Functional Quantum Programming - cs.ioc.ee › ~tarmo › pls06 › altenkirch-slides.pdf ·...

Functional QuantumProgramming

Thorsten Altenkirch

University of Nottingham

based on joint work with Jonathan Grattage

supported by EPSRC grant GR/S30818/01

Tallinn Feb 06 – p.1/44

Background

Simulation of quantum systems is expensive:PSPACE complexity for 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!


BackgroundSimulation of quantum systems is expensive:PSPACE complexity for polynomial circuits.



Grover: Blind search in





BackgroundSimulation of quantum systems is expensive:PSPACE complexity for 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 functional language for quantum computationson 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 functional language for quantum computationson finite types.



Analogy with classical computation

Finite classical computations





QMLQML: a functional language for quantum computationson finite types.



Analogy with classical computation� � �

Finite classical computations� �





Example: Hadamard operation

Matrix

QML



Matrix

� ��

� ��

QML



Matrix

� ��

� ��

QML �� !" #$ � % � &(') " *

� +�, � �� !" # � &(') " *


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E \ C] �O W XZ U H KL >E \ C] NGO P Q RSU V �_^ ` � O W XZ U [

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K] b FTallinn Feb 06 – p.6/44

Overview

1. Finite classical computation

2. Finite quantum computation

3. QML basics

4. Compiling QML

5. Conclusions and further work


1. Semantics



3. QML basics

4. Compiling QML



Something we know well . . .

Start with 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 ,




c def � g h �


an initial heap ,


a bijection ,




c def � g h �


an initial heapikj ,


a bijection ,




c def � g h �


an initial heapikj ,


a bijection j l m l ,


Composing classical computations

�

>>>>

>>>>

8888

888 �

�

��

��

Theorem:U U U



n d on �

>>>>

>>>>

8888

888 n�

o �

��

�� o�

o � n

Theorem:U U U



n d on �

>>>>

>>>>

8888

888 n�

o �

��

�� o�

o � nTheorem:

U

$ pq % �$ U

% p$ Uq %


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 computationq �$ r i r r %induces a function Uq j by

l s // ltvu

��

wyx{z | }OO

U n //

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


Extensional equality . . .

Theorem:

U

$ pq % �$ U

% p$ Uq %

Hence, classical computations uptoextensional equality give rise to the category

.

Theorem: Any function on finitesets can be realized by a computation.

Translation for Category Theoreticians:U is full and faithful.


Extensional equality . . .

Theorem:

U

$ pq % �$ U

% p$ Uq %

Hence, classical computations uptoextensional equality give rise to the category

.

Theorem: Any function j on finitesets r can be realized by a computation.

Translation for Category Theoreticians:U is full and faithful.


Example � :function ~�� j $ � r � % �

~�� $ � r � % � �

computation

�


Example � :function ~�� j $ � r � % �

~�� $ � r � % � �

computation � �

� �e��


Example :

function �j � $ � r � %

�� $ � r� %

computation

� '&%$ !"#


Example :

function �j � $ � r � %

�� $ � r� %

computation � � � � � � �

� � � � '&%$ !"# � � �

� � j $ � r � % $ � r � %

�$ � r� % �$ � r� %

�$ � r� % �$ � r � � %





3. QML basics

4. Compiling QML



Linear algebra revision

Given a finite set (the base)is a Hilbert space.

Linear operators:induces .

we writeNorm of a vector:

,Unitary operators:A unitary operator unitary is a linear iso-morphism that preserves the norm.



Given a finite set (the base)� � is a Hilbert space.

Linear operators:induces .

we writeNorm of a vector:

,Unitary operators:A unitary operator unitary is a linear iso-morphism that preserves the norm.



Given a finite set (the base)� � is a Hilbert space.Linear operators:j induces

�j � �

.we write j �

Norm of a vector:,

Unitary operators:A unitary operator unitary is a linear iso-morphism that preserves the norm.




