Quantum Computation and Statistical MechanicsQuantum Computation and Statistical Mechanics Maarten...
Transcript of Quantum Computation and Statistical MechanicsQuantum Computation and Statistical Mechanics Maarten...
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Quantum Computation
and Statistical Mechanics
Maarten Van den Nest
Max Planck Institute for Quantum Optics
El Escorial, July 11th
2011
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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
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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
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Outline
Part 0: Classical spin models
Part I: Stat Mech and Quantum Circuits
Part II: Stat Mech and Measurement-Based QC
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0.
Classical spin models
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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
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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
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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
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I.
Stat Mech
and Quantum circuits
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I.A.
Mappings
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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
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⟨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
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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
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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
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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?
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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
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I.B.
Applications
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Application 1: Quantum Algorithms
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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
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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
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Application 2: Classical simulations
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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
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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
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II.
Stat Mech
and Measurement-based QC
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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
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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?
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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
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II.A.
Mappings
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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
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These states belong to familyof stabilizer states
–
i.e. eigenstates
of sets ofPauli operators
Relation to the stabilizer formalism
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These states belong to familyof stabilizer states
–
i.e. eigenstates
of sets ofPauli operators
1 Pauli operator per vertex
Relation to the stabilizer formalism
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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
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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
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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⟩
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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
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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
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II.B.
Applications
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Application 1: Classical simulations & universality of MBQC
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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⟩
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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
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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
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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
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Application 2: Completeness
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= 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
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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
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Thank you very much!