Quantum Computation

and Statistical Mechanics

Maarten Van den Nest

Max Planck Institute for Quantum Optics

El Escorial, July 11th

2011

Recent investigations establish and exploit mappings betweenclassical statistical mechanics an quantum information & computation:

–

Quantum algorithms–

Measurement-based QC –

Strongly correlated systems & PEPS–

Quantum error-correction & fault tolerance–

…

Such mappings allow to

–

interchange techniques

between these two fields–

Formulate quantum algorithms

for problems in Stat Mech

Statistical Mechanics and QIT

Overview of connections between Stat Mech and Quantum Computation

Emphasis on:

–

Power of quantum versus classical computation–

Classical simulation of quantum computation–

Computational models: circuit model, measurement-based QC–

Complexity of classical spin systems

…Not so much: phase transitions, critical behavior, etc..

Scope of this talk

Outline

Part 0: Classical spin models

Part I: Stat Mech and Quantum Circuits

Part II: Stat Mech and Measurement-Based QC

0.

Classical spin models

Edge models

Ising model: 2-level spins sa = 1, -1

Potts model: q-level spins sa = 0, ..., q-1

With/without magnetic fields, e.g.

Spins located at vertices, interactions along edges

H({s}) = -

Σ

Jab

sa

sb

H({s}) = -

Σ

Jab

δ(sa

. sb

)

-

Σ

hab

sa

sa

sb haJab

Vertex models

Spins located at edges, interactions at vertices

Local energy H(s,t,u,v) associated to each configuration around vertex

Six-vertex model:

1 2

3 4

5 6

7 8

0 00 00 0

0 0

ϖ ϖϖ ϖϖ ϖ

ϖ ϖ

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤ =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

stuvW

Eight-vertex model:

–

2-state spins on 2D lattice

–

Boltzmann weights Wst

= exp[-βH(s,t,u,v)]uv

2 70ϖ ϖ= =

s t

uw

Partition function

E.g. Ising model partition function

For each spin configuration {s} on the lattice, compute the product of all local Boltzmann weights

Partition function Z is the sum, over all spin configurations, of such products

Z contains info about thermodynamical properties of system

Z = Σ ∏

e-βJabsasb

{sa} edges

I.

Stat Mech

and Quantum circuits

I.A.

Mappings

Consider a poly-size quantum circuit C acting nearest-neighbor q-level quantum systems

Consider standard basis states |L⟩ and |R⟩ (e.g. |L⟩ = |R⟩ = |00…0⟩)

We will identify ⟨L|C|R⟩ with partition function of classical spin model

C =

Nearest-neighbor gate acting on (at most) two

systems

Quantum circuits

⟨L|C|R⟩ = tensor network

Graphical representation:

–

Each gate

becomes a vertex

with 4 incident edges:

–

Contraction

of two indices = gluing

together corresponding edges

abcdU

a c

b d

αα

α∑ ab b'

c c'd'U V

a c

b α 'c

'b 'd

Quantum circuits as tensor networks

For the whole circuit we find a 2D square lattice:

–

At each edge of 2D lattice sits an index variable which takes q values–

The variables left/right are fixed in boundary conditions L

and R–

Each configuration a, b, c, d

around a vertex is given a weight Uab

–

⟨L|C|R⟩

is given as sum, over all configurations, of products of Uabcd

cd

Quantum circuits as tensor networks

a c

b d

1Ls

2Ls

LNs

1Rs

2Rs

RNs

.∑

a,b,c,d,..⟨L|C|R⟩

=

Uabcd

We can interpret:

–

the variables a, b, c, d

as q-level classical spins and –

the weights Uab

= exp[-βH(a,b,c,d)]

as Boltzman

weights

Corresponds to vertex model:

⟨L|C|R⟩

= partition function of vertex model

on (tilted)

2D square lattice with left and right boundary conditions

cd

Tensor networks as partition functions

⟨L|C|R⟩

= ∑

a,b,c,d,...

∏vertices of 2D latt

abcd

ice

U

2-level classical spin at each vertex of 2D latticeSpins at the left/right are fixed in boundary conditions L and RTwo spins i, j at an edge have Boltzmann weight wij = exp[-βH(i,j)]

1Ls

2Ls

LNs

1Rs

2Rs

RNs

ijw

jkw

i j

k

What about edge models?

