quantum circuits via dependent linear types

> under construction

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###Context###
#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Computing
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[[!include constructivism - contents]]
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#Quantum circuits via dependent linear types#
* table of contents
{:toc}

## Introduction

We follow [SS23-QM](#SS23QM).

### The problem

In the context of [[quantum computation]], a pure *[[quantum circuit diagram]]* (originally: "quantum computational network", [Deutsch 1989](quantum+circuit#Deutsch89)) is a kind of [[string diagram]] ([Abramsky & Coecke 2004](quantum+circuit#AbramskyCoecke04)) in [[finite-dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]], typically used to express a sequence of low-level [[quantum logic gate|quantum gates]] mapping between [[space of quantum states|spaces of]] [[pure quantum state|pure]] [[quantum states]].

A generic pure [[quantum circuit]] might look as follows:

\begin{imagefromfile}
    "file_name": "QuantumCircuitScheme-221025c.jpg",
    "width": "600",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 10,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

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 [[composition|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](#NielsenChuang00); [Miszczak 2011, §3](#Miszczak11), [2012, §4.3](#Miszczak12); [Beneti & Casati 2018, §3.2](#BenetiCasati18)).

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

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

\linebreak

**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 cross the [[quantum-classical divide]], in that they may involve:

1. *[[quantum measurement]]*,

1. *[[quantum state preparation]]*,

1. *[[classically controlled quantum gates]]*.

All three of these appear already in basic examples such as the [[quantum teleportation protocol]]:

\begin{imagefromfile}
    "file_name": "QBitQuantumTeleportationProtocol-221109.jpg",
    "width": "830",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Yet more generally --- notably for any plausible implementation of [[quantum error correction]] --- one needs to allow for [[quantum measurement]]-results to be fed back during runtime ("[[dynamic lifting]]") into [[algorithms]] running on a universal classical computer which schedule further [[quantum circuit]]-execution, and so forth:

\begin{imagefromfile}
    "file_name": "SQRAMschema-WithDynLifting.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "SQRAM model adapted from [Nagarajan, Papanikolaou & Williams 2007, Fig 1](#NagarajanPapanikolaouWilliams07)"
\end{imagefromfile}

The conceptual subtlety with fully formalizing these common ideas is that [[quantum measurement]], in particular, is not only non-[[reversible computation|reversible]] (due to the [[quantum state collapse]] involved) but also [[nondeterministic computation|non-deterministic]], meaning that it is not immediately represented by a fixed linear map at all -- at least not between the original [[Hilbert spaces]].

Concretely, the "[[projection postulate]]" of [[quantum physics]] asserts ([von Neumann 1932](quantum+measurement#vonNeumann32); [Lüders 1951](quantum+measurement#Lüders51)) that:

1. measurement of [[quantum states]] is *with respect* to a choice of [[orthonormal linear basis]] $\big\{\vert \psi_b \rangle \big\}_{b : B}$ of the given [[Hilbert space]] $\mathscr{H}$ [[space of quantum states|of]] [[pure quantum states]];

1. the result of measurement on [[pure quantum states]] $\vert \psi \rangle \;\in\; \mathscr{H}$ is

   1. a random value $b \in B$;

   1. the "[[collapse of the wavefunction|collapse]]" of the [[quantum state]] being measured by [[orthogonal linear basis|orthogonal]] [[projection operator|projection]] to the the [[linear span]] of the $b$th basis state.

      $$
        \array{
          P_b
            &\colon&
          \mathscr{H}
            &\xrightarrow{\phantom{---}}&
          \mathscr{H}_B \hookrightarrow \mathscr{H}
          \\
          &&
          \vert \psi \rangle
          &\mapsto&
          P_b \vert \psi \rangle
          \mathrlap{
            =
            \vert \psi_b \rangle
            \langle \psi_b \vert \psi \rangle
          }
        }
      $$

There are different ways to *[[type theory|type]]* the quantum measurement, taking into account the non-deterministic nature of its outcome:

1. Regarding the [[direct sum]] $\bigoplus_{b \colon B} \mathscr{H}$ of [[Hilbert spaces]] as the [[logical disjunction]] ("or") of [[quantum logic]], one may regard measurement as being the [[linear map]] into the direct sum whose $b$th component is $P_b$.

   > This choice of typing appears (briefly) in [Selinger 2004](quantum+measurement#Selinger04), [p. 39](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf#page=39), in a precursor discussion that led to the formulation of the [[quantum programming language]] *[[Quipper]]*.

1. Regarding the measurement outcome $b \in B$ as the observed [[context]] of the actual quantum collapse, one may regard the collapse projection as [[dependent type theory|dependently typed]].

   > Getting from previous option back to this one is known in the the [[Quipper]]-community as *[dynamic lifting](Quipper#ReferencesDynamicLifting)* (namely "of the measured bits back into the context")

\begin{imagefromfile}
    "file_name": "PossibleTypingsOfQuantumMeasurement-221025.jpg",
    "width": "560",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Both of these options naturally emerge and are naturally unified in the "[Quantum Modal Logic](necessity+and+possibility#ModalQuantumLogic)" inherent to [[dependent linear type theory]]: This is what we now discuss.

### A solution

The following discussion hupf indicates how a natural [[quantum programming language]] for [[quantum information]]-processing protocols --- hence  for [[quantum circuits]] consisting of general [[quantum gates]] with classical control, including [[decoherence|de-coherent]] [[quantum measurement]]-gates and subsequent coherent quantum gates depending on these --- is elegantly subsumed within any [[dependent linear type theory|dependent linearly typed]] [[programming language]] which expresses the "[[yoga of six operations]]", such as is the case in the [[linear homotopy type theory]] of [Riley 2022](dependent+linear+type+theory#Riley22Thesis).

This works by regarding classically controlled [[quantum states]] as having [[linear type|linear data types]] [[dependent type|dependent]] on classical control parameters and on measurement results, and then making the following key observations on **[modal quantum logic](necessity+and+possibility#ModalQuantumLogic)** with [[dependent linear types]]:

1. The "[[necessity]]" [[modality]] $\Box_C \coloneqq (p_C)^\ast (p_C)_\ast$ on [[linear types]] dependent on a given classical [[context]] $C$ (which includes classical control parameters as well as eventual [[quantum measurement]]-outcomes on the same footing) knows all about controlled [[quantum gates]]: These are identified just with the [[morphisms]] of the $\Box_C$-[[co-Kleisli category]], where the $\Box_C$-[[counit of a comonad|counit]] (taking a linear type "out of the comonad") reflects the [[quantum state collapse]] of [[quantum measurement]], and where the $\Box_C$-[[comultiplication]] encodes the [[superposition]]-principle.

1. The [[external tensor product]] of dependent linear types gives the compilation of [[quantum gates]] into [[quantum circuits]], essentially as familiar from the quantum [[string diagram]]-calculus of [Abramsky & Coecke 2004](finite+quantum+mechanics+in+terms+of+dagger-compact+categories#AbramskyCoecke04) with respect to the plain [[tensor product]], but now taking into account dependency on classical control and potential measurement outcomes.

All constructions and proofs here flow readily from just the [[six operation yoga]] and ([[co-Kleisli category|co-]])[[Kleisli category|Kleisli theory]]. One key point is (eq:NecessityRespectsExternalTensorProduct) that the [[necessity]] [[modality]] respects the [[external tensor product]] -- so that these classically controlled quantum circuits are unambiguously defined --, under the assumption that it coincides with the [[formal duality|dual]] *[[possibility]]* modality $\lozenge_C \mathscr{H}_C$  on the quantum data types $\mathcal{H}_\bullet$ involved -- hence (in full generality) when the linear types are [[dualizable object|dualizable]] in the sense of [[ambidextrous adjunction|ambidexterity]] and specifically (in the traditional models) when [[space of quantum states|quantum state spaces]] are [[finite-dimensional vector space|finite dimensional]], as is the case in [[quantum information theory]].

Aspects of the following have a resemblance to some variants and categorical interpretations of the [[quantum programming language]] "[[Quipper]]" (see the references [there](Quipper#ReferencesDependentLinearTypesAndCategoricalSemantics)), but avoids (see [below](#NoticeTheTypingOfAQbit)) Quipper's problem of needing "dynamic lifting" (see [there](Quipper#ReferencesDynamicLifting)) of "measurement bits back into the context": The linear [[necessity]]-modality knows right away that measurement outcomes are recorded in the context.

\linebreak

## Idea: Modal quantum logic via linear homotopy types

We assume a [[dependent linear type theory]] which [[functor|functorially]] obtains (as anticipated in [Schreiber 2014](dependent+linear+type+theory#Schreiber14), [§3.2](https://arxiv.org/pdf/1402.7041.pdf#page=39) and established in [Riley 2022](dependent+linear+type+theory#Riley22Thesis), [§2.4](https://digitalcollections.wesleyan.edu/islandora/object/ir:3269)):

1. from a classical ("intuitionistic") [[type]] $X : Type$ a (large) type $LinType_X$ of [[linear types]] [[dependent type|dependent]] on $X$ (cf. *[[quantum sets]]*), equipped with [[symmetric monoidal category|symmetric]] [[closed monoidal category|closed monoidal structure]] $(-) \otimes (-)$ ([[tensor product]]) and $[-,\,-]$ ([[internal hom]]);

1. from a [[function]] $f : X_1 \xrightarrow{\;} X_2$ between classical types a [[yoga of six operations]] (specifically: a "[[Wirthmüller context]]") between their types of dependent linear types, hence a [[contravariant functor|contravariant]] [[substitution]]/[[pullback]] operation $f^\ast$ which is [[strong monoidal functor|strong monoidal]], [[strong closed functor|strong closed]] and extends to a [[base change]] [[adjoint triple]] (with [[left adjoint]] $f_!$ a version of [[dependent sum]] and [[right adjoint]] a version of [[dependent product]]):

\begin{imagefromfile}
    "file_name": "QCwthLHT-BaseChange-221001.jpg",
    "width": "730",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "CharacteristicOfLinearTypes-221108jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

### Classical modal logic
 {#ClassicalModalLogic}

First observe that (without imposing a further condition) also classical ("non-linear") [[dependent type theory]] provides an example of this [[yoga of six operations]] (actually five distinct operations, in the present case of [[Wirthmüller contexts]]), with:

1. [[tensor product]] given by [[product types]],

1. [[internal hom]] given by [[function types]],

1. $f^\ast$ given by [[context extension]],

1. $f_!$ given by [[dependent coproduct]],

1. $f_\ast$ given by [[dependent product]].

In this classical situation, the ([[comonad|co]]-)[[monad]] [[adjoint pair]] $\lozenge_C \dashv \Box_C$ induced by the [[adjoint triple]] is readily found (see [here](necessity+and+possibility#InFirstOrderLogicAndTypeTheory)) to reflect the [[necessity]]-[[modality]] and the [[possibility]]-[[modality]] of a kind of [[modal logic]] which regards the potential measurement outcomes $c : C$ as "[[possible worlds semantics|possible worlds]]":

\begin{imagefromfile}
    "file_name": "QCwthLHT-ClassicalNecessity-221003.jpg",
    "width": "900",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In fact there is (see [here](necessity+and+possibility#RelationToPotentiality)) a quadruple of ([[comonad|co-]])[[monad|monadic]] [[modalities]] associated with any [[base change]]:

Here

1. the definiteness-modality $\star$ is also known as the *[[writer comonad]]*;

1. the randomness-modality $\bigcirc$ is also known as the *[[reader monad]]*.

We recover the meaning of these expressions in terms of [[quantum measurement]] and [[quantum state preparation]], [below](#QuantumMeasurement).

(In general there may be richer dependency on classical contexts. In the next simplest case the [[base change]] is along the [[projection]] map out of a [[Cartesian product]] of classical types, which makes one of them label potential upcoming measurement results as before, while the other encodes further dependency on classical parameters, such as on previous measurement results:

\begin{imagefromfile}
    "file_name": "QCwthLHT-RelativeBaseChange-221003.jpg",
    "width": "670",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This plays a role for classically-controlled quantum gates, [below](#ClassicallyControlledQuantumGate)).

