Towards a quantum lambda-calculus with quantum control

Towards a quantum λ-calculus
with quantum control
arXiv:1601.04294
Alejandro Díaz-Caro
UNIVERSIDAD NACIONAL DE QUILMES
Joint work with
Gilles Dowek
Inria & ENS-Cachan
V Congreso Latinoamericano de Matemáticos
Logic and Computability Session
Barranquilla, Colombia, July 14, 2016

Goal
We want a pure functional
extension of lambda calculusi.e. we do not want clasical control / quantum data
Alejandro Díaz-Caro (UNQ) Towards a quantum lambda calculus with quantum control 2 / 26

Overview
Some quantum properties (with dead and alive cats)
Projective measurement
Destructive interference
No-cloning
Entanglement and separability
Expressing those properties in the lambda-calculus
Superpositions, no-cloning and measurement
Examples
Deutsch algorithm
Teleportation algorithm

α + β

α + β
|α|2
|β| 2

Probabilistic vs. Quantum
Probabilistic
+

Probabilistic
a + b
(a + b = 1)

Probabilistic
a + b
(a + b = 1)
Quantum
α + β
α|2
+ |β
2
= 1

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1
1
2
1
2
+
1
2
+
1
2

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1
1
2
1
2
+
1
2
+
1
2
α + β

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1
1
2
1
2
+
1
2
+
1
2
3
4
+
1
4

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1
1
2
1
2
+
1
2
+
1
2
3
4
+
1
4
5
8
+
3
8

Probabilistic
a + b
(a + b = 1)
Quantum
α − β
α|2
+ |−β
2
= 1
1
2
1
2
+
1
2
+
1
2
3
4
+
1
4
5
8
+
3
8
1
√
2
1
√
2
+
1
√
2
+
1
√
2
1
√
2
−
1
√
2

No-cloning
Superpositions vs. basis states
There is no universal cloning machine
for quantum states

No-cloning
for quantum states
α + β

No-cloning
for quantum states
α + β
α + β
α ⊗ + β ⊗
=
α + β ⊗ α + β
α2
⊗ +αβ ⊗ +βα ⊗ +β2
⊗

Example 1
α ⊗ +β ⊗
=
α + β
Superposed state
⊗
Basis state

Example 1
α ⊗ +β ⊗
=
α + β
Superposed state
⊗
Basis stateExample 2
α1α2 ⊗ +α1β1 ⊗ +β2α2 ⊗ +β1β2 ⊗
α1 + β1
Superposed state
⊗ α2 + β2
Superposed state

Example 1
α ⊗ +β ⊗
=
α + β
Superposed state
⊗
Basis stateExample 2
α1α2 ⊗ +α1β1 ⊗ +β2α2 ⊗ +β1β2 ⊗
α1 + β1
Superposed state
⊗ α2 + β2
Superposed state
Example 3
α ⊗ + β ⊗
Entangled (and superposed) state

Logical linearity vs. algebraic linearity
No-cloning =⇒ logical-linear terms
e.g. λx.x ⊗ x forbidden

Another way
No-cloning =⇒ algebraic-linear operators
e.g. M(α. |0 + β. |1 ) → α.M |0 + β.M |1

Another way
e.g. M(α. |0 + β. |1 ) → α.M |0 + β.M |1
What about measurement?
(λx.πx) (α. |0 + β. |1 )
(Measurement operator)
α.(λx.πx) |0 + β.(λx.πx) |1 ← Wrong!

Another way
e.g. M(α. |0 + β. |1 ) → α.M |0 + β.M |1
What about measurement?
(λx.πx) (α. |0 + β. |1 )
(Measurement operator)
α.(λx.πx) |0 + β.(λx.πx) |1 ← Wrong!
We can use a combination of both:
Logical-linear for abstractions taking superpositions
Algebraic-linear for abstractions taking basis states

Key point
We need to distinguish
superposed states
from
basis states
Basis states can be cloned
Superposed states cannot

Grammars
First version, without tensor
Types
Ψ := Q | S(Ψ) Qubit types
A := Ψ | Ψ ⇒ A | S(A) Types
Terms
b := x | λxΨ
.t | |0 | |1 Basis terms
v := b | v + v | α.v | 0S(A) Values
t := v | tt | t + t | α.t | πt | ?· Terms
where α ∈ C

Two types of linearity
(λxQ
.t) b
Q
→ t[b/x] call-by-base
(λxS(Ψ)
.t)
linear abstraction
u
S(Ψ)
→ t[u/x] call-by-name
(λxQ
.t) (b1 + b2)
S(Q)
→ (λxQ
.t) b1
Q
+(λxQ
.t) b2
Q
linear distribution

