# nLab quantum circuit diagram

Contents

### Context

#### Computation

intuitionistic mathematics

### Computability

#### Monoidal categories

monoidal categories

# Contents

## Idea

In the context of quantum computation, a quantum circuit diagram (originally: “quantum computational network”, Deutsch 1989) is a kind of string diagram (Abramsky & Coecke 2004) in finite-dimensional Hilbert spaces, typically used to express a sequence of low-level quantum gates mapping between spaces of pure quantum states.

A generic pure quantum circuit might look as follows:

Here, for instance, the “quantum gate$U_{34}$ is a linear operator on the tensor product Hilbert space $\mathscr{H} \oplus \mathscr{H}$.

Notice that the quantum circuit is understood to be the actual string diagram, up to the usual admissible topological deformations, not just the composite linear transformation which it encodes: The compositeness of the diagram encodes how (e.g. in which order) available quantum gate-operations would be operated on an actual quantum computer, and the composite linear map only encodes the inpute-to-output-specification of the resulting quantum computation. In this sense, quantum circuits constitute a quantum programming language and one also speaks of the “quantum circuit model of quantum computation” (e.g. Nielsen & Chuang 2000, §II.4.6; Miszczak 2011, §3, 2012, §4.3; Beneti & Casati 2018, §3.2).

Accordingly, many existing quantum programming languages are actually domain specific programming languages for quantum circuits, and as such they are often embedded into ambient type theories (for instance Quipper and QWIRE).

Often the basic Hilbert space building block here is taken to be complex 2-dimensional, $\mathscr{H} \,\coloneqq\, \mathbb{C}^2$, in which case one speaks of a quantum circuit on q-bits.

Sometimes all quantum gates involved, and hence also the resulting composite linear maps, are assumed to be invertible, as befits reversible computation by unitary quantum state evolution in quantum mechanics.

But more generally and certainly in the broader context of quantum information theory (such as in formulating the quantum teleportation protocol and similar), quantum circuits are admitted to also contain crucial classical/quantum mechanics-interaction, in the form of:

1. quantum measurement-gates,

2. quantum state preparation-gates,

3. classically controlled quantum gates.

Here quantum measurement, in particular, is not only non-reversible (due to the quantum state collapse involved) but also non-deterministic, meaning that it is not immediately represented by a fixed linear map at all – at least not between the original Hilbert spaces.

One may model the $B$-tuple of projection operators corresponding to the possible set $B$ of measurement outcomes as a linear map between the original Hilbert space $\mathscr{H}$ and its $B$-indexed direct sum (one rare place where this is made explicit, albeit without comment, is Selinger 2004, p. 39):

$\array{ meas & \colon & \mathcal{H} & \xrightarrow{\phantom{--}} & \underset{ b \colon B }{\oplus} \mathcal{H} \\ && \vert \psi \rangle &\mapsto& \underset{b \colon B}{\oplus} \, P_{b} \vert \psi \rangle } \,.$

Therefore, as soon as these three operations crossing the classical/quantum physics-boundary are involved (and made explicit), quantum circuits are no longer just string diagrams in finite-dimensional HilbertSpaces, but something richer, involving besides tensor products also some kind of reflection of direct sums of spaces of quantum states.

For a critical assessment of the (lack of) literature on this issue see Gurevich & Blass 2021.

One approach for handling these more general quantum circuits is (Aharonov, Kitaev & Nisan 1998, Selinger 2004) to understand them, in the tradition on quantum probability theory, as diagrams of “quantum operations” on spaces of mixed quantum states, namely as a kind of string diagrams of completely positive maps between convex spaces of density matrices.

Other approaches for giving categorical semantics for quantum circuits in this general sense have been proposed (and implemented) in the discussion of the quantum programming language Quipper (see the references there). A natural formulation in terms of Quantum Modal Logic in terms of dependent linear type theory is discussed at:

These turn out to be related: From a suitably rich formulation of quantum circuits of pure states with measurement and classical control, the density matrix-formulation may be recovered (…).

At the time of writing (2021) most of the actual programming of experimental quantum computers is conceived through quantum circuit diagrams, while more high-level quantum programming languages are are awaiting the rise of more powerful quantum hardware.