Corresponding quantum circuit C:–

Weights on horizontal

edges become single-qubit gates Σ

wij

|i ⟩⟨j|

–

Weights on vertical

edges become diagonal 2-qubit gates: Σ

wjk

|jk⟩⟨jk|–

Boundary conditions become computational basis states

Similar to before, ⟨L|C|R⟩ is partition function of edge model

1Ls

2Ls

LNs

2Rs

RNs

1Rs

ijw

jkw

1Ls

2Ls

LNs

1Rs

2Rs

RNs

jkw

i j

k

ijw

From edge models to quantum circuits

I.B.

Applications

Application 1: Quantum Algorithms

Immediate quantum algorithm for Z = ⟨L|C|R⟩ whenever Boltzmann weights are chosen such that C is unitary

circuit

–

standard method: Hadamard

test

This yields (additive) approximation: algo returns number c such that (with probability exponentially close to 1) one has

If Boltzmann weights correspond to universal quantum gate set, the corresponding partition function problem is BQP-complete

E.g. 6-vertex model and Ising model on 2D lattice are BQP-complete

≤1| - L C R | c

poly(N)

Quantum algorithms

Some remarks

Since quantum circuit is to be unitary, one generally needs complexBoltzmann weights (= unphysical)

This family of Q algo’s allows to efficiently approximate Z. However the exact evaluation of Z is #P-hard and thus generally intractable

–

also for quantum computers

–

Bonus:

using mappings to Q circuits one can in fact prove

#P-hardness of Z !

BQP completeness is strong indication that no classical algo exists

–

Note: BQP completeness e.g. includes Shor’s

factoring algo

Application 2: Classical simulations

Evaluation of ⟨L|C|R⟩ corresponds to evaluation of Z of lattice model

If lattice model is exactly solvable (i.e. Z can be computed efficiently)then ⟨L|C|R⟩

can be computed efficiently

However, watch out for

–

Translation invariance–

Complex couplings–

Finite lattice size–

Boundary conditions

Nevertheless, certain solvable models can be translated into classically simulatable quantum circuits

Classical simulations via solvable models

For example, the eight-vertex model

is solvable for finite dimensions, complex non-translation-invariant couplings, whenever the condition

is fulfilled. The solution is given by mapping to free fermions.

Correspondingly, Q circuits composed of such 2-qubit gates have efficient classical simulations

(quantum matchgate

circuits)

1 2

3 4

5 6

7 8

0 00 00 0

0 0

ϖ ϖϖ ϖϖ ϖ

ϖ ϖ

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤ =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

abcdW

1 8 2 7 3 6 4 5ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ− = −

Example: 8-vertex model and matchgates

II.

Stat Mech

and Measurement-based QC

Measurement-Based QC (= One-way QC)

Preparation of 2D cluster state: universal resource state Sequence of adaptive single-qubit measurementsRemaining qubits are (up to local basis change) in desired output state

Understanding the power of MBQC

2D cluster states are universal quantum computers via measurements

We want to understand where this exceptional computational power originates

–

Which other resource states are universal?–

Is there an essential feature that makes 2D cluster states universal?

–

Which states are efficiently simulatable?–

What is role of entanglement?–

…

?

?

?

?

? ?

?

?

?

?

1D cluster universal?

Mapping MBQC to Stat Mech

What will follow: map lattice models to entangled resource states à

la cluster states

Properties of lattice model are reflected in computational powerof resource state

II.A.

