Entanglement entropy and the simulation of quantum systems Open discussion with pde2007

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Entanglement entropyandthe simulation of
quantum systemsOpen discussion with pde2007

• José Ignacio Latorre
• Universitat de Barcelona
• Benasque, September 2007

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Physics
Theory 1
Theory 2
Exact solution
Approximated methods
Simulation
Classical Simulation
Quantum Simulation
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Introduction
Introduction

• Classical Theory
• Classical simulation
• Quantum simulation
• Quantum Mechanics
• Classical simulation
• Quantum simulation

Classical computer
?
Quantum computer
Classical simulation of Quantum Mechanics is
related to our ability to support large
entanglement Classical simulation may be enough
to handle e.g. ground states MPS, PEPS,
MERA Quantum simulation needed for time
evolution of quantum systems and for non-local
Hamiltonians
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Introduction
Introduction
Is it possible to classically simulate faithfully
a quantum system?
Quantum Ising model
represent
evolve
read
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Introduction
The lowest eigenvalue state carries a large
superposition of product states
Ex. n3
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Introduction
Introduction
Is it possible to classically simulate faithfully
a quantum system?

• Naïve answer NO
• Exponential growth of Hilbert space

computational basis
n
Classical representation requires dn complex
coefficients

• A random state carries maximum entropy

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Introduction
Introduction

• Refutation
• Realistic quantum systems are not random
• symmetries (translational invariance, scale
  invariance)
• local interactions
• little entanglement
• We do not have to work on the computational
  basis
• use an entangled basis

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Plan
Measures of entanglement Efficient description
of slight entanglement Entropy physics vs.
simulation New ideas MPS, PEPS, MERA
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Measures of entanglement
Measures of entanglement
One qubit
Quantum superposition
Two qubits
Quantum superposition several parties
entanglement
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Measures of entanglement
Measures of entanglement

• Separable states

e.g.

• Entangled states

Local realism is dropped Quantum non-local
correlations
e.g.
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Measures of entanglement
Measures of entanglement
Measures of entanglement
Pure states Schmidt decomposition Singular
Value Decomposition
A B
Diagonalise A
? min(dim HA, dim HB) is the Schmidt number
Separable state
Entangled state
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Measures of entanglement
Measures of entanglement
Von Neumann entropy of the reduced density matrix
Product state
large
large
Very entangled state
e-bit
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Measures of entanglement
Measures of entanglement

• Maximum Entropy for n-qubits
• Strong subadditivity theorem
• implies entropy concavity on a chain of spins

Smaxn
SLM
SL
SL-M
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Efficient description
Efficient description
Efficient description for slightly entangled
states
Schmidt decomposition
A B
Retain eigenvalues and changes of basis
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Efficient description
Vidal 03 Iterate this process

• Slight entanglement iff ??poly(n)ltlt dn
• Representation is efficient
• Single qubit gates involve only local updating
• Two-qubit gates reduces to local updating
• Readout is efficient

efficient simulation
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Efficient description
Graphic representation of a MPS
Efficient computation of scalar products
operations
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Efficient description
Efficient computation of a local action
U
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Efficient description
Efficient description
Matrix Product States
i
a
canonical form PVWC06
Approximate physical states with a finite ? MPS
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Efficient description
Intelligent way to represent, manipulate,
read-out entanglement
Adaptive representation for correlations among
parties
Classical simplified analogy I want to send
16,24,36,40,54,60,81,90,100,135,150,225,250,375,62
5 Instruction take all 4 products of 2,3,5
MPS compression algorithm
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Efficient description
Spin-off Image compression
i1 ?
i2 i1 ?
105 2,1 ?
i11 i12 i13
i14
RG addressing
pixel address
level of grey
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Efficient description

• QPEG
• Read image by blocks
• Fourier transform
• RG address and fill
• Set compression level ?
• Find optimal
• gzip (lossless, entropic compression)
• (define discretize Gs to improve gzip)
• diagonal organize the frequencies and use 1d RG
• work with diferences to a prefixed table

Max ? 81
? 1 PSNR17
? 4 PSNR25
? 8 PSNR31
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Efficient description
Efficient description
Efficient description
Spin-off Differential equations
Note classical problems with a direct product
structure!
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Efficient description
Matrix Product States for continuous variables
Harmonic chains
MPS handles entanglement
Product basis
Truncate ?tr d tr
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Efficient description
Nearest neighbour interaction
Minimize by sweeps
Choose Hermite polynomials for local basis
optimize over a
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Efficient description
Results for n100 harmonic coupled
oscillators (lattice regularization of a quantum
field theory)
Error in Energy
dtr3 ?tr3
dtr4 ?tr4
dtr5 ?tr5
dtr6 ?tr6
Newton-raphson on a
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Physics vs. simulation
Physics vs. simulation
Back to the central idea entanglement support
Success of MPS will depend on how much
entanglement is present in the physical state
Physics
Simulation
If
MPS is in very bad shape
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Physics vs. simulation
Physics vs. simulation
Exact entropy for a reduced block in spin chains
At Quantum Phase Transition
Away from Quantum Phase Transition
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Physics vs. simulation
Physics vs. simulation
Maximum entropy support for MPS
Maximum supported entanglement
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Physics vs. simulation
Physics vs. simulation
Faithfullness Entanglement support
MPS
Spin chains
Spin networks
PEPS
Area law
Computations of entropies are no longer academic
exercises but limits on simulations
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NP-complete
Entanglement for NP-complete problems
0 1 1 0 0 1 1 0
instance
For every clause, one out of eight options is
rejected
3-SAT is NP-complete k-SAT is hard for k gt
2.41 3-SAT with m clauses easy-hard-easy around
m4.2n

