Towards a High Level Quantum Programming Language
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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 Towards ahigh LevelQuantum Programming Language p.1/36
2 Background Towards ahigh LevelQuantum Programming Language p.2/36
3 Background Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Towards ahigh LevelQuantum Programming Language p.2/36
4 Background Simulation of quantum systems is expensive: PSPACE complexity for polynomial circuits. Feynman: Can we exploit this fact to perform computations more efficiently? Towards ahigh LevelQuantum Programming Language p.2/36
5 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. Towards ahigh LevelQuantum Programming Language p.2/36
6 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 Towards ahigh LevelQuantum Programming Language p.2/36
7 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? Towards ahigh LevelQuantum Programming Language p.2/36
8 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. Towards ahigh LevelQuantum Programming Language p.2/36
9 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! Towards ahigh LevelQuantum Programming Language p.2/36
10 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! Assumption: Nature is fair... Towards ahigh LevelQuantum Programming Language p.2/36
11 The quantum software crisis Towards ahigh LevelQuantum Programming Language p.3/36
12 The quantum software crisis Quantum algorithms are usually presented using the circuit model. Towards ahigh LevelQuantum Programming Language p.3/36
13 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. Towards ahigh LevelQuantum Programming Language p.3/36
14 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! Towards ahigh LevelQuantum Programming Language p.3/36
15 QML Towards ahigh LevelQuantum Programming Language p.4/36
16 QML QML: a functional language for quantum computations on finite types. Towards ahigh LevelQuantum Programming Language p.4/36
17 QML QML: a functional language for quantum computations on finite types. Quantum control and quantum data. Towards ahigh LevelQuantum Programming Language p.4/36
18 QML QML: a functional language for quantum computations on finite types. Quantum control and quantum data. Design guided by semantics Towards ahigh LevelQuantum Programming Language p.4/36
19 QML QML: a functional language for quantum computations on finite types. Quantum control and quantum data. Design guided by semantics Analogy with classical computation Finite classical computations Finite quantum computations Towards ahigh LevelQuantum Programming Language p.4/36
20 QML QML: a functional language for quantum computations 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 Towards ahigh LevelQuantum Programming Language p.4/36
21 QML QML: a functional language for quantum computations 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 Draft paper available (Google:Thorsten,functional,quantum) Towards ahigh LevelQuantum Programming Language p.4/36
22 QML QML: a functional language for quantum computations 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 Draft paper available (Google:Thorsten,functional,quantum) Compiler under construction (Jonathan) Towards ahigh LevelQuantum Programming Language p.4/36
23 Example: Hadamard operation Towards ahigh LevelQuantum Programming Language p.5/36
24 Example: Hadamard operation Matrix Towards ahigh LevelQuantum Programming Language p.5/36
25 ( ( Example: Hadamard operation Matrix QML $&%' # "! $&%'! )* Towards ahigh LevelQuantum Programming Language p.5/36
26 Overview 1. Semantics of finite classical and quantum computation 2. QML basics 3. Compiling QML 4. Conclusions and further work Towards ahigh LevelQuantum Programming Language p.6/36
27 1. Semantics 1. Semantics of finite classical and quantum computation 2. QML basics 3. Compiling QML 4. Conclusions and further work Towards ahigh LevelQuantum Programming Language p.7/36
28 Something we know well... Towards ahigh LevelQuantum Programming Language p.8/36
29 Something we know well... Start with classical computations on finite types. Towards ahigh LevelQuantum Programming Language p.8/36
