Interacting Frobenius Algebras are Hopf R. Duncan Summary - - PowerPoint PPT Presentation
Interacting Frobenius Algebras are Hopf R. Duncan Summary Quantum theory in categorical form Frobenius algebras and their phase groups Complementarity and strong complementarity Bialgebras and Hopf algebras Classical
Classical waves can exhibit superposition too!
unless and are orthogonal [Wooters & Zurek 1982] Theorem: There are no unitary operations D such that
D : |ψ⌅ ⇤⇥ |ψ⌅ |ψ⌅ D : |φ⌅ ⇤⇥ |φ⌅ |φ⌅
unless and are orthogonal [Pati & Braunstein 2000] Theorem: There are no unitary operations E such that
E : |ψ⇤ ⇥ |0⇤ E : |φ⇤ ⇥ |0⇤ |ψ |ψ |φ |φ
Separate classical and quantum data in a hybrid machine Linear types have been proposed* to capture this:
But we need additional axioms to get the QM-like behaviour
*vanTonder 2003, Selinger and Valiron 2005, Arrighi and Dowek 2003, Altenkirch & Grattage 2005 ** Hensinger et al Nature 2005
* Aspect et al PRL 1981
* Aspect et al PRL 1981
"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more."
λx.λy.λz.xz(yz)
λx.λy.λz.xz(yz)
Hilbert space, unitary transforms, self-adjoint
λx.λy.λz.xz(yz)
Hilbert space, unitary transforms, self-adjoint
This is an 8-bit adder
Quantum theory as an internal theory in a monoidal category
A ⊗ A
A
abstraction
Quantum states are represented by unit vectors* in a complex Hilbert space.
|0 |1 |0 + eiα |1
* actually : rays = 1-dim subspaces = unit vectors modulo scalar factors
|0i , |1i 2 C2
two or more systems is the tensor product of their individual state spaces
|00i , |11i , |00i + |11i p 2 2 C2 ⌦ C2
For each discrete time step, an undisturbed quantum system evolves according to a unitary operator acting
The collection of possible evolutions forms a group.
|0 |1
Zα = ✓ 1 eiα ◆
Non-degenerate, projective observables
i.e. Orthonormal bases
Outcome probability:
Possible outcomes:
symmetric monoidal category.
ψ : I → A
† : Cop → C
* Technically: the base category should be †-symmetric monoidal.
f † = f −1
“Unitary” :
Theorem 1: in fdHilb orthonormal bases are in bijection with †-special commutative Frobenius algebras.
: A → A ⊗ A µ : A ⊗ A → A ✏ : A → I ⌘ : I → A
:: |aii ! |aii ⌦ |aii µ = † ✏ :: |aii ! 1 ⌘ = ⌘†
Observables are represented by †- special commutative Frobenius algebras.
unless and are orthogonal [Wooters & Zurek 1982] Theorem: There are no unitary operations D such that
D : |ψ⌅ ⇤⇥ |ψ⌅ |ψ⌅ D : |φ⌅ ⇤⇥ |φ⌅ |φ⌅
unless and are orthogonal [Pati & Braunstein 2000] Theorem: There are no unitary operations E such that
E : |ψ⇤ ⇥ |0⇤ E : |φ⇤ ⇥ |0⇤ |ψ |ψ |φ |φ
When can a quantum state be treated as if classical?
states; In other words:
eigenstate of some known observable. We’ll use this property to formalise observables in terms of copying and deleting operations.
δ = = =
Comonoid Laws
= = =
δ = δ† = † = = =
Comonoid Laws
= = =
δ = δ† = † = = = = =
Monoid Laws
=
δ = δ† = † =
Frobenius Law Isometry Law
= = = =
δ = δ† = † =
=
This defines a special commutative †-Frobenius algebra aka Classical Structure
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Coecke & Paquette 2006
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Coecke & Paquette 2006
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Coecke & Paquette 2006
Theorem: any maps constructed from δ and ε , and their adjoints, whose graph is connected, is determined uniquely by the number of inputs and outputs.
Coecke & Paquette 2006
Corollary: if A supports a †SFCA then the monoidal sub-category generated by A is self-dual compact.
Let A support a †-SCFA in a †-category C, and let f : An ⟶ Am be a morphism.
f = f
f = (f †) = (f )† ;
is
is
is
if f = f ,
if f † = f . E , as is the sy
Defn: a point 𝜔:I⟶A is called unbiased if the map
α = α
ψ
‘æ
ψ
Λ(ψ) : ψ 7! µ (ψ ⌦ id)
unbiased point 𝜔:I⟶A such that:
unbiased point 𝜔:I⟶A such that:
Proposition 6:
Thm: Any connected monochrome diagram with phases is determined completely by its arity and the sum of its phases.