### Quantum modal logic
 {#QuantumModalLogic}

This general [[modal logic]] of [[possible worlds]] [found inside](necessity+and+possibility#InFirstOrderLogicAndTypeTheory) [[dependent type theory]] becomes subject to one curious further rule as we pass to genuinely [[linear types]], such as in the sense of [[stable homotopy types]] in [[linear homotopy type theory]]:

Namely, assuming that the potential measurement outcomes $c \colon C$ form a [[finite set]] --- which is the case of relevance in [[quantum information theory]] ---, the linearity or *[[stable (infinity,1)-category|stability]]* axiom implies that the [[base change]] down from the context $C$ becomes [[ambidextrous adjunction|ambidextrous]], in that on dependent linear types $\mathscr{H}_\bullet$ the [[coproduct]] computed by the  [[possibility]]-[[modality]] and the [[product]] computed by the [[necessity]]-[[modality]] *agree* (the situation of a "[[biproduct]]"):

$$
  BC : FinSet
  ,\;\;
  \mathscr{H}_\bullet
  \,\colon\,
  LinType_B
  \;\;\;\;\;
  \vdash
  \;\;\;\;\;
  \lozenge \mathscr{H}_\bullet
    \;\simeq\;
  \Box \mathscr{H}_\bullet
  \,.
$$

Curiously, in terms of [[modal logic]], this is the *[[Gell-Mann principle]]* of quantum physics:

$$
  \array{
    \llap{\text{The}}\; \text{possible}
    &
    \text{is}
    &
    \text{necessary}
    &
    \text{and hence}
    &
    \text{actualized}
    \\
    \lozenge \mathcal{H}_\bullet
    &
    \simeq
    &
    \Box \mathcal{H}_\bullet
    &
    \xrightarrow{\phantom{--} \epsilon \phantom{--}}
    &
    \mathcal{H}_{
      \bullet
      \;
      \mathrlap{\text{with some probability}}
    }
  }
$$

Concretely, the [[linear homotopy type theory]] of [Riley 2022](dependent+linear+type+theory#Riley22Thesis) should have [[categorical semantics]] in the [[(infinity,1)-topos|$\infty$-topos]] of [[parameterized spectrum|parameterized]] [[Eilenberg-MacLane spectrum|$H\mathbb{C}$]]-[[module spectra]] ([[schreiber:Quantization via Linear homotopy types|S. 2014, Ex. 3.11]]) and [[complex vector spaces]] embed into this [[stable homotopy theory]] under passage to their [[Eilenberg-MacLane spectra]] (see the [[stable Dold-Kan correspondence]]).

\[
  \label{NotationalConvention}
\text{It is convenient and meaningful to declare the notational conventions to:}
  \phantom{--------------------}
\]
1. abbreviate $\mathscr{H} \,\coloneqq\, p_! \mathscr{H}_\bullet \,\simeq\, p_\ast \mathscr{H}_\bullet$

1. notationally suppress any outer application of $(p_B)^\ast \colon LinType \to LinType_B$;

Thereby, the definiteness/randomness modalities in relation to [[necessity and possibility]] of classical modal logic ([above](#ClassicalModalLogic)) may be expressed in quantum modal logic by:

\begin{imagefromfile}
    "file_name": "DefinitelyRandomlyViaPossiblyNecessarily.jpg",
    "width": "180",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This way, the default [[categorical semantics]] for modal quantum logic is that of [[bundles]] of [[finite dimensional vector spaces]] (finite-dimensional complex [[Hilbert spaces]]) over [[finite sets]]: In this case both $(p_C)_!$ as well as $(p_C)_\ast$ yield the ordinary [[direct sum]] of the [[fibers]] (a [[biproduct]]). Since this is the [[categorical semantics]] under which the following type theory describes the traditional [[quantum circuits]], we will adopt our notation to this case and denote the generic [[dependent linear type]] by $\mathcal{H}_\bullet : LinType_C$ and its generic [[term]] by the usual notation for [[quantum states]]:

$$
  c : C
  \;\;\;
  \vdash
  \;\;\;
  \vert \psi_c \rangle \;\colon\; \mathcal{H}_c
  \,.
$$

one finds for the base change [[adjoint triple]] of [[dependent linear types]] along [[family of finite types|finite fibers]] (eq:FiniteBaseType)
that the interpretation of their ([[counit of a comonad|co-]])[[unit of a monad|units]] is as follows:

By the assumption that $B$ is [[finite set|finite]] and since [[Vect]] has [[biproducts]], the [[colimit]]/[[coproduct]]

$$
  (p_B)_! \mathscr{V}_\bullet
  \;=\;
  \coprod_{b : B} \mathscr{V}_b
$$

and the [[limit]]/[[product]]

$$
  (p_B)_\ast \mathscr{V}_\bullet
  \;=\;
  \prod_{b : B} \mathscr{V}_b
$$

agree (we have an [[ambidextrous adjunction]]) and are both equivalent to the [[direct sum]]:

$$
  (p_B)_! \mathscr{V}_\bullet
    \;\simeq\;
  (p_B)_\ast \mathscr{V}_\bullet
  \;=\;
    \underset{b \colon B}{\bigoplus}
  \mathcal{V}_b
  \,.
$$

Accordingly, the underlying [[endofunctors]] of the induced [[monad]] and [[comonad]] on $k Vec_B$

$$
  \Box_B
  \;\coloneqq\;
  (p_B)^\ast (p_B)_\ast
  \;\;\vdash\;\;
  \lozenge_B
  \;\coloneqq\;
  (p_B)^\ast (p_B)_!
  \;\;\colon\;\;
  k Vec_B
  \longrightarrow
  k Vec_B
$$

agree, to make a [[self-adjoint functor]]

$$
  (p_B)^\ast
  \bigotimes_B
  \;\;\vdash\;\;
  (p_B)^\ast
  \bigotimes_B
  \;\;\;\colon\;\;\;
  k Vec
  \longrightarrow
  k Vec
$$

which carries the [[structure]] both of a [[monad]] and of a [[comonad]].

This way, one finds that the four (co)unit operations of quantum modal logic are given as follows:

\[
  \label{QuantumModalUnits}
\]
\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-230806.jpg",
    "width": "840",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
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\end{imagefromfile}

### Quantum modality
 {#TheQuantumModality}

In addition to the above [[axiom scheme|schemata]] of ([[comonad|co]])[[monads]] acting on the ([[dependent linear type|dependent]]) [[linear type|linear]] [[type systems]], we invoke the [[relative monad]]

$$
  \array{
    \mathrm{Q}
    &\colon&
    Type &\longrightarrow&
    LType
    \\
    &&
    B &\mapsto& \underset{B}{\bigoplus} \mathbb{1}
  }
$$

which sends a [[finite type]] (in particular) to the [[direct sum]] that it [[linear span|spans]] (a detailed description of this relative monad is [here](relative+monad#LinearSpan)).

This is in fact [[idempotent monad|idempotent]], in that the actual [[monad]] on the full [[dependent linear type|dependent linear]] [[type system]] that this relative monad is induced by is the [[idempotent monad]] which witnesses a [[reflective subcategory]]-inclusion of the linear types into dependent linear types:

$$
  \mathrm{Q}
  \;\colon\;
  Type
  \xrightarrow{ (-) \times \mathbb{1} }
  DLType
  \twoheadrightarrow
  LType
  \xhookrightarrow{ \phantom{---} }
  DLType
  \,.
$$

This implies that whenever $D \;\colon\; LType$,  we may describe morphisms of the form $\mathrm{Q}B \to LType$ by [[Kleisli morphisms]] for $\mathrm{Q}$.

For example for $B = Bit$ the type of classical [[bits]], we have that

$$
  \mathrm{Q}(Bit)
  \;=\;
  QBit
$$

is the usual type of [[q-bits]].

### Quantum gates
 {#QuantumGates}

With the above modal quantum logic, we may naturally type ([[controlled quantum gates|controlled]]) [[quantum gates]] as follows:

\begin{imagefromfile}
    "file_name": "ControlledQuantumGateTyping-230808.jpg",
    "width": "750",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

As indicated at the bottom, we may understand classically [[controlled quantum gates]] as $\bigcirc$-effectful programs (in the sense of [[monads in computer science]]).

(In the bottom left we assume that the state spaces  themselves are independent of $w \colon W$, hence $\mathscr{H}_\bullet \,=\, \mathscr{H}$, which is typically the case in application.)

Next we see that the $\bigcirc$-effect handling, in this sense, is nothing but [[quantum measurement]]:

### Quantum measurement
 {#QuantumMeasurement}

Specifically, $\Box_C$-**actualization is the measurement process** --  in that the $\Box_C$-[[counit of a comonad|counit]] implements exactly the rule for [[wavefunction collapse]] (due to [Lüders 1951](wave+function+collapse#Lüders51), following [von Neumann 1932, §III.3 and §VI](wave+function+collapse#vonNeumann32)) with regards to [[quantum  measurements]] in the $C$-indexed basis of the total Hilbert space
$
  \mathcal{H} \coloneqq \Box_C \mathcal{H}_\bullet
$
which is encoded by its direct sum decomposition $\mathcal{H}_\bullet$:

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumNecessity-221021.jpg",
    "width": "900",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Hence we may type quantum measurement gates as follows:

As indicated at the bottom, we may understand quantum measurement in this form as the *[effect handling](monad+in+computer+science#MonadModulesInIdeaSection)* (in the sense of [[monads in computer science]]) of indefiniteness effects (in the sense of [potentiality modal logic](necessity+and+possibility#RelationToPotentiality)):

\begin{imagefromfile}
    "file_name": "QMeasurementViaIndefinitenessHandling-221109c.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Notice that this process may be understood as the "[[dynamic lifting]]" of [[quantum measurement]]-results into the [[context]] (here: $B = $ [[Bool]]) of the ambient [[dependent linear type theory]].

\linebreak

There is another possible typing of quantum measurement, where a measurement lands in the random [[reader monad]]:

\begin{imagefromfile}
    "file_name": "PossibleTypingsOfQuantumMeasurement-221028.jpg",
    "width": "600",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

These two are related by the following [[commuting diagram]] (being the [[naturality square]] of the necessity-counit on the possibility-unit, see at *[[S5 modal logic]]*, [this remark](S5+modal+logic#HaxagonOfEpistemicEntailments)):

\[
  \label{MeasurementViaWritedComonad}
\]

\linebreak

Given all this, the [Kleisli equivalence](Kleisli+category#KleisliEquivalence) for the [[necessity]] [[comonad]] $\Box_B$  is the **[[deferred measurement principle]]** for [[quantum circuits]]:

\begin{imagefromfile}
    "file_name": "DeferredMeasurementAsKleisliEquiv-221027b.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

## Example: Modal typing of basic QBit gates
 {#QCwthLHT-QBits}

We spell out in terms of the above [[modal type theory|modal]] [[quantum logic]] found inside [[dependent linear type theory]] the traditional zoo of [[quantum gates]] and basic [[quantum circuits]] acting on [[tensor products]] of [[qbits]]. On the one hand this serves to show how all the expected constructions are naturally recovered, on the other hand it enhances these standard concepts by what is de-facto a [[quantum programming language]] for their classically-controlled versions (as such somewhat akin to parts of [[Quipper]])

\linebreak

The typing of the [[CNOT gate]] in its two incarnations as a classically or a quantumly [[controlled quantum gate]]:

\begin{imagefromfile}
    "file_name": "CNOTgate-typing-230809.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Here the [[quantum measurement]] on one $QBit$:

\begin{imagefromfile}
    "file_name": "QBitMeasurementTyping-230809.jpg",
    "width": "400",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and their combination, in accord with the [[deferred measurement principle]]:

\begin{imagefromfile}
    "file_name": "CNOT-DeferredMeasurement-230809.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Quantum Programming Language
 {#TheQuantumProgrammingLanguage}

With all elements of [[quantum circuits]] formulated via ([[comonad|co]])[[monads]] constructed inside a language of [[dependent linear types]] as above, we obtain an [[domain specific embedded programming language|embedded]] [[quantum programming language]] for quantum circuits with classical control and effects.