(λxQ
.t) b
Q
(λxS(Ψ)
.t)
linear abstraction
u
S(Ψ)
(λxQ
.t) (b1 + b2)
S(Q)
→ (λxQ
.t) b1
Q
+(λxQ
.t) b2
Q
linear distribution
Problem?
λxQ⇒Q
.x(|0 +|1 ) : (Q ⇒ Q) ⇒ Q Non-linear! (not a superposition)

(λxQ
.t) b
Q
(λxS(Ψ)
.t)
linear abstraction
u
S(Ψ)
(λxQ
.t) (b1 + b2)
S(Q)
→ (λxQ
.t) b1
Q
+(λxQ
.t) b2
Q
linear distribution
Problem?
λxQ⇒Q
No problem
It is a function which produces a superposition, is not a superposition
It can be cloned

(λxQ
.t) b
Q
(λxS(Ψ)
.t)
linear abstraction
u
S(Ψ)
(λxQ
.t) (b1 + b2)
S(Q)
→ (λxQ
.t) b1
Q
+(λxQ
.t) b2
Q
linear distribution
Problem?
λxQ⇒Q
No problem
It can be cloned
What about λyS(Q)
.λxQ
.xy ?

(λxQ
.t) b
Q
(λxS(Ψ)
.t)
linear abstraction
u
S(Ψ)
(λxQ
.t) (b1 + b2)
S(Q)
→ (λxQ
.t) b1
Q
+(λxQ
.t) b2
Q
linear distribution
Problem?
λxQ⇒Q
No problem
It can be cloned
What about λyS(Q)
.λxQ
.xy ?
Ok, let’s stay in ﬁrst order for now

Typing applications
Γ t : Ψ ⇒ A ∆ u : Ψ
Γ, ∆ tu : A
⇒E

Typing applications
Γ t : Ψ ⇒ A ∆ u : Ψ
Γ, ∆ tu : A
⇒E
What about (λxQ
.t) (b1 + b2)
S(Q)
?

Typing applications
Γ t : Ψ ⇒ A ∆ u : Ψ
Γ, ∆ tu : A
⇒E
What about (λxQ
.t) (b1 + b2)
S(Q)
?
Γ t : Ψ ⇒ A ∆ u : S(Ψ)
Γ, ∆ tu : S(A)

Typing applications
Γ t : Ψ ⇒ A ∆ u : Ψ
Γ, ∆ tu : A
⇒E
What about (λxQ
.t) (b1 + b2)
S(Q)
?
Γ t : Ψ ⇒ A ∆ u : S(Ψ)
Γ, ∆ tu : S(A)
What about ((λxQ
.t) + (λyQ
.u))
S(Q⇒A)
v?

Typing applications
Γ t : Ψ ⇒ A ∆ u : Ψ
Γ, ∆ tu : A
⇒E
What about (λxQ
.t) (b1 + b2)
S(Q)
?
Γ t : Ψ ⇒ A ∆ u : S(Ψ)
Γ, ∆ tu : S(A)
What about ((λxQ
.t) + (λyQ
.u))
S(Q⇒A)
v?
Γ t : S(Ψ ⇒ A) ∆ u : S(Ψ)
Γ, ∆ tu : S(A)
⇒ES

Example
f : Q ⇒ A g : Q ⇒ A
f + g : S(Q ⇒ A)
S+
I
|0 : Q
Ax|0
|0 : S(Q)
(f + g) |0 : S(A)
⇒ES
⇓
f : Q ⇒ A |0 : Q
Ax|0
f |0 : A
⇒E
g : Q ⇒ A |0 : Q
Ax|0
g |0 : A
⇒E
f |0 + g |0 : S(A)
S+
I

Measurement
π(
n
i=1
[αi.]bi) −→ |αk|2
n
i=1 |αi|2
bk
∀i, bi = |0 or bi = |1 .
n
i=1 αi .bi is normal (and hence 1 ≤ n ≤ 2).
k ≤ n
Example
π(i. |0 + 2. |1 )
|0
|1
1
3
2
3

Adding tensor products
Intepretation of types
S(Q) vs. Q
Q = {|0 , |1 } ⊆ C2
A ⊗ B = A × B
S(A) = G A

Adding tensor products
Intepretation of types
S(Q) vs. Q
Q = {|0 , |1 } ⊆ C2
A ⊗ B = A × B
S(A) = G A
Examples
(1/
√
2. |0 + 1/
√
2. |1 ) ⊗ |0 ∈ S(Q) ⊗ Q
= G({|0 , |1 }) × {|0 , |1 }
= C2
× {|0 , |1 }
1/
√
2. |0 ⊗ |0 + 1/
√
2. |1 ⊗ |1 ∈ S(Q ⊗ Q)
= G({|0 , |1 } × {|0 , |1 })
= G({|0 ⊗ |0 , |0 ⊗ |1 , |1 ⊗ |0 , |1 ⊗ |1 })
= C2
⊗ C2