## Examples

### Bell state preparation

In qbit-based quantum computation, the elementary Bell state is usually prepared via the following small quantum circuit consisting of a Hadamard gate followed by a quantum CNOT gate:

### Quantum teleportation

The standard (cf. GLRSV13, p. 5) qbit-form of the quantum teleportation-protocol is the following quantum ciruit on qbits (where “$H$” denotes the Hadamard gate, the circles denote the quantum CNOT gates and the boxed pointers denote quantum measurement in the qbit-basis):

## References

### Quantum circuit diagrams

The notion of quantum gates and quantum circuits acting on pure quantum states originates with:

On quantum circuits for mixed quantum states (with density matrices) and rigorous propsals for the classical/quantum interaction:

Standard textbook accounts:

Standard lecture notes:

• John Preskill, Classical and quantum circuits (pdf), Chapter 5 in: Quantum Computation, lecture notes (web)

• Ryan O’Donnell, Introduction to the Quantum Circuit Model (2015) (pdf)

• Scott Aaronson, §4.2 in: Introduction to Quantum Information Science (2018) [pdf, webpage]

The (dagger-compact monoidal) category theoretic formulation is due to:

with exposition and further development in:

For more on this see at finite quantum mechanics in terms of dagger-compact categories.

Survey, examples, and implementations:

With an eye towards quantum complexity theory:

• Richard Cleve, Section 1.2 in: An Introduction to Quantum Complexity Theory (pdf)

Formal quantum programming language-perspective on quantum circuits:

### Quantum information theory via String diagrams

The observation that a natural language for quantum information theory and quantum computation, specifically for quantum circuit diagrams, is that of string diagrams in †-compact categories (see quantum information theory via dagger-compact categories):

On the relation to quantum logic/linear logic:

Early exposition with introduction to monoidal category theory:

• Bob Coecke, Kindergarten quantum mechanics $[$arXiv:quant-ph/0510032$]$

• Bob Coecke, Introducing categories to the practicing physicist $[$arXiv:0808.1032$]$

• John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone in: New Structures for Physics, Bob Coecke (ed.), Lecture Notes in Physics 813, Springer (2011) 95-174 $[$arxiv/0903.0340$]$

• Bob Coecke, Eric Oliver Paquette, Categories for the practising physicist $[$arXiv:0905.3010$]$

• Bob Coecke, Quantum Picturalism $[$arXiv:0908.1787$]$

With further emphasis on quantum computation:

• Jamie Vicary, The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$arXiv:1209.3917, doi:10.1109/LICS.2013.14$]$

Generalization to quantum operations on mixed states (completely positive maps of density matrices):

Formalization of quantum measurement via Frobenius algebra-structures:

Textbook accounts (with background on relevant monoidal category theory):

• Bob Coecke, Aleks Kissinger, Picturing Quantum Processes – A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press (2017) $[$ISBN:9781107104228$]$

• Chris Heunen, Jamie Vicary, Categories for Quantum Theory, Oxford University Press 2019 $[$ISBN:9780198739616$]$

based on:

Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) $[$pdf, pdf$]$

Refinement to the ZX-calculus for specific control of quantum circuit-diagrams:

Relating the ZX-calculus to braided fusion categories for anyon braiding:

### Quantum programming languages

The first proposal towards formalizing quantum programming languages was:

General:

Surveys of existing languages:

• Simon Gay, Quantum programming languages: Survey and bibliography, Mathematical Structures in Computer Science16(2006) (doi:10.1017/S0960129506005378, pdf)

• Sunita Garhwal, Maryam Ghorani , Amir Ahmad, Quantum Programming Language: A Systematic Review of Research Topic and Top Cited Languages, Arch Computat Methods Eng 28, 289–310 (2021) (doi:10.1007/s11831-019-09372-6)

• B. Heim et al., Quantum programming languages, Nat Rev Phys 2 (2020) 709–722 $[$doi:10.1038/s42254-020-00245-7$]$

Quantum programming via quantum logic understood as linear type theory interpreted in symmetric monoidal categories:

The corresponding string diagrams are known in quantum computation as quantum circuit diagrams:

QPL:

QML:

quantum lambda calculus?:

Quipper:

QWIRE:

CoqQ:

• Li Zhou, Gilles Barthe, Pierre-Yves Strub, Junyi Liu, Mingsheng Ying, CoqQ: Foundational Verification of Quantum Programs $[$arXiv:2207.11350$]$

On Q Sharp:

• Kartik Singhal, Kesha Hietala, Sarah Marshall, Robert Rand, Q# as a Quantum Algorithmic Language, Proceedings of Quantum Physics and Logic (2022) $[$arXiv:2206.03532$]$

Quantum programming via dependent linear type theory/indexed monoidal (∞,1)-categories:

specifically with Quipper:

On quantum software verification:

with Quipper:

• Linda Anticoli, Carla Piazza, Leonardo Taglialegne, Paolo Zuliani, Towards Quantum Programs Verification: From Quipper Circuits to QPMC, In: Devitt S., Lanese I. (eds) Reversible Computation. RC 2016. Lecture Notes in Computer Science, vol 9720. Springer, Cham (doi:10.1007/978-3-319-40578-0_16)

with QWIRE:

Last revised on November 7, 2022 at 09:10:37. See the history of this page for a list of all contributions to it.