### Quantum/classical types and modalities

We use the following notation for the [[type system]] and the quantum/classical [[modalities]] on these.

\begin{imagefromfile}
    "file_name": "QS_QuantumClassicalDataTypes-221123.jpg",
    "width": "520",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Here the expressions in square brackets may be thought of as the corresponding [[categorical semantics]] as [[bundles]] of linear types (for instance semantics [[internalization|in]] the category [[VectBund]] of [[complex vector bundles]] over [[Sets]], or more generally, in the [[(infinity,1)-category|$\infty$-category]] of [[parameterized spectrum|parametrized]] [[Eilenberg-MacLane spectrum|$H\mathbb{C}$-]][[module spectra]]).

The classical modality $\natural$ and the quantum modality  $\mathrm{Q}$ send a bundle type to its base and to its "linearized total space", respectively:

\begin{imagefromfile}
    "file_name": "QS_QuantumClassicalDivide-221123.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

(The quantum monad $\mathrm{Q}$ is a [[relative monad]]; its [[categorical semantics]] in [[VectBund]] is discussed [there](relative+monad#LinearSpan).)

The classical [[internal hom]] and its quantum version relative to the ("[[external tensor product|external]]") linear [[tensor product]] (for more on this see the discussion [here](VectBund#ClosedMonoidalStructures) at *[[VectBund]]*):

\begin{imagefromfile}
    "file_name": "QS_ProductAndFunctionTypes-221123.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Notice that all these types are bundles "*in themselves*", in that we did not invoke (yet) any [[dependent types]] (semantically we speak about a total [[tangent (infinity,1)-topos|tangent $\infty$-topos]], not its system of [[slice (infinity,1)-category|slice $\infty$-categories]]).

Instead, the actual $B$-[[dependent types]] that we need to reflect ("[[dynamic lifting]]" of) [[quantum measurement]] will all be viewed, indirectly, through the lens of $\bigcirc_B$-modales on the non-dependent types (view the [[comparison functor]] which here is an equivalence).

The only explicit [[context]] that we do need to consider is -- besides the trivial dependency on the classical [[unit type]] $\ast$, its quantum analog namely: -- dependency on the [[tensor unit]] $\mathbb{1}$. This only exception to the rule we *sugar away* by the following "circle-colon"-notation (which is meant to rhyme on the "$\multimap$"-notation for [[linear implication]]):

\begin{imagefromfile}
    "file_name": "QS_QuantumDataDeclaration-221123.jpg",
    "width": "480",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

### Sugaring modal dependent linear types to a quantum language

We now obtain  a quantum programming language from [[modal type theory|modal]] [[dependent linear type theory]] simply by meticulously [[syntactic sugar|sugaring]]:

[(1)](#ForDoNotation) the standard `do`-notation for [[monads in computer science]],

[(2)](#NotationForPureEffects) the ([[counit of a comonad|co]])[[unit of a monad|pure/return]]-functions of the various [[monads]].

#### Syntax for effect binding
 {#ForDoNotation}

We code maps in the [[Kleisli category]] of an [[monad in computer science|effect monad]] $\mathcal{E}$ by this schema:

\begin{imagefromfile}
    "file_name": "QS_ForDoNotation-221123b.jpg",
    "width": "440",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This syntax is to closely reflect the fact that

* *for* an input of the form $\return^{\mathcal{E}}_D(d) \colon \mathcal{E}D$

* which may appear as a contribution *in* the effectful input data,

* the operation $\bind^{\mathcal{E}}_{\mathrm{prog}}$ *does* produce the output $\mathrm{prog}(d)$,

which prescription completely defines it.

\begin{remark}
Beware that common classical notation for exactly the same costruction is a little different:

\begin{imagefromfile}
    "file_name": "ClassicalMonadicDoNotation-221123.jpg",
    "width": "280",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This classical notation is meant to suggest that pure data $d \colon D$ may be "read out" from effectful data $e \colon \mathcal{E}D$. While this is suggestive for the [[list monad]] and its common relatives in classical programming, it is misleading in [[linear type theory]] and notably so for the quantum monad $\mathrm{Q}$: Here the effectful input $e = \vert \psi \rangle$ is a quantum state like a [[q-bit]], in which case $d \colon Bit$ is a classical [[bit]], whence the classical notation "$d \leftarrow \vert \psi \rangle$" could only be suggestive of performing a [[quantum measurement]] -- in contradiction to the actual nature of the resulting $\bind^{\mathrm{Q}}_{\mathrm{prog}}$-operation constituting a coherent non-measurement quantum gate.

Instead, what really happens in Kleisli formalism is that operations are defined on *generators* for effectful data types $\mathcal{E}(D)$, namely on data of the form $\return^{\mathcal{E}}_D(d)$. For example, the space of [[qbits]] $\vert \psi \rangle \colon \QBit$ is generated (here: [[linear span|linearly spanned]]) by the [[linear basis|basis]] [[qbits]] $\vert 0 \rangle$ and $\vert 1 \rangle$, where we may naturally identify the ket-notation $\vert - \rangle$ as the unit/return operation which regards a classical bit $b$ as the corresponding basis quantum state $\vert b \rangle$.
\end{remark}

Moreover, we write an analogous `for...do`-code for maps out of [[tensor products]], using that these are generated from the homogeneous tensor products.

\begin{imagefromfile}
    "file_name": "QS_TensorProductForDo-221123.jpg",
    "width": "500",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

(...)

#### Syntax for pure effects
 {#NotationForPureEffects}

Finally, the monad-[[unit of a monad|return operations]] "$`return`^{\mathcal{E}}(-)$" (and the comonad [[counit of a comonad|extract operations]]) we sugar according to the following table:

\begin{imagefromfile}
    "file_name": "QS_SyntaxForPurePotentialityEffects-221123.jpg",
    "width": "540",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In the same vein we sugar the constructor for terms in the [[quantum reader monad]] $\bigcirc_B$ as follows:

\begin{imagefromfile}
    "file_name": "QS_MeasurementTermConstruction-221123.jpg",
    "width": "490",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

### Example pseudo-code

#### Basic qbit circuit ingredients

Basic [[quantum logic gates]] on [[qbits]]:

\linebreak

The quantum [[NOT]]/[[Pauli matrix|Pauli X]]-gate

\begin{imagefromfile}
    "file_name": "QS_PauliXGate-221123.jpg",
    "width": "170",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

The [[Hadamard gate]]:

\begin{imagefromfile}
    "file_name": "QS_HadamardGate-221123.jpg",
    "width": "270",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

The [[CNOT gate]]:

\begin{imagefromfile}
    "file_name": "QS_CNOTGate-221123.jpg",
    "width": "300",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

#### Quantum teleportation protocol

The [[quantum teleportation protocol]]:

\begin{imagefromfile}
    "file_name": "QBitQuantumTeleportationProtocol-221109.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

is given by the following code:

\begin{imagefromfile}
    "file_name": "QS_QuantumTeleportationProtocol-221123.jpg",
    "width": "650",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

#### Quantum bit flip code

The [[quantum bit flip code]] for 3-qbit encoding:

\begin{imagefromfile}
    "file_name": "QS_3BitFlipCodeIngredients-221125.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "QS_3BitFlipCode-221125.jpg",
    "width": "715",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

## References
 {#References}

* {#CQTS22} [[CQTS]], *[Quantum Data Types via Linear HoTT](https://ncatlab.org/schreiber/show/QDataInLHoTT#QTML2022)*, talk at *[Workshop on Quantum Software](https://quasar.unina.it/qtml2022/workshop.php)* @ *[QTML 2022](https://quasar.unina.it/qtml2022.html)* (Nov 2022) [[pdf](/schreiber/files/QuantumDataInLHoTT-221117.pdf)]

* {#SS23QM} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:The Quantum Monadology]]* [[arXiv:2310.15735](https://arxiv.org/abs/2310.15735)]

The above figure of an SQRAM scheme is taken from:

* {#NagarajanPapanikolaouWilliams07} Rajagopal Nagarajan, Nikolaos Papanikolaou, David Williams, *Simulating and Compiling Code for the Sequential Quantum Random Access Machine*, Electronic Notes in Theoretical Computer Science Volume 170, 6 March 2007, Pages 101-124 [[doi:10.1016/j.entcs.2006.12.014](https://doi.org/10.1016/j.entcs.2006.12.014)]

(...)


quantum cohomology as Pontrjagin rings -- references

### Quantum cohomology as Pontrjagin rings
 {#ReferencesOnQuantumCohomologyAsPontrjaginRings}

On the relation between [[quantum cohomology rings]], hence of [[Gromov-Witten invariants]] in [[topological string theory]], for [[flag manifold]] [[target spaces]] (such as the [[CP^1 sigma-model|$\mathbb{C}P^1$-sigma model]]) and [[Pontrjagin rings]] ([[homology]]-[[Hopf algebras]] of [[based loop spaces]]):

That the [[Pontryagin ring]]-[[structure]] on the [[ordinary homology]] of the [[based loop space]] of a [[simply-connected topological space|simply-connected]] [[compact Lie group]] $K$ is essentially  the [[quantum cohomology ring]] of the [[flag variety]] of its [[complexification]] by its [[Borel subgroup]] is attributed ("Peterson isomorphism") to

* [[Dale H. Peterson]] (notes by [[Arun Ram]]), *Quantum Cohomology of $G/H$*, MIT (1997) $[$[web](http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html), [pdf](http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf), [[Peterson-QuantumCohomology.pdf:file]]$]$

see also

* Yasha Savelyev, *Quantum characteristic classes and the Hofer metric*, Geom. Topol. **12** (2008) 2277-2326 $[$[arXiv:0709.4510](https://arxiv.org/abs/0709.4510), [doi:10.2140/gt.2008.12.2277](https://doi.org/10.2140/gt.2008.12.2277)$]$

and proven in

* [[Thomas Lam]], [[Mark Shimozono]], §6.2 in: *Quantum cohomology of $G/P$ and homology of affine Grassmannian*, Acta Mathematica **204** (2010) 49–90 $[$[arXiv:0705.1386](https://arxiv.org/abs/0705.1386), [doi:10.1007/s11511-010-0045-8](https://doi.org/10.1007/s11511-010-0045-8)$]$

* [[Chi Hong Chow]], *Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula* $[$[arXiv:2110.09985](https://arxiv.org/abs/2110.09985)$]$

* {#Chow22} [[Chi Hong Chow]], *On D. Peterson's presentation of quantum cohomology of $G/P$* $[$[arXiv:2210.17382](https://arxiv.org/abs/2210.17382)$]$

reviewed in

* [[Jimmy Chow]], *Homology of based loop groups and quantum cohomology of flag varieties*, talk at *Western Hemisphere Virtual Symplectic Seminar* (2021) $[$[pdf](https://freemath.xyz/chowslides.pdf), [[Chow-QuantumCohomologyOfFlagVarieties.pdf:file]], video:[YT](https://www.youtube.com/watch?v=ovbQBj7t3eU)$]$

with further discussion in:

* [[Naichung Conan Leung]], [[Changzheng Li]], *Gromov-Witten invariants for $G/B$ and Pontryagin product for $\Omega K$*, Transactions of the American Mathematical Society **364** 5 (2012) 2567-2599  $[$[web](https://www.ams.org/journals/tran/2012-364-05/S0002-9947-2012-05438-9/), [arXiv:0810.4859](https://arxiv.org/abs/0810.4859), [jstor:41524936](https://www.jstor.org/stable/41524936)$]$

On the variant for Pontryagin products not on [[ordinary homology]] but in [[topological K-theory|topological]] [[K-homology]]:

* {#LamLiMihalceaShimozono18} [[Thomas Lam]], [[Changzheng Li]], [[Leonardo C. Mihalcea]], [[Mark Shimozono]], *A conjectural Peterson isomorphism in K-theory*, Journal of Algebra **513** (2018) 326-343 $[$[doi:10.1016/j.jalgebra.2018.07.029](https://doi.org/10.1016/j.jalgebra.2018.07.029), [arXiv:1705.03435](https://arxiv.org/abs/1705.03435)$]$

* Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno, *Peterson Isomorphism in K-theory and Relativistic Toda Lattice*, International Mathematics Research Notices **2020** 19 (2020) 6421–6462 $[$[arXiv:1703.08664](https://arxiv.org/abs/1703.08664), [doi:10.1093/imrn/rny051](https://doi.org/10.1093/imrn/rny051)$]$

* [[Syu Kato]], *Loop structure on equivariant K-theory of semi-infinite flag manifolds* $[$[arXiv:1805.01718](https://arxiv.org/abs/1805.01718)$]$

* {#Kato19} [[Syu Kato]], *On quantum $K$-groups of partial flag manifolds* $[$[arXiv:1906.09343](https://arxiv.org/abs/1906.09343)$]$

* [[Syu Kato]], *Darboux coordinates on the BFM spaces* $[$[arXiv:2008.01310](https://arxiv.org/abs/2008.01310)$]$

* {#Kato21} [[Syu Kato]], *Quantum $K$-groups on flag manifolds*, talk at *IMPANGA 20* (2021) $[$[pdf](https://www.impan.pl/konferencje/bcc/2020/20-impanga/impanga2021.pdf)$]$

On the example of the [[CP^1 sigma-model]]: [LLMS18, §4.1](#LamLiMihalceaShimozono18), [Kato21 p. 17](#Kato21), [Chow22 Exp. 1.4 ](#Chow22).