Some information is lost on reduction
Subtyping
{|0 , |1 } ⊂ C2
then Q ≤ S(Q)
G(GA) = GA then S(S(Q)) ≤ S(Q)

Subtyping
{|0 , |1 } ⊂ C2
then Q ≤ S(Q)
{|0 , |1 } × C2
⊂ C2
⊗ C2
then Q ⊗ S(Q) ≤ S(Q ⊗ Q)

Subtyping
{|0 , |1 } ⊂ C2
then Q ≤ S(Q)
{|0 , |1 } × C2
⊂ C2
⊗ C2
|0 ⊗ (|0 + |1 ) : Q ⊗ S(Q)
|0 ⊗ |0 + |0 ⊗ |1 : S(Q ⊗ Q)

Subtyping
{|0 , |1 } ⊂ C2
then Q ≤ S(Q)
{|0 , |1 } × C2
⊂ C2
⊗ C2
|0 ⊗ (|0 + |1 ) : Q ⊗ S(Q)
|0 ⊗ |0 + |0 ⊗ |1 : S(Q ⊗ Q)
Same happens in math!
(X − 1)(X − 2) −→ X2
− 3X + 2
we lost the information that it was a product

Subtyping
{|0 , |1 } ⊂ C2
then Q ≤ S(Q)
{|0 , |1 } × C2
⊂ C2
⊗ C2
|0 ⊗ (|0 + |1 ) : Q ⊗ S(Q)
|0 ⊗ |0 + |0 ⊗ |1 : S(Q ⊗ Q)
Same happens in math!
(X − 1)(X − 2) −→ X2
− 3X + 2
we lost the information that it was a product
Solution: casting
|0 ⊗ (|0 + |1 ) |0 ⊗ |0 + |0 ⊗ |1
⇑
S(Q⊗Q)
Q⊗S(Q) |0 ⊗ (|0 + |1 ) → |0 ⊗ |0 + |0 ⊗ |1

Full grammars
Types
Q := Q | Q ⊗ Q Basis qubit types
Ψ := Q | S(Ψ) | Ψ ⊗ Ψ Qubit types
A := Ψ | Ψ ⇒ A | S(A) | A ⊗ A Types
Terms
b := x | λxΨ
.t | |0 | |1 | b ⊗ b Basis terms
v := b | v + v | α.v | 0S(A) | v ⊗ v Values
t := v | tt | t + t | α.t | πj t | ?·
| t ⊗ t | head t | tail t | ⇑
S(B⊗C)
S(A) t
Terms
where α ∈ C

Measurement of the ﬁrst j qubits
πj (
n
i=1
[αi .](b1i ⊗ · · · ⊗ bmi ))
−→


i∈P



|αi |2
n
i=1
|αi |2






j
h=1
bhk ⊗
i∈P




αi
i∈P
|αi |2



 .(bj+1,i ⊗ · · · ⊗ bmi )
P ⊆ N≤n
, such that
∀i ∈ P, ∀h ≤ j,
bhi = bhk .
Example
π2( 2.(|0 ⊗ |1 ⊗ |1 ) + |0 ⊗ |1 ⊗ |0 + 3.(|1 ⊗ |1 ⊗ |1 ) )
|0 ⊗ |1 ⊗ ( 2√
5
. |1 + 1√
5
. |0 ) |1 ⊗ |1 ⊗ (1. |1 )

Deutsch algorithm
Preliminaries
Hadamard
H |0 =
1
√
2
|0 +
1
√
2
|1 H |1 =
1
√
2
|0 −
1
√
2
|1

Deutsch algorithm
Preliminaries
Hadamard
H |0 =
1
√
2
|0 +
1
√
2
|1 H |1 =
1
√
2
|0 −
1
√
2
|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )

Deutsch algorithm
Preliminaries
Hadamard
H |0 =
1
√
2
|0 +
1
√
2
|1 H |1 =
1
√
2
|0 −
1
√
2
|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )
Oracle
A “black box” implementing a function f : {0, 1} → {0, 1}
Uf (|x ⊗ |y ) = |x ⊗ |y ⊕ f (x)

Deutsch algorithm
Preliminaries
Hadamard
H |0 =
1
√
2
|0 +
1
√
2
|1 H |1 =
1
√
2
|0 −
1
√
2
|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )
Oracle
Uf (|x ⊗ |y ) = |x ⊗ |y ⊕ f (x)
not = λxQ
.x?|0 ·|1

