towards a high level quantum programming language

Towards aHigh Level

Quantum Programming LanguageThorsten Altenkirch

University of Nottingham

based on joint work with Jonathan Grattage

and discussions with V.P. Belavkin

Towards aHigh LevelQuantum Programming Language – p.1/36

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!

Assumption: Nature is fair. . .

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!

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

Draft paper available(Google:Thorsten,functional,quantum)

Compiler under construction (Jonathan)

QMLQML: a functional language for quantum computationson finite types.

Analogy with classical computation� � �

Finite classical computations� �

Compiler under construction (Jonathan)Towards aHigh LevelQuantum Programming Language – p.4/36

Example: Hadamard operation

Matrix

� ��

Matrix

� ��

QML �� !" � # � $&%' (

� )�* � �� ! � $&%' (

Overview

1. Semantics of finite classical and quantumcomputation

2. QML basics

3. Compiling QML

4. Conclusions and further work

1. Semantics

2. QML basics

3. Compiling QML

Something we know well . . .

Start with classical computationson 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 �

an initial heap ,

a bijection ,

+ ,-. � / 0 �

an initial heap ,

a bijection ,

+ ,-. � / 0 �

an initial heap132 ,

a bijection ,

+ ,-. � / 0 �

a bijection ,

+ ,-. � / 0 �

a bijection 2 4 5 4 ,

Semantics

A classical computation

induces a function U by

Theorem Any function (on finitesets ) can be realized by a quantumcomputation.

Semantics

A classical computation6 �" 7 7 7 1 2 7 7 2 4 5 4 #

induces a function U 6 2 by

U 68 � 9;: " 1 7 8 #

Theorem Any function (on finitesets ) can be realized by a quantumcomputation.

Semantics

A classical computation6 �" 7 7 7 1 2 7 7 2 4 5 4 #

induces a function U 6 2 by

U 68 � 9;: " 1 7 8 #

Theorem Any function 2 (on finitesets 7 ) can be realized by a quantumcomputation.

Composing classical computations

888 �

��

Theorem:U U U

< , =< �

888 <�

��

�� =�

= � <

Theorem:U U U

< , =< �

888 <�

��

�� =�

= � <Theorem:

" > 6 # �" U

# >" U 6 #

Coming next: Quantum computations

Develop analogously to . . .

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:2 induces

?2 .we write 2 �

Norm of a vector:,

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

?2 .we write 2 �Norm of a vector:@�A @ � B C D" A 8 # E" A 8 # 2 F

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

?2 .we write 2 �Norm of a vector:@�A @ � B C D" A 8 # E" A 8 # 2 F

,Unitary operators:A unitary operator 2 �

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 A 2 withunit norm

@�A @ � �

A reversible computation is given by aunitary operator unitary .

A pure state over is a vector A 2 withunit norm

@�A @ � �

A reversible computation is given by aunitary operator 2 �

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):

+ ,-. � / 0 �

a heap initialisation vector

+ ,-. � / 0 �

a unitary operator 2 �

unitary .Towards aHigh LevelQuantum Programming Language – p.15/36

Composing quantum computations

888 �

��

Composing quantum computations

< , =< �

888 <�

��

�� =�

= � <

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.

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

There is no (sensible) operator on vectorspaces, replacing 9 : 2 4 .

Density matrices and superoperators

Mixed states are represented by densitymatrices.

Operations on mixed states (i.e. densitymatrices) are represented by superoperators.

Every unitary operator gives rise to asuperoperator .

There is an operator

called partial trace.

Every unitary operator gives rise to asuperoperator

There is an operatorHJILKNM O 2 �

called partial trace.Towards aHigh LevelQuantum Programming Language – p.18/36

Semantics

Every quantum computation gives rise to asuperoperator U super

��

Theorem: Every superoperator super

(on finite Hilbert spaces) comes from a quantumcomputation.