• Exact Cover
• A clause is accepted if 001 or 010 or 100
• Exact Cover is NP-complete

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NP-complete

• Adiabatic quantum evolution (Farhi,Goldstone,Gutma
  nn)

t
s(T)1
H(s(t)) (1-s(t)) H0 s(t) Hp
s(0)0
Inicial hamiltonian
Problem hamiltonian
Adiabatic theorem
if
E
E1
gmin
E0
t
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NP-complete

• Adiabatic quantum evolution for exact cover

0gt
0gt
0gt
1gt
1gt
1gt
1gt
0gt
(0gt1gt)
(0gt1gt)
(0gt1gt)
(0gt1gt)
.
NP problem as a non-local two-body hamiltonian!
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n100 right solution found with MPS among
1030 states
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Physics vs. simulation
Physics vs. simulation
Non-critical spin chains S ct
Critical spin chains S log2 n
Spin chains in d-dimensions S nd-1/d
Fermionic systems? S n log2 n
NP-complete problems 3-SAT Exact Cover S .1 n
Shor Factorization S r n
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New ideas
New ideas
Recent progress on the simulation side
MPS using Schmidt decompositions
(iTEBD) Arbitrary manipulations of 1D systems
PEPS 2D, 3D systems MERA Scale
invariant 1D, 2D, 3D systems
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MPS
2. Euclidean evolution
Non-unitary evolution entails loss of norm
are sums of commuting pieces
Trotter expansion
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MPS
Ex iTEBD (infinite time-evolving block
decimation)
A
A
A
B
B
A
B
even
B
A
A
B
odd
A ? B
Translational invariance is momentarily broken
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MPS
i)
ii)
iii)
iv)
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MPS
Schmidt decomposition produces orthonormal L,R
states
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MPS
Moreover, sequential Schmidt decompositions
produce isometries
are isometries

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MPS
Read out
Energy
Entropy for half chain
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New ideas
New ideas
Heisenberg model
Trotter 2 order, ?.001
?2 -.42790793 S.486
?4 -.44105813 S.764
?6 -.44249501 S.919
?8 -.44276223 S.994
?16 -.443094 S1.26
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MPS
Convergence
entropy
energy
Local observables are much easier to get than
global entanglement properties
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MPS
S
Perfect alignment
M
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New ideas
New ideas
PEPS Projected Entangled Pairs
physical index
ancillae
Good PEPS support an area law!! Bad
Contraction of PEPS is P
New results beat Monte Carlo simulations
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PEPS
Entropy is proportional to the boundary
B
A
Contour A L
Area law
Some violations of the area law have been
identified
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PEPS
Contraction of PEPS is P
Building physical PEPS would solve NP-complete
problems
As the contraction proceeds, the number of open
indices grows as the area law
2D seemed out of reach to any efficient
representation
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PEPS
PEPS
PEPS
E
Yet, for translational invariant systems, it
comes down to iTEBD !!
E becomes a non-unitary gate
E
E
Comparable to quantum Monte Carlo?
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PEPS
Results for 2D Quantum Ising model (JOVVC07)
PEPS
MC
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MERA
MERA
MERA Multiscale Entanglement Renormalization
Ansatz
Intrinsic support for scale invariance!!
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All entanglemnent on one line
MERA
MERA
All entanglemnent distributed on scales
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MERA
Update
U
Contraction Identity
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Physics vs. simulation

• Physics
• Scaling of entropy Area law ltlt Volume law
• Translational symmetry and locality reduce
  dramatically the amount
• of entanglement
• Worst case (max entropy) remains at phase
  transition points

• If MPS, PEPS, MERA are a good representation of
  QM
• Approach hard problems
• Precision
• Can we simulate better than Monte Carlo?
• Are MPS, PEPS and MERA the best simulation
  solution?
• Spin-off?

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QMA
QMA
Quantum Complexity Classes
L is in QMA if there exists a fixed ? and a
polynomial time verifier (V) such that
What is the QMA-complete problem?
Feynman idea (shaped by Kitaev)
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QMA
QMA-complete problem
Given H on n-party decide if

• Log-local hamiltonian
• 5-body
• 3-body
• 2-body (non-local interactions)
• 2-body (nearest neighbor 12 levels interaction)!

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Open problems

• Separability problem (classification of
  completely positive maps)
• Classification of entanglement (canonical form
  of arbitrary tensors)
• Better descriptions of quantum many-body systems
• Spin-off of MPS??
• Rigorous results for PEPS, MERA
• Need for theorems for gaps/correlation
  length/size of approximation
• Exact diagonalisation of dilute quantum gases
  (BEC)
• Classification of Quantum Computational
  Complexity classes
• .