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CLARENDON LABORATORY
PHYSICS DEPARTMENT
UNIVERSITY OF OXFORD
and
CENTRE FOR QUANTUM TECHNOLOGIES
NATIONAL UNIVERSITY OF SINGAPORE
Quantum Simulation
Dieter Jaksch
Outline

Lecture 1: Introduction


Lecture 2: Optical lattices


Analogue simulation: Bose-Hubbard model and artificial gauge fields.
Digital simulation: using cold collisions or Rydberg atoms.
Lecture 4: Tensor Network Theory (TNT)
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
Bose-Einstein condensation, adiabatic loading of an optical lattice.
Hamiltonian
Lecture 3: Quantum simulation with ultracold atoms


What defines a quantum simulator? Quantum simulator criteria. Strongly
correlated quantum systems.
Tensors and contractions, matrix product states, entanglement properties
Lecture 5: TNT applications

TNT algorithms, variational optimization and time evolution
Quantum simulation
INTRODUCTION
Early simulators
T.H. Johnson, S.R. Clark and DJ, EPJ Quantum Technology 1, 10 (2014)
Orrery simulating the solar system
Differential analyzer in 1938
Difficult simulations
© Prof Michael Engel
Quantum systems as simulators
Cold atoms
Trapped ions
Superconductors
© Nature publishing group
“More is different.”
P.W. Anderson, 1972
“... we never experiment with single particles
inasmuch as we cannot raise an
Ichtyosaurus in the zoo.”
E. Schrödinger, 1952
Quantum simulation
WHAT IS QUANTUM SIMULATION
What are simulators?
What are simulators?
Physical System
Simulation
The process of simulation
Physical System
Quantitative Model
Physical Simulation
What is a Quantum Simulator?

Every system is quantum if considered at sufficiently short length
and/or time scales
Diagonal density matrix

For classicality we require the density matrix to be diagonal in a
single particle basis
𝜌(𝑡) =
𝑝 𝑁𝑠,𝑖 , 𝑡
𝑁𝑠,𝑖 , 𝑡
𝑁𝑠,𝑖 , 𝑡
𝑠,𝑖

Here | 𝑁𝑠,𝑖 , 𝑡〉 is a Fock state with 𝑁𝑠,𝑖 particles of species 𝑠
occupying the single particle mode 𝑖.

𝑝( 𝑁𝑠,𝑖 , 𝑡) is the probability for such a mode to be occupied.

The modes can be extended in space so this definition allows for
single particle tunnelling of “classical” degrees of freedom.

The definition does allow for natural entanglement between the
modes of identical particles.
A beam splitter
(a)
(b)
photon
photon
|𝑢〉
BS
|𝑒〉
|𝑔〉
atom 1
BS
|𝑑〉
𝜓 ∝ 𝑢 + |𝑑〉
|𝑒〉
𝜓 ∝ |𝑒𝑔〉 + |𝑔𝑒〉
|𝑔〉
atom 2
(c)
photon 2
|𝑢〉
BS
|𝑑〉
photon 1
𝜓 ∝ 𝑢𝑢 + |𝑑𝑑〉
All of these are quantum systems
But only (b) and (c) are quantum
states that could be used to violate
a Bell inequality
When are simulators trustworthy?

Extension of the spring changes like

Time of fall for frictionless ball to fall goes like

Can work out

Use a pocket calculator or laptop to work out a
of the temperature.
of the fallen height.
using an abacus.
.
Simulation part of scientific method
Physical System
Simulator
TRUST
Quantitaive Model
Only require model for system of interest and simulator to be falsifiable
Quantum simulator criteria
J. Ignacio Cirac and Peter Zoller, Nature Physics 8, 264 (2010)

Quantum system


Initialization



Prepare a known quantum state, pure or mixed, e.g. thermal
Hamiltonian engineering

Set of interactions with external fields or between different particles

Interactions either local or of longer range
Detection


Large number of degrees of freedom, lattice system or confined in space
Perform measurement on the system, particles individually or collectively.
Single shot which can be repeated several times.
Verification