Semantics

Every quantum computation 6 gives rise to asuperoperator U 6 2 �

PRQ //S;TU

��

V W XZYOO

U < //

Theorem: Every superoperator super

Semantics

Every quantum computation 6 gives rise to asuperoperator U 6 2 �

PRQ //S;TU

��

V W XZYOO

U < //

Theorem: Every superoperator 2 �

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

finite sets

finite dimensional Hilbert spaces

classical quantum

bijections

unitary operators

classical quantum

cartesian product ( [ )

tensor product ( )

classical quantum

cartesian product ( [ ) tensor product ( \)

classical quantum

cartesian product ( [ ) tensor product ( \)functions

superoperators

classical quantum

cartesian product ( [ ) tensor product ( \)functions superoperators

classical quantum

projections

partial trace

classical quantum

Decoherence

� '&%$ !"# �

Classically

Quantum

input:

output:

Decoherence

� ] �

^ � '&%$ !"# �

-`_ -bac

Classically

Quantum

input:

output:

Decoherence

� ] �

^ � '&%$ !"# �

-`_ -bacClassically 9L: > d � e

Quantum

input:

output:

Decoherence

� ] �

^ � '&%$ !"# �

Quantum

input:

output:

Decoherence

� ] �

^ � '&%$ !"# �

Quantum

input:

� :f � ! ^ g : f � ! ^ g (

output:

Decoherence

� ] �

^ � '&%$ !"# �

Quantum

input:

� :f � ! ^ g : f � ! ^ g (

output:: � � ! ^ g ( : � � ! � g (

2. QML basics

3. 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

Qbytes.

QML basics

QML types are

� 7 h i 7 h i

Qbytes.

QML basics

QML types are

� 7 h i 7 h iQbits � � � �

Qbytes.

QML basics

QML types are

� 7 h i 7 h iQbits � � � �Qbytesj� � � � � � � � � �.

QML basics . . .

A QML program is an expression in a contextof typed variables, e.g.�k l $� � � ��k l $� � ��

� )* � � $&%'

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.�k l $� � � ��k l $� � ��

� )* � � $&%' We can compile QML programs into quantumcomputations (i.e. quantum circuits).

QML basics . . .

Forgetting variables has to be explicit.

is illegal,but

is ok.

QML basics . . .

Forgetting variables has to be explicit.E.g.� � $m� � � � �� $" � 7 n # � �

is illegal,

is ok.

QML basics . . .

Forgetting variables has to be explicit.E.g.� � $m� � � � �� $" � 7 n # � �

is illegal,but � � $m� � � � �� $" � 7 n # � � � n (

is ok.

QML basics . . .There are two different if-then-else (or moregenerally case) constructs.

is just the identity, but

introduces a measurement (end hencedecoherence).

QML basics . . .There are two different if-then-else (or moregenerally case) constructs.o � � � �o � � � � � �� $&%'

� )* � � � � � is just the identity,

introduces a measurement (end hencedecoherence).

QML basics . . .There are two different if-then-else (or moregenerally case) constructs.o � � � �o � � � � � �� $&%'

� )* � � � � � is just the identity, butp � � � � � �p � � � � �� $&%'

� )�* � � � ��

introduces a measurement (end hencedecoherence). Towards aHigh LevelQuantum Programming Language – p.26/36

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.q �r �s � � � � � � � �q �r �s " � 7 n # q � �� q�� " n 7� #

� )* � " � 7 n #is illegal,

is ok.

��

is only allowed, if the branches areorthogonal, i.e. observable different.q �r �s � � � � � � � �q �r �s " � 7 n # q � �� q�� " n 7� #

� )* � " � 7 n #is illegal, butq �r �s � � � � � � � " � � #q �r �s " � 7 n # q � �� q�� " � $&%' 7" n 7� # #

� )* � " � � � � 7" � 7 n # #

is ok.Towards aHigh LevelQuantum Programming Language – p.27/36

QML basics . . .

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

� )�* � �� ! � $&%' (

However, the terms in the superposition haveto be orthogonal.

QML basics . . .

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

� )�* � �� ! � $&%' (

However, the terms in the superposition haveto be orthogonal.

3. Compiling QML

2. QML basics

3. 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

t uv � h i

7 w � h 7 x � i u�y � z {|~}t u�� " w 7 x # �v �� y �

Compilation

t uv � h i

7 w � h 7 x � i u�y � z {|~}t u�� " w 7 x # �v �� y �

-elim t uv � h i

7 w � h 7 x � i u�y � z { |}t u�� " w 7 x # �v � � y �

��

-elim t uv � h i

7 w � h 7 x � i u�y � z { |}t u�� " w 7 x # �v � � y �

t ��

��M � � � ��

��

� �

;; ��

� �

��

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.

Compiler

4. Conclusions

2. QML basics

3. Compiling QML

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.

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.

Further work

The end

Thank you for your attention.

towards a high level quantum programming language

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