Functional Quantum Programming Thorsten Altenkirch University of - - PowerPoint PPT Presentation

Functional Quantum Programming

Thorsten Altenkirch University of Nottingham based on joint work with Jonathan Grattage supported by EPSRC grant GR/S30818/01

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Background

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Background

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

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently?

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time.

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time. Grover: Blind search in

• ✁

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time. Grover: Blind search in

• ✁

Can we build a quantum computer?

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time. Grover: Blind search in

• ✁

Can we build a quantum computer?

yes We can run quantum algorithms.

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Background

Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations 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!

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The quantum software crisis

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The quantum software crisis

Quantum algorithms are usually presented using the circuit model.

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The quantum software crisis

Quantum algorithms are usually presented using the circuit model. Nielsen and Chuang, p.7, Coming up with good quantum algorithms is hard.

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The quantum software crisis

Quantum algorithms are usually presented using the circuit model. Nielsen and Chuang, p.7, Coming up with good quantum algorithms is hard. Richard Josza, QPL 2004: We need to develop quantum thinking!

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QML

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QML

QML: a functional language for quantum computations

• n finite types.

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QML

QML: a functional language for quantum computations

• n finite types.

Quantum control and quantum data.

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QML

QML: a functional language for quantum computations

• n finite types.

Quantum control and quantum data. Design guided by semantics

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QML

QML: a functional language for quantum computations

• n finite types.

Quantum control and quantum data. Design guided by semantics Analogy with classical computation

Finite classical computations

Finite quantum computations

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QML

QML: a functional language for quantum computations

• n 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

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QML

QML: a functional language for quantum computations

• n 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)

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Example: Hadamard operation

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Example: Hadamard operation

Matrix

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Example: Hadamard operation

Matrix

QML

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Deutsch algorithm

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Overview

• 1. Finite classical computation
• 2. Finite quantum computation
• 3. QML basics
• 4. Compiling QML
• 5. Conclusions and further work

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• 1. Semantics
• 1. Finite classical computation
• 2. Finite quantum computation
• 3. QML basics
• 4. Compiling QML
• 5. Conclusions and further work

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Something we know well . . .

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Something we know well . . .

Start with classical computations on finite types.

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Something we know well . . .

Start with classical computations on finite types. Quantum mechanics is time-reversible. . .

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Something we know well . . .

Start with classical computations on finite types. Quantum mechanics is time-reversible. . . . . . hence quantum computation is based on reversible

• perations.

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Something we know well . . .

Start with classical computations on finite types. Quantum mechanics is time-reversible. . . . . . hence quantum computation is based on reversible

• perations.

However: Newtonian mechanics, Maxwellian electrodynamics are also time-reversible. . .

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Something we know well . . .

Start with classical computations on finite types. Quantum mechanics is time-reversible. . . . . . hence quantum computation is based on reversible

• perations.

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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Classical computation ( )

Given finite sets (input) and (output):

• ❣

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Classical computation ( )

Given finite sets (input) and (output):

• ❣

• a finite set of initial heaps

,

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Classical computation ( )

Given finite sets (input) and (output):

• ❣

• a finite set of initial heaps

, an initial heap

,

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Classical computation ( )

Given finite sets (input) and (output):

• ❣

• a finite set of initial heaps

, an initial heap

, a finite set of garbage states ,

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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

,

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Composing classical computations

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Composing classical computations

• ♥
• ♦
• ♦
• ♦

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Composing classical computations

• ♥
• ♦
• ♦
• ♦

Theorem: U

U

U

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Extensional equality

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Extensional equality

A classical computation

induces a function U

by

• ❧

• ✇②①④③

• U

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Extensional equality

A classical computation

induces a function U

by

• ❧

• ✇②①④③

• U

• We say that two computations are

extensionally equivalent, if they give rise to the same function.

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Extensional equality . . .

Theorem:

U

U

U

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Extensional equality . . .

Theorem:

U

U

U

Hence, classical computations upto extensional equality give rise to the category .

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Extensional equality . . .

Theorem:

U

U

U

Hence, classical computations upto extensional equality give rise to the category . Theorem: Any function

• n finite

sets

can be realized by a computation.

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Extensional equality . . .

Theorem:

U

U

U

Hence, classical computations upto extensional equality give rise to the category . Theorem: Any function

• n finite

sets

can be realized by a computation. Translation for Category Theoreticians:

U is full and faithful.

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Example

:

function

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Example

:

function

computation

• ❡❸❷

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Example :

function

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Example :

function

computation

• ❻

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• 2. Finite quantum computation
• 1. Finite classical computation
• 2. Finite quantum computation
• 3. QML basics
• 4. Compiling QML
• 5. Conclusions and further work

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Linear algebra revision

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Linear algebra revision

Given a finite set (the base)

is a Hilbert space.