Increase confidence about result, benchmark by running known limiting
cases, run backward and forward, adjust time in adiabatic simulations.
Digital quantum simulation

Assume a discrete lattice in space time
Creation operator
a† |  = | 
time
a† |  = 0
site empty
site filled
space
Destruction operator
a| = | 
a| =0
Counting operator
Any quantum system can be simulated
by a Hamiltonian which couples these
sites locally!
n| =1| 
n| =0| 
n= a†a

First envisaged by R. Feynman in 1982
Feynman on simulation in 1982
using a classical computer:
‘… 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.’
using a quantum computer:
‘… I believe it is rather simple to answer that question and to
find the class, but I just haven’t done it.’
Done: S. Lloyd, Science 273, 1073 (1996)
Analogue quantum simulator
TRUST
Mathematical Model 𝐻
Why do we need quantum simulators?

The curse of dimensionality in quantum systems
# config = 4,722,366,482,869,645,213,696

Hard classical problems
Travelling salesman
Ising model
Quantum simulation
STRONGLY CORRELATED SYSTEMS
Strong correlations?
Can we view “correlations” as a synonym for “interactions”?
weak interactions
• can extrapolate en masse
behaviour from one particle
strong interactions
• particles do not move independently
Dramatic consequences
Interplay of microscopic interactions with external influences can
lead to abrupt macroscopic changes …
-
Simple “classical” Ising magnet:
+
+ -
+
1
+
0
1
Strong correlations
Beyond physics strong correlations appear pervasive …
Add quantum mechanics …
Major interest in quantum many-body problems arises in lattice systems – trying to
understand the remarkable properties of electrons in some materials …
•
High-temperature superconductivity
What is the pairing
mechanism?
•
Quantum Hall effect
What are the topological
properties of fractional
QH states?
These are seminal strongly-correlated phenomena.
Quantum simulation
THE MANY BODY PROBLEM
Strong or weak correlations?
It would seem from the offset that condensed matter physics should be a very strongly
correlated quantum problem?
–
–
+
+
–
+
+
+
–
–
–
+
–
–
–
+
–
+
+
+
–
+
–
+
+
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–
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+
A problem described by the “theory
of everything”?
+
–
+
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–
Weak correlations
Paradoxically, many solid state systems, like metals, display weak correlations despite being
composed of strongly interacting particles.
Why? First, the Born-Oppenheimer approx. decouples ions and e’s
• Electrons see fixed periodic
potential of the ions
.
+
+
+
• Neighbouring localised orbitals
overlap.
Locality means that only these single-particle elements are relevant:
on-site potential:
n.n. hopping:
Weak correlations
Moving to many non-interacting electrons (add spin and 2nd quantise):
in real and momentum space
Ground state simply fills up single-particle states:
Band structure explains metals,
semiconductors and insulators:
Weak correlations
The concept of a Fermi surface is crucial to non-interacting fermions:
Landau’s Fermi-liquid theory
shows that the Fermi surface
survives non-zero interactions
System is weakly correlated and described by non-interacting quasi-particles, i.e.
renormalised non-interacting electrons:
–
Strong correlations
This picture fails for some materials like transition metal oxides,
and the CuO2 planes in high-Tc superconductors. The reason is:
•
“core-like” d or f valence orbitals
•
small overlap = narrow bands
•
confinement = large repulsion
+
+
+
Gives the Hubbard model:
Interactions significant – no
simple quasi-particle picture.
Half-filled strong-coupling limit is
a Heisenberg anti-ferromagnet:
What do we want to compute?
Given a Hamiltonian
describing our strongly-correlated system
we typically want to compute:
gapped or gapless?
(1) Solve
Find ground state and low-lying
excitations (also thermal):
(2) Solve
Dynamics of quenches, driving, …
In both cases, we want to simulate local observables and longranged correlations and other properties.
These reduce to linear algebra problems we’ll review shortly …
Analogue quantum simulation
Such model Hamiltonian are now accurately realisable with cold atoms:
Time of flight imaging:
BEC
Lasers
Optical Lattice
Bose Hubbard model
Mott-Insulator :
Superfluid :
quantum phase transition