Quantum programming

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Quantum programming is a set of computer programming languages that allow the expression of quantum algorithms using high-level constructs.[1] 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 reason formally about quantum algorithms.

One can single out two main groups of quantum programming languages: imperative quantum programming languages and functional quantum programming languages.

The most prominent representatives of the first group are QCL[2] and LanQ.[3]

Efforts are underway to develop functional programming languages for quantum computing. Examples include Selinger's QPL,[4] and the Haskell-like language QML by Altenkirch and Grattage.[5][6] Higher-order quantum programming languages, based on lambda calculus, have been proposed by van Tonder,[7] Selinger and Valiron [8] and by Arrighi and Dowek.[9]

Simon Gay's Quantum Programming Languages Survey provides information on the state of research and a comprehensive bibliography of resources about quantum programming as of 2007.

Imperative quantum programming languages[edit]

Quantum pseudocode[edit]

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

Quantum computing language[edit]

QCL (Quantum Computation Language) is one of the first implemented quantum programming languages. Its syntax resembles the syntax of the C programming language and its classical data types are similar to primitive data types in C. One can combine classical code and quantum code in the same program.

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 realized 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.

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
    • Quantum - qureg, quvoid, quconst, quscratch, qucond
    • Classical - int, real, complex, boolean, string, vector, matrix, tensor
  • Function types
    • qufunct - Pseudo-classic operators. Can only change the permutation of basic states.
    • operator - General unitary operators. Can change the amplitude.
    • procedure - Can call measure, print, and dump inside this function. This function is non-invertible.
  • Built-in functions
    • 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
    • Classical
      • Arithmetic - sin, cos, tan, log, sqrt, ...
      • Complex - Re, Im, conj

Q language[edit]

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

The computation process is executed using a 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 programs specification.

Functional quantum programming languages[edit]

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[edit]

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.[5] 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 to , and is not to be confused with the impossible operation of cloning; the authors claim it is akin to how sharing is modeled 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.[10]


LIQUi|> (pronounced 'liquid') is a quantum simulation extension on the F# programming language. It is currently being developed by the Quantum Architectures and Computation Group (QuArC) [11] part of the StationQ efforts at Microsoft Research. LIQUi|> seeks to allow theorists to experiment with quantum algorithm design before physical quantum computers are available for use.

It includes a programming language, optimization and scheduling algorithms, and quantum simulators. LIQUi|> can be used to translate a quantum algorithm written in the form of a high-level program into the low-level machine instructions for a quantum device. [12]

Quantum lambda calculi[edit]

Quantum lambda calculi are extensions of the classical 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.[13] 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.[14]

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


Quipper was published in 2013.[15] It is implemented as an embedded language, using Haskell as the host language.[16] For this reason, quantum programs written in Quipper are written in Haskell using provided libraries. For example, the following code implements preparation of a superposition

   import Quipper
   spos :: Bool -> Circ Qubit
   spos b = do q <- qinit b
               r <- hadamard q
               return r


  1. ^ Jarosław Adam Miszczak. "High-level Structures in Quantum Computing". Retrieved 12 December 2015. 
  2. ^ Bernhard Omer. "The QCL Programming Language". 
  3. ^ Hynek Mlnařík. "LanQ – a quantum imperative programming language". 
  4. ^ Peter Selinger, "Towards a quantum programming language", Mathematical Structures in Computer Science 14(4):527-586, 2004.
  5. ^ a b Jonathan Grattage: QML Research (website)
  6. ^ T. Altenkirch, V. Belavkin, J. Grattage, A. Green, A. Sabry, J. K. Vizzotto, QML: A Functional Quantum Programming Language (website)
  7. ^ Andre van Tonder, "A Lambda Calculus for Quantum Computation", SIAM J. Comput., 33(5), 1109–1135. (27 pages), 2004. Also available from arXiv:quant-ph/0307150
  8. ^ Peter Selinger and Benoît Valiron, "A lambda calculus for quantum computation with classical control", Mathematical Structures in Computer Science 16(3):527-552, 2006.
  9. ^ Pablo Arrighi, Gilles Dowek, "Linear-algebraic lambda-calculus: higher-order, encodings and confluence", 2006
  10. ^ Jonathan Grattage, QML: A Functional Quantum Programming Language (compiler), 2005–2008
  11. ^ Quantum Architectures and Computation Group (QuArC), https://www.microsoft.com/en-us/research/group/quantum-architectures-and-computation-group-quarc/, 2011
  12. ^ Language-Integrated Quantum Operations: LIQUi|>, [1], 2016
  13. ^ Philip Maymin, "Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms", 1996
  14. ^ André van Tonder. "A lambda calculus for quantum computation (website)". 
  15. ^ Alexander S. Green; Peter LeFanu Lumsdaine; Neil J. Ross; Peter Selinger; Benoît Valiron. "The Quipper Language (website)". 
  16. ^ Alexander S. Green; Peter LeFanu Lumsdaine; Neil J. Ross; Peter Selinger; Benoît Valiron (2013). "An Introduction to Quantum Programming in Quipper". arXiv:1304.5485Freely accessible. 

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