α1 α5 α3 α2 α4 = q
i αi
Non-degenerate, projective observables
i.e. Orthonormal bases
Z =
−1 ⇥ X =
1 ⇥
α |0⇥ + β |1⇥ |0⇥
= α
2
|1⇥
|+⇥
p=(α+β)/22 ⇥
|⇥
p=(α−β)/22
⇥
We can measure the spin of qubit |ψ = α |0 + β |1
Z =
−1 ⇥ X =
1 ⇥
We can measure the spin of qubit |ψ = α |0 + β |1
|0⇥ |0⇥
= 1
|1⇥
|+⇥
p=1/2
⇥ |⇥
p = 1 / 2
⇥
Measuring A then B may give a different answer than measuring B first then A!
A state is unbiased for a basis A if
|ψi
|hai | ψi| = 1 p d
|0 |1 |0 |1 |0 + eiα |1
Zα = ✓ 1 eiα ◆
a0
a1 b1
b0
a0
a1 b1
b0 |ai | bj⇥| = 1 ⌅ D
a0
a1 b1
b0 |ai | bj⇥| = 1 ⌅ D “Mutually Unbiased Bases”
Two observables A and B are complementary if
subgroup of the phase group .
|hai | bji| = 1 p d 8i, j
ΦA {Ubi}i
Theorem 3: Two observables are strongly complementary iff they form a Hopf algebra
Theorem 3: Two observables are strongly complementary iff they form a Hopf algebra
Frobenius Frobenius
Theorem 3: Two observables are strongly complementary iff they form a Hopf algebra
Hopf Hopf
Theorem: In fdHilb, strongly complementary
... but in big enough dimension, it is possible to construct MUBs which are not strongly complementary.
Defn: A bialgebra is 4-tuple: where: is a monoid
= = =
=
Defn: A Hopf algebra is a bialgebra with a map s : A ⟶ A satisfying
s =
(H)
= = =
=
s =
is a monoid is a comonoid
Given an orthonormal basis. |0i , . . . , |d 1i
” :: |nÍ ‘æ |nÍ ¢ |nÍ µ :: |nÍ ¢ |mÍ ‘æ ; |nÍ if n = m 0 otherwise ‘ = q
nœZD Èn|
÷ = q
nœZD |nÍ
µ :: |nÍ ¢ |mÍ ‘æ |n + mÍ ” :: |nÍ ‘æ q
m+mÕ=n
|mÍ ¢ |mÕÍ ÷ = |0Í ‘ = È0|
Given an orthonormal basis. |0i , . . . , |d 1i
” :: |nÍ ‘æ |nÍ ¢ |nÍ µ :: |nÍ ¢ |mÍ ‘æ ; |nÍ if n = m 0 otherwise ‘ = q
nœZD Èn|
÷ = q
nœZD |nÍ
µ :: |nÍ ¢ |mÍ ‘æ |n + mÍ ” :: |nÍ ‘æ q
m+mÕ=n
|mÍ ¢ |mÕÍ ÷ = |0Í ‘ = È0|
In complex Hilbert space all strongly complementary pairs are group algebras of abelian groups
Coecke, Duncan, Kissinger, Wang, “Strong Complementarity and Non-locality in Categorical Quantum Mechanics”,
Oh uh:
|hai | bji| = 1 p d 8i, j
Defn: A scaled bialgebra is 4-tuple: where: is a †SCFA
= = = (B)
Defn: A scaled bialgebra is 4-tuple: where: is a †SCFA
= = = (B)
=
Not this one:
The antipode can be defined as:
Theorem: the scaled bialgebra is Hopf iff
:=
et (” , ‘ , µ , ÷ ) i .e. = = (+)
is a †SCFA is a †SCFA
= =
= = =
Proposition: Let s be the antipode of a commutative Hopf algebra; then
morphism f between Hopf algebras H and K
Corollary: (1) s is a self-adjoint unitary (2) (3) s is the antipode of (4)
= s
ve f = f .
Corollary: (1) s is a self-adjoint unitary (2) (3) s is the antipode of (4)
= s
ve f = f .
f = (s⊗m) f (s†⊗n) = (s†⊗m) f (s⊗n) = f . Proof(4):
Lemma: if f commutes with s then
;
f
=
f
=
f
=
f
We can use the bialgebra to define a convolution:
C f + f Õ := f Õ f 0 :=
Lemma: Let
:= n + 1 = n
n
n m = nm and n m = n + m bialgebra morphism for ( ).
