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

|ki|�ki<latexit sha1_base64="PBJu4rfOT0d/LlFezfoK9cLv1io=">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</latexit>

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