�j � �

.we write j �Norm of a vector:�� $ �� % �$ �� % j �

,

Unitary operators:A unitary operator unitary is a linear iso-morphism that preserves the norm.




�j � �

.we write j �Norm of a vector:�� $ �� % �$ �� % j �

,Unitary operators:A unitary operator j �

unitary is a linear iso-morphism that preserves the norm.


Basics of quantum computation

A pure state over is a vector withunit norm .

A reversible computation is given by aunitary operator unitary .



A pure state over is a vector � j �with

unit norm

��

.

A reversible computation is given by aunitary operator unitary .



A pure state over is a vector � j �with

unit norm

��

.

A reversible computation is given by aunitary operator j �

unitary .


Quantum computations ( )


� �

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 .


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

c def � g h �


a heap initialisation vector ,





c def � g h �


a heap initialisation vector

i j �

,





c def � g h �


a heap initialisation vector

i j �

,


a unitary operator j �

unitary .Tallinn Feb 06 – p.19/44

Composing quantum computations

�

>>>>

>>>>

8888

888 �

�

��

��


Composing quantum computations

n d on �

>>>>

>>>>

8888

888 n�

o �

��

�� o�

o � n


Semantics of quantum computations. . .

. . . is a bit more subtle.

There is no (sensible) operator on vectorspaces, replacing .

Indeed: Forgetting part of a pure stateresults in a mixed state.


Semantics of quantum computations. . .

. . . is a bit more subtle.

There is no (sensible) operator on vectorspaces, replacing ~ � j l .

Indeed: Forgetting part of a pure stateresults in a mixed state.


Density matrizes

Mixed states can be represented by densitymatrizes .

Eigenvalues represent probabilities

System is in state with prob.Eigenvalues have to be positive and their sum(the trace) is .


Density matrizes

Mixed states can be represented by densitymatrizes �j � .


System is in state with prob.Eigenvalues have to be positive and their sum(the trace) is .


Density matrizes



� �� System is in state

�� with prob.

�

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


Density matrizes



� �� System is in state

�� with prob.

�

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

�.


Example: forgetting a qbit

EPR is represented by:



EPR is represented by�j � � � � �:

��< � � �<� � � �

� � � ��< � � �<��



EPR is represented by�j � � � � �:

��< � � �<� � � �

� � � ��< � � �<��

�$ �� # � � � �� # � � � % � �� # � � � � � � # � � �


Example: forgetting a qbit . . .

After measuring one qbit we obtain� j � � �: �< �� <


Example: forgetting a qbit . . .

After measuring one qbit we obtain� j � � �: �< �� <

� # � � � �� # � �

� # � � � �� # � �


Superoperators

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

completely positive, andtrace preserving

Every unitary operator gives rise to asuperoperator .


Superoperators

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

completely positive, andtrace preserving

Every unitary operator gives rise to asuperoperator

¡.


Superoperators. . .

There is an operator&' �z ¢ j �super

called partial trace.

E.g. super isrepresented by a matrix.


Superoperators. . .

There is an operator&' �z ¢ j �super

called partial trace.

E.g. &('¤£ ¥z £ ¥ j � � �super

� isrepresented by a

�¦ l §matrix.


Semantics

Every quantum computation gives rise to asuperoperator U super

//

��

OO

U//

Theorem: Every superoperator super

(on finite Hilbert spaces) comes from a quantumcomputation.


Semantics

Every quantum computationq gives rise to asuperoperator Uq j �

super

¨s //©�ª«

��

x ¬ |OO

U n //

Theorem: Every superoperator super



Semantics

Every quantum computationq gives rise to asuperoperator Uq j �

super

¨s //©�ª«

��

x ¬ |OO

U n //

Theorem: Every superoperator j �

super



Classical vs quantum

classical ( ) quantum ( )

finite sets finite dimensional Hilbert spaces

cartesian product ( ) tensor product ( )

bijections unitary operators

functions superoperators

injective functions ( ) isometries ( )

projections partial trace



classical (

� � �

) quantum (

� �

)









classical (

� � �

) quantum (

� �

)

