a functional quantum programming languagepsztxa/talks/slides-tfp04.pdfwhat you always wanted to know...

A Functional QuantumProgramming Language

Thorsten Altenkirch

University of Nottingham

based on joint work with Jonathan Grattage

and discussions with V.P. Belavkin

supported by EPSRC grant GR/S30818/01

A Functional Quantum Programming Language – p.1/??

Alternative title

What you always wanted to know

about quantum computation

but never dared to ask.

A Functional Quantum Programming Language – p.2/??

Another alternative title

Quantum programming

for the

lazy functional programmer

A Functional Quantum Programming Language – p.3/??

Background

Simulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in� � � �

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in� � � �

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in� � � �

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in� � � �

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

BackgroundSimulation of quantum systems is expensive:Exponential time to simulate polynomial circuits.

Feynman: Can we exploit this fact to performcomputations more efficiently?

Shor: Factorisation in quantum polynomial time.

Grover: Blind search in� � � �

Can we build a quantum computer?

yes We can run quantum algorithms.

no Nature is classical after all!

Assumption: Nature is fair. . .

A Functional Quantum Programming Language – p.4/??

The quantum software crisis

Quantum algorithms are usually presentedusing the circuit model.

Nielsen and Chuang, p.7, Coming up withgood quantum algorithms is hard.Richard Josza, QPL 2004: We need todevelop quantum thinking!

A Functional Quantum Programming Language – p.5/??

The quantum software crisis

Quantum algorithms are usually presentedusing the circuit model.

Nielsen and Chuang, p.7, Coming up withgood quantum algorithms is hard.Richard Josza, QPL 2004: We need todevelop quantum thinking!

A Functional Quantum Programming Language – p.5/??

The quantum software crisis

Quantum algorithms are usually presentedusing the circuit model.

Nielsen and Chuang, p.7, Coming up withgood quantum algorithms is hard.

Richard Josza, QPL 2004: We need todevelop quantum thinking!

A Functional Quantum Programming Language – p.5/??

The quantum software crisis

Quantum algorithms are usually presentedusing the circuit model.

Nielsen and Chuang, p.7, Coming up withgood quantum algorithms is hard.Richard Josza, QPL 2004: We need todevelop quantum thinking!

A Functional Quantum Programming Language – p.5/??

QML

QML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation

Finite classical computations

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation

Finite classical computations

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation

Finite classical computations

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation

Finite classical computations

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation� � �

Finite classical computations� �

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation� � �

Finite classical computations� �

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

QMLQML: a first-order functional language for quantumcomputations on finite types.

Quantum control and quantum data.

Design guided by semantics

Analogy with classical computation� � �

Finite classical computations� �

Finite quantum computations

Important issue: control of decoherence

Compiler under construction (Jonathan)

A Functional Quantum Programming Language – p.6/??

Example: Hadamard operation

Matrix

QML

A Functional Quantum Programming Language – p.7/??

Example: Hadamard operation

Matrix

� ��

� �� �

QML

A Functional Quantum Programming Language – p.7/??

Example: Hadamard operation

Matrix

� ��

� �� �

QML �� �� � � ��� �� � �� � ������ ��� ��� �!" #$ � % � &(') " *

� +�, � ��� ��� �!" # � &(') " *

A Functional Quantum Programming Language – p.7/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Something we know well . . .

Classical computations on finite types.

Quantum mechanics is time-reversible. . .

. . . hence quantum computation is based on reversibleoperations.

However: Newtonian mechanics, Maxwellianelectrodynamics are also time-reversible. . .

. . . hence classical computation should be based onreversible operations.

A Functional Quantum Programming Language – p.8/??

Classical computation ( )

Given finite sets (input) and (output):

� �

a finite set of initial heaps ,

an initial heap ,

a finite set of garbage states ,

a bijection ,

A Functional Quantum Programming Language – p.9/??

Classical computation ( )

Given finite sets (input) and (output):

- ./

0 � 1 2 �

a finite set of initial heaps ,

an initial heap ,

a finite set of garbage states ,

a bijection ,

A Functional Quantum Programming Language – p.9/??

Classical computation ( )

Given finite sets (input) and (output):

- ./

0 � 1 2 �

a finite set of initial heaps ,

an initial heap354 ,

a finite set of garbage states ,

a bijection 4 6 7 6 ,

A Functional Quantum Programming Language – p.9/??

Composing computations

�

>>>>

>>>>

8888

888 �

�

��������

������� �

A Functional Quantum Programming Language – p.10/??

Composing computations

8. 98 �

>>>>

>>>>

8888

888 8�

9 �

��������

������� 9�

9 � 8

A Functional Quantum Programming Language – p.10/??