See also:

* [[Constantin Teleman]], *Gauge theory and mirror symmetry* $[$[arXiv:1404.6305](https://arxiv.org/abs/1404.6305)$]$

Relation to [[chiral rings]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Constantin Teleman]], *Coulomb branches for quaternionic representations* $[$[arXiv:2209.01088](https://arxiv.org/abs/2209.01088)$]$


quantum cohomology ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Certain [[topological string theories]] (2d [[topological field theory|topological]] [[sigma-models]] [[Witten 1988](#Witten88)]), with [[target space]] a suitable [[symplectic manifold]] $X$, have as [[spaces of quantum states]] the [[ordinary cohomology]] ([[real cohomology|real]]/[[complex cohomology|complex]]/[[de Rham cohomology]]) $H^\bullet(X)$ of the target manifold, but such that the [[genus of a surface|genus]]$=0$ [[worldsheet]] [[3-point function]] (a [[Gromov-Ruan-Witten invariant]]) equips the [[underlying]] [[vector space]] of the [[cohomology groups]] with a new product $H^\bullet(X) \otimes H^\bullet(X) \to H^\bullet(X)$ that [[deformation|deforms]] ("[[quantization|quantizes]]") the ordinary [[cup product]]/[[wedge product]]-[[cohomology ring]] to a non-commutative ring [[Witten (1990), §3](#Witten90)], whence called a *quantum cohomology ring* [[Vafa (1992)](#Vafa92)].

Beware that later authors often abbreviate the term from *quantum cohomology ring* to just *quantum cohomology*, which is however a misnomer since it is not the underlying notion of ([[ordinary cohomology|ordinary]]) [[cohomology]] that is being deformed/quantized here (just the [[coefficients]] are typically enlarged), but the multiplicative *[[ring]]-[[structure]]* [[cohomology ring|on cohomology]] is deformed -- indeed the title of [Vafa (1992)](#Vafa92) speaks more properly of *quantum rings*.

## Examples

\begin{example}
\label{QuantumCohomologyOfComplexProjectiveSpace}
**(quantum cohomology ring of complex projective space)**
\linebreak
The (small) quantum cohomology ring of [[complex projective space]] $\mathbb{C}P^{N-1}$, $N \geq 2$, should be
$$
  QH^\bullet\big(
    \mathbb{C}P^{N-1}
    ;\,
    \mathbb{C}
  \big)
  \;\simeq\;
  \mathbb{C}\big[
    a_2,\, b_{2N}
  \big]/(a_2^N - b_{2N})
  \,.
$$
and as such a "[[deformation]]" of the ordinary [[cohomology ring]], which is (see [there](complex+projective+space#eq:CohomologyRing)):
$$
  H^\bullet\big(
    \mathbb{C}P^{N-1}
    ;\,
    \mathbb{C}
  \big)
  \;\simeq\;
  \mathbb{C}\big[
    a_2
  \big]/(a_2^N)
  \,.
$$
\end{example}
([Witten 1990, p. 275](#Witten90); [Cecotti & Vafa 1992, p. 3](#CecottiVafa92); [Cecotti & Vafa 1993 p. 89](#CecottiVafa93); [Bourdeau & Douglas 1994 3.4](#BourdeauDouglas94); see also [Dorfmeister, Guest & Rossman 2010, p .1](#DorfmeisterGuestRossman10))

## Related concepts

* Quantum cohomology arises as [[chiral rings]] in [[super Yang-Mills theory]].

* [[Pontryagin ring]]

## References

### Quantum cohomology rings

The notion of quantum cohomology originates as a model for certain [[topological string]] [[n-point functions]] in:

* {#Vafa92} [[Cumrun Vafa]], §4 in: *Topological mirrors and quantum rings*, in [[Shing-Tung Yau]] (ed.) *Essays on mirror manifolds*, International Press (1992), republished in *Mirror Symmetry I*, AMS/IP Studies in Advanced Mathematics **9** (1998) [[arXiv:hep-th/9111017](https://arxiv.org/abs/hep-th/9111017), [doi:10.1090/amsip/009](https://doi.org/10.1090/amsip/009)]

* {#Witten90} [[Edward Witten]],  §3 in: *Two-dimensional gravity and intersection theory on moduli space*,  Surveys in Differential Geometry **1** (1990) 243-310 [[doi:10.4310/SDG.1990.v1.n1.a5](https://dx.doi.org/10.4310/SDG.1990.v1.n1.a5), [inspire:307956](https://inspirehep.net/literature/307956), [SemanticScholar](https://www.semanticscholar.org/paper/Two-dimensional-gravity-and-intersection-theory-on-Witten/e5935bb91e79d8247468b19c310a201b8e41e808)]

motivated by

* {#Witten88} [[Edward Witten]], *Topological sigma models*,  Comm. Math. Phys. **118** 3 (1988) 411-449 [[euclid:cmp/1104162092](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-118/issue-3/Topological-sigma-models/cmp/1104162092.full)]

See also:

* [[Eric Zaslow]], *Topological Orbifold Models and Quantum Cohomology Rings*, Commun. Math. Phys. **156** (1993) 301-331 [[arXiv:hep-th/9211119](https://arxiv.org/abs/hep-th/9211119), [doi:10.1007/BF02098485](https://doi.org/10.1007/BF02098485)]

  > (targets [[orbifolds]] of [[Riemann sphere|$\mathbb{C}P^1$]])

The rigorous mathematical formulation in [[differential geometry|differential]] [[symplectic geometry]] via [[Gromov-Ruan-Witten invariants]] is due to:

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, Mathematical Research Letters **1** 2 (1994) 269-278 [[doi:10.4310/MRL.1994.v1.n2.a15](https://doi.org/10.4310/MRL.1994.v1.n2.a15), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0002/MRL-1994-0001-0002-a015.pdf)]

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, J. Diff. Geometry __42__ 2 (1995) 259-367 [[doi:10.4310/jdg/1214457234](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-2/A-mathematical-theory-of-quantum-cohomology/10.4310/jdg/1214457234.full)]

with early computations in

* [[Bernd Siebert]], [[Gang Tian]], *On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator*,  Asian Journal of Mathematics **1** 4 (1997) 679 – 695 [[arXiv:alg-geom/9403010](https://arxiv.org/abs/alg-geom/9403010), [doi:10.4310/AJM.1997.v1.n4.a2](https://dx.doi.org/10.4310/AJM.1997.v1.n4.a2)]

and in terms of [[algebraic geometry]] and via [[Frobenius manifolds]] due to:

* [[Maxim Kontsevich]], [[Yuri Manin]], *Gromov-Witten classes, quantum cohomology, and enumerative geometry*, Commun. Math. Phys. __164__ (1994) 525-562 [[doi:10.1007/BF02101490](https://doi.org/10.1007/BF02101490), [hep-th/9402147](https://arxiv.org/pdf/hep-th/9402147)]

* [[Maxim Kontsevich]], [[Yuri Manin]], [[Ralph Kaufmann]], _Quantum cohomology of a product_,  Invent. Math. __124__ (1996) 313-339 [[doi:10.1007/s002220050055](https://doi.org/10.1007/s002220050055) arXiv:[q-alg/9502009](https://arxiv.org/abs/q-alg/9502009)]

* [[Yuri I. Manin]], *Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces*, Colloquium Publications **47** (1999) [[ISBN:978-0-8218-1917-3](https://bookstore.ams.org/coll-47), [mpim-eprint:1598](https://archive.mpim-bonn.mpg.de/id/eprint/1598)]

See also:

* [[Boris Dubrovin]], *Geometry and Analytic Theory of Frobenius Manifolds*, Extra Volume ICM II (1998) 315–326 [[arXiv:math/9807034](https://arxiv.org/abs/math/9807034)]

* [[Ron Donagi]], [[Josh Guffin]], [[Sheldon Katz]], [[Eric Sharpe]], *A Mathematical Theory of Quantum Sheaf Cohomology*, Asian J. Math. **18** (2014) 387-418 [[arXiv:1110.3751](https://arxiv.org/abs/1110.3751), [doi:10.4310/AJM.2014.v18.n3.a1](https://doi.org/10.4310/AJM.2014.v18.n3.a1)]

More on the history:

* [[Maxim Kontsevich]], *On the History of quantum cohomology and homological mirror symmetry* (2021) [video:[YT](https://youtu.be/Ml2x5NnEQ1I)]

Introduction and review:

* [[Martin A. Guest]], *Introduction to Quantum Cohomology*, Vietnam Journal of Mathematics **33** SI (2005) 29–59 [[pdf](http://www.math.ac.vn/publications/vjm/VJM_33/Pdf_files_DB_2005/Bai3_Guest.pdf)]

* [[Joachim Kock]], [[Israel Vainsencher]], *An Invitation to Quantum Cohomology -- Kontsevich's Formula for Rational Plane Curves*, Birkhäser (2007) [[doi:10.1007/978-0-8176-4495-6](https://doi.org/10.1007/978-0-8176-4495-6)]

* [[Martin A. Guest]], *Quantum cohomology: is it still relevant?* [[arXiv:2210.05413](https://arxiv.org/abs/2210.05413)]

* Josh Guffin, _Quantum sheaf cohomology, a précis_, Matemática Contemporânea **41** (2012) 17-26 [[arxiv:1101.1305](https://arxiv.org/abs/1101.1305)]

* Tom Coates, *An Introduction to Quantum Cohomology* [[pdf](https://homepages.abdn.ac.uk/kedra/pages/lms/tom.pdf), [[Coates-QuantumCohomology.pdf:file]]]

* [[Alexander Givental]], *A tutorial on Quantum Cohomology* [[pdf](https://math.berkeley.edu/~giventh/papers/lqc.pdf), [[Givental-QuantumCohomologyTutorial.pdf:file]]]

See also:

* Wikipedia, *[Quantum cohomology](https://en.wikipedia.org/wiki/Quantum_cohomology)*

Relation to [[chiral algebras]]:

* [[Fyodor Malikov]], [[Vadim Schechtman]], *Deformations of chiral algebras and quantum cohomology of toric varieties* [[arXiv:math/0001170](https://arxiv.org/abs/math/0001170)]

Slides of a talk for an audience of mathematical [[string theory|string theorists]] are

* [[Eric Sharpe]],

  _Quantum sheaf cohomology_ ([pdf](http://www.phys.vt.edu/~ersharpe/brandeis-mar10-2.pdf)) Brandeis university (2010)

  _Quantum sheaf cohomology I_ ([pdf](http://www.phys.vt.edu/~ersharpe/banff-mar10.pdf))

### Of flag varieties

On quantum cohomology rings of [[flag varieties]] --- "quantum [[Schubert calculus]]":

* [[Dale H. Peterson]] (notes by [[Arun Ram]]), *Quantum Cohomology of $G/H$*, MIT (1997) [[web](http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html), [pdf](http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf), [[Peterson-QuantumCohomology.pdf:file]]]

* {#LeungLi14} [[Naichung Conan Leung]], [[Changzheng Li]], *An update of quantum cohomology of homogeneous varieties* [[arXiv:1407.5905](https://arxiv.org/abs/1407.5905)]

> (e.g. for $\mathbb{C}P^1 = SL(2,\mathbb{C})/P$)

Specifically the case of [[complex projective space|$\mathbb{C}P^N$]] ([[CP^N sigma-model|$\mathbb{C}P^N$ sigma-model]]):

