# Notes 2026-08-18

## (Quantum 1) Quantum computing modalities, physical systems, and encodings

### General picture

It is useful to distinguish several levels:

$$
\text{physical platform}
\rightarrow
\text{physical system/device}
\rightarrow
\text{physical Hilbert space}
\rightarrow
\text{encoding}
\rightarrow
\text{logical qubit}.
$$

A logical qubit always has Hilbert space

$$
\mathcal H_L \cong \mathbb C^2.
$$

An encoding specifies how this abstract two-dimensional system is represented inside a physical Hilbert space:

$$
V:\mathbb C^2\hookrightarrow\mathcal H_{\mathrm{phys}}.
$$

The corresponding code/computational subspace is

$$
\mathcal C
=
V(\mathbb C^2)
=
\operatorname{span}\{|0_L\rangle,|1_L\rangle\}.
$$

Thus:

- **basis** — coordinates used to describe a physical Hilbert space;
- **encoding** — map specifying which physical states represent logical states;
- **code space** — image of that encoding;
- **logical qubit** — abstract two-dimensional quantum system represented by the code space.

---

### Physical platforms

Major physical approaches include:

- superconducting circuits;
  - transmons;
  - fluxonium;
  - microwave resonators;
- trapped ions;
- neutral atoms / Rydberg atoms;
- photonic systems;
- semiconductor spin qubits;
- defect centers such as NV centers;
- topological systems.

These describe primarily **what physical quantum system is being controlled**, rather than how logical information is encoded in it.

"Platform" and "device" are not rigidly standardized terms. Roughly, a platform is a technological family, while a device is a particular realization within that family:

$$
\text{superconducting circuits}
\rightarrow
\text{transmon}.
$$

---

## Transmon qubit

The transmon Hamiltonian is

$$
H
=
4E_C(\hat n-n_g)^2
-
E_J\cos\hat\phi.
$$

It can first be represented in the charge basis

$$
\{|n\rangle\}_{n\in\mathbb Z}.
$$

Diagonalizing the Hamiltonian gives

$$
H|E_j\rangle=E_j|E_j\rangle,
$$

where

$$
|E_j\rangle
=
\sum_n c_n^{(j)}|n\rangle.
$$

Thus

$$
|n\rangle
\quad\longrightarrow\quad
|E_j\rangle
$$

is merely a **change of basis**.

The ordinary transmon encoding then chooses

$$
|0_L\rangle=|E_0\rangle,
\qquad
|1_L\rangle=|E_1\rangle.
$$

Hence

$$
\mathcal C_{\mathrm{transmon}}
=
\operatorname{span}\{|E_0\rangle,|E_1\rangle\}.
$$

States such as

$$
|E_2\rangle,|E_3\rangle,\ldots
$$

are outside the computational subspace. Population entering them constitutes **leakage**.

The distinction is therefore

$$
\boxed{\text{diagonalization}=\text{change of basis}}
$$

versus

$$
\boxed{\text{encoding}=\text{choice of physical representation of logical information}}.
$$

---

## Bosonic systems

A bosonic mode has Fock space

$$
\mathcal F
=
\operatorname{span}
\{|0\rangle,|1\rangle,|2\rangle,\ldots\}
$$

with ladder operators

$$
[a,a^\dagger]=1.
$$

For a harmonic oscillator,

$$
H
=
\hbar\omega
\left(
a^\dagger a+\frac12
\right).
$$

Electromagnetic field modes are bosonic modes, so this description applies both to optical photons and microwave cavity photons.

A **bosonic code** uses the larger oscillator Hilbert space to encode a smaller logical system:

$$
\mathbb C^2\hookrightarrow\mathcal F.
$$

Important bosonic codes include:

- cat codes;
- GKP codes;
- binomial codes.

These are alternative ways of encoding logical information into bosonic oscillator states.

---

## Cat code

A coherent state satisfies

$$
a|\alpha\rangle=\alpha|\alpha\rangle
$$

and has Fock expansion

$$
|\alpha\rangle
=
e^{-|\alpha|^2/2}
\sum_{n=0}^{\infty}
\frac{\alpha^n}{\sqrt{n!}}|n\rangle.
$$

Cat states are superpositions such as

$$
|C_\alpha^\pm\rangle
=
\mathcal N_\pm
\left(
|\alpha\rangle\pm|-\alpha\rangle
\right).
$$

For example, one possible logical basis is

$$
|0_L\rangle=|C_\alpha^+\rangle,
\qquad
|1_L\rangle=|C_\alpha^-\rangle.
$$

Thus a logical state is encoded across many oscillator occupation numbers.

---

## GKP code

The GKP code uses oscillator quadratures

$$
\hat q
=
\frