Thorsten Altenkirch- Designing a Quantum Programming Language

8/3/2019 Thorsten Altenkirch- Designing a Quantum Programming Language

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Designing aQuantum Programming Language

Thorsten AltenkirchUniversity of Nottingham

based on joint work with Jonathan Grattage

supported by EPSRC grant GR/S30818/01

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Why . . .

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Why . . .

. . . design a quantum programming language?

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Why . . .


Better intuitive understanding of quantumalgorithms.

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Why . . .



Simplify the design of quantum programs.

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Why . . .




Develop formal reasoning principles for

quantum programs.

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Why . . .




Develop formal reasoning principles for

quantum programs.Understanding the computational aspect ofquantum physics.

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QML

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QMLQML: a first order functional quantum

programming language for finite types

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QML

QML: a first order functional quantum

programming language for finite typesReversible and irreversible quantum programs

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QML



Quantum data and control

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QML



Quantum data and controlOperational semantics: quantum circuits

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QML



Quantum data and controlOperational semantics: quantum circuits

Denotational semantics: superoperators

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Example: Hadamard operation

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Matrix

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Matrix

QML

"

#$

%&

'

(

)1

$

2

3

4

"

#$

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(

)1

$

2

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Related Work

P. Zuliani, PhD 2001, Quantum Programming (QGCL)

P. Selinger, MSCS 2003, Towards a Quantum

Programming Language(QPL)

A. van Tonder, SIAM 2003, A Lambda Calculus for

Quantum Computation

A. Sabry, Haskell 2003, Modeling quantum computing

in Haskell

P. Selinger and B. Valiron, TLCA 2005, A lambda

calculus for quantum computation with classical control

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Overview

1. Finite classical computation

2. Finite quantum computation

3. QML

4. Conclusions and further work

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3. QML


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Classical computations on finite types

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Quantum mechanics is time-reversible. . .

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. . . hence quantum computation is based on reversible

operations.

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operations.

However: Newtonian mechanics, Maxwellian

electrodynamics are also time-reversible. . .

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operations.

However: Newtonian mechanics, Maxwellian

electrodynamics are also time-reversible. . .

. . . hence classical computation should be based on

reversible operations.

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Classical computation ( )

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Given finite sets (input) and (output):

1

1



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Given finite sets (input) and (output):

1

1

a finite set of initial heaps ,

an initial heap

,

a finite set of garbage states ,

a bijection

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Composing computations



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Composing computations

1

bbbb

bbbb

VVVV

VVV

1

1

1



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Extensional equality



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A classical computation &

'

induces a function U

by

j

//

k

l

n

o

OO

U

//



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A classical computation &

'

induces a function U

by

j

//

k

l

n

o

OO

U

//

We say that two computations areextensionally equivalent, if they give rise tothe same function.



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Extensional equality . . .

Theorem:

U&

'

&

U'

&

U

'



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Theorem:

U&

'

&

U'

&

U

'

Theorem: Any function

on finite

sets can be realized by a computation.



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Example :

function

&

'

&

z

'



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Example :

function

&

'

&

z

'

computation

1

{

}



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Example :

function~

&

'

~

&

'



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Example :

function~

&

'

~

&

'

computation

1 '&%$!"#

&

' &

'

&

'

&

'

&

'

&

'



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3. QML basics

4. Compiling QML5. Conclusions and further work



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Pure quantum values



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Pure quantum values

A pure quantum value over a finite set is

given by by a vector represented as

with unit norm:% %

% %

%

%



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Qbits



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Qbits

Qbits are represented as

which

we write as&

' %

&

' %



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Qbits

Qbits are represented as

which

we write as&

' %

&

' %

Exambles of Qbits

%

%

%

%

%

%



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Tensor products



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Tensor products

Tensor product = product of bases:


T d


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Tensor products


E.g.


T d t


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Tensor products


E.g.

Examples of :

EPR

%

%

&

'

%

%

%

%


R ibl t ti


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Reversible quantum operations


R ibl t ti


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Reversible operations on pure quantum

values are represented by unitary operators.


R ibl t ti


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id


R sibl t ti s


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id

E.g.

&

'

&

'

&

'

&

'

&

&

' '

&

&

' '

&

&

' '

&

&

' '

where&

'

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Reversible quantum operations. . .