* {#Witten90} [[Edward Witten]], *On the structure of the topological phase of two-dimensional gravity*, Nuclear Physics B **340** 2–3 (1990) 281-332 [doi:10.1016/0550-3213(90)90449-N]

* {#CecottiVafa92} [[Sergio Cecotti]], [[Cumrun Vafa]], *Exact Results for Supersymmetric Sigma Models*, Phys. Rev. Lett. **68** (1992) 903-906 [[arXiv:hep-th/9111016](https://arxiv.org/abs/hep-th/9111016), [doi:10.1103/PhysRevLett.68.903](https://doi.org/10.1103/PhysRevLett.68.903)]

* {#CecottiVafa93} [[Sergio Cecotti]], [[Cumrun Vafa]], *On Classification of $N=2$ Supersymmetric Theories*, Comm. Math. Phys. **158** (1993) 569-644 [[arXiv:hep-th/9211097](https://arxiv.org/abs/hep-th/9211097), [doi:10.1007/BF02096804](https://doi.org/10.1007/BF02096804)]

* {#BourdeauDouglas94} [[M. F. Bourdeau]], [[Michael R. Douglas]], *Topological-Antitopological Fusion and the Large $N$ $\mathbb{C}P^N$ Model*, Nucl. Phys. B **420** (1994) 243-267 [[doi:hep-th/9312095](https://arxiv.org/abs/hep-th/9312095), doi:10.1016/0550-3213(94)90380-8]

And specifically the case of [[Riemann sphere|$\mathbb{C}P^1$]]:

* {#DorfmeisterGuestRossman10} Josef F. Dorfmeister, [[Martin A. Guest]], Wayne Rossman, *The $t t^\ast$ Structure of the Quantum Cohomology of $\mathbb{C}P^1$ from the Viewpoint of Differential Geometry*, Asian Journal of Mathematics **14** 3 (2010) 417-438 [[doi:10.4310/AJM.2010.v14.n3.a7](https://dx.doi.org/10.4310/AJM.2010.v14.n3.a7)]

### Quantum K-cohomology rings
 {#ReferencesQuantumKTheoryRings}

The idea of an analogous quantum deformation of [[topological K-theory]]-rings originates around:

* [[Alexander B. Givental]], *On the WDVV-equation in quantum K-theory*, Michigan Math. J. **48** 1 (2000) 295-304 [[arXiv:math/0003158](https://arxiv.org/abs/math/0003158), [doi:10.1307/mmj/1030132720](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-48/issue-1/On-the-WDVV-equation-in-quantum-K-theory/10.1307/mmj/1030132720.full)]

* Y.-P. Lee, *Quantum K-Theory I: Foundations*, Duke Math. J. **121** 3 (2004) 389-424 [[arXiv:math/0105014](https://arxiv.org/abs/math/0105014), [doi:10.1215/S0012-7094-04-12131-1](https://projecteuclid.org/journals/duke-mathematical-journal/volume-121/issue-3/Quantum-K--theory-I-Foundations/10.1215/S0012-7094-04-12131-1.short)]

and for some form of [[equivariant K-theory]] in:

* [[Alexander B. Givental]], *Permutation-equivariant quantum K-theory* [[webpage](https://math.berkeley.edu/~giventh/perm/perm.html)]

  *I. Definitions. Elementary K-theory of $\overline{\mathcal{M}}_{0,n}/S_n$* [[arXiv:1508.02690](https://arxiv.org/abs/1508.02690)]

  *II. Fixed point localization* [[arXiv:1508.04374](https://arxiv.org/abs/1508.04374)]

  *III. Lefschetz' formula on $\overline{\mathcal{M}}_{0,n}/S_n$ and adelic characterization* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]

  *IV. $D_q$-modules* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]

  *V. Toric $q$-hypergeometric functions* [[arXiv:1509.03903](https://arxiv.org/abs/1509.03903)]

  *VI. Mirrors* [[arXiv:1509.07852](https://arxiv.org/abs/1509.07852)]

  *VII. General theory* [[arXiv:1510.03076](https://arxiv.org/abs/1510.03076)]

  *VIII. Explicit reconstruction* [[arXiv:1510.06116](https://arxiv.org/abs/1510.06116)]

  *IX. Quantum Hirzebruch-Riemann-Roch in all genera* [[arXiv:1709.03180](https://arxiv.org/abs/1709.03180)]

  *X. Quantum Hirzebruch-Riemann-Roch in genus 0*, SIGMA **16** 031 (2020) [[arXiv:1710.02376](https://arxiv.org/abs/1710.02376), [doi:10.3842/SIGMA.2020.031](https://doi.org/10.3842/SIGMA.2020.031)]

  *XI. Quantum Adams-Riemann-Roch* [[arXiv:1711.04201](https://arxiv.org/abs/1711.04201)]

See also:

* [[Anders S. Buch]], [[Leonardo C. Mihalcea]], *Quantum K-theory of Grassmannians*, Duke Math. J. **156** 3 (2011) 501-538 [[arXiv:0810.0981](https://arxiv.org/abs/0810.0981), [doi:10.1215/00127094-2010-218](https://doi.org/10.1215/00127094-2010-218)]

Relation to [[D=3 N=2 super Yang-Mills theory]]:

* [[Hans Jockers]], [[Peter Mayr]], *Quantum K-Theory of Calabi-Yau Manifolds*, J. High Energ. Phys. **2019** 11 (2019) [[arXiv:1905.03548](https://arxiv.org/abs/1905.03548), doi:10.1007/JHEP11(2019)011]

* [[Hans Jockers]], [[Peter Mayr]], *A 3d Gauge Theory/Quantum K-Theory Correspondence*, Advances in Theoretical and Mathematical Physics **24** 2 (2020) [[arXiv:1808.02040](https://arxiv.org/abs/1808.02040), [doi:10.4310/ATMP.2020.v24.n2.a4](https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a4)]

* Cyril Closset, Osama Khlaif, *Grothendieck lines in 3d $\mathcal{N}=2$ SQCD and the quantum K-theory of the Grassmannian* [[arXiv:2309.06980](https://arxiv.org/abs/2309.06980)]

For more references see also those listed in

* W. Gu, [[Leonardo C. Mihalcea]], [[Eric Sharpe]], H. Zou, *Quantum K theory of symplectic Grassmannians* [[arXiv:2008.04909](https://arxiv.org/abs/2008.04909)

[[!include quantum cohomology as Pontrjagin rings -- references]]

[[!redirects quantum cohomology rings]]

[[!redirects quantum cohomology]]
[[!redirects quantum sheaf cohomology]]

[[!redirects quantum Schubert calculus]]

[[!redirects quantum K-theory]]
[[!redirects quantum K-theory ring]]
[[!redirects quantum K-theory rings]]


quantum communication

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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[quantum information]]

* [[quantum computation]]

* [[quantum sensing]]

* [[quantum logic]]

## References

* Sumeet Khatri, Mark M. Wilde, *Principles of Quantum Communication Theory: A Modern Approach* [[arXiv:2011.04672](https://arxiv.org/abs/2011.04672)]

* [[Koji Azuma]], Stefan Bäuml, Tim Coopmans, [[David Elkouss]], Boxi Li, *Tools for quantum network design* [[arXiv:2012.06764](https://arxiv.org/abs/2012.06764)]

See also:

* Wikipedia, *[Quantum network](https://en.wikipedia.org/wiki/Quantum_network)*

* [PicoQuant](https://www.picoquant.com/), *[Quantum Communication](https://www.picoquant.com/applications/category/quantum-optics/quantum-communication)*

On quantum repeaters for quantum networks :

* [[Koji Azuma]], [[Sophia E. Economou]], [[David Elkouss]], Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, Ilan Tzitrin, *Quantum repeaters: From quantum networks to the quantum internet* [[arXiv:2212.10820](https://arxiv.org/abs/2212.10820)]

[[!redirects quantum communication theory]]

[[!redirects quantum network]]
[[!redirects quantum networks]]

[[!redirects quantum repeater]]
[[!redirects quantum repeaters]]


quantum complexity theory

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=--
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#Contents#
* table of contents
{:toc}

## Idea

[[complexity theory]] for [[quantum systems]] and [[quantum computation]]

## Quantum complexity classes

Quantum [[complexity classes]]:

* [[BQP]]

* [[PostBQP]]

* [[DQC1]] ("one clean qbit")

* ...

## Related concepts

* [[quantum supremacy]]

## References

Original articles:

* [[Ethan Bernstein]], [[Umesh Vazirani]], *Quantum Complexity Theory*, Siam Journal of Computation **26** 5 (1997) 1411-1473 [[doi:10.1137/S0097539796300921](https://doi.org/10.1137/S0097539796300921), [pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.655.1186&rep=rep1&type=pdf)]

Introduction and review:

* [[Umesh Vazirani]], *A survey of quantum complexity theory*, in: [[Samuel J. Lomonaco]] (ed.): *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

* [[Richard Cleve]], *An Introduction to Quantum Complexity Theory*, p. 103-127 of: *Quantum Computation and Quantum Information Theory*, World Scientific (2001) [[arXiv:quant-ph/9906111](https://arxiv.org/abs/quant-ph/9906111), [doi:10.1142/9789810248185_0004]
(https://doi.org/10.1142/9789810248185_0004)]

Lecture notes:

* [[Scott Aaronson]], Section 24.3 of: *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

See also

* Wikipedia, *[Quantum complexity theory](https://en.wikipedia.org/wiki/Quantum_complexity_theory)*

[[!redirects quantum complexity]]


quantum computation

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#Contents#
* table of contents
{:toc}

## Idea

### General
 {#IdeaGeneral}

_Quantum computation_ is [[computation]] in terms of [[quantum information theory]], possibly implemented on _quantum computers_, hence on [[physical systems]] for which phenomena of [[quantum mechanics]] are not negligible: [[quantum systems]] such as consisting of [[fundamental particles]] like [[photons]] or [[electrons]] but also often of quantum-[[quasiparticles]] such as [[Cooper pairs]] in [[superconductors]].

In terms of the [[computational trilogy]], quantum computation is the computation corresponding to (some kind of) [[quantum logic]]/([[dependent linear type theory|dependent]]) [[linear type theory]].

To the extent that quantum computation involves [[quantum measurement]] (generally to read out results, but often also as a kind of [[quantum logic gate]] in its own right) it is a form of [[nondeterministic computation]], due to the intrinsic [[probability theory|stochastic]] nature of the [[quantum state collapse]] during [[quantum measurement]].

However, away from [[quantum measurement]], [[quantum gates]] are understood to implement [[unitary operators]] on a given [[Hilbert space]] [[space of quantum states|of quantum states]] (fundamentally due to the unitary nature of time-evolution as given by the [[Schrödinger equation]]), for which quantum computation is hence not only deterministic but in fact a form of *[[reversible computation]]*. This relates to its susceptibility to noise and the practical need for [[quantum error correction]] and/or [[topological quantum computation|topological quantum stabilization]].

Specifically, _[[topological quantum computation]]_ is (or is meant to be) quantum computation implemented on [[physical systems]] governed by [[topological quantum field theory]], such as [[Chern-Simons theory]], thought to describe for instance [[anyon]] [[defects]] in [[topological order|topologically ordered]] [[mass gap|gapped]] [[ground states]] of [[quantum materials]] in [[topological phases of matter]].

### Classical control, quantum data
 {#ClassicalControlQuantumData}

Any practical quantum computer will be *[[classically controlled quantum computation|classically controlled]]* ([Knill 96](#Knill96), [Ömer 03](#Omer03), [Nagarajan, Papanikolaou & Williams 07](#NagarajanPapanikolaouWilliams07), [Devitt 14](#Devitt14), in line with the general idea of *[[parameterized quantum systems]]*):

\begin{imagefromfile}
    "file_name": "ClassicallyControlledQuantumComputation.jpg",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Miszczak 11](#Miszczak11)"
\end{imagefromfile}

{#DynamicLifting} But in general the classical control itself may be conditioned on [[quantum measurement]]-outcomes available only during runtime, a more subtle mechanism that (at least in the [[Quipper]]-community) has come to be called "[[dynamic lifting]]":

\begin{imagefromfile}
    "file_name": "SQRAMschema-WithDynLifting.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "SQRAM model adapted from [Nagarajan, Papanikolaou & Williams 2007, Fig 1](#NagarajanPapanikolaouWilliams07)"
\end{imagefromfile}

See the list of references [below](#ReferencesClassicallyControlledQuantumComputing).

The paradigm of [[classically controlled quantum computation]] applies in particular ([Kim & Swingle 17](#KimSwingle17)) to the currently and near-term available noisy intermediate-scale quantum (NISQ) computers ([Preskill 18](#Preskill18), see the references [below](#ReferencesNISQ)), which are useful for highly specialized tasks (only) and need to be embedded in and called from a more comprehensive classical computing environment.

This, in turn, applies particularly to applications like [[quantum machine learning]] ([Benedetti, Lloyd, Sack & Fiorentini 19](#BenedettiLloydSackFiorentini19), [TensorFlow Quantum](#TensorFlow), for more see the references [there](machine+learning#ReferencesQuantumMachineLearning)).

### Quantum languages and quantum circuits

A natural way to understand (via [[computational trilogism]]) [[quantum programming languages]] is as  [[linear logic]]/[[linear type theory]] ([Pratt 92](#Pratt92), for more see at *[[quantum logic]]*) with [[categorical semantics]] in non-[[Cartesian monoidal category|cartesian]] [[symmetric monoidal categories]] ([Abramsky & Coecke 04](#AbramskyCoecke04), [Abramsky & Duncan 05](#AbramskyDuncan05), [Duncan 06](#Duncan06), [Lago-Faffian 12](#LagoFaffian12)).  .

The corresponding [[string diagrams]] are known as *[[quantum circuit diagrams]]*.

In fact, languages for [classically controlled quantum computation](#ClassicalControlQuantumData)  should be based on [[dependent linear type theory]] ([Vakar 14](dependent+linear+type+theory#Vakar14), [Vakar 15](dependent+linear+type+theory#Vakar15), [Vakar 17, Sec. 3](dependent+linear+type+theory#Vakar17), [Lundfall 17](dependent+linear+type+theory#Lundfall17), [Lundfall 18](dependent+linear+type+theory#Lundfall18), following [Schreiber 14](dependent+linear+type+theory#Schreiber14)) with [[categorical semantics]] in [[indexed monoidal categories]]:

| classical control | quantum data |
|-------------------|--------------|
| [[intuitionistic type theory|intuitionistic types]] | [[dependent linear type theory|dependent linear types]]

This idea of classically controlled quantum programming via [[dependent linear type theory]] has been implemented in [[QML]] and in *[[Quipper]]* in [FKS 20](#FKS20), [FKRS 20](#FKRS20).

## Examples

* [[QML]]

* [[Quipper]]

## Related entries

* [[quantum simulation]]

* [[quantum sensing]]

* [[quantum parallelism]], [[parallel computing]]

* [[Bloch region]], [[quantum operation]]

* [[Grover's algorithm]], [[Shor's algorithm]]

* [[Solovay-Kitaev theorem]]

* [[quantum information theory]]

* [[quantum communication]]

* [[quantum sensing]]

* [[qbit]], [[qtrit]], [[qdit]]

* [[quantum logic gate]], [[quantum circuit]]

* [[quantum error correction]]

* [[quantum logic]], [[linear logic]]

* [[adiabatic quantum computation]]

* [[topological quantum computation]]

* [[measurement-based quantum computation]]

* [[quantum Hall effect]], [[Chern-Simons theory]].

* [[superoperator]]

* [[computable physics]]

* [[quantum probability]]

## References

### General

The idea of quantum computation was first expressed in:

* {#Manin1980} [[Yuri I. Manin]], Introduction to: *Computable and Uncomputable*, Sov. Radio (1980) [Russian original: [[Manin-1980.pdf:file]]], Enlish translation on p. 69-77 of *Mathematics as Metaphor: Selected essays of Yuri I. Manin*, Collected Works **20**, AMS (2007) [[ISBN:978-0-8218-4331-4](https://bookstore.ams.org/cworks-20/)]

  > Perhaps, for a better understanding of [molecular biology], we need a mathematical theory of quantum automata.

* [[Paul Benioff]], *The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines*, J Stat Phys 22, 563–591 (1980) [[doi:10.1007/BF01011339](https://doi.org/10.1007/BF01011339)]

* [[Paul Benioff]], *Quantum Mechanical Models of Turing Machines That Dissipate No Energy*, Phys. Rev. Lett. **48** 1581  (1982) [[doi:10.1103/PhysRevLett.48.1581](https://doi.org/10.1103/PhysRevLett.48.1581)]

* {#Feynman82} [[Richard Feynman]], *Simulating physics with computers*,  Int J Theor Phys **21** (1982) 467–488 [[doi:10.1007/BF02650179](https://doi.org/10.1007/BF02650179)]

  > "because nature isn't classical, dammit, if you want to make a simulation of nature, you'd better make it quantum mechanical"

First ideas of a [[quantum Turing machine]]:

* David Z. Albert, *On quantum-mechanical automata*, Physics Letters A **98** 5–6 (1983) 249-252 [doi:10.1016/0375-9601(83)90863-0]

* [[David Deutsch]], *Quantum theory, the Church–Turing principle and the universal quantum computer*, Proceedings of the Royal Society of London A **400** 1818 (1985) 97-117 [[doi:10.1098/rspa.1985.0070](https://doi.org/10.1098/rspa.1985.0070), [pdf](https://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf)]

The notion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]] and [[quantum circuits]] was first made explicit in:

* [[Richard Feynman]], *Quantum mechanical computers*, Foundations of Physics **16** (1986) 507–531 [[doi:10.1007/BF01886518](https://doi.org/10.1007/BF01886518)]

* [[David E. Deutsch]], *Quantum computational networks*, Proceedings of the Royal Society A, **425** 1868 (1989) 73-90 [[doi:10.1098/rspa.1989.0099](https://doi.org/10.1098/rspa.1989.0099), [jstor:2398494](https://www.jstor.org/stable/2398494)]

The terminology *[[q-bit]]* goes back to:

* [[Benjamin Schumacher]], *Quantum coding*, Phys. Rev. A **51** (1995) 2738 [[doi:10.1103/PhysRevA.51.2738](https://doi.org/10.1103/PhysRevA.51.2738)]

The potential practical relevance of quantum computation was highlighted early on via toy examples

such as the [[Deutsch-Jozsa algorithm]]:

* [[David Deutsch]], [[Richard Jozsa]], *Rapid solution of problems by quantum computation*. Proc. R. Soc. Lond. A **439** 1907 (1992) 553-558 [[pdf](https://www.isical.ac.in/~rcbose/internship/lectures2016/rt08deutschjozsa.pdf), [doi:10.1098/rspa.1992.0167](https://doi.org/10.1098/rspa.1992.0167)]

and became widely appreciated in 1996/1997 with the discovery of several practically interesting [[algorithms]] exhibiting "[[quantum supremacy]]", such as:

in [[quantum chemistry]]

* [[Seth Lloyd]], *Universal Quantum Simulators*, Science New Series **273** 5278 (1996) 1073-1078 [[jstor:2899535](https://www.jstor.org/stable/2899535), [doi:10.1126/science.273.5278.1073](https://doi.org/10.1126/science.273.5278.1073)]

in search algorithms ([[Grover's algorithm]])

* [[Lov K. Grover]], *A fast quantum mechanical algorithm for database search*, STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing (1996) 212-219 [[arXiv:quant-ph/9605043](https://arxiv.org/abs/quant-ph/9605043), [doi:10.1145/237814.237866](https://doi.org/10.1145/237814.237866)]

and in ([[quantum cryptography|quantum]]) [[cryptography]] ([[Shor's algorithm]])

* [[Daniel R. Simon]], *On the power of quantum computation*, SIAM Journal on Computing **26** 5 (1997) [[doi:10.1137/S0097539796298637](https://doi.org/10.1137/S0097539796298637), [pdf](https://courses.cs.washington.edu/courses/cse599/01wi/papers/simon_qc.pdf)]

* [[Peter W. Shor]], *Algorithms for quantum computation: discrete logarithms and factoring*,  Proceedings 35th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press: 124-134 (1994) [[doi:10.1109/SFCS.1994.365700](https://doi.org/10.1109/SFCS.1994.365700)]

Review:

* Renato Portugal, *Basic Quantum Algorithms* [[arXiv:2201.10574](https://arxiv.org/abs/2201.10574)]

* Kamil Khadiev, *Lecture Notes on Quantum Algorithms* [[arXiv:2212.14205](https://arxiv.org/abs/2212.14205)]

* [[Alessandro Luongo]], *[Quantum algorithms for data analysis](https://quantumalgorithms.org)*

Quantum computation became widely thought to be not just a theoretical but a plausible practical possibility with the theoretical understanding of [[quantum error correction]] (which however remains to be implemented scalably):

* {#Shor95} [[Peter W. Shor]], *Scheme for reducing decoherence in quantum computer memory*, Phys. Rev. A 52, R2493(R) 1995 ([doi:10.1103/PhysRevA.52.R2493](https://doi.org/10.1103/PhysRevA.52.R2493))

* {#Steane96} [[Andrew M. Steane]], *Error Correcting Codes in Quantum Theory*, Phys. Rev. Lett. 77, 793 1996 ([doi:10.1103/PhysRevLett.77.793](https://doi.org/10.1103/PhysRevLett.77.793))

* {#CalderbankShor96} [[Robert Calderbank]], [[Peter W. Shor]], *Good Quantum Error-Correcting Codes Exist*, Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996 ([doi:10.1103/PhysRevA.54.1098](https://doi.org/10.1103/PhysRevA.54.1098))

Early review:

* [[Adriano Barenco]], *Quantum Physics and Computers*,  Contemp. Phys. **37** (1996) 375-389 [[arXiv:quant-ph/9612014](https://arxiv.org/abs/quant-ph/9612014), [doi:10.1080/00107519608217543](https://doi.org/10.1080/00107519608217543)]

* [[Yuri I. Manin]], *Classical computing, quantum computing, and Shor's factoring algorithm*, Astérisque, **266** Séminaire Bourbaki 862 (2000) 375-404 [[arXiv:quant-ph/9903008](https://arxiv.org/abs/quant-ph/9903008), [numdam:SB_1998-1999__41__375_0](http://www.numdam.org/item/?id=SB_1998-1999__41__375_0)]

An alternative/parallel approach to avoiding errors due to [[decoherence]] is [[topological quantum computation]], see [there](topological+quantum+computation#References) for references.

Influential list of criteria necessary for physical realization of quantum computation:

* {#DiVincenzo00} [[David P. DiVincenzo]], *The Physical Implementation of Quantum Computation*, Fortschr. Phys. **48** 9-11 (2000) 771-783 [[arXiv:quant-ph/0002077](https://arxiv.org/abs/quant-ph/0002077), doi:10.1002/1521-3978(200009)48:9/11%3C771::AID-PROP771%3E3.0.CO;2-E]

Simple quantum computers have been realized (first with [[NMR]] technology, see [references below](#SpinResonanceQBitsReferences), then with [[superconductor]] technology, see [references further below](#SuperconductingQBitsReferences)) and have been argued to demonstrate *[[quantum supremacy]]* (see [there](quantum+supremacy#References) for references) but existing machines remain "noisy and intermediate scale" (see the references [below](#ReferencesNISQ)).