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Linear algebra revision

Given a finite set (the base)

is a Hilbert space. Linear operators:

induces

. we write

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Linear algebra revision

Given a finite set (the base)

is a Hilbert space. Linear operators:

induces

. we write

Norm of a vector:

,

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Linear algebra revision

Given a finite set (the base)

is a Hilbert space. Linear operators:

induces

. we write

Norm of a vector:

, Unitary operators: A unitary operator

unitary

is a linear iso- morphism that preserves the norm.

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Basics of quantum computation

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Basics of quantum computation

A pure state over is a vector

with unit norm

.

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Basics of quantum computation

A pure state over is a vector

with unit norm

. A reversible computation is given by a unitary operator

unitary

.

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Quantum computations ( )

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Quantum computations ( )

Given finite sets (input) and (output):

• ❣

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Quantum computations ( )

Given finite sets (input) and (output):

• ❣

• a finite set

, the base of the space of initial heaps,

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Quantum computations ( )

Given finite sets (input) and (output):

• ❣

• a finite set

, the base of the space of initial heaps, a heap initialisation vector

,

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Quantum computations ( )

Given finite sets (input) and (output):

• ❣

• a finite set

, the base of the space of initial heaps, a heap initialisation vector

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

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Quantum computations ( )

Given finite sets (input) and (output):

• ❣

• a finite set

, the base of the space of initial heaps, a heap initialisation vector

, a finite set , the base of the space of garbage states, a unitary operator

unitary

.

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Composing quantum computations

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Composing quantum computations

• ♥
• ♦
• ♦
• ♦

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Semantics of quantum computations. .

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Semantics of quantum computations. .

. . . is a bit more subtle.

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Semantics of quantum computations. .

. . . is a bit more subtle. There is no (sensible) operator on vector spaces, replacing

.

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Semantics of quantum computations. .

. . . is a bit more subtle. There is no (sensible) operator on vector spaces, replacing

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

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Density matrizes

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Density matrizes

Mixed states can be represented by density matrizes

.

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Density matrizes

Mixed states can be represented by density matrizes

. Eigenvalues represent probabilities

System is in state

with prob.

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Density matrizes

Mixed states can be represented by density matrizes

. Eigenvalues represent probabilities

System is in state

with prob.

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

.

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Example: forgetting a qbit

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Example: forgetting a qbit

EPR is represented by

:

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Example: forgetting a qbit

EPR is represented by

:

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Example: forgetting a qbit . . .

After measuring one qbit we obtain

:

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Example: forgetting a qbit . . .

After measuring one qbit we obtain

:

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Superoperators

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Superoperators

Morphisms on density matrizes are called superoperators, these are linear maps, which are completely positive, and trace preserving

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Superoperators

Morphisms on density matrizes are called superoperators, these are linear maps, which are completely positive, and trace preserving Every unitary operator gives rise to a superoperator

.

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• Superoperators. . .

There is an operator

super

called partial trace.

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• Superoperators. . .

There is an operator

super

called partial trace. E.g.

super

is represented by a

matrix.

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Semantics

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Semantics

Every quantum computation

gives rise to a superoperator U

super

• ➞⑩➟

• ①

• U

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Semantics

Every quantum computation

gives rise to a superoperator U

super

• ➞⑩➟

• ①

• U

• Theorem: Every superoperator

super

(on finite Hilbert spaces) comes from a quantum computation.

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Classical vs quantum

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Classical vs quantum

classical (

) quantum (

)

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Classical vs quantum

classical (

) quantum (

) finite sets

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

)

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

)

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators functions

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators functions superoperators

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators functions superoperators injective functions (

)

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators functions superoperators injective functions (

) isometries (

)

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Classical vs quantum

classical (

) quantum (

) finite sets finite dimensional Hilbert spaces cartesian product (

) tensor product (

) bijections unitary operators functions superoperators injective functions (

) isometries (

) projections

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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

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Decoherence

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Decoherence

• ❡➧➦

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Decoherence

• ❡➧➦

Classically

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Decoherence

• ❡➧➦

Classically

Quantum

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Decoherence

• ❡➧➦

Classically

Quantum input:

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Decoherence

• ❡➧➦

Classically

Quantum input:

• utput:

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• 3. QML basics
• 1. Finite classical computation
• 2. Finite quantum computation
• 3. QML basics
• 4. Compiling QML
• 5. Conclusions and further work

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QML basics

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QML basics

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

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QML basics

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

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QML basics

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

Qbytes

.