30 Something we know well... Start with classical computations on finite types. Quantum mechanics is time-reversible... Towards ahigh LevelQuantum Programming Language p.8/36
31 Something we know well... Start with classical computations on finite types. Quantum mechanics is time-reversible hence quantum computation is based on reversible operations. Towards ahigh LevelQuantum Programming Language p.8/36
32 Something we know well... Start with 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... Towards ahigh LevelQuantum Programming Language p.8/36
33 Something we know well... Start with 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. Towards ahigh LevelQuantum Programming Language p.8/36
34 Classical computation ( ) Towards ahigh LevelQuantum Programming Language p.9/36
35 . / + -, 0 Classical computation ( ) Given finite sets (input) and (output): Towards ahigh LevelQuantum Programming Language p.9/36
36 . / + -, 0 Classical computation ( ) Given finite sets (input) and (output): a finite set of initial heaps, Towards ahigh LevelQuantum Programming Language p.9/36
37 . / + -, 0 Classical computation ( ) Given finite sets (input) and (output): a finite set of initial heaps, an initial heap 132, Towards ahigh LevelQuantum Programming Language p.9/36
38 . / + -, 0 Classical computation ( ) Given finite sets (input) and (output): a finite set of initial heaps, an initial heap 132 a finite set of garbage states,, Towards ahigh LevelQuantum Programming Language p.9/36
39 . 2 / + 4-5, 0 4 Classical computation ( ) Given finite sets (input) and (output): a finite set of initial heaps, an initial heap 132 a finite set of garbage states, a bijection,, Towards ahigh LevelQuantum Programming Language p.9/36
40 Semantics Towards ahigh LevelQuantum Programming Language p.10/36
41 # " " 86 Semantics A classical computation 1 2 induces a function U by U 9;: # Towards ahigh LevelQuantum Programming Language p.10/36
42 # " " 86 2 Semantics A classical computation 1 2 induces a function U by U 9;: # Theorem Any function (on finite sets 7) can be realized by a quantum computation. Towards ahigh LevelQuantum Programming Language p.10/36
43 Composing classical computations Towards ahigh LevelQuantum Programming Language p.11/36
44 = < <, = = < Composing classical computations = < Towards ahigh LevelQuantum Programming Language p.11/36
45 = < " <, " # 6 > = " # > 6 # = < Composing classical computations = < Theorem: U U U Towards ahigh LevelQuantum Programming Language p.11/36
46 Coming next: Quantum computations Develop analogously to... Towards ahigh LevelQuantum Programming Language p.12/36
47 Linear algebra revision Towards ahigh LevelQuantum Programming Language p.13/36
48 Linear algebra revision Given a finite set (the base) is a Hilbert space. Towards ahigh LevelQuantum Programming Language p.13/36
49 2 2 Linear algebra revision Given a finite set (the base) is a Hilbert space. Linear operators: induces we write? 2. Towards ahigh LevelQuantum Programming Language p.13/36
50 2 2 E"# F " A 8 A 8 D C B Linear algebra revision Given a finite set (the base) is a Hilbert space. Linear operators: induces we write Norm of # 2? 2. Towards ahigh LevelQuantum Programming Language p.13/36
51 2 2 E"# F " A 8 A 8 D C B 2 Linear algebra revision Given a finite set (the base) is a Hilbert space. Linear operators: induces we write Norm of a vector:, A unitary operator # 2? 2 unitary is a linear isomorphism that preserves the norm.. Towards ahigh LevelQuantum Programming Language p.13/36
52 Basics of quantum computation Towards ahigh LevelQuantum Programming Language p.14/36
53 2 A Basics of quantum computation A pure state over is a vector with Towards ahigh LevelQuantum Programming Language p.14/36
54 2 A 2 Basics of quantum computation A pure state over is a vector A reversible computation is given by a unitary operator unitary. with Towards ahigh LevelQuantum Programming Language p.14/36
55 Quantum computations ( ) Towards ahigh LevelQuantum Programming Language p.15/36
56 . / + -, 0 Quantum computations ( ) Given finite sets (input) and (output): Towards ahigh LevelQuantum Programming Language p.15/36