Textbook accounts on quantum computation:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* [[Samuel J. Lomonaco]] (ed.), *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

* [[Alexei Y. Kitaev]], A. H. Shen, M. N. Vyalyi, *Classical and Quantum Computation*, Graduate Studies in Mathematics **47** (2002) [[doi:10.1090/gsm/047](http://dx.doi.org/10.1090/gsm/047)]

* [[Ranee K. Brylinski]], [[Goong Chen]] (eds.), *Mathematics of Quantum Computation*, Chapman and Hall/CRC 2002 ([doi:10.1201/9781420035377](https://doi.org/10.1201/9781420035377))

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], *Principles of Quantum Computation and Information*, World Scientific (2004, 2018)  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

* [[Goong Chen]], [[Louis Kauffman]], [[Samuel J. Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Chapman and Hall/CRC (2007) (ISBN:9780367388614, [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007))

* [[Eleanor Rieffel]], [[Wolfgang Polak]], *Quantum Computing -- A gentle introduction*, MIT Press (2011) [[ISBN:9780262526678](https://mitpress.mit.edu/9780262526678/quantum-computing/), [pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608150244118.pdf)]

* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

* [[Steven Duplij]], [[Raimund Vogl]], *Innovative Quantum Computing*, IOP (2023) [[ISBN:978-0-7503-5281-9](https://iopscience.iop.org/book/mono/978-0-7503-5281-9)]

Introducing the notion of [[quantum supremacy]] and highlighting its relation to [[quantum entanglement]]:

* [[John Preskill]]: *Quantum computing and the entanglement frontier*: pp. 63-80 in: *The Theory of the Quantum World -- Proceedings of the 25th Solvay Conference on Physics*, World Scientific (2013) [[arXiv:1203.5813](https://arxiv.org/abs/1203.5813), [doi:10.1142/8674](https://doi.org/10.1142/8674), slides: [pdf](https://simons.berkeley.edu/sites/default/files/docs/394/preskilljohn.pdf)]

Further introduction and survey:

* [[Alexei Kitaev]],  *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), pdf]

* [[Eleanor Rieffel]], [[Wolfgang Polak]],  *An Introduction to Quantum Computing for Non-Physicists*, ACM Comput. Surveys **32** (2000) 300-335 [[arXiv:quant-ph/9809016](https://arxiv.org/abs/quant-ph/9809016), [doi:10.1145/367701.367709](https://dl.acm.org/doi/10.1145/367701.367709)]

* [[John Preskill]], *Quantum Computation*, lecture notes, since 2004 [[web](http://theory.caltech.edu/~preskill/ph229/)]

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

* [[Jens Eisert]], M. M. Wolf, _Quantum computing_, In: _Handbook of Nature-Inspired and Innovative Computing_, Springer 2006 ([arXiv:quant-ph/0401019](https://arxiv.org/abs/quant-ph/0401019))

* [[Scott Aaronson]], _Quantum Computing Since Democritus_, Lecture notes (2006) [[web](http://www.scottaaronson.com/democritus/)]

* [[Phillip Kaye]], [[Raymond Laflamme]], [[Michele Mosca]], *An Introduction to Quantum Computing*, Oxford University Press (2007) [[pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149125645.pdf), [ISBN:9780198570493](https://global.oup.com/academic/product/an-introduction-to-quantum-computing-9780198570493?cc=at&lang=en&)]

* {#Loceff13} Michael Loceff, _A course in quantum computing_, 2013 ([pdf](https://www.fgamedia.org/faculty/loceff/cs_courses/cs_83a/Intro_to_QC_Vol_1_Loceff.pdf))

* [[Scott Aaronson]], *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

  *Introduction to Quantum Information Science II* (2022) [[pdf](https://www.scottaaronson.com/qisii.pdf)]

* National Academies of Sciences, Engineering, and Medicine, _Quantum Computing: Progress and Prospects_, The National Academies Press 2019 ([doi:10.17226/25196](https://doi.org/10.17226/25196))

* Qiang Zhang, Feihu Xu, Li Li, Nai-Le Liu, Jian-Wei Pan, *Quantum information research in China*, Quantum Sci. Technol. 4 040503 ([doi:10.1088/2058-9565/ab4bea](https://iopscience.iop.org/article/10.1088/2058-9565/ab4bea))

* Farzan Jazaeri, Arnout Beckers, Armin Tajalli, Jean-Michel Sallese, *A Review on Quantum Computing: Qubits, Cryogenic Electronics and Cryogenic MOSFET Physics*, 2019 MIXDES - 26th International Conference "Mixed Design of Integrated Circuits and Systems", 2019, pp. 15-25, ([arXiv:1908.02656](https://arxiv.org/abs/1908.02656), [doi:10.23919/MIXDES.2019.8787164](https://doi.org/10.23919/MIXDES.2019.8787164))

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

* Jiajun Chen, *Review on Quantum Communication and Quantum Computation*,  Journal of Physics: Conference Series, Volume 1865, 2021 International Conference on Advances in Optics and Computational Sciences (ICAOCS) 2021 21-23 January 2021, Ottawa, Canada ([doi:10.1088/1742-6596/1865/2/022008](https://iopscience.iop.org/article/10.1088/1742-6596/1865/2/022008))

* David Matthews,  *How to get started in quantum computing*, Nature **591** March  2021 ([nature:d41586-021-00533-x](https://www.nature.com/articles/d41586-021-00533-x), [pdf](https://media.nature.com/original/magazine-assets/d41586-021-00533-x/d41586-021-00533-x.pdf))

* Christine Middleton, *What’s under the hood of a quantum computer?*, Physics Today, March 2021 ([doi:10.1063/PT.6.1.20210305a](https://physicstoday.scitation.org/do/10.1063/PT.6.1.20210305a))

* [[John Preskill]], *The Physics of Quantum Information*, talk at *The Physics of Quantum Information*, 28th Solvay Conference on Physics (2022) [[arXiv:2208.08064](https://arxiv.org/abs/2208.08064)]

* Oswaldo Zapata, *A Short Introduction to Quantum Computing for Physicists* [[arXiv:2306.09388](https://arxiv.org/abs/2306.09388)]

* [[Peter Shor]], *Quantum Computation*, Lecture notes (2022) [[web](https://math.mit.edu/~shor/435-LN/)]

* Anton Frisk et al., *Lecture notes on quantum computing* [[arXiv:2311.08445](https://arxiv.org/abs/2311.08445)]

On "continuous variable" analog quantum computation:

* Sophie Choe, *Quantum computing overview: discrete vs. continuous variable models* [[arXiv:2206.07246](https://arxiv.org/abs/2206.07246)]

* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]

See also:

* Wikipedia, _[Quantum computation](http://en.wikipedia.org/wiki/Quantum_computation)_

* [The Net Advance of Physics](http://web.mit.edu/redingtn/www/netadv/): *[Quantum Computation](http://web.mit.edu/redingtn/www/netadv/Xqucomputa.html)*

* [Quantiki](https://www.quantiki.org/) -- Quantum Information Portal and Wiki

* idquantique, *Quantum Computing Review*:

  * [q2 2020](https://www.idquantique.com/quantum-computing-review-q2-2020/)

  * [q3 2020](https://www.idquantique.com/quantum-computing-review-q3-2020/)

  * [q4 2020](https://www.idquantique.com/quantum-computing-review-q4-2020/)

  * [q2 2021](https://www.idquantique.com/quantum-computing-review-q2-2021/)

  * [q3 2021](https://www.idquantique.com/quantum-computing-review-q3-2021/)

  * [q4 2021](https://www.idquantique.com/quantum-computing-review-q4-2021/)

  * [q1 2022](https://www.idquantique.com/quantum-computing-review-q1-2022/)

Experimental demonstration of "[[quantum supremacy]]" ("quantum advantage"):

* F. Arute et al. *Quantum supremacy using a programmable superconducting processor*, Nature **574** (2019) 505–510 ([doi:10.1038/s41586-019-1666-5](https://doi.org/10.1038/s41586-019-1666-5))

* Han-Sen Zhong et al., *Quantum computational advantage using photons*, Science  **370** 6523 (2020) 1460-1463 ([doi:10.1126/science.abe8770](https://science.sciencemag.org/content/370/6523/1460))

Review:

* Adrian Cho, *Google claims quantum computing milestone*, Science **365** 6460 (2019) 1364 ([doi:10.1126/science.365.6460.1364](https://science.sciencemag.org/content/365/6460/1364))

* Philip Ball, *Physicists in China challenge Google’s "quantum advantage"*, Nature **588** 380 (2020) ([doi:10.1038/d41586-020-03434-7](https://doi.org/10.1038/d41586-020-03434-7))

* {#DasSarma22} [[Sankar Das Sarma]], *Quantum computing has a hype problem*, [MIT Technology Review (March 2022)](https://www.technologyreview.com/2022/03/28/1048355/quantum-computing-has-a-hype-problem/)

  > The [[qubit]] systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. $[\cdots]$ What is missing is the breakthrough $[\cdots]$ bypassing [[quantum error correction]] by using far-more-stable qubits, in an approach called [[topological quantum computing]].

Potential relation to the [[Unruh effect]]:

* Eric W. Aspling, John A. Marohn, Michael J. Lawler, *Design Constraints for Unruh-DeWitt Quantum Computers* [[arXiv:2210.12552](https://arxiv.org/abs/2210.12552)]

[[!include quantum programming languages -- references]]

### Classically controlled quantum computing
 {#ReferencesClassicallyControlledQuantumComputing}

Theory of [[classically controlled quantum computing]] and [[parameterized quantum circuits]]:

* {#Knill96} [[Emanuel Knill]], *Conventions for quantum pseudocode*, Los Alamos Technical Report (1996) [[LA-UR-96-2724](https://www.osti.gov/biblio/366453-conventions-quantum-pseudocode), [doi:10.2172/366453](https://doi.org/10.2172/366453), [pdf](https://www.osti.gov/servlets/purl/366453)]

* {#Omer03} [[Bernhard Ömer]], *Structured Quantum Programming*, 2003/2009 ([pdf](http://www.itp.tuwien.ac.at/~oemer/doc/structquprog.pdf))

* Simon Perdrix, Philippe Jorrand, *Classically-Controlled Quantum Computation*, Math. Struct. in Comp. Science, 16:601-620, 2006 ([arXiv:quant-ph/0407008](https://arxiv.org/abs/quant-ph/0407008))

* {#NagarajanPapanikolaouWilliams07} Rajagopal Nagarajan, Nikolaos Papanikolaou, David Williams, *Simulating and Compiling Code for the Sequential Quantum Random Access Machine*, Electronic Notes in Theoretical Computer Science Volume 170, 6 March 2007, Pages 101-124 ([doi:10.1016/j.entcs.2006.12.014](https://doi.org/10.1016/j.entcs.2006.12.014))

* {#Miszczak11} [[Jarosław Adam Miszczak]], *Models of quantum computation and quantum programming languages*, Bull. Pol. Acad. Sci.-Tech. Sci., Vol. 59, No. 3 (2011), pp. 305-324 ([arXiv:1012.6035](https://arxiv.org/abs/1012.6035))

* {#Devitt14} [[Simon J. Devitt]], *Classical Control of Large-Scale Quantum Computers*, In: S. Yamashita, S. Minato  (eds.) *Reversible Computation* Lecture Notes in Computer Science, vol 8507. Springer 2014 ([arXiv:1405.4943](https://arxiv.org/abs/1405.4943), [doi:10.1007/978-3-319-08494-7_3](https://doi.org/10.1007/978-3-319-08494-7_3))

* Jarrod R McClean, Jonathan Romero, Ryan Babbush and Alán Aspuru-Guzik, *The theory of variational hybrid quantum-classical algorithms*, New Journal of Physics, Volume 18, February 2016 ([doi:10.1088/1367-2630/18/2/023023](https://iopscience.iop.org/article/10.1088/1367-2630/18/2/023023))

* Murray Thom (D-Wave), *[Three Truths and the Advent of Hybrid Quantum Computing](https://medium.com/d-wave/three-truths-and-the-advent-of-hybrid-quantum-computing-1941ba46ff8c)*, Jun 25, 2019

Application of classically controlled quantum computation:

* {#KimSwingle17} Isaac H. Kim, [[Brian Swingle]], *Robust entanglement renormalization on a noisy quantum computer* ([arXiv:1711.07500](https://arxiv.org/abs/1711.07500))

  > (in terms of [[holographic entanglement entropy|holographic]] [[tensor network]] states)

* Sukin Sim, Peter D. Johnson, Alan Aspuru-Guzik, *Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms*, Adv. Quantum Technol. 2 (2019) 1900070 ([arXiv:1905.10876](https://arxiv.org/abs/1905.10876))

* Mateusz Ostaszewski, Edward Grant, Marcello Benedetti, *Structure optimization for parameterized quantum circuits*, Quantum 5, 391 (2021) ([arXiv:1905.09692](https://arxiv.org/abs/1905.09692))

* {#ShaydulinEtAl19} Ruslan Shaydulin et al., *A Hybrid Approach for Solving Optimization Problems on Small Quantum Computers* ([doi:10.1109/MC.2019.2908942](https://doi.ieeecomputersociety.org/10.1109/MC.2019.2908942))

* Eneko Osaba, Esther Villar-Rodriguez, Izaskun Oregi, Aitor Moreno-Fernandez-de-Leceta, *Focusing on the Hybrid Quantum Computing -- Tabu Search Algorithm: new results on the Asymmetric Salesman Problem* ([arXiv:2102.05919](https://arxiv.org/abs/2102.05919))

in particular in [[quantum machine learning]]:

* {#BenedettiLloydSackFiorentini19} Marcello Benedetti, Erika Lloyd, Stefan Sack, Mattia Fiorentini, *Parameterized quantum circuits as machine learning models*, Quantum Science and Technology 4, 043001 (2019) ([arXiv:1906.07682](https://arxiv.org/abs/1906.07682))

* D. Zhu et al. *Training of quantum circuits on a hybrid quantum computer*, Science Advances, 18 Oct 2019: Vol. 5, no. 10, eaaw9918 ([doi: 10.1126/sciadv.aaw9918](https://advances.sciencemag.org/content/5/10/eaaw9918))

* {#MariBromleyIzaacSchuldKilloran20} Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, Nathan Killoran,  *Transfer learning in hybrid classical-quantum neural networks*, Quantum 4, 340 (2020) ([arXiv:1912.08278](https://arxiv.org/abs/1912.08278))

* Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, Koen Bertels, *Evaluation of Parameterized Quantum Circuits: on the relation between classification accuracy, expressibility and entangling capability*, Quantum Machine Intelligence volume 3, Article number: 9 (2021) ([arXiv:2003.09887](https://arxiv.org/abs/2003.09887), [doi:10.1007/s42484-021-00038-w](https://doi.org/10.1007/s42484-021-00038-w))

* {#TensorFlow} [TensorFlow.org](https://www.tensorflow.org/) (Google), *[Hybrid quantum classical models](https://www.tensorflow.org/quantum/concepts#hybrid_quantum-classical_models)*

### Noisy intermediate-scale quantum computing
 {#ReferencesNISQ}

* Nikolaj Moll et al., *Quantum optimization using variational algorithms on near-term quantum devices*, Quantum Science and Technology, Volume 3, Number 3 2018 ([arXiv:1710.01022](https://arxiv.org/abs/1710.01022))

* {#Preskill18} [[John Preskill]], *Quantum Computing in the NISQ era and beyond*, Quantum  **2** 79 (2018) [[arXiv:1801.00862](https://arxiv.org/abs/1801.00862), [doi:10.22331/q-2018-08-06-79](https://doi.org/10.22331/q-2018-08-06-79)]

* Daniel Koch, Brett Martin, Saahil Patel, Laura Wessing, and Paul M. Alsing, *Demonstrating NISQ era challenges in algorithm design on IBM's 20 qubit quantum computer*, AIP Advances 10, 095101 (2020) ([doi:10.1063/5.0015526](https://doi.org/10.1063/5.0015526))

* Frank Leymann, Johanna Barzen, *The bitter truth about gate-based quantum algorithms in the NISQ era*, Quantum Sci. Technol. 5 044007 (2020) ([doi:10.1088/2058-9565/abae7d](https://iopscience.iop.org/article/10.1088/2058-9565/abae7d))

* Olivier Ezratty, *Where are we heading with NISQ?* [[arXiv:2305.09518](https://arxiv.org/abs/2305.09518)]

  > "no one has yet successfully implemented a use case matching the original definition of the NISQ regime."

### Quantum programming via monads
 {#ReferencesInTermsOfMonads}

Discussion of aspects of quantum programming in terms of [[monad (in computer science)|monads]] in [[functional programming]] are in

* [[Thorsten Altenkirch]], Alexander Green, _The quantum IO monad_, in _Semantic Techniques in Quantum Computation_, January 2009, appeared in 2010 ([pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [talk slides](http://www.cs.nott.ac.uk/~txa/talks/qnet06.pdf))

* J. K. Vizzotto, [[Thorsten Altenkirch]], A. Sabry, _Structuring quantum effects: superoperators as arrows_ ([arXiv:quant-ph/0501151](http://arxiv.org/abs/quant-ph/0501151))

### As linear logic
 {#ReferencesAsLinearLogic}

Discussion of quantum computation as the [[internal logic|internal]] [[linear logic]]/[[linear type theory]] of [[compact closed categories]] is in

* {#AbramskyDuncan05} [[Samson Abramsky]], [[Ross Duncan]], _A Categorical Quantum Logic_, Mathematical Structures in Computer Science, 2006 ([arXiv:quant-ph/0512114](http://arxiv.org/abs/quant-ph/0512114))

* {#Duncan06} [[Ross Duncan]], _Types for quantum mechanics_, 2006 ([pdf](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf), [slides](http://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf))

* {#LagoFaffian12} [[Ugo Dal Lago]], [[Claudia Faggian]], _On Multiplicative Linear Logic, Modality and Quantum Circuits_, EPTCS 95, 2012, pp. 55-66 ([arXiv:1210.0613](http://arxiv.org/abs/1210.0613))

An exposition along these lines is in

* [[John Baez]], [[Mike Stay]], _Physics, topology, logic and computation: a rosetta stone_, [arxiv/0903.0340](http://arxiv.org/abs/0903.0340); in "New Structures for Physics", ed. Bob Coecke, Lecture Notes in Physics __813__, Springer, Berlin, 2011, pp. 95-174

[[!include quantum information theory via string diagrams -- references]]

### Topological quantum computing

[[topological quantum computation]] is discussed in

* [[Michael Freedman]], [[Alexei Kitaev]], Michael J. Larsen, Zhenghan Wang, _Topological Quantum Computation_ ([arXiv:quant-ph/0101025](http://arxiv.org/abs/quant-ph/0101025))

* Zhenghan Wang, _Topologization of electron liquids with Chern-Simons theory and quantum computation_ ([arXiv:cond-mat/0601285](http://arxiv.org/abs/cond-mat/0601285))

* {#FreedmanLarsenWang00} [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics **227** (2002) 605–622 [[arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108), [doi:10.1007/s002200200645](https://doi.org/10.1007/s002200200645)]

* [[Alexei Kitaev]], _Fault-tolerant quantum computation by anyons_, Annals Phys. 303 (2003) 2-30 ([arXiv:quant-ph/9707021](http://arxiv.org/abs/quant-ph/9707021))

### Relation to tensor networks

Relation to [[tensor networks]], specifically [[matrix product states]]:

* Yiqing Zhou, E. Miles Stoudenmire, Xavier Waintal, _What limits the simulation of quantum computers?_, [arXiv:2002.07730](https://arxiv.org/abs/2002.07730)

[[!include spin resonance qbits -- references]]

[[!include superconducting qbits -- references]]

### Photonic qbits

Realization in photonics:

* Han-Shen Zhong et al. _Quantum computational advantage using photons_, Science 370, n. 6523 (2020) 1460-1463 [doi](https://doi.org/10.1126/science.abe8770)

### Commercial quantum computers
 {#CommercialQuantumComputers}

Since 2019, NISQ quantum computers can be accessed through cloud services:

* New Scientist, *[IBM unveils its first commercial quantum computer](https://www.newscientist.com/article/2189909-ibm-unveils-its-first-commercial-quantum-computer/)* (Jan 2019)

* [IBM Quantum](https://quantum-computing.ibm.com/) (Cloud service)

  [IBM Quantum compute resources](https://quantum-computing.ibm.com/lab/docs/iql/manage/systems/)

  [Build and deploy quantum programs with Qiskit Runtime](https://cloud.ibm.com/quantum)

There exist toy desktop quantum computers for educational purposes, operating on a couple of [[nuclear magnetic resonance]] [[qbits]] at room temperature :

* [CIQTEK](https://en.ciqtek.com/): *[Diamond Quantum Computer for Education](https://en.ciqtek.com/products/detail/diamond-quantum-computer-for-education.html?id=644102ee-942e-4568-aacd-20241a1a2469&id=644102ee-942e-4568-aacd-20241a1a2469)*

* [SpinQ](https://www.spinquanta.com/): *SpinQ Triangulum: a commercial three-qubit desktop quantum computer* [[arXiv:2202.02983](https://arxiv.org/abs/2202.02983)]

category: computer science, physics

[[!redirects quantum computing]]
[[!redirects quantum computations]]

[[!redirects quantum computer]]
[[!redirects quantum computers]]

[[!redirects classically controlled quantum computation]]
[[!redirects parameterized quantum computation]]

[[!redirects NISQ]]
[[!redirects noisy intermediate-scale quantum computer]]
[[!redirects noisy intermediate-scale quantum computer]]
[[!redirects noisy intermediate-scale quantum computing]]


quantum contextuality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In *[[quantum physics]]* and in particular in the context of [[hidden variable theories]] (cf. [[interpretation of quantum mechanics]]) the term (quantum) *contextuality* refers to the (assumed) dependence of the value of [[quantum observables]] not *just* on (hypothetical) [[hidden variable theory|hidden variables]] but also on a specified "context" of other compatible (commuting) observables.

The notion originates in the discussion of the [[Bell-Kochen-Specker theorem]] which *rules out* the assumption that [[quantum observables]] all at once jointly dependent (only) on "hidden variables". However, for any chosen *commutative* subalgebra of observables the assumption is valid (cf. *[[Bohr topos]]*), whence one may consider the idea that observables are defined by hidden variables *and* the specification of such "contexts", hence that quantum observables are "contextual".

## Related concepts

* [[Kochen-Specker theorem]]

## References

Textbook accoutn:

* {#Moretti16} [[Valter Moretti]], Chapter 5 of: *Mathematical Foundations of Quantum Mechanics: An Advanced Short Course*, Int. J. Geom. Methods Mod. Phys. **13** Supp. 1 (2016) 1630011 [[arXiv:1508.06951](http://arxiv.org/abs/1508.06951), [doi:10.1142/S0219887816300117](https://doi.org/10.1142/S0219887816300117)]

See also:

* Wikipedia, *[Quantum contextuality](https://en.wikipedia.org/wiki/Quantum_contextuality)*

[[sheaf theory|Sheaf-theoretic]] aspects:

* {#AbramskyBrandenburger11} [[Samson Abramsky]], [[Adam Brandenburger]], _The Sheaf-Theoretic Structure Of Non-Locality and Contextuality_, New Journal of Physics **13** (2011) 113036 [[arXiv:1102.0264](https://arxiv.org/abs/1102.0264), [doi:10.1088/1367-2630/13/11/113036](https://doi.org/10.1088/1367-2630/13/11/113036)]

Cohomological aspects:

* [[Samson Abramsky]], Shane Mansfield, Rui Soares Barbosa, *The Cohomology of Non-Locality and Contextuality* [[arXiv:1111.3620](https://arxiv.org/abs/1111.3620)]

and via ([[singular cohomology]] of) [[simplicial sets]]:

* {#OkayRobertsBartlettRaussendorf17} [[Cihan Okay]], [[Sam Roberts]], [[Stephen D. Bartlett]], [[Robert Raussendorf]], *Topological proofs of contextuality in quantum mechanics*, Quantum Information and Computation **17** (2017) 1135-1166 [[arXiv:1701.01888](https://arxiv.org/abs/1701.01888), [doi:10.26421/QIC17.13-14-5](https://doi.org/10.26421/QIC17.13-14-5)]

* {#OkayKharoofIpek22} [[Cihan Okay]], Aziz Kharoof, Selman Ipek, *Simplicial quantum contextuality*, Quantum **7** (2023) 1009 [[arXiv:2204.06648](https://arxiv.org/abs/2204.06648), [doi:10.22331/q-2023-05-22-1009](https://doi.org/10.22331/q-2023-05-22-1009)]

* Aziz Kharoof, [[Cihan Okay]], *Homotopical characterization of strongly contextual simplicial distributions on cone spaces* [[arXiv:2311.14111](https://arxiv.org/abs/2311.14111)]

exposition:

* [[Cihan Okay]], *Simplicial Distributions and Contextuality*, [talk at](CQTS#OkayOct2023) [[CQTS]] (11 Oct 2023) [video:[YT](https://youtu.be/pgKYGUWl1kU)]

Probabilistic aspects:

* Ehtibar Dzhafarov, Janne Kujala, _Probabilistic Foundations of Contextuality_ [[arXiv:1604.08412](https://arxiv.org/abs/1604.08412)]