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Quantum programming

Quantum programming is a set of computer programming languages that allow the expression of quantum algorithms using high-level constructs. The point of quantum languages is not so much to provide a tool for programmers, but to provide tools for researchers to understand better how quantum computation works and how to formally reason about quantum algorithms.

Efforts are underway to develop functional programming languages for quantum computing. Examples include Selinger's QPL [1], and the Haskell-like language QML by Altenkirch and Grattage [2][3]. Higher-order quantum programming languages, based on lambda calculus, have been proposed by van Tonder [4], Selinger and Valiron [5] and by Arrighi and Dowek[6].

Simon Gay's Quantum Programming Languages Survey has more information on the current state of research and a comprehensive bibliography of resources.

Imperative quantum programming languages

Quantum pseudocode

Quantum pseudocode proposed by E. Knill is the first formalised language for description of quantum algorithms was introduced and, moreover, it was tightly connected with a model of quantum machine called Quantum Random Access Machine (QRAM).

Quantum computing language

QCL (Quantum Computer Language) is one of the first implemented quantum programming languages. Its syntax resembles syntax of the C programming language and classical data types are similar to data types in C. You can combine a classical code and a quantum code in one source code.

The basic built-in quantum data type in QCL is the qureg (quantum register). It can be interpreted as an array of qubits (quantum bits).

qureg x1[2]; // 2-qubit quantum register x1
qureg x2[2]; // 2-qubit quantum register x2
H(x1); // Hadamard operation on x1
H(x2[1]); // Hadamard operation on the first qubit of the register x2

Since the qcl interpreter uses qlib simulation library, it is possible to observe the internal state of the quantum machine during execution of the quantum program.

qcl> dump
: STATE: 4 / 32 qubits allocated, 28 / 32 qubits free
0.35355 |0> + 0.35355 |1> + 0.35355 |2> + 0.35355 |3>
+ 0.35355 |8> + 0.35355 |9> + 0.35355 |10> + 0.35355 |11>

Note that the dump operation is different from measurement, since it does not influence the state of the quantum machine and can be realised only using a simulator.

The QCL standard library provides standard quantum operators used in quantum algorithms such as:

* controlled-not with many target qubits,
* Hadamard operation on many qubits,
* parse and controlled phase.

But the most important feature of QCL is the support for user-defined operators and functions. Like in modern programming languages, it is possible to define new operations which can be used to manipulate quantum data. For example:

operator diffuse (qureg q) {
H(q); // Hadamard Transform
Not(q); // Invert q
CPhase(pi, q); // Rotate if q=1111..
!Not(q); // undo inversion
!H(q); // undo Hadamard Transform

defines inverse about the mean operator used in Grover's algorithm. This allows one to define algorithms on a higher level of abstraction and extend the library of functions available for programmers.


* Data types
o Quantum - qureg, quvoid, quconst, quscratch, qucond
o Classical - int, real, complex, boolean, string, vector, matrix, tensor
* Function types
o qufunct - Pseudo-classic operators. Can only change the permutation of basic states.
o operator - General unitary operators. Can change the amplitude.
o procedure - Can call measure, print, and dump inside this function. This function is non-invertible.
* Build-in functions
o Quantum
+ qufunct - Fanout, Swap, Perm2, Perm4, Perm8, Not, CNot
+ operator - Matrix2x2, Matrix4x4, Matrix8x8, Rot, Mix, H, CPhase, SqrtNot, X, Y, Z, S, T
+ procedure - measure, dump, reset
o Classical
+ Arithmetic - sin, cos, tan, log, sqrt, ...
+ Complex - Re, Im, conj

Q language

Q Language is the second implemented imperative quantum programming language.

Q Language was implemented as an extension of C++ programming language. It provides classes for basic quantum operations like QHadamard, QFourier, QNot, and QSwap, which are derived from the base class Qop. New operators can be defined using C++ class mechanism.

Quantum memory is represented by class Qreg.

Qreg x1(); // 1-qubit quantum register with initial value 0
Qreg x2(2,0); // 2-qubit quantum register with initial value 0

Computation process is executed using provided simulator. Noisy environment can be simulated using parameters of the simulator.


Quantum Guarded Command Language (qGCL) was defined by P. Zuliani in his PhD thesis. It is based on Guarded Command Language created by Edsger Dijkstra.

It can be described as a language of quantum programmes specification.

Functional quantum programming languages

During the last few years many quantum programming languages based on the functional programming paradigm were proposed. Functional programming languages are well-suited for reasoning about programs.


QFC and QPL are two closely related quantum programming languages defined by Peter Selinger. They differ only in their syntax: QFC uses a flow chart syntax, whereas QPL uses a textual syntax. These languages have classical control flow, but can operate on quantum or classical data. Selinger gives a denotational semantics for these languages in a category of superoperators.


QML is a Haskell-like quantum programming language by Altenkirch and Grattage[2][7]. Unlike Selinger's QPL, this language takes duplication, rather than discarding, of quantum information as a primitive operation. Duplication in this context is understood to be the operation that maps |\phi\rangle to |\phi\rangle\otimes|\phi\rangle, and is not to be confused with the impossible operation of cloning; the authors claim it is akin to how sharing is modelled in classical languages. QML also introduces both classical and quantum control operators, whereas most other languages rely on classical control.

An operational semantics for QML is given in terms of quantum circuits, while a denotational semantics is presented in terms of superoperators, and these are shown to agree. Both the operational and denotational semantics have been implemented (classically) in Haskell[8].

Quantum lambda calculi

Quantum lambda calculi are extensions of the lambda calculus, introduced by Alonzo Church and Stephen Cole Kleene in the 1930s. The purpose of quantum lambda calculi is to extend quantum programming languages with a theory of higher-order functions.

The first attempt to define a quantum lambda calculus was made by Philip Maymin in 1996. His lambda-q calculus is powerful enough to express any quantum computation. However, this language can efficiently solve NP-complete problems, and therefore appears to be strictly stronger than the standard quantum computational models (such as the quantum Turing machine or the quantum circuit model). Therefore, Maymin's lambda-q calculus is probably not implementable on a physical device.

In 2003, André van Tonder defined an extension of the lambda calculus suitable for proving correctness of quantum programs. He also provided an implementation in the Scheme programming language[9].

In 2004, Selinger and Valiron defined a strongly typed lambda calculus for quantum computation with a type system based on linear logic.


1. ^ Peter Selinger: Papers
2. ^ a b Jonathan Grattage: QML Research
3. ^ QML: A Functional Quantum Programming Language
4. ^ A Lambda Calculus for Quantum Computation
5. ^ Peter Selinger: Papers
6. ^ Linear-algebraic lambda-calculus: higher-order, encodings and confluence
7. ^ "QML: A Functional Quantum Programming Language". http://sneezy.cs.nott.ac.uk/qml.
8. ^ QML: A Functional Quantum Programming Language
9. ^ André van Tonder. "A lambda calculus for quantum computation". http://www.het.brown.edu/people/andre/qlambda.

External links

* 5th International Workshop on Quantum Physics and Logic
* 4th International Workshop on Quantum Programming Languages
* 3rd International Workshop on Quantum Programming Languages
* 2nd International Workshop on Quantum Programming Languages
* Bibliography on Quantum Programming Languages
* Quantum programming language in Quantiki

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