Extensional equality

A classical computationinduces a function U by

//

��

OO

U//

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

A Functional Quantum Programming Language – p.11/??

Extensional equality

A classical computation : �$ ; 3 ; ; %

induces a function U : 4 by

6 < // 6=?>

��

@BADC E FOO

U 8 //

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

A Functional Quantum Programming Language – p.11/??

Extensional equality

A classical computation : �$ ; 3 ; ; %

induces a function U : 4 by

6 < // 6=?>

��

@BADC E FOO

U 8 //

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

A Functional Quantum Programming Language – p.11/??

Extensional equality . . .

Theorem:

U

$ G : % �$ U

% G$ U : %

Hence, classical computations upto extensionalequality give rise to the category .

Theorem: Any function on finite setscan be realized by a computation.

Translation for Category Theoreticans:U is full and faithful.

A Functional Quantum Programming Language – p.12/??

Extensional equality . . .

Theorem:

U

$ G : % �$ U

% G$ U : %

Hence, classical computations upto extensionalequality give rise to the category .

Theorem: Any function on finite setscan be realized by a computation.

Translation for Category Theoreticans:U is full and faithful.

A Functional Quantum Programming Language – p.12/??

Extensional equality . . .

Theorem:

U

$ G : % �$ U

% G$ U : %

Hence, classical computations upto extensionalequality give rise to the category .

Theorem: Any function4

on finite sets; can be realized by a computation.

Translation for Category Theoreticans:U is full and faithful.

A Functional Quantum Programming Language – p.12/??

Extensional equality . . .

Theorem:

U

$ G : % �$ U

% G$ U : %

Hence, classical computations upto extensionalequality give rise to the category .

Theorem: Any function4

on finite sets; can be realized by a computation.

Translation for Category Theoreticans:U is full and faithful.

A Functional Quantum Programming Language – p.12/??

Example � :function HJI 4 $ � ; � % �

HKI $ � ; L % � �

computation

�

A Functional Quantum Programming Language – p.13/??

Example � :function HJI 4 $ � ; � % �

HKI $ � ; L % � �

computation � �

� �

/NMO

A Functional Quantum Programming Language – p.13/??

Example :

function P4 � $ � ; � %

P� �$ � ;� %

computation

� '&%$ !"#

A Functional Quantum Programming Language – p.14/??

Example :

function P4 � $ � ; � %

P� �$ � ;� %

computation Q � � R Q � �

S � � � '&%$ !"# Q � �

T T 4 $ � ; � % $ � ; � %

T$ S ;� % �$ S ;� %

T$ � ;� % �$ � ; U � %

A Functional Quantum Programming Language – p.14/??

Classical vs quantum

classical ( ) quantum ( )

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets

finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections

unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V )

tensor product ( )

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V ) tensor product ( W)

functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V ) tensor product ( W)functions

superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V ) tensor product ( W)functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V ) tensor product ( W)functions superoperators

projections

partial trace

A Functional Quantum Programming Language – p.15/??

Classical vs quantum

classical (

� � �

) quantum (� �

)

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( V ) tensor product ( W)functions superoperators

projections partial trace

A Functional Quantum Programming Language – p.15/??

� X , classically

HDI G P� � �

� '&%$ !"# �

A Functional Quantum Programming Language – p.16/??

� X , classically

HDI G P� � �Q � � R Q � �

S � � � '&%$ !"# �

/ZY /NMO

A Functional Quantum Programming Language – p.16/??

� X , classically

HDI G P� � �Q � � R Q � �

S � � � '&%$ !"# �

/ZY /NMO[

� �

A Functional Quantum Programming Language – p.16/??

� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MO

input:

output:

Decoherence!

A Functional Quantum Programming Language – p.17/??

� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MOinput:

� I ^ � # S _ I^ � # � _ *

output:

Decoherence!

A Functional Quantum Programming Language – p.17/??

� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MOinput:

� I ^ � # S _ I^ � # � _ *output:

I � � # S _ * I � � # � _ *

Decoherence!

A Functional Quantum Programming Language – p.17/??

� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MOinput:

� I ^ � # S _ I^ � # � _ *output:

I � � # S _ * I � � # � _ *

Decoherence!

A Functional Quantum Programming Language – p.17/??

Control of decoherence

QML is based on strict linear logic

Contraction is implicit and realized by .

Weakening is explicit and leads todecoherence.

A Functional Quantum Programming Language – p.18/??

Control of decoherence

QML is based on strict linear logic

Contraction is implicit and realized by .

Weakening is explicit and leads todecoherence.

A Functional Quantum Programming Language – p.18/??

Control of decoherence

QML is based on strict linear logic

Contraction is implicit and realized by T.

Weakening is explicit and leads todecoherence.