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QML basics . . .

A QML program is an expression in a context

• f typed variables, e.g.

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QML basics . . .

A QML program is an expression in a context

• f typed variables, e.g.

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

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QML basics . . .

Forgetting variables has to be explicit.

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QML basics . . .

Forgetting variables has to be explicit. E.g.

is illegal,

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QML basics . . .

Forgetting variables has to be explicit. E.g.

is illegal, but

is ok.

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QML basics . . .

There are two different if-then-else constructs.

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QML basics . . .

There are two different if-then-else constructs.

is just the identity,

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QML basics . . .

There are two different if-then-else constructs.

is just the identity, but

introduces a measurement (end hence decoherence).

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QML basics . . .

Using

is only allowed, if the branches are

• rthogonal, i.e. observable different.

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QML basics . . .

Using

is only allowed, if the branches are

• rthogonal, i.e. observable different.

is illegal,

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QML basics . . .

Using

is only allowed, if the branches are

• rthogonal, i.e. observable different.

is illegal, but

is ok.

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QML basics . . .

We can introduce superpositions, e.g.

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QML basics . . .

We can introduce superpositions, e.g.

However, the terms in the superposition have to be orthogonal.

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• 4. Compiling QML
• 1. Finite classical computation
• 2. Finite quantum computation
• 3. QML basics
• 4. Compiling QML
• 5. Conclusions and further work

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Compilation

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Compilation

Correct QML programs are defined by typing rules, e.g.

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Compilation

Correct QML programs are defined by typing rules, e.g.

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

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• elim

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• elim

• Ø

• Ø
• Ü

• Ü
• Ù
• Ù
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Compiler

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Compiler

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

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Compiler

A compiler is currently being implemented by my student Jonathan Grattage (in Haskell). The output of the compiler are quantum circuits which can be simulated by a quantum circuit simulator.

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Compiler

A compiler is currently being implemented by my student Jonathan Grattage (in Haskell). The output of the compiler are quantum circuits which can be simulated by a quantum circuit simulator. Amr Sabry and Juliana Vizotti (Indiana University) embarked on an independent implementation of QML based on our paper.

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• 5. Conclusions
• 1. Semantics of finite classical and quantum

computation

• 2. QML basics
• 3. Compiling QML
• 4. Conclusions and further work

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Conclusions

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Conclusions

Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs.

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Conclusions

Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs. Our analysis also highlights the differences between classical and quantum programming.

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Conclusions

Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs. Our analysis also highlights the differences between classical and quantum programming. We have developed an algebra of quantum programs which for pure programs is complete wrt the semantics and a normalisation algorithm.

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Conclusions

Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs. Our analysis also highlights the differences between classical and quantum programming. We have developed an algebra of quantum programs which for pure programs is complete wrt the semantics and a normalisation algorithm. Quantum programming introduces the problem of control of decoherence, which we address by making forgetting variables explicit and by having different if-then-else constructs.

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Further work

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Further work

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

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Further work

We have to analyze more quantum programs to evaluate the practical usefulness of our approach. We should be able to extend our algebra and normalisation to the full language (including measurements).

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Further work

We have to analyze more quantum programs to evaluate the practical usefulness of our approach. We should be able to extend our algebra and normalisation to the full language (including measurements). Are we able to come up with completely new algorithms using QML?

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Further work

We have to analyze more quantum programs to evaluate the practical usefulness of our approach. We should be able to extend our algebra and normalisation to the full language (including measurements). Are we able to come up with completely new algorithms using QML? How to deal with higher order programs?

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Further work

We have to analyze more quantum programs to evaluate the practical usefulness of our approach. We should be able to extend our algebra and normalisation to the full language (including measurements). Are we able to come up with completely new algorithms using QML? How to deal with higher order programs? How to deal with infinite datatypes?

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Further work

We have to analyze more quantum programs to evaluate the practical usefulness of our approach. We should be able to extend our algebra and normalisation to the full language (including measurements). Are we able to come up with completely new algorithms using QML? How to deal with higher order programs? How to deal with infinite datatypes? Investigate the similarities/differences between FCC and FQC from a categorical point of view.

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The end Thank you for your attention.

Papers, available from //www.cs.nott.ac.uk/˜txa/publ/

A functional quantum programming language LICS 2005

with J.Grattage

Structuring Quantum Effects: Superoperators as Arrows

MFCS 2006 with J.Vizzotto and A.Sabry

An Algebra of Pure Quantum Programming QPL 2005

with J.Grattage, J.Vizzotto and A.Sabry

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