finite sets

finite dimensional Hilbert spaces








classical (

� � �

) quantum (

� �

)









classical (

� � �

) quantum (

� �

)


cartesian product ( ® )

tensor product ( )







classical (

� � �

) quantum (

� �

)


cartesian product ( ® ) tensor product ( ¯)







classical (

� � �

) quantum (

� �

)


cartesian product ( ® ) tensor product ( ¯)bijections

unitary operators






classical (

� � �

) quantum (

� �

)


cartesian product ( ® ) tensor product ( ¯)bijections unitary operators






classical (

� � �

) quantum (

� �

)



functions

superoperators





classical (

� � �

) quantum (

� �

)








classical (

� � �

) quantum (

� �

)




injective functions (� � � M)

isometries ( )




classical (

� � �

) quantum (

� �

)




injective functions (� � � M) isometries (

� � M

)




classical (

� � �

) quantum (

� �

)





� � M

)

projections

partial trace



classical (

� � �

) quantum (

� �

)





� � M

)



Decoherence

� '&%$ !"# �

Classically

Quantum

input:

output:


Decoherence

� � �

� � '&%$ !"# �

e±° e��

Classically

Quantum

input:

output:


Decoherence

� � �

� � '&%$ !"# �

e±° e��Classically ~� p � � ²

Quantum

input:

output:


Decoherence

� � �

� � '&%$ !"# �


Quantum

input:

� �� # � � � � � # � � *

output:


Decoherence

� � �

� � '&%$ !"# �


Quantum

input:

� �� # � � � � � # � � *

output:� � � # � � * � � � # � � *


3. QML basics



3. QML basics

4. Compiling QML



QML basics

QML is a first order functional languages, i.e.programs are well-typed expressions.

QML types are

Qbytes.


QML basics


QML types are

� r ³ ´ r �

Qbytes.


QML basics


QML types are

� r ³ ´ r �Qbytesµ� � � � � � � � � �.


QML basics . . .

A QML program is an expression in a contextof typed variables, e.g.¶·¸ ¹� � � �¶·¸ ¹� � �� !"� +, � � &(') "

We can compile QML programs into quantumcomputations (i.e. quantum circuits).


QML basics . . .

A QML program is an expression in a contextof typed variables, e.g.¶·¸ ¹� � � �¶·¸ ¹� � �� !"� +, � � &(') "We can compile QML programs into quantumcomputations (i.e. quantum circuits).


QML basics . . .

Forgetting variables has to be explicit.

E.g.

is illegal,but

is ok.


QML basics . . .

Forgetting variables has to be explicit.E.g. ¶º ¹»� � � � �¶ º ¹$ � r � % � �

is illegal,

but

is ok.


QML basics . . .

Forgetting variables has to be explicit.E.g. ¶º ¹»� � � � �¶ º ¹$ � r � % � �

is illegal,but ¶º ¹»� � � � �¶ º ¹$ � r � % � � � � *

is ok.


QML basics . . .There are two different if-then-else constructs.

is just the identity, but

introduces a measurement (end hencedecoherence).


QML basics . . .There are two different if-then-else constructs.¼ � � � �¼ � � � � � �� &(') "� +, � � � � �! "is just the identity,

but



QML basics . . .There are two different if-then-else constructs.¼ � � � �¼ � � � � � �� &(') "� +, � � � � �! "is just the identity, but½¾ � º � � � �½¾ � º � � �� &(') "� +�, � � � � !"



QML basics . . .Using

��

is only allowed, if the branches areorthogonal, i.e. observable different.

is illegal, but

is ok.



��

is only allowed, if the branches areorthogonal, i.e. observable different.¿º À�Á � � � � � � � �¿º À�Á $ � r � % ¿ � �� ¿�� $ � r� %

� +, � $ � r � %is illegal,

but

is ok.



