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.


Another alternative title

Quantum programming

for the

lazy functional programmer


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. . .


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



Grover: Blind search in






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



Grover: Blind search in� � � �






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!




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

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




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


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)


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



Analogy with classical computation

Finite classical computations





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



Analogy with classical computation� � �

Finite classical computations� �





Example: Hadamard operation

Matrix

QML



Matrix

� ��

� ��

QML



Matrix

� ��

� ��

QML �� !" #$ � % � &(') " *

� +�, � �� !" # � &(') " *


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.


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 ,




- ./

0 � 1 2 �


an initial heap ,


a bijection ,




- ./

0 � 1 2 �


an initial heap354 ,


a bijection 4 6 7 6 ,


Composing computations

�

>>>>

>>>>

8888

888 �

�

��

��


Composing computations

8. 98 �

>>>>

>>>>

8888

888 8�

9 �

��

�� 9�

9 � 8


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.


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.


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.


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.


Example � :function HJI 4 $ � ; � % �

HKI $ � ; L % � �

computation

�


Example � :function HJI 4 $ � ; � % �

HKI $ � ; L % � �

computation � �

� �

/NMO


Example :

function P4 � $ � ; � %

P� �$ � ;� %

computation

� '&%$ !"#


Example :

function P4 � $ � ; � %

P� �$ � ;� %

computation Q � � R Q � �

S � � � '&%$ !"# Q � �

T T 4 $ � ; � % $ � ; � %

T$ S ;� % �$ S ;� %

T$ � ;� % �$ � ; U � %


Classical vs quantum

classical ( ) quantum ( )

finite sets finite dimensional Hilbert spaces

bijections unitary operators

cartesian product ( ) tensor product ( )

functions superoperators

projections partial trace



classical (

� � �

) quantum (� �

)








classical (

� � �

) quantum (� �

)

finite sets

finite dimensional Hilbert spaces







classical (

� � �

) quantum (� �

)








classical (

� � �

) quantum (� �

)


bijections

unitary operators






classical (

� � �

) quantum (� �

)








classical (

� � �

) quantum (� �

)



cartesian product ( V )

tensor product ( )





classical (

� � �

) quantum (� �

)



cartesian product ( V ) tensor product ( W)





classical (

� � �

) quantum (� �

)



cartesian product ( V ) tensor product ( W)functions

superoperators




classical (

� � �

) quantum (� �

)



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




classical (

� � �

) quantum (� �

)




projections

partial trace



classical (

� � �

) quantum (� �

)






� X , classically

HDI G P� � �

� '&%$ !"# �


� X , classically

HDI G P� � �Q � � R Q � �

S � � � '&%$ !"# �

/ZY /NMO


� X , classically

HDI G P� � �Q � � R Q � �

S � � � '&%$ !"# �

/ZY /NMO[

� �


� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MO

input:

output:

Decoherence!


� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MOinput:

� I ^ � # S _ I^ � # � _ *

output:

Decoherence!


� X , quantum

Q � � R Q � �

S � � � '&%$ !"# �

/\Y /]MOinput:

� I ^ � # S _ I^ � # � _ *output:

I � � # S _ * I � � # � _ *

Decoherence!


Control of decoherence

QML is based on strict linear logic

Contraction is implicit and realized by .

Weakening is explicit and leads todecoherence.




Contraction is implicit and realized by T.

Weakening is explicit and leads todecoherence.




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


QML overview

Types

Terms


QML overview

Types ` � � # ` a # ` a

Terms


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 *


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 *


QML overview . . .

Typing judgementsprograms

strict programs

Semantics


QML overview . . .

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

Semantics


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


The let-rule

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

;;;;

;

�

��

�

;;;;

;;

9999

99�

�

��

��


The let-rule

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

o |~}�� ;;

;;;

�

��C � � � ��

� �

;;;;

;;

9999

99 ��

� �

��

��


on contexts

if dom

�


on contexts

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

domR �

�


on contexts

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

domR �

o |~}�� o

�C � �


Another source of decoherence

mentions

but doesn’t use it.

Hence, it has to measure it!


Another source of decoherence

�� b mentions�� b� � � �

�� b� � �� &') " � +, � � &(') "

but doesn’t use it.

Hence, it has to measure it!


-elim

::::

:

�

��

�

�

DDDD

DD�

�

zzzzzz �


-elim

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

� W � /�¶¸·º¹ » �::

:::

/½¼¾ ¿ÁÀ Â1 �ÄÃ � � � �� /ÄÅ� Æ � �

Ç¸È ÉËÊ�1Å �

DDDD

DD

2�1¾¸Ì À �

zzzzzz

2ÍÅ�


-elim decoherence-free

::::

:

�

��

�

DDDD

DD�

�

zzzzzz �



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

::::

:

�

��

�

DDDD

DD�

�

zzzzzz �



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

� W � /Ä¶¸·º¹ » �

::::

:

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

ÇØÈ ÇØÈ1Ô �

DDDD

DD

2�1Ò Ì Ó �

zzzzzz

2ÙÔ�


Ú G

This program has a type error, because.

This program typechecks, because.


Ú G�� b l� � � �

�� b l� � �� &') " � +�, � � &') "

This program has a type error, because.



Ú G�� b l� � � �

�� b l� � �� &') " � +�, � � &') "

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



Ú G�� b l� � � �

�� b l� � �� &') " � +�, � � &') "


Þß � b� � � �

Þß � b� � �� !" � +�, � � &') "



Ú G�� b l� � � �

�� b l� � �� &') " � +�, � � &') "


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.


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.


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.


The end

Thank you for your attention.


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

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