57 . / + -, 0 Quantum computations ( ) Given finite sets (input) and (output): a finite set heaps,, the base of the space of initial Towards ahigh LevelQuantum Programming Language p.15/36
58 . / + -, 0 Quantum computations ( ) Given finite sets (input) and (output): a finite set, the base of the space of initial heaps, a heap initialisation vector 1 2, Towards ahigh LevelQuantum Programming Language p.15/36
59 . / + -, 0 Quantum computations ( ) Given finite sets (input) and (output): a finite set, the base of the space of initial heaps, a heap initialisation vector 1 2 a finite set, the base of the space of garbage states,, Towards ahigh LevelQuantum Programming Language p.15/36
60 . / + 2 -, 0 Quantum computations ( ) Given finite sets (input) and (output): a finite set, the base of the space of initial heaps, a heap initialisation vector 1 2 a finite set, the base of the space of garbage states, a unitary operator, unitary. Towards ahigh LevelQuantum Programming Language p.15/36
61 Composing quantum computations Towards ahigh LevelQuantum Programming Language p.16/36
62 = < <, = = < Composing quantum computations = < Towards ahigh LevelQuantum Programming Language p.16/36
63 Semantics of quantum computations.. Towards ahigh LevelQuantum Programming Language p.17/36
64 Semantics of quantum computations..... is a bit more subtle. Towards ahigh LevelQuantum Programming Language p.17/36
65 2 4 Semantics of quantum computations..... is a bit more subtle. There is no (sensible) operator on vector spaces, replacing. 9: Towards ahigh LevelQuantum Programming Language p.17/36
66 2 4 Semantics of quantum computations..... is a bit more subtle. There is no (sensible) operator on vector spaces, replacing. 9: Indeed: Forgetting part of a pure state results in a mixed state. Towards ahigh LevelQuantum Programming Language p.17/36
67 Density matrices and superoperators Towards ahigh LevelQuantum Programming Language p.18/36
68 Density matrices and superoperators Mixed states are represented by density matrices. Towards ahigh LevelQuantum Programming Language p.18/36
69 Density matrices and superoperators Mixed states are represented by density matrices. Operations on mixed states (i.e. density matrices) are represented by superoperators. Towards ahigh LevelQuantum Programming Language p.18/36
70 G Density matrices and superoperators Mixed states are represented by density matrices. Operations on mixed states (i.e. density matrices) are represented by superoperators. Every unitary operator superoperator. gives rise to a Towards ahigh LevelQuantum Programming Language p.18/36
71 G 2 O Density matrices and superoperators Mixed states are represented by density matrices. Operations on mixed states (i.e. density matrices) are represented by superoperators. Every unitary operator superoperator. There is an operator gives rise to a HJILKNM super called partial trace. Towards ahigh LevelQuantum Programming Language p.18/36
72 Semantics Towards ahigh LevelQuantum Programming Language p.19/36
73 26 V W < 6 Semantics Every quantum computation superoperator U super gives rise to a PRQ S;TU XZY U Towards ahigh LevelQuantum Programming Language p.19/36
74 26 V W < 6 2 Semantics Every quantum computation superoperator U super gives rise to a PRQ S;TU XZY U Theorem: Every superoperator super (on finite Hilbert spaces) comes from a quantum computation. Towards ahigh LevelQuantum Programming Language p.19/36
75 Classical vs quantum Towards ahigh LevelQuantum Programming Language p.20/36
76 Classical vs quantum classical quantum Towards ahigh LevelQuantum Programming Language p.20/36
77 Classical vs quantum classical finite sets quantum Towards ahigh LevelQuantum Programming Language p.20/36
78 Classical vs quantum classical finite sets quantum finite dimensional Hilbert spaces Towards ahigh LevelQuantum Programming Language p.20/36
79 Classical vs quantum classical finite sets bijections quantum finite dimensional Hilbert spaces Towards ahigh LevelQuantum Programming Language p.20/36
80 Classical vs quantum classical finite sets bijections quantum finite dimensional Hilbert spaces unitary operators Towards ahigh LevelQuantum Programming Language p.20/36
81 [ Classical vs quantum classical finite sets bijections cartesian product ( ) quantum finite dimensional Hilbert spaces unitary operators Towards ahigh LevelQuantum Programming Language p.20/36
82 \ [ Classical vs quantum classical finite sets bijections cartesian product ( quantum finite dimensional Hilbert spaces unitary operators ) tensor product ( ) Towards ahigh LevelQuantum Programming Language p.20/36
83 \ [ Classical vs quantum classical finite sets bijections cartesian product ( functions quantum finite dimensional Hilbert spaces unitary operators ) tensor product ( ) Towards ahigh LevelQuantum Programming Language p.20/36
84 \ [ Classical vs quantum classical finite sets bijections cartesian product ( functions quantum finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators Towards ahigh LevelQuantum Programming Language p.20/36
85 \ [ Classical vs quantum classical finite sets bijections cartesian product ( functions projections quantum finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators Towards ahigh LevelQuantum Programming Language p.20/36
86 \ [ Classical vs quantum classical finite sets bijections cartesian product ( functions projections quantum finite dimensional Hilbert spaces unitary operators ) tensor product ( ) superoperators partial trace Towards ahigh LevelQuantum Programming Language p.20/36
87 Decoherence Towards ahigh LevelQuantum Programming Language p.21/36
88 ^ ] Decoherence -bac -`_ Towards ahigh LevelQuantum Programming Language p.21/36
89 ^ ] d e Decoherence -bac -`_ Classically > 9L: Towards ahigh LevelQuantum Programming Language p.21/36
90 ^ ] d e Decoherence -bac -`_ Classically > 9L: Quantum Towards ahigh LevelQuantum Programming Language p.21/36
91 ^ g^! ] ( g^! d e Decoherence -bac -`_ Classically > 9L: Quantum input: : f : f Towards ahigh LevelQuantum Programming Language p.21/36
92 ^ g^! ] ( g^! d e Decoherence -bac -`_ Classically > 9L: Quantum input: : f : f output: ( g! : ( g^! : Towards ahigh LevelQuantum Programming Language p.21/36
93 2. QML basics 1. Semantics of finite classical and quantum computation 2. QML basics 3. Compiling QML 4. Conclusions and further work Towards ahigh LevelQuantum Programming Language p.22/36
94 QML basics Towards ahigh LevelQuantum Programming Language p.23/36
95 QML basics QML is a first order functional languages, i.e. programs are well-typed expressions. Towards ahigh LevelQuantum Programming Language p.23/36
96 i QML basics QML is a first order functional languages, i.e. programs are well-typed expressions. QML types are h 7 h i 7 Towards ahigh LevelQuantum Programming Language p.23/36
97 i QML basics QML is a first order functional languages, i.e. programs are well-typed expressions. QML types are Qbits h 7 h i 7 Towards ahigh LevelQuantum Programming Language p.23/36
98 i QML basics QML is a first order functional languages, i.e. programs are well-typed expressions. QML types are Qbits Qbytes j h 7 h i 7. Towards ahigh LevelQuantum Programming Language p.23/36
99 l k k QML basics... A QML program is an expression in a context of typed variables, e.g. $ $ l $&%' ) * Towards ahigh LevelQuantum Programming Language p.24/36
100 l k k QML basics... A QML program is an expression in a context of typed variables, e.g. $ $ l $&%' ) * We can compile QML programs into quantum computations (i.e. quantum circuits). Towards ahigh LevelQuantum Programming Language p.24/36
101 QML basics... Forgetting variables has to be explicit. Towards ahigh LevelQuantum Programming Language p.25/36
102 # " $ n 7 QML basics... Forgetting variables has to be explicit. E.g. $m is illegal, Towards ahigh LevelQuantum Programming Language p.25/36
103 # " $ n 7 ( # " $ n n 7 QML basics... Forgetting variables has to be explicit. E.g. $m is illegal, but $m is ok. Towards ahigh LevelQuantum Programming Language p.25/36
104 QML basics... There are two different if-then-else (or more generally case) constructs. Towards ahigh LevelQuantum Programming Language p.26/36
105 o o QML basics... There are two different if-then-else (or more generally case) constructs. $&%' ) * is just the identity, Towards ahigh LevelQuantum Programming Language p.26/36