0 = 1 0 0 1 0 0 1 0 0 1 = 1 0 0 0 1 0 0 0 1 2 = 1 0 0 0 0 1 0 1 0
In
3
Lemma: Let f be a bialgebra morphism; then we have: – f◦(g + h)=(f◦g)+(f◦h), – (g + h)◦f =(g◦f)+(h◦f), and – f +(s◦f) = 0.
R, and the bialgebra morphisms of do too!
f (g + h) =
f g h
=
f f g h
= (f g) + (f h) 2. (g + h) f =
g h f
=
g h f f
= (g f) + (h f) 3. f + (f s) =
f f
=
f
=
f
= =
Lemma: Let a,b ∈ such that ab = 1. Then b = a†. Proof: basic idea is show that (-b) + a† = 0
f 0
=
f f 0
=
f f f 0
=
f f f 0
=
f f f 0
=
f
=
f
=
f
=
f
=
f
=
f
algebra form an abelian group, with h-1 = sh
they are a subgroup of the phase group.
h
=
h h
is
is
Classical points Unbiased points π
Classical points Unbiased points π
π
|0 = |1 =
eiα ⇥
=
Classical points Unbiased points π
π
π
|0 = |1 =
eiα ⇥
=
Classical points Unbiased points π
π
π
|0 = |1 =
eiα ⇥
=
Classical points Unbiased points π
π
π
|0 = |1 =
eiα ⇥
=
=
2
i sin α
2
i sin α
2
cos α
2
⇥
π
|+ = |⇥ =
By definition every -set-like element in determines a - phase map: we call these -classical maps.
(1) k is a coalgebra morphism for (2) k† = sks
is
is
is
δX = X = δZ = Z =
i
i i j
j j
i j i j i j
i
i i j
j j
i j i j i j
structures are closed
i
δX = X = δZ = Z =
maps are comonoid homomorphisms
i i
δX = X = δZ = Z =
maps are comonoid homomorphisms
δX = X = δZ = Z =
satisfy canonical commutation relations
i
j
δX = X = δZ = Z =
structures form a bialgebra
then is a -phase. Proof:
k k =
α k
k†αk
is
is
is
Proof :
k
=
k α
=
k α
=
k k α
α k k =
α k
Corollary 6.8. For HK the group of -set-like elements and the group of -phases there is a group action:
◊ æ α
k
‘æ
α k
is
is
HK × Φ → Φ
k n = · · · k n = · · · k k k n = · · · kn n = kn n
Let X be some set of points; we say X is enough points when, for all maps f,g :
elements are linearly independent.
(∀x:X fx = gx) ⇒ f = g
Theorem: suppose Hk is finite, with exponent d. Then if Hk is enough points, internal ring R is isomorphic to . Proof: use nk = knn.
(scaled) axioms of Bonchi et al’s theory as a sub theory.
d
The Pauli Z and X observables are strongly complementary, with some additional features:
π
π
π
π
= =
Good Points: + Universal + Derived from the basic algebra of complementarity + Powerful algebraic theory + Can represent almost anything Bad point:
Meh Point:
Defn: A diagram is an undirected open graph generated by the above vertices.
α ∈ [0, 2π)
|+i⊗n 7! |+i⊗m |i⊗n 7! eiα |i⊗m |0i⊗n 7! |0i⊗m |1i⊗n 7! eiα |1i⊗m
Theorem: Let U be a unitary map on n qubits; then there exists a ZX-calculus term D such that:
JDK = U
Theorem: Let U be a unitary map on n qubits; then there exists a ZX-calculus term D such that:
JDK = U 7! 7! 7!