Example :

:




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Example :

:

%

%

&

%

%

'

%

%

&

%

%

'




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Example :

:

%

%

&

%

%

'

%

%

&

%

%

'

On finite-dimensional spaces:

Unitary = norm-preserving isomorphismQUOXIC Feb 05 p.21/4

Quantum computations ( )


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Quantum computations ( )Given finite sets (input) and (output):

1 1

a finite set , the base of the space of initialheaps,

a heap initialisation vector

,a finite set , the base of the space ofgarbage states,

a unitary operator

unitary .QUOXIC Feb 05 p.22/4

Composing quantum computations


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1

bbbb

bbbb

VVVV

VVV

1

1

1


Extensional equality?


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. . . is a bit more subtle.




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There is no (sensible) operator on vectorspaces replacing

.




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There is no (sensible) operator on vectorspaces replacing

.

Indeed: Forgetting part of a pure state

results in a mixed state.


Density matrizes


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Density matrizes


Density matrizes


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e s ty at es

Mixed states can be represented by densitymatrizes

.


Density matrizes


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y


.

Eigenvalues represent probabilities

System is in state

with prob.


Density matrizes


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y


.

Eigenvalues represent probabilities

System is in state

with prob.

Eigenvalues have to be positive and their sum(the trace) is

.


Example: forgetting a qbit


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p g g q




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p g g q

EPR is represented by

:

P

P

P

P




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p g g q

EPR is represented by

:

P

P

P

P

&

%

%

'

%

%


Example: forgetting a qbit . . .


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After measuring one qbit we obtain

:

P

P


Example: forgetting a qbit . . .


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After measuring one qbit we obtain

:

P

P

%

%

%

%


Superoperators


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Superoperators


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Morphisms on density matrizes are calledsuperoperators, these are linear maps, whichare

completely positive, and

trace preserving


Superoperators


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Morphisms on density matrizes are calledsuperoperators, these are linear maps, whichare

completely positive, and

trace preservingEvery unitary operator gives rise to a

superoperator

.


Superoperators. . .


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There is an operator

(

)

super

called partial trace.


Superoperators. . .


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There is an operator

(

)

super

called partial trace.

E.g.(

)

super

isrepresented by a

matrix.




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A quantum computation

givesrise to a superoperator U

super

j

//

o

OO

U

//




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A quantum computation

givesrise to a superoperator U

super

j

//

o

OO

U

//

We say that two computations areextensionally equivalent, if they give rise tothe same superoperator.




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Theorem:

U

&

'

&

U

'

&

U

'




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Theorem:

U

&

'

&

U

'

&

U

'

Theorem (?): Every superoperator

super (on finite dimensional Hilbertspaces) comes from a quantum computation.


Classical vs quantum


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classical (

) quantum (

)




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classical (

) quantum (

)

finite sets




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classical (

) quantum (

)

finite sets finite dimensional Hilbert spac




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classical (

) quantum (

)

finite sets finite dimensional Hilbert spaccartesian product (

)




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classical (

) quantum (

)


) tensor product (

)




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classical (

) quantum (

)


) tensor product (

)

bijections




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classical (

) quantum (

)


) tensor product (

)

bijections unitary operators




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classical (

) quantum (

)


) tensor product (

)


functions




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classical (

) quantum (

)


) tensor product (

)


functions superoperators




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classical (

) quantum (

)

finite sets finite dimensional Hilbert spacecartesian product (

) tensor product (

)



injective functions (

i

)




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classical (

) quantum (

)


) tensor product (

)




i

) isometries (

i

)




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classical (

) quantum (

)


) tensor product (

)




i

) isometries (

i

)

projections




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classical (

) quantum (

)


) tensor product (

)




i

) isometries (

i

)

projections partial trace


, classically


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~


, classically


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~

1 '&%$!"# 1

-

{

}


, classically


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~

1 '&%$!"# 1

-

{

}


, quantum


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1 '&%$!"# 1

-

{

}


, quantum


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1 '&%$!"# 1

-

{

}

input:

%

%

2


, quantum


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1 '&%$!"# 1

-

{

}

input:

%

%

2

output:

%

2

%

2


, quantum


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1 '&%$!"# 1

-

{

}

input:

%

%

2

output:

%

2

%

2

Decoherence!


Control of decoherence


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QML is based on strict linear logic




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Contraction is implicit and realized by

.




