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
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.
easier quantum simulations
1. solve trivial special case(e.g. non-interacting theory)
2. treat corrections to theoryas perturbations
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.
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]
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)
phase transitions?
Complexity smoothly increases with-entanglement-correlation length-# of non-stabilizer gates
Complexity jumps discontinuously with-noise rate
Today: what about circuit depth?
task chosen to favor quantum computers and clear comparison
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.
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, …]
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)
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
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)
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.
cheaper tensor contractionT=O(1)! × ! grid effective
timeevolution
! qubitsevolving for! time
Ui
sideways gatesgenerically notunitary
≈ U
weakmeasurement
Approximate simulationEn
tang
lemen
t E
acro
ss c
ut
Matrixproductstate
bonddimensionexp(E)
Example:
| i =X
i,j
ci,j |ii ⌦ |ji
=X
k
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X
k
|ki|↵ki<latexit sha1_base64="cqrXNo0l/0MI6oVF9ceciPqUnh8=">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</latexit>
X
k
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X
k
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Simulation algorithm:• Do tensor contraction• Truncate bonds to dim
exp(O(E)).Run-time is N2O(E).
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
stat mech model
U
X
�,⌧2Sk<latexit 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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)
random tensor networksX
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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
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?
Thanks!
FernandoBrandão
AlexDalzell Rolando
La Placa John Napp
arXiv:2001.00021
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