# Categories for Quantum#

Alexis Toumi & Giovanni de Felice

*5th May 2021, Tallcat*

Categorical quantum mechanics

the miracle of scalars

the way of the dagger

braids and symmetry

compact closed categories

special commutative Frobenius algebras

bialgebras & Hopf algebras

Qubit quantum computing

the circuit model

the ZX-calculus

Applications

optimisation

compilation

extraction

natural language processing

## Categorical quantum mechanics#

The traditional formulation of quantum theory happens in \(\mathbf{Hilb}\), the monoidal category of Hilbert spaces and bounded linear maps with the tensor product.

Categorical quantum mechanics (CQM) seeks to make explicit precisely what structure is needed of \(\mathbf{Hilb}\) in order to recover quantum theory.

**References:**

*A categorical semantics of quantum protocols*, Abramsky & Coecke (arXiv:0402130)*Categorical quantum mechanics*, Abramsky & Coecke (arXiv:0808.1023)*Picturing quantum processes: a first course in quantum theory and diagrammatic reasoning*, Coecke & Kissinger (2017)*Categories for quantum theory: an introduction*, Heunen & Vicary (2019)

In the process, we get a diagrammatic language for quantum processes.

We also get a new intuition for monoidal categories:

**Systems**are objects \(A, B\), composition of systems is given by their tensor product \(A \otimes B\).**Processes**are arrows \(f : A \to B\), \(g : B \to C\), with their sequential and parallel compositions \(g \circ f\) and \(f \otimes g\).**States**are arrows from the unit \(v : 1 \to A\).**Effects**are arrows into the unit \(e : A \to 1\).**Scalars**are endomorphisms of the unit \(s : 1 \to 1\).

### The miracle of scalars#

**Lemma:** Tensoring with a scalar is a natural transformation.

*Proof:*

```
[9]:
```

```
from discopy.monoidal import Ty, Box
from discopy.drawing import Equation
x, y, z = Ty(*"xyz")
f, g = Box('f', x, y), Box('g', y, z)
a, b, c, d = (Box(x, Ty(), Ty()) for x in 'abcd')
assert a @ x >> f == (f >> a @ y).normal_form()
Equation(a @ x >> f, f >> a @ y).draw(figsize=(5, 2))
```

**Lemma:** In a monoidal category, scalars form a commutative monoid.

*Proof:* Eckmann-Hilton argument.

```
[10]:
```

```
from discopy import drawing
assert (a @ b).interchange(0, 1) == b @ a
assert (a @ b).interchange(0, 1).interchange(0, 1) == a @ b
drawing.to_gif(a @ b, b @ a, loop=True, figsize=(.5, 1), path='../_static/monoidal/Eckmann-Hilton.gif')
```

```
[10]:
```

```
[10]:
```

**Definition:** A monoidal category is enriched in commutative monoids (com.mon. enriched) if every homset is a commutative monoid such that composition and tensor are monoid homomorphisms.

```
[11]:
```

```
from discopy.monoidal import Sum
zero = Sum((), Ty(), Ty())
assert zero + a == Sum((a, )) == a + zero
assert (a + b) + c == a + b + c == a + (b + c)
assert (a + b) @ (c + d) == (a @ c) + (a @ d) + (b @ c) + (b @ d)
assert (a + b) >> (c + d) == (a >> c) + (a >> d) + (b >> c) + (b >> d)
```

**Lemma:** In a com.mon. enriched monoidal category, the scalars form a commutative semiring.

**Definition:** A commutative semiring is a com.mon. enriched monoidal category with one object.

**Lemma:** Given a commutative semiring \(\mathbb{S}\), the category \(\mathbf{Mat}_\mathbb{S}\) of \(\mathbb{S}\)-valued matrices is monoidal with the Kronecker product.

**Lemma:** \(\mathbf{FHilb} \simeq \mathbf{Mat}_\mathbb{C}\).

**Lemma:** \(\mathbf{FRel} \simeq \mathbf{Mat}_\mathbb{B}\).

Com.mon. enrichment allows us to talk about **superposition**, **basis** and **dimension** in an abstract setting.

**Definition:** Given two states \(v, w : 1 \to A\), their superposition is a linear combination \(a \otimes v + b \otimes w : 1 \to A\) for two scalars \(a, b : 1 \to 1\).

**Definition:** A basis for \(A\) is a minimal set of states \(\{\vert i \rangle : 1 \to A\}\) such that every state \(w : 1 \to A\) is a linear combination \(w = \sum_i a_i \otimes \vert i \rangle\) for some scalars \(a_i : 1 \to 1\).

**Definition:** The dimension of a system is the cardinality of its smallest basis.

### The way of the dagger#

**Definition:** A dagger is a contravariant involutive identity-on-objects monoidal functor.

In terms of diagrams, the dagger is the vertical reflexion.

We may draw asymmetric boxes to distinguish a generator from its dagger.

```
[12]:
```

```
Equation(f, f[::-1], symbol="$\mapsto$").draw(asymmetry=.25, figsize=(3, 1))
```

```
[13]:
```

```
from discopy.monoidal import Id
assert Id(x)[::-1] == Id(x)
assert (f >> g)[::-1] == g[::-1] >> f[::-1]
assert (f @ g)[::-1].normal_form() == f[::-1] @ g[::-1]
Equation((f @ g)[::-1], f[::-1] @ g[::-1]).draw(figsize=(5, 2))
```

The dagger allows us to formulate the notion of inner product in an abstract setting.

**Definition:** The inner product of two states \(w, w' : 1 \to A\) is the scalar \(w^\dagger \circ w' : 1 \to 1\).

**Definition:** The squared norm of a state \(w : 1 \to A\) is the scalar \(w^\dagger \circ w\). It is normalised when it has norm \(1\), the identity of the monoidal unit.

**Definition:** Two states \(w, w' : 1 \to A\) are orthogonal when \(w^\dagger \circ w' = 0\).

**Definition:** An isometry is a norm-preserving process \(f : A \to B\), i.e. \(f^\dagger \circ f = \text{id}_A\).

**Definition:** A unitary is an isomorphism \(f : A \to B\) with inverse \(f^\dagger\), i.e. \(f^\dagger \circ f = \text{id}_A\) and \(f \circ f^\dagger = \text{id}_B\).

**Definition:** A basis is orthonormal if \(\langle i | j \rangle = \delta_{ij}\).

It also allows us to talk about quantum measurements, the Born rule and even a simple form of Heisenbergâ€™s uncertainty principle!

**Definition:** A projector is a process \(p : A \to A\) such that \(p = p^\dagger\) and \(p \circ p = \text{id}_A\).

**Example:** Take \(w \circ w^\dagger : A \to A\) for any normalised state \(w : 1 \to A\).

**Definition:** A (projection-valued) measurement is a set of projectors \(\{p_i : A \to A\}\) such that \(\sum p_i = \text{id}_A\).

**Example:** Take an orthonormal basis \(\{\vert i \rangle\}\) then \(\{\vert i \rangle \langle i \vert\}\) is a measurement.

**Lemma:** A measurement is a choice of basis and a partition of it.

**Definition (Born rule):** Given a normalised state \(w : 1 \to A\) and a measurement \(\{p_i : A \to A\}\), the probability of measuring outcome \(i\) and getting the state \(p_i(w)\) as a result is \(P(i \vert w) = w^\dagger \circ p_i \circ w\).

**Example:** Take \(p_i = v \circ v^\dagger\) then we get the squared amplitude \(P(v \vert w) = \langle w \vert v \rangle^2\).

**Definition:** Two bases are complementary (a.k.a. mutually unbiased) if \(P(i \vert j) = P(i' \vert j')\) for all \(i, i'\) and \(j, j'\).

**Example:** Take \(\{\vert 0 \rangle, \vert 1 \rangle\}\) and \(\{\vert + \rangle, \vert - \rangle\}\) the two bases of \(\mathbb{C}^2\) with:

.