A Functional Quantum Programming Language – p.18/??

Control of decoherence

QML is based on strict linear logic

Contraction is implicit and realized by T.Weakening is explicit and leads todecoherence.

A Functional Quantum Programming Language – p.18/??

QML overview

Types

Terms

A Functional Quantum Programming Language – p.19/??

QML overview

Types ` � � # ` a # ` a

Terms

A Functional Quantum Programming Language – p.19/??

QML overview

Types ` � � # ` a # ` aTerms b �� # +� �� � b �� c # � �d# $ % #$ b ; c % # +� �$ � ; L % � b �� c# � egf b # � e f ' c#ih j, � b(k � ��� e f � c # � e f ' L c l *

#ih j, � � b(k � ��� e f � c # � e f ' L c l *

# �$ m % b # $�n % c *

A Functional Quantum Programming Language – p.19/??

Qbits

� � � �

� &') " � � egf $ %

� ��� �!" � � egf ' $ %

�� b ����� c � +, � c l

�h j, � ��� e f c # � e f ' c l *

�� � b ���� � c � +, � c l

�h j, � � ��� e f c # � e f ' c l *

A Functional Quantum Programming Language – p.20/??

QML overview . . .

Typing judgementsprograms

strict programs

Semantics

A Functional Quantum Programming Language – p.21/??

QML overview . . .

Typing judgementso pq � ` programso p � q � ` strict programs

Semantics

A Functional Quantum Programming Language – p.21/??

QML overview . . .

Typing judgementso pq � ` programso p � q � ` strict programs

Semantics o pq � `rq s 4 r o s r ` s

o p � q � `rq s 4 � r o s r ` s

A Functional Quantum Programming Language – p.21/??

The let-rule

o pq � `; Q � ` p�t � a " &o pvu w x Q �q y{z t � a

;;;;

;

�

�����

�

;;;;

;;

9999

99�

�

������

������ �

A Functional Quantum Programming Language – p.22/??

The let-rule

o pq � `; Q � ` p�t � a " &o pvu w x Q �q y{z t � a

o |~}�� � �;;

;;;

�

��C � � � ����� � � �

� �

;;;;

;;

9999

99 ��

� �

������

������ ��

A Functional Quantum Programming Language – p.22/??

on contexts

if dom

�

A Functional Quantum Programming Language – p.23/??

on contexts

o ; Q � ` ; Q � ` � $ o % ; Q � `o ; Q � ` � $ o % ; Q � ` if Q �4

domR �

�

A Functional Quantum Programming Language – p.23/??

on contexts

o ; Q � ` ; Q � ` � $ o % ; Q � `o ; Q � ` � $ o % ; Q � ` if Q �4

domR �

o |~}�� � o

�C � �

A Functional Quantum Programming Language – p.23/??

Another source of decoherence

mentions

but doesn’t use it.

Hence, it has to measure it!

A Functional Quantum Programming Language – p.24/??

Another source of decoherence

��� � b mentions���� � b� � � �

��� � b� � �� � ����� � &') " � +, � � &(') "

but doesn’t use it.

Hence, it has to measure it!

A Functional Quantum Programming Language – p.24/??

Another source of decoherence

��� � b mentions���� � b� � � �

��� � b� � �� � ����� � &') " � +, � � &(') "

but doesn’t use it.

Hence, it has to measure it!

A Functional Quantum Programming Language – p.24/??

Another source of decoherence

��� � b mentions���� � b� � � �

��� � b� � �� � ����� � &') " � +, � � &(') "

but doesn’t use it.

Hence, it has to measure it!

A Functional Quantum Programming Language – p.24/??

-elim

::::

:

�

�����

�

�

DDDD

DD�

�

zzzzzz �

A Functional Quantum Programming Language – p.25/??

-elim

� ���� � ����� �� � ��� ��� �� � ��� � � �¡  ¢£¥¤� W � ��¦§ ¨© � ª « ¬®­°¯ ± � ²� ³ ­°¯ ´ � ² � µ� �

::::

:

�

�����

�

�

DDDD

DD�

�

zzzzzz �

A Functional Quantum Programming Language – p.25/??

-elim

� ���� � ����� �� � ��� ��� �� � ��� � � �¡  ¢£¥¤� W � ��¦§ ¨© � ª « ¬®­°¯ ± � ²� ³ ­°¯ ´ � ² � µ� �

� W � /�¶¸·º¹ » �::

:::

/½¼¾ ¿ÁÀ Â1 �Äà � � � ����� /ÄÅ� Æ � �

Ç¸È ÉËÊ�1Å �

DDDD

DD

2�1¾¸Ì À �

zzzzzz

2ÍÅ�

A Functional Quantum Programming Language – p.25/??

-elim decoherence-free

::::

:

�

�����

�

DDDD

DD�

�

zzzzzz �

A Functional Quantum Programming Language – p.26/??

-elim decoherence-free

� �Î �� � � ��� �� � �Ï �� ��� �� � �Ï � � � � Ð� ��Ñ   ¢£¥¤ Ï� W � �Î ¦§ ¨ © Ï � ª « ¬®­°¯ ± � ²� ³ ­ ¯ ´ � ² � µ� �

::::

:

�

�����

�

DDDD

DD�

�

zzzzzz �

A Functional Quantum Programming Language – p.26/??

-elim decoherence-free

� �Î �� � � ��� �� � �Ï �� ��� �� � �Ï � � � � Ð� ��Ñ   ¢£¥¤ Ï� W � �Î ¦§ ¨ © Ï � ª « ¬®­°¯ ± � ²� ³ ­ ¯ ´ � ² � µ� �

� W � /Ķ¸·º¹ » �

::::

:

/½¼Ò ¿ÁÓ Â1 �Äà � � � ����� /ÕÔ� Æ � Ö /¾ ×À �

ÇØÈ ÇØÈ1Ô �

DDDD

DD

2�1Ò Ì Ó �

zzzzzz

2ÙÔ�

A Functional Quantum Programming Language – p.26/??

Ú G

This program has a type error, because.

This program typechecks, because.

A Functional Quantum Programming Language – p.27/??

Ú G��� � b l� � � �

��� � b l� � �� � � ����� � &') " � +�, � � &') "

This program has a type error, because.

This program typechecks, because.

A Functional Quantum Programming Language – p.27/??

Ú G��� � b l� � � �

��� � b l� � �� � � ����� � &') " � +�, � � &') "

This program has a type error, becauseÛ xDÜÝ w Û x ÜÝ w.

This program typechecks, because.

A Functional Quantum Programming Language – p.27/??

Ú G��� � b l� � � �

��� � b l� � �� � � ����� � &') " � +�, � � &') "

This program has a type error, becauseÛ xDÜÝ w Û x ÜÝ w.

Þß � b� � � �

Þß � b� � �� � ����� � ��� �!" � +�, � � &') "

This program typechecks, because.

A Functional Quantum Programming Language – p.27/??

Ú G��� � b l� � � �

��� � b l� � �� � � ����� � &') " � +�, � � &') "

This program has a type error, becauseÛ xDÜÝ w Û x ÜÝ w.

Þß � b� � � �

Þß � b� � �� � ����� � ��� �!" � +�, � � &') "

This program typechecks, becauseÛ àÁá u â w Û xDÜÝ w.A Functional Quantum Programming Language – p.27/??

Conclusions

Our semantic ideas proved useful whendesigning a quantum programming language,analogous concepts are modelled by thesame syntactic constructs.

Our analysis also highlights the differencesbetween classical and quantumprogramming.

Quantum programming introduces theproblem of control of decoherence, which weaddress by making forgetting variablesexplicit and by having different if-then-elseconstructs.

A Functional Quantum Programming Language – p.28/??

ConclusionsOur semantic ideas proved useful whendesigning a quantum programming language,analogous concepts are modelled by thesame syntactic constructs.

Our analysis also highlights the differencesbetween classical and quantumprogramming.

Quantum programming introduces theproblem of control of decoherence, which weaddress by making forgetting variablesexplicit and by having different if-then-elseconstructs.

A Functional Quantum Programming Language – p.28/??

ConclusionsOur semantic ideas proved useful whendesigning a quantum programming language,analogous concepts are modelled by thesame syntactic constructs.

Our analysis also highlights the differencesbetween classical and quantumprogramming.

Quantum programming introduces theproblem of control of decoherence, which weaddress by making forgetting variablesexplicit and by having different if-then-elseconstructs.

A Functional Quantum Programming Language – p.28/??

ConclusionsOur semantic ideas proved useful whendesigning a quantum programming language,analogous concepts are modelled by thesame syntactic constructs.

Our analysis also highlights the differencesbetween classical and quantumprogramming.

Quantum programming introduces theproblem of control of decoherence, which weaddress by making forgetting variablesexplicit and by having different if-then-elseconstructs.

A Functional Quantum Programming Language – p.28/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

Further work

We have to analyze more quantum programsto evaluate the practical usefulness of ourapproach.

Are we able to come up with completely newalgorithms using QML?

How to deal with higher order programs?

How to deal with infinite datatypes?

Investigate the similarities/differencesbetween FCC and FQC from a categoricalpoint of view.

A Functional Quantum Programming Language – p.29/??

The end

Thank you for your attention.

A Functional Quantum Programming Language – p.30/??