Mappings

Start with graph

Place qubit at each edge

Place qubit at each site

Define

Cluster-type states

Vertex

qubits

Edge

qubits

|ψ⟩

= ⊗|sa

⟩ ⊗ |sa

+ sb

⟩Σ{sa}

a ab

These states belong to familyof stabilizer states

–

i.e. eigenstates

of sets ofPauli operators

Relation to the stabilizer formalism

These states belong to familyof stabilizer states

–

i.e. eigenstates

of sets ofPauli operators

1 Pauli operator per vertex

Relation to the stabilizer formalism

These states belong to familyof stabilizer states

–

i.e. eigenstates

of sets ofPauli operators

1 Pauli operator per vertex

1 Pauli operator per edge

Relation to the stabilizer formalism

These states belong to familyof stabilizer states

–

i.e. eigenstates

of sets ofPauli operators

1 Pauli operator per vertex

1 Pauli operator per edge

|ψ⟩ is unique joint eigenstate

Relation to the stabilizer formalism

2D Ising model with external fields

Partition function = overlap

Z =

⟨α|ψ⟩

The mapping

Product state containing info of temperature β and couplings Jab and ha

Edges ab

Vertices a|α⟩

= ⊗

|αa

⟩ ⊗ |αab

⟩a ab

|αab

⟩

= e-βJab|0⟩

+ eβJab|1⟩

|αa

⟩

= e-βha|0⟩

+ eβha

|1⟩

2D Ising model with external fields

Partition function = overlap

Z =

⟨α|ψ⟩

The mapping

Product state containing info of temperature β and couplings Jab and ha

Edges ab

Vertices a|α⟩

= ⊗

|αa

⟩ ⊗ |αab

⟩a ab

|αab

⟩

= e-βJab|0⟩

+ eβJab|1⟩

|αa

⟩

= e-βha|0⟩

+ eβha

|1⟩

Similar mappings for arbitrary graphs, Ising without fields, q-state Potts model, etc

1D Ising

model with

fields 1D cluster state

1D Ising

model without

fields

–

Open boundary conditions

Product state

–

Periodic BCs

GHZ state

2D Ising

model with

fields 2D cluster state

2D Ising

model without

fields Planar code state

Examples

ISING MODEL QUANTUM STATE

II.B.

Applications

Application 1: Classical simulations & universality of MBQC

Mapping connects computational power of resource state with hardness of computing Z

of corresponding spin model

Exactly solvable model gives rise to classically simulatable resource state

Intractable model (probably) gives rise to powerful resource state

Understanding the power of MBQC

Partition function = overlap

Z =

⟨α|ψ⟩

1D Ising

model with

fields 1D cluster state

1D Ising

model without

fields

–

Open boundary conditions

Product state

–

Periodic BCs

GHZ state

2D Ising

model with fields 2D cluster state

2D Ising

model without

fields Planar code state

Exactly solvable models

ISING MODEL QUANTUM STATE

1D Ising

model with fields 1D cluster state

1D Ising

model without fields

–

Open boundary conditions

Product state

–

Periodic BCs

GHZ state

2D Ising

model with fields 2D cluster state

2D Ising

model without fields Planar code state

An NP-hard model

ISING MODEL QUANTUM STATE

1D Ising

model with fields 1D cluster state

1D Ising

model without fields

–

Open boundary conditions

Product state

–

Periodic BCs

GHZ state

2D Ising

model with fields 2D cluster state

2D Ising

model without fields Planar code state

An NP-hard model

ISING MODEL QUANTUM STATE

Universal

state

Application 2: Completeness

= overlap between cluster state |ϕ2D⟩ and product state

Consider |ψ⟩ associated with some classical model:

2D cluster state is universal resource for MQC. Therefore:

|ϕ2D

⟩

= 2D Cluster state on N qubits|β⟩

= product state on measured qubits

Therefore

Universal MQC and the Ising

model

{ }α ϕ⊗ 2Dψ = ' I

αGZ ({J}) = ψ

{ }'α α α ϕ IsingG 22D D= ψ = Z ({J}) Z ({J, = J'})

Ising2D Z

The 2D Ising

model is complete

Z of 2D Ising specializes to Z of any (reasonable) classical spin model

–

E.g. D-dimensional, q-level, vertex-

or edge model, etc..

Ising coupling strengths need to be complex. This problem can be resolved by considering instead Z2

lattice gauge theory in 4D

IsingG 2DZ ({J}) = Z ({J, J'})

- Initial model on n spins

- couplings {J}-

2D Ising

model on N = poly(n) spins

-

Couplings {J} on original spins

-

Couplings {J’} on additional spins

Thank you very much!