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Contraction is implicit and realized by

.Weakening is explicit and leads todecoherence.


3. QML

Fi i l i l i


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2. Finite quantum computation3. QML



QML overview


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QML overview

T


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Types

%

%


QML overview

T


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Types

%

%

Terms

%

3

%

% & ' %&

' %

3

&

z

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%

)

%

4

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) z

2

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4

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2

% &

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Qbits


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(

)1

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& '

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#$

)

& '

3

4

4

"

%

)

2

3

4

4

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QML overview . . .


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QML overview . . .

Typing judgements


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Typing judgements

programs

strict programs


QML overview . . .

Typing judgements


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Typing judgements

programs

strict programs

Semantics


The let-rule


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"

$

(


The let-rule


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"

$

(

YYYY

Y

1

1

YYYYYY

WWWWWW

1

1

1


on contexts


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on contexts


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&

'

&

'

if

dom


on contexts


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&

'

&

'

if

dom

1


Another source of decoherence


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mentions


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mentions

(

)1$

3

4

(

)1

$



mentions


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e t o s

(

)1$

3

4

(

)1

$

but doesnt use it.



mentions


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(

)1$

3

4

(

)1

$

but doesnt use it.

Hence, it has to measure it!


-elim


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-elim


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-elim


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XXXX

X

!

"

$

1 $

%

&

'

(

0

1

2

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%

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hhhh

hh51

!

6

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zzzzzz

5

%

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-elim decoherence-free


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8


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9

9

@

9

8

9



8


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9

9

@

9

8

9

XXXX

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zzzzzz5

C

1


I

P


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I

P

QRS T

U

V

W

Y

Y


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QRS T

U

V

a

b

cd

e

a

f

g

hp

q

r

su

v h

w

x h

q

r

su

v


I

P

QRS T

U

V

W

Y

Y


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QRS T

U

V

a

b

cd

e

a

f

g

hp

q

r

su

v h

w

x h

q

r

su

v

This program has a type error, becausey

y

.


I

P

QRS T

U

V

W

Y

Y


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QRS T

U

V

a

b

cd

e

a

f

g

hp

q

r

su

v h

w

x h

q

r

su

v


y

.

Q

U

W

Y

Y

Q

U

a

b

cd

e

a

f

g

hp

q

v h

w

x h

q

r

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v


I

P

QRS T

U

V

W

Y

Y


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QRS T

U

V

a

b

cd

e

a

f

g

hp

q

r

su

v h

w

x h

q

r

su

v


y

.

Q

U

W

Y

Y

Q

U

a

b

cd

e

a

f

g

hp

q

v h

w

x h

q

r

su

v

This program typechecks, becausey

y

.


4. Conclusions



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3. QML



Conclusions


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Conclusions

Our semantic ideas proved useful whendesigning a quantum programming language,analogous concepts are modelled by the


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a a ogous co cepts a e ode ed by t e

same syntactic constructs.


Conclusions



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g p y

same syntactic constructs.Our analysis also highlights the differencesbetween classical and quantum

programming.


Conclusions



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g p y

same syntactic constructs.Our analysis also highlights the differencesbetween classical and quantum

programming.Quantum programming introduces theproblem of control of decoherence, which we

address by making forgetting variablesexplicit and by having different if-then-elseconstructs.


Further work


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Further work

Show that the operational semantics iscompositional by showing that it agrees with


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compositional by showing that it agrees with

the denotational semantics.


Further work



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the denotational semantics.mostly done


Further work



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An algebra of quantum programs, which is

complete wrt. the denotational semantics.


Further work



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complete wrt. the denotational semantics.in progress


Further work



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Add higher order types.


Further work



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p y g g




Add higher order types.sketchy


Further work



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p y g g





Indexed types and programs


Further work



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p y g g





Indexed types and programs

sketchy QUOXIC Feb 05 p.48/4

Thank you for your attention.

Draft papers, available from//www.cs.nott.ac.uk/txa/publ/


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A functional quantum programming language

with J.Grattage

Structuring Quantum Effects: Superoperators as Arrowswith J.Vizzotto and A.Sabry

A compiler for a functional quantum programming langua

with J.GrattageQML: Quantum data and control with J.Grattage


Thorsten Altenkirch- Designing a Quantum Programming Language

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