MGS Xmas 04 p.1/?? Towards a High Level Quantum Programming 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
Background MGS Xmas 04 p.2/??
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate polynomial circuits.
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently?
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time.
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Shor: Factorisation in quantum polynomial time. Grover: Blind search in
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate 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?
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate 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.
MGS Xmas 04 p.2/?? Background Simulation of quantum systems is expensive: Exponential time to simulate 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!
The quantum software crisis MGS Xmas 04 p.3/??
MGS Xmas 04 p.3/?? The quantum software crisis Quantum algorithms are usually presented using the circuit model.
MGS Xmas 04 p.3/?? 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.
MGS Xmas 04 p.3/?? 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!
QML MGS Xmas 04 p.4/??
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types.
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types. Design based on semantic analogy: Finite classical computations Finite quantum computations
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types. Design based on semantic analogy: Finite classical computations Finite quantum computations Quantum control and quantum data.
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types. Design based on semantic analogy: Finite classical computations Finite quantum computations Quantum control and quantum data. Contraction is interpreted as sharing not cloning.
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types. Design based on semantic analogy: Finite classical computations Finite quantum computations Quantum control and quantum data. Contraction is interpreted as sharing not cloning. Control of decoherence, hence no implicit weakening.
MGS Xmas 04 p.4/?? QML QML: a first-order functional language for quantum computations on finite types. Design based on semantic analogy: Finite classical computations Finite quantum computations Quantum control and quantum data. Contraction is interpreted as sharing not cloning. Control of decoherence, hence no implicit weakening. Compiler under construction (Jonathan)
Example: Hadamard operation MGS Xmas 04 p.5/??
MGS Xmas 04 p.5/?? Example: Hadamard operation Matrix
MGS Xmas 04 p.5/?? Example: Hadamard operation Matrix QML * " &(') % $#!" * " &(') #!" +,
-./ 4.- A5 A @ 4 > > D CB 9U S X 9U > E S X 9 > 9 S X 4 M 9U > > S X 4 M 9 > > Z U4 B Deutsch algorithm 021 3 021 3 021 6 576 =<?> 8:9 ; S OQPR M N F JLKM GIH E=F D CB 4 T ; OQPR M > F T ; 5 OQPR M > F OQPR M F WV N JLKM GIH E=F 5 JLKM > GIH F OQPR M N F JLKM GIH F WV 8:Y 9 5 D CB J KM > GIH 8:Y 9 F OQPR M F OQPR N F J KM G H F E WV T ; J KM G H F OQPR F WV N J KM G H E=F 8:Y 9 [ < MGS Xmas 04 p.6/??
MGS Xmas 04 p.7/?? Overview 1. Finite classical computation 2. Finite quantum computation 3. QML 4. Conclusions and further work
MGS Xmas 04 p.8/?? 1. Finite classical computation 1. Finite classical computation 2. Finite quantum computation 3. QML 4. Conclusions and further work
Classical computations on finite types MGS Xmas 04 p.9/??
MGS Xmas 04 p.9/?? Classical computations on finite types Quantum mechanics is time-reversible...
MGS Xmas 04 p.9/?? Classical computations on finite types Quantum mechanics is time-reversible...... hence quantum computation is based on reversible operations.
MGS Xmas 04 p.9/?? Classical computations on finite types Quantum mechanics is time-reversible...... hence quantum computation is based on reversible operations. However: Newtonian mechanics, Maxwellian electrodynamics are also time-reversible...
MGS Xmas 04 p.9/?? Classical computations on finite types Quantum mechanics is time-reversible...... hence quantum computation is based on reversible operations. However: Newtonian mechanics, Maxwellian electrodynamics are also time-reversible...... hence classical computation should be based on reversible operations.
Classical computation ( ) MGS Xmas 04 p.10/??
_ ` \ ^ ] a MGS Xmas 04 p.10/?? Classical computation ( ) Given finite sets (input) and (output):
_ c ` \ e ^ f ] a e MGS Xmas 04 p.10/?? Classical computation ( ) Given finite sets (input) and (output): a finite set of initial heaps, an initial heap bdc a finite set of garbage states, a bijection,,
Composing computations MGS Xmas 04 p.11/??
h g g ] h h g MGS Xmas 04 p.11/?? Composing computations h g
Extensional equality MGS Xmas 04 p.12/??
e ci k g $ i j MGS Xmas 04 p.12/?? Extensional equality A classical computation induces a function U by % j b j e lnm ut oqpsr U
e ci k g $ i j MGS Xmas 04 p.12/?? Extensional equality A classical computation induces a function U by % j b j e lnm ut oqpsr U We say that two computations are extensionally equivalent, if they give rise to the same function.