106 o o p p introduces a measurement (end hence Towards ahigh LevelQuantum Programming Language p.26/36 decoherence). QML basics... There are two different if-then-else (or more generally case) constructs. $&%' ) * is just the identity, but $&%' )*
107 QML basics... Using is only allowed, if the branches are orthogonal, i.e. observable different. Towards ahigh LevelQuantum Programming Language p.27/36
108 s rq # " q q n s rq 7 # " # n 7 QML basics... Using is only allowed, if the branches are orthogonal, i.e. observable different. n7 " ) * is illegal, Towards ahigh LevelQuantum Programming Language p.27/36
109 # " q n s rq 7 # # " n 7 # " # " q n s rq 7 ## " 7 " ## " n 7 7 QML basics... Using is only allowed, if the branches are orthogonal, i.e. observable different. s rq q " n7 ) * is illegal, but s rq q $&%' " n7 ) * is ok. Towards ahigh LevelQuantum Programming Language p.27/36
110 ( ( QML basics... We can introduce superpositions, e.g. $&%' # "! $&%'! )* Towards ahigh LevelQuantum Programming Language p.28/36
111 ( ( QML basics... We can introduce superpositions, e.g. # $&%' "! $&%'! )* However, the terms in the superposition have to be orthogonal. Towards ahigh LevelQuantum Programming Language p.28/36
112 3. Compiling QML 1. Semantics of finite classical and quantum computation 2. QML basics 3. Compiling QML 4. Conclusions and further work Towards ahigh LevelQuantum Programming Language p.29/36
113 Compilation Towards ahigh LevelQuantum Programming Language p.30/36
114 t i h 7 z v # " t Compilation Correct QML programs are defined by typing rules, e.g. uv uy i x h 7 w { ~} u y ƒ x w 7 Towards ahigh LevelQuantum Programming Language p.30/36
115 t i h 7 z v # " t Compilation Correct QML programs are defined by typing rules, e.g. uv uy i x h 7 w { ~} u y ƒ x w 7 For each rule we can construct a quantum computation, i.e. a circuit. Towards ahigh LevelQuantum Programming Language p.30/36
116 i 7 v # " t uv t h uy i x h 7 w { z} u y ƒ x w 7 Towards ahigh LevelQuantum Programming Language p.31/36 -elim
117 i 7 v # " t t Œ M Ž uv t h uy i x h 7 w { z} u y ƒ x w 7 ˆ Š -elim Ž Ž Towards ahigh LevelQuantum Programming Language p.31/36
118 Compiler Towards ahigh LevelQuantum Programming Language p.32/36
119 Compiler A compiler is currently being implemented by my student Jonathan Grattage (in Haskell). Towards ahigh LevelQuantum Programming Language p.32/36
120 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. Towards ahigh LevelQuantum Programming Language p.32/36
121 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. Towards ahigh LevelQuantum Programming Language p.32/36
122 4. Conclusions 1. Semantics of finite classical and quantum computation 2. QML basics 3. Compiling QML 4. Conclusions and further work Towards ahigh LevelQuantum Programming Language p.33/36
123 Conclusions Towards ahigh LevelQuantum Programming Language p.34/36
124 Conclusions Our semantic ideas proved useful when designing a quantum programming language, analogous concepts are modelled by the same syntactic constructs. Towards ahigh LevelQuantum Programming Language p.34/36
125 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. Towards ahigh LevelQuantum Programming Language p.34/36
126 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. Towards ahigh LevelQuantum Programming Language p.34/36
127 Further work Towards ahigh LevelQuantum Programming Language p.35/36
128 Further work We have to analyze more quantum programs to evaluate the practical usefulness of our approach. Towards ahigh LevelQuantum Programming Language p.35/36
129 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? Towards ahigh LevelQuantum Programming Language p.35/36
130 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? Towards ahigh LevelQuantum Programming Language p.35/36
131 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? Towards ahigh LevelQuantum Programming Language p.35/36
132 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. Towards ahigh LevelQuantum Programming Language p.35/36
133 The end Thank you for your attention. Towards ahigh LevelQuantum Programming Language p.36/36
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