Deutsch algorithm
Preliminaries
Hadamard
H |0 =
1
√
2
|0 +
1
√
2
|1 H |1 =
1
√
2
|0 −
1
√
2
|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )
Oracle
Uf (|x ⊗ |y ) = |x ⊗ |y ⊕ f (x)
not = λxQ
.x?|0 ·|1
Uf = λxQ⊗Q
.(head x) ⊗ ((tail x)?not(f (head x))·f (head x))

Deutsch algorithm
Goal:
Given an oracle Uf determine whether f is constant or not

Deutsch algorithm
Goal:
|0 H
Uf
H
|1 H

Deutsch algorithm
Goal:
|0 H
Uf
H ± |f (0) ⊕ f (1)
|1 H
1√
2
|0 − 1√
2
|1

Deutsch algorithm
Goal:
|0 H
Uf
H ± |f (0) ⊕ f (1)
|1 H
1√
2
|0 − 1√
2
|1
If f constant, f (0) ⊕ f (1) = 0
± |0 ⊗
1
√
2
|0 −
1
√
2
|1
If f not constant, f (0) ⊕ f (1) = 1
± |1 ⊗
1
√
2
|0 −
1
√
2
|1

Deutsch in λ
|0 H
Uf
H ± |f (0) ⊕ f (1)
|1 H
1√
2
|0 − 1√
2
|1
not = λxQ
.x?|0 ·|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )
Hboth = λxQ⊗Q
.(H(head x)) ⊗ (H(tail x))
Uf = λxQ⊗Q
H1 = λxQ⊗Q
.(H(head x)) ⊗ (tail x)
Deutschf = π1(⇑
S(Q⊗Q)
S(S(Q)⊗Q) H1(Uf ⇑
S(Q⊗Q)
S(Q⊗S(Q))⇑
S(Q⊗S(Q))
S(S(Q)⊗S(Q)) Hboth(|0 ⊗ |1 )

Deutsch in λ
|0 H
Uf
H ± |f (0) ⊕ f (1)
|1 H
1√
2
|0 − 1√
2
|1
not = λxQ
.x?|0 ·|1
H = λxQ
.1/
√
2.(|0 + x?−|1 ·|1 )
Hboth = λxQ⊗Q
.(H(head x)) ⊗ (H(tail x))
Uf = λxQ⊗Q
H1 = λxQ⊗Q
.(H(head x)) ⊗ (tail x)
Deutschf = π1(⇑
S(Q⊗Q)
S(S(Q)⊗Q) H1(Uf ⇑
S(Q⊗Q)
S(Q⊗S(Q))⇑
S(Q⊗S(Q))
S(S(Q)⊗S(Q)) Hboth(|0 ⊗ |1 )
Deutschf : Q ⊗ Q ⇒ Q ⊗ S(Q)
Deutschid −→∗
(1) π1(1/
√
2. |1 ⊗ |0 − 1/
√
2. |1 ⊗ |1 )
−→(1) |1 ⊗ (1/
√
2. |0 − 1/
√
2. |1 )

Teleportation
Goal:
To send a qubit, using an entangled pair, by sending only two bits
of information
Alice
|ψ • H
β00
Zb1
notb2 |ψ
Bob
where β00 = 1√
2
|0 ⊗ |0 + 1√
2
|1 ⊗ |1

Teleportation in λ
Alice
|ψ • H
β00
Zb1
notb2 |ψ
Bob
Teleportation : S(Q) ⇒ S(Q)
Teleportation q −→(1) q
Alice =
λxS(Q)⊗S(Q⊗Q)
.π2(⇑
S(Q⊗Q⊗Q)
S(S(Q)⊗Q⊗Q) H3
1 (cnot3
12 ⇑
S(Q⊗Q⊗Q)
S(Q⊗S(Q⊗Q))⇑
S(Q⊗S(Q⊗Q))
S(S(Q)⊗S(Q⊗Q)) x))
Ub
= λbQ
.λxQ
.b?Ux·x
Bob = λxQ⊗Q⊗Q
.Zhead x
nothead tail x
.(tail tail x)
β00 = 1/
√
2. |0 ⊗ |0 + 1/
√
2. |1 ⊗ |1
Teleportation = λqS(Q)
.Bob (⇑
S(Q⊗Q⊗Q)
S(Q⊗Q⊗S(Q)) Alice x ⊗ β00)

Summarising
Extension of (pure) ﬁrst-order lambda-calculus for
quantum computing
Logical-linearity and algebraic-linearity both used for
no-cloning
Denotational semantics:
Types: sets of vectors or vector spaces
Terms: vectors
If Γ t : A then t φΓ
⊆ A

Towards a quantum lambda-calculus with quantum control

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