Generalised Spider
“Strong Complementarity”
· · · · · ·
π 2
α + nπ
2 π 2 π 2 π 2
· · · · · ·
− π
2
α − π
2
− π
2
− π
2
=
(colour change)
· · · · · ·
π 2
α + nπ
2 π 2 π 2 π 2
· · · · · ·
− π
2
α − π
2
− π
2
− π
2
=
(colour change)
A weird one specific to ZX
π 2 π 2 π 2
H = 1 √ 2 3 1 1 1 −1 4 = J K
π 2 π 2 π 2
H
:=
H H H H
· · · · · ·
α
H H H H
· · · · · ·
α
· · · · · ·
α + nπ
2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
=
H H H H
· · · · · ·
α
· · · · · ·
α + nπ
2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
=
· · · · · ·
α
π 2
π 2
π 2
π 2
− π
2
− π
2
− π
2
− π
2
=
H H H H
· · · · · ·
α
· · · · · ·
α + nπ
2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
π 2
=
· · · · · ·
α
π 2
π 2
π 2
π 2
− π
2
− π
2
− π
2
− π
2
=
· · · · · ·
α
=
Corollary: total symmetry between red and green
H H
:= H H
H H
H
=
= = = = = = =
controlled gate
Zπ/2
j1 = |1 j0 = |0
input qubits
H
−π/4
H
π/4
H
−π/4
H
π/4
H
−π/4
H
π/4
H
−π/4
H
π/4
H
−π/4
π/4
−π/4
π/4
−π/4
π/4
−π/4
π/4
π/4 π/4
π/2
qubits
j0 = |0 + e
iπ 2 |1
j1 = |0 + eiπ |1
π/2
QC is going to be — not quantum circuits!
deal with this extra layer of complexity
their implementation — powerful and flexible verification formalisms will be needed!
1WQC is a quantum computer design based on single qubit projective measurements on a graph state.
circuit model
Let G = (V,E) be a simple, undirected graph. Then define:
|Gi = O
(v,u)∈E
CZvu O
v∈V
|+i
H H H H H
... ... ... ...
Let G = (V,E) be a simple, undirected graph. Then define:
O
(v,u)∈E
CZvu O
v∈V
|+i
H H H H H
... ...
Let G = (V,E) be a simple, undirected graph. Then define:
O
(v,u)∈E
CZvu O
v∈V
|+i
Let G = (V,E) be a simple, undirected graph. Then define:
|Gi = O
(v,u)∈E
CZvu O
v∈V
|+i
A graph state, coupled to some input qubits: The only operation is to measure single qubits in the basis:
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state. Finally, any output qubits can be corrected by a local Pauli Z
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
A graph state, coupled to some input qubits: * Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state. Finally, any output qubits can be corrected by a local Pauli
Raussendorf and Briegel. PRL (86) 2001 Raussendorf, Browne, Briegel J. Mod. Opt. (46) 2003
Non-determinism of measurements leads to probabilistic branching
Yay! Boo!
Yay! Boo!
Yay! Boo! Yay! Boo! Yay! Boo!Yay! Boo!
Attempt to control branching by using adaptive measurements:
(choice of later measurements depends on the outcome of earlier ones)
Non-determinism of measurements leads to probabilistic branching
Yay! Boo!
Yay! Boo!
Yay! Boo! Yay! Boo! Yay! Boo!Yay! Boo!
Attempt to control branching by using adaptive measurements:
(choice of later measurements depends on the outcome of earlier ones)
Can we carry out measurement-based computation deterministically?
Measurement Pattern
“low-level program”
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Flow and GFlow
“correction strategy” can possess
Uniformly Deterministic Pattern
jointly determine
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Flow and GFlow
“correction strategy” can possess
Uniformly Deterministic Pattern
jointly determine
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Flow and GFlow
“correction strategy” can possess not necessarily the same!
Uniformly Deterministic Pattern
jointly determine
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Flow and GFlow
“correction strategy” can possess not necessarily the same!
Quantum Circuit
translates to
Uniformly Deterministic Pattern
jointly determine with ancilla qubits
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state” implicitly defines
Flow and GFlow
“correction strategy” can possess not necessarily the same!
Quantum Circuit
translates to
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state”
Flow and GFlow
“correction strategy”
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state”
Flow and GFlow
“correction strategy”
Graphical form
“direct translation”
Minimal Graphical Form
“annotated geometry” rewrites to translation included in
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state”
Flow and GFlow
“correction strategy”
Quantum Circuit
if original pattern is deterministic “flow strategy”
Graphical form
“direct translation”
Minimal Graphical Form
“annotated geometry” rewrites to translation included in
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state”
Flow and GFlow
“correction strategy”
Circuit-like form
“gflow strategy” has flow
Quantum Circuit
if original pattern is deterministic “flow strategy”
Graphical form
“direct translation”
Minimal Graphical Form
“annotated geometry” rewrites to translation included in
Measurement Pattern
“low-level program”
Geometry
“entangled resource” “graph state”
Flow and GFlow
“correction strategy”
Thm: any Hopf algebra expression can be put into normal form:
Result: complicated MBQC implementation to simpler circuit specification.
ZX-calculus can demonstrate the correctness Quantum Error Correcting Codes: