Case Study - an algebra of qubit strings

Construction

Let $H$ be a 2-dimensional complex vector space, we choose some basis that we label as $\{\uparrow ,\downarrow \}$.

Consider the algebra

$$\mathbb{C}⟨\uparrow ,\downarrow ⟩:=Fre{e}_{\mathbb{C}}^{(1)}(H)$$

the free associative algebra (tensor algebra) on two elements $\{\uparrow ,\downarrow \}$, in $Vec{t}_{\mathbb{C}}$.

Explicitly, the underlying vector space is

$$\underline{\mathbb{C}⟨\uparrow ,\downarrow ⟩}\simeq \mathbb{C}\oplus H\oplus H^{\otimes 2}\oplus H^{\otimes 3}\oplus \cdots \simeq \underset{n}{⨁}{H}^{\otimes n}$$

A basis is indexed by

$$\left\{\vec{a}\text{, finite length bit string in {uparrow, downarrow}}\right\}$$

The multiplicative structure in the algebra give maps ${\mu }_n:{\underline{\mathbb{C}⟨\uparrow ,\downarrow ⟩}}^{\otimes n}\to \underline{\mathbb{C}⟨\uparrow ,\downarrow ⟩}$ act on the basis by concatenating bit strings. The multiplicative structure encodes coherences of such concactenations (although thus far we’re in ordinary (non-homotopical) algebra so associativity is a condition).

Example

An example of an element in $\mathbb{C}⟨\uparrow ,\downarrow ⟩$ is

$$(1+2i)|\downarrow ⟩+|\uparrow \uparrow ⟩+(e^{3/4\pi i})|\uparrow \uparrow \downarrow ⟩+(e^{1/4\pi i})|\downarrow \uparrow \downarrow ⟩+|\uparrow \downarrow \downarrow \uparrow \downarrow \downarrow ⟩$$

we see that $\underline{\mathbb{C}⟨\uparrow ,\downarrow ⟩}$ is the vector space encoding non-normalized superpositions of many different qubit strings of different lengths.

The algebra structure acts on such a superposition by producing a new superposition encoding all possible concactentations of the input superposition of bit strings, in a $\mathbb{C}$-weighted way.

For example, two superpositions

$$|\varphi ⟩:=({\varphi }_0|\overrightarrow{a_0}⟩+{\varphi }_1|\overrightarrow{a_1}⟩),|\psi ⟩:=({\psi }_0|\overrightarrow{b_0}⟩+{\psi }_1|\overrightarrow{b_1}⟩)$$

multiply to

${\varphi }_0{\psi }_0|\overrightarrow{a_0{b}_0}⟩+{\varphi }_0{\psi }_1|\overrightarrow{a_0{b}_1}⟩+{\varphi }_1{\psi }_0|\overrightarrow{a_1{b}_0}⟩+{\varphi }_1{\psi }_1|\overrightarrow{a_1{b}_1}⟩$ Hence, we can think of $\mathbb{C}⟨\uparrow ,\downarrow ⟩$ as an object encoding all states of a quantum computer, where the computer itself is thought of as a quantum object - that is, allowed to be in superposition.

Remark The way we’ll handle normalization is to remember an action by the stable unitary group $U$.

Dual interpretation Now, there’s a dual interpretation of this data. An associative algebra is a discrete $E_1$ algebra. An $E_1$-algebra is a locally-constant factorization algebra on the 1-dimensional line. Let’s talk about this in very concrete physical terms.

Imagine a quantum system on an open interval (open here just means we’re not talking about boundary conditions, yet). By a quantum system, we means something like the continuum limit of a 1-dimensional spin chain.

From this perspective, the data of an $E_1$-multiplicative structure is instruction on how to build a state $|\varphi ⟩$ out of a length $n$-sequence of states $|{\varphi }_i{⟩}_{1\le i\le n}$,

$|\varphi ⟩\simeq \mu (|{\varphi }_1⟩,|{\varphi }_2⟩,|{\varphi }_3⟩,\cdots )$ In the explicit example that $|\varphi ⟩\in \mathbb{C}⟨\uparrow ,\downarrow ⟩$, the multiplicative structure provide instructions like

$$|010⟩+|011⟩\simeq \mu (|01⟩,|0⟩+|1⟩)$$

This gives us a natural setting to measure complexity of quantum states.

Some notation We’ve so far introduced two ways of thinking about the algebra $(\mathbb{C}⟨\uparrow ,\downarrow ⟩,{\mu }^{\bullet })$

1. As (states of a quantum computer, concatenation).
   1. We can think of this as “bottom-up”: how to assemble states
2. As (states of a 1-dimensional material, forming tensor product states).
   1. We can think of this as “top-down”: how to decompose states.