��

is only allowed, if the branches areorthogonal, i.e. observable different.¿º À�Á � � � � � � � �¿º À�Á $ � r � % ¿ � �� ¿�� $ � r� %

� +, � $ � r � %is illegal, but¿º À�Á � � � � � � � $ � � %

¿º À�Á $ � r � % ¿ � �� ¿�� $ � &(') " r$ � r� % %

� +, � $ � � � !" r$ � r � % %

is ok.Tallinn Feb 06 – p.35/44

QML basics . . .

We can introduce superpositions, e.g.�� !" #$ � % � &(') " *

� +�, � �� !" # � &(') " *

However, the terms in the superposition haveto be orthogonal.


4. Compiling QML



3. QML basics

4. Compiling QML



Compilation

Correct QML programs are defined by typingrules, e.g.

For each rule we can construct a quantumcomputation, i.e. a circuit.


Compilation

Correct QML programs are defined by typingrules, e.g.

Â ÃÄ � ³ ´

r � � ³ r Å � ´ Ã�Æ � " ÇÉÈÂ ÃËÊ Ì Í $ � r Å % �Ä ÎÐÏ Æ �

For each rule we can construct a quantumcomputation, i.e. a circuit.


-elim Â ÃÄ � ³ ´

r � � ³ r Å � ´ Ã�Æ � " Ç ÈÂ ÃÊ Ì Í $ � r Å % �Ä Î Ï Æ �

;;;;

;

�

��

�

;;;;

;;

;;;;

;;�

�

��

��


-elim Â ÃÄ � ³ ´

r � � ³ r Å � ´ Ã�Æ � " Ç ÈÂ ÃÊ Ì Í $ � r Å % �Ä Î Ï Æ �

Â ÑÓÒÕÔ Ö ×

;;;;

;Ø

ÙÚz Û � Ø ��Ü

ÝÞ ß

Ü �

;;;;

;;

;;;;

;; Ü�

Ù �

��

�� Ù�


Compiler

A compiler is currently being implemented bymy student Jonathan Grattage (in Haskell).

The output of the compiler are quantumcircuits which can be simulated by a quantumcircuit simulator.

Amr Sabry and Juliana Vizotti (IndianaUniversity) embarked on an independentimplementation of QML based on our paper.


5. Conclusions

1. Semantics of finite classical and quantumcomputation

2. QML basics

3. Compiling QML



Conclusions

Our semantic ideas proved useful when designing aquantum programming language, analogous conceptsare modelled by the same syntactic constructs.

Our analysis also highlights the differences betweenclassical and quantum programming.

We have developed an algebra of quantum programswhich for pure programs is complete wrt the semanticsand a normalisation algorithm.

Quantum programming introduces the problem ofcontrol of decoherence, which we address by makingforgetting variables explicit and by having differentif-then-else constructs.


ConclusionsOur semantic ideas proved useful when designing aquantum programming language, analogous conceptsare modelled by the same syntactic constructs.

Our analysis also highlights the differences betweenclassical and quantum programming.

We have developed an algebra of quantum programswhich for pure programs is complete wrt the semanticsand a normalisation algorithm.

Quantum programming introduces the problem ofcontrol of decoherence, which we address by makingforgetting variables explicit and by having differentif-then-else constructs.


Further work

We have to analyze more quantum programs toevaluate the practical usefulness of our approach.

We should be able to extend our algebra andnormalisation to the full language (includingmeasurements).

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/differences between FCCand FQC from a categorical point of view.


Further workWe have to analyze more quantum programs toevaluate the practical usefulness of our approach.

We should be able to extend our algebra andnormalisation to the full language (includingmeasurements).

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/differences between FCCand FQC from a categorical point of view.


The end

Thank you for your attention.Papers, available from//www.cs.nott.ac.uk/˜txa/publ/

A functional quantum programming language LICS 2005with J.Grattage

Structuring Quantum Effects: Superoperators as Arrows

MFCS 2006with J.Vizzotto and A.Sabry

An Algebra of Pure Quantum Programming QPL 2005with J.Grattage, J.Vizzotto and A.Sabry Tallinn Feb 06 – p.44/44

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