$ $ % i v $ % v i % MGS Xmas 04 p.13/?? Extensional equality... Theorem: U U U
$ $ % i v $ % v i % MGS Xmas 04 p.13/?? Extensional equality... Theorem: U U U Hence, classical computations upto extensional equality give rise to the category.
$ $ % i v c $ % v i % MGS Xmas 04 p.13/?? Extensional equality... Theorem: U U U Hence, classical computations upto extensional equality give rise to the category. Theorem: Any function on finite sets jcan be realized by a computation.
$ $ % i v c $ % v i % MGS Xmas 04 p.13/?? Extensional equality... Theorem: U U U Hence, classical computations upto extensional equality give rise to the category. Theorem: Any function on finite sets jcan be realized by a computation. Translation for Category Theoreticians: U is full and faithful.
% $ c j % $ { j MGS Xmas 04 p.14/?? : Example function wyx wzx
% $ c j % $ { j MGS Xmas 04 p.14/?? ^} ~ : Example function wyx wzx computation
$ % $ j j % MGS Xmas 04 p.15/?? Example : function c
% $ j % $ j ƒ % $ c ƒ j j % $ % $ ƒ % $ % $ ƒ MGS Xmas 04 p.15/?? Example : function c computation $ % j j j j
MGS Xmas 04 p.16/?? 2. Finite quantum computation 1. Finite classical computation 2. Finite quantum computation 3. QML basics 4. Compiling QML 5. Conclusions and further work
Pure quantum values MGS Xmas 04 p.17/??
# # cˆ # MGS Xmas 04 p.17/?? Pure quantum values A pure quantum value over a finite set given by with unit norm: c is # ˆ # #
# # cˆ # MGS Xmas 04 p.17/?? Pure quantum values A pure quantum value over a finite set given by with unit norm: c is # ˆ # # is monadic, giving rise to the category of (finite dimensional) vector spaces.
$ $ $ š # Vector spaces as a monad Š Œ c Ž Ž +, Ž Ž % c % % $ %?œ Ž š MGS Xmas 04 p.18/??
Reversible quantum operations MGS Xmas 04 p.19/??
MGS Xmas 04 p.19/?? Reversible quantum operations Reversible operations on pure quantum values are given by unitary operators.
MGS Xmas 04 p.19/?? Reversible quantum operations Reversible operations on pure quantum values are given by unitary operators. On finite dimensional vector spaces: unitary = norm preserving linear iso.
ž Ÿ % % MGS Xmas 04 p.19/?? Reversible quantum operations Reversible operations on pure quantum values are given by unitary operators. On finite dimensional vector spaces: unitary = norm preserving linear iso. The inverse is given by the adjoint: $ žc $ ž
Quantum computations ( ) MGS Xmas 04 p.20/??
_ ` \ ^ ] a MGS Xmas 04 p.20/?? Quantum computations ( ) Given finite sets (input) and (output):
_ ` \ c ^ ] a Quantum computations ( ) Given finite sets (input) and (output): a finite set, the base of the space of initial heaps, a heap initialisation vector b c a finite set, the base of the space of garbage states, a unitary operator, unitary. MGS Xmas 04 p.20/??
Composing quantum computations MGS Xmas 04 p.21/??
h g g ] h h g MGS Xmas 04 p.21/?? Composing quantum computations h g
Extensional equality? MGS Xmas 04 p.22/??
MGS Xmas 04 p.22/?? Extensional equality?... is a bit more subtle.
c e MGS Xmas 04 p.22/?? Extensional equality?... is a bit more subtle. There is no (sensible) operator on vector spaces replacing. wx
c e MGS Xmas 04 p.22/?? Extensional equality?... is a bit more subtle. There is no (sensible) operator on vector spaces replacing. wx Indeed: Forgetting part of a pure state results in a mixed state.
Density operators MGS Xmas 04 p.23/??
c MGS Xmas 04 p.23/?? Density operators Mixed states are represented by density operators (positive operators with unit trace).
c MGS Xmas 04 p.23/?? Density operators Mixed states are represented by density operators (positive operators with unit trace). is interpreted as the system is in the pure state with probability.
Superoperators MGS Xmas 04 p.24/??
MGS Xmas 04 p.24/?? Superoperators Morphisms on mixed states are completely positive linear operators on the space of density operators, called superoperators.
MGS Xmas 04 p.24/?? Superoperators Morphisms on mixed states are completely positive linear operators on the space of density operators, called superoperators. Every unitary operator superoperator. gives rise to a
r c MGS Xmas 04 p.24/?? Superoperators Morphisms on mixed states are completely positive linear operators on the space of density operators, called superoperators. Every unitary operator superoperator. There is an operator gives rise to a & ' super called partial trace.
Extensional equality MGS Xmas 04 p.25/??
p «g ci ci MGS Xmas 04 p.25/?? Extensional equality A quantum computation rise to a superoperator U super gives k z ª t U
p «g ci ci MGS Xmas 04 p.25/?? Extensional equality A quantum computation rise to a superoperator U super gives k z ª t U We say that two computations are extensionally equivalent, if they give rise to the same superoperator.
$ $ % i v $ % v i % MGS Xmas 04 p.26/?? Extensional equality... Theorem: U U U
$ $ % i v $ % v i % MGS Xmas 04 p.26/?? Extensional equality... Theorem: U U U Hence, quantum computations upto extensional equality give rise to the category.
$ $ % i v $ % v c i % MGS Xmas 04 p.26/?? Extensional equality... Theorem: U U U Hence, quantum computations upto extensional equality give rise to the category. Theorem: Every superoperator super (on finite Hilbert spaces) comes from a quantum computation.
$ $ % i v $ % v c i % MGS Xmas 04 p.26/?? Extensional equality... Theorem: U U U Hence, quantum computations upto extensional equality give rise to the category. Theorem: Every superoperator super (on finite Hilbert spaces) comes from a quantum computation. (U is full and faithful).
Classical vs quantum MGS Xmas 04 p.27/??
MGS Xmas 04 p.27/?? Classical vs quantum classical ( ) quantum ( )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets ) quantum ( )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets ) quantum ( finite dimensional Hilbert spaces )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections ) quantum ( finite dimensional Hilbert spaces )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections ) quantum ( finite dimensional Hilbert spaces unitary operators )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( ) quantum ( finite dimensional Hilbert spaces unitary operators ) )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( ) quantum ( finite dimensional Hilbert spaces unitary operators ) tensor product ( ) )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( functions ) quantum ( finite dimensional Hilbert spaces unitary operators ) tensor product ( ) )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( functions ) quantum ( finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( functions projections ) quantum ( finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators )
MGS Xmas 04 p.27/?? Classical vs quantum classical ( finite sets bijections cartesian product ( functions projections ) quantum ( finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators partial trace )
MGS Xmas 04 p.28/??, classically v wsx
MGS Xmas 04 p.28/??, classically v wsx ^} ~ ^±
² MGS Xmas 04 p.28/??, classically v wsx ^} ~ ^±
MGS Xmas 04 p.29/??, quantum ^ ~ ^³
# # MGS Xmas 04 p.29/??, quantum ^ ~ ^³ * x µ x µ input:
# # # # MGS Xmas 04 p.29/??, quantum ^ ~ ^³ * x µ x µ input: * x * x output:
# # # # MGS Xmas 04 p.29/??, quantum ^ ~ ^³ * x µ x µ input: * x * x output: Decoherence!
Control of decoherence MGS Xmas 04 p.30/??
MGS Xmas 04 p.30/?? Control of decoherence QML is based on strict linear logic
ƒ MGS Xmas 04 p.30/?? Control of decoherence QML is based on strict linear logic Contraction is implicit and realized by.
ƒ MGS Xmas 04 p.30/?? Control of decoherence QML is based on strict linear logic Contraction is implicit and realized by Weakening is explicit and leads to decoherence..
MGS Xmas 04 p.31/?? 3. QML 1. Finite classical computation 2. Finite quantum computation 3. QML 4. Conclusions and further work
QML overview MGS Xmas 04 p.32/??
# # MGS Xmas 04 p.32/?? QML overview Types
# # # % $ { j # % À QML overview Types Terms ¹ # + # % $# % $# + j º» ' # º¼» * { º» ' # º» (¾,½ # * { º» ' # º» (¾,½ # * % #$Á $ # MGS Xmas 04 p.32/??
,½ MGS Xmas 04 p.33/?? * * Qbits % $ º¼» " & ') % $ º¼» '!" +, º» ' # º»,½ +, º» ' # º»
QML overview... MGS Xmas 04 p.34/??
  MGS Xmas 04 p.34/?? QML overview... Typing judgements ÃÄ ÃÄ programs strict programs
Æ Æ Å Â Å Æ Æ Å Â Å QML overview... Typing judgements programs strict programs ÃÄ Â ÃÄ Â Semantics Ä Ã Â ÃÄ Â Æ c Å Ä Æ c Å Ä MGS Xmas 04 p.34/??
j & " Â Ä Ë Ê MGS Xmas 04 p.35/?? ÃÄ Â ÃÇ ÃÉÈ Ç ÌÎÍ The let-rule
j & " Â Ä Ë Ê Â Ö Õ Ö Ù Ø Ü Û r Ú Ú Ú MGS Xmas 04 p.35/?? ÃÄ Â ÃÇ ÃÉÈ Ç ÌÎÍ The let-rule Ô ÏÑÐÓÒ
on contexts MGS Xmas 04 p.36/??
% j j Ý c if % j dom MGS Xmas 04 p.36/?? on contexts  $  j  $  j
j Ý c Â Â Ù Ø r dom MGS Xmas 04 p.36/?? on contexts % j  $  j if % j  $  j Ô ÏÑÐÓÒ
Another source of decoherence MGS Xmas 04 p.37/??
MGS Xmas 04 p.37/?? Another source of decoherence mentions Ž Ž " &(') +, " & ') Ž
MGS Xmas 04 p.37/?? Another source of decoherence mentions Ž Ž " &(') +, " & ') Ž but doesn t use it.
Ž Ž Ž MGS Xmas 04 p.37/?? Another source of decoherence mentions " &(') +, & ') " but doesn t use it. Hence, it has to measure it!
-elim MGS Xmas 04 p.38/??
ë é â ë ì å â û ú ë é â äæå ã ßáàâ Þ ßê â ã ç è ßáíâ ç è ñòôó îðï í ì ý ÿ ê üþý ÿ à ùø MGS Xmas 04 p.38/?? -elim ßáö ç Þnõ
ë é â ë ì å â û ú ë é â ç ë $ äæå ã ßáàâ Þ ßê â ã ç è ßáíâ ç è ñòôó îðï í ì ý ÿ ê üþý ÿ à ùø!#" $& MGS Xmas 04 p.38/?? -elim ßáö ç Þnõ Þõ %
-elim decoherence-free MGS Xmas 04 p.39/??
ë é â ê ë ì å â û ú à ( ùø ë é â -elim decoherence-free ä å ã ß' àâ Þ ß(ê â ã ç è ) í ß( íâ ç è ( ñòôó ï ä+* í ì ý ÿ ê üþý ÿ ' ß ö ç Þõ MGS Xmas 04 p.39/??
ë é â ê ë ì å â û ú à ( ùø ë é â ç ë 0 -elim decoherence-free ä å ã ß' àâ Þ ß(ê â ã ç è ) í ß( íâ ç è ( ñòôó ï ä+* í ì ý ÿ ê üþý ÿ ' ß ö ç Þõ Þõ 1 -, 2 2 /.. $3. -, % MGS Xmas 04 p.39/??
5 4 MGS Xmas 04 p.40/??
: 9 876 : 9 876 A O KMLNJ PHQ G GO KML J N MGS Xmas 04 p.40/?? >? > ;=< FHGI E D @ CB ;@ 5 4
: 9 876 : 9 876 A W W 5 4 >? > ;=< O KMLNJ PHQ G GO KML J N FHGI E D @ CB ;@ This program has a type error, because. STV R SUTV R MGS Xmas 04 p.40/??
: 9 876 : 9 876 A W W A 5 4 >? > ;=< O KMLNJ PHQ G GO KML J N FHGI E D @ CB ;@ This program has a type error, because. STV R SUTV R >? > : 6< YX O KMLNJ PHQ G GO ]H^ Z\[ J FHGI E D @ CB : @ 6 YX MGS Xmas 04 p.40/??
: 9 876 : 9 876 A W W A W 5 4 >? > ;=< O KMLNJ PHQ G GO KML J N FHGI E D @ CB ;@ This program has a type error, because. STV R SUTV R >? > : 6< YX O KMLNJ PHQ G GO ]H^ Z\[ J FHGI E D @ CB : @ 6 YX This program typechecks, because. SUTV R W amb _` R MGS Xmas 04 p.40/??
MGS Xmas 04 p.41/?? 4. QML 1. Finite classical computation 2. Finite quantum computation 3. QML 4. Conclusions and further work
Conclusions MGS Xmas 04 p.42/??
Conclusions Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs. MGS Xmas 04 p.42/??
MGS Xmas 04 p.42/?? 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.
MGS Xmas 04 p.42/?? 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. 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.
Further work MGS Xmas 04 p.43/??
MGS Xmas 04 p.43/?? Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach.
MGS Xmas 04 p.43/?? Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach. Are we able to come up with completely new algorithms using QML?
MGS Xmas 04 p.43/?? Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach. Are we able to come up with completely new algorithms using QML? How to deal with higher order programs?
MGS Xmas 04 p.43/?? Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach. 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?
MGS Xmas 04 p.43/?? Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach. 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.
MGS Xmas 04 p.44/?? The end Thank you for your attention. Draft paper: quant-ph/0409065 from arxiv.org