , || Wg – I ||op ≤ k2 / q
U ≈ Uk2 ≪ q
![Page 20: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/20.jpg)
Open questions
• Rigorously prove the location of the phase transition and the correctness of the algorithm.
• Random tensor networks with low bond dimension
• Universality classes in random circuits?
• (time-independent) Hamiltonian versions?
• Where exactly is the boundary between easy and hard?
![Page 21: Phase transitions in the complexity of simulating random shallow … · 2020-06-15 · Phase transitions in the complexity of simulating random shallow quantum circuits John Napp,](https://reader034.fdocuments.net/reader034/viewer/2022050603/5faa63429178252b8e65b4ed/html5/thumbnails/21.jpg)
Thanks!
FernandoBrandão
AlexDalzell Rolando
La Placa John Napp
arXiv:2001.00021