### Braids and symmetry#

**Definition:** A (strict) monoidal category is braided if for every objects \(A, B\) we have swaps:

such that

**Definition:** A braided monoidal category is symmetric if

or equivalently

**Lemma:** In a braided monoidal category, we can prove the Yang-Baxter equation. *Proof:* Naturality + Hexagon.

### Compact closed categories#

**Recall:** In a right-rigid monoidal category, for every \(L\) there is a right adjoint \(R\) and a pair of arrows:

such that

**Lemma:** In a braided monoidal category, left and right adjoints collapse.

*Proof:* Define

Then from naturality we get:

**Definition:** A compact closed category is a rigid symmetric monoidal category.

**Lemma:** In a compact closed category, the following equations hold:

**Definition:** The transpose of a process is given by:

**Definition:** A \(\dagger\) compact closed category is a compact closed category with a dagger, such that swaps are unitaries and

In a \(\dagger\) compact closed category, we get a \(\mathbb{Z}_2 \times \mathbb{Z}_2\) action on processes:

Compact closed categories allow to formulate the notion of **entanglement** in an abstract setting.

**Definition:** A state \(w : 1 \to A \otimes B\) is separable when it can be written as a tensor \(w = v \otimes v'\) for some \(v : 1 \to A\) and \(v' 1 \to B\).

**Definition:** A state \(w : 1 \to A \otimes B\) is entangled if it is not separable.

**Lemma:** The cup of a non-trivial system is entangled.

*Proof:* Suppose the cup is separable, then the identity factors through the unit.

**Lemma (Teleportation):**

*Proof:*

We can also formulate an abstract version of the **no-deleting** and **no-cloning** theorem.

**Definition:** Deleting is a natural transformation \(e_A : A \to 1\) with \(e_1 = 1\).

**Lemma:** A monoidal category has deleting if and only if \(1\) is terminal.

**Theorem:** If a compact closed category has deleting, then it is a preorder.

*Proof:* Take two parallel arrows \(f, g : A \to B\), then their coname \(\hat f, \hat g : A \otimes B^\star \to 1\) are equal, hence \(f = g\).

**Definition:** Copying is an commutative, associative natural transformation \(d_A : A \to A \otimes A\) such that:

**Lemma:** If a braided monoidal category has copying, then:

*Proof:*

**Theorem:** If a compact closed category has copying, then:

*Proof:*

### Special commutative Frobenius algebras#

**Definition:** A Frobenius algebra is a pair of a monoid and comonoid such that:

**Definition:** A Frobenius algebra is dagger when the dagger of multiplication is comultiplication, and the dagger of unit is counit.

**Definition:** A \(\dagger\) Frobenius algebra is commutative when its monoid is commutative.

**Definition:** A \(\dagger\) Frobenius algebra is special when

**Lemma:** Frobenius algebras are self-adjoint.

*Proof:* Define cups and caps

then

**Lemma:** Homomorphisms of Frobenius algebras are isomorphisms, with inverse given by

*Proof:* Suppose \(f\) is a homomorphism of Frobenius algebras.

**Theorem (Spider Fusion):** Every connected diagram \(d : A^{\otimes n} \to A^{\otimes m}\) generated by the co/monoid of a Frobenius algebra is equal to a spider-shaped diagram

*Proof:* By induction, we move all the black comonoid below the white monoid.

**Theorem (Artinâ€“Wedderburn):** in \(\mathbf{FHilb}\), \(\dagger\) special commutative Frobenius algebras are in one-to-one correspondance with orthonormal bases.

*Proof:* \(C^\star\) algebra magic.

The correspondance sends a \(\dagger\) SCFA to the basis of copyable states:

It sends a basis to the Frobenius algebra given by:

Frobenius algebras not only characterise bases in \(\mathbf{Hilb}\), they also allow us to formulate an abstract notion of **phases**.

**Definition:** Given a \(\dagger\)SCFA, a phase is a state \(a : 1 \to A\) such that:

It defines a phase shift map, a.k.a. diagonal unitary matrices:

**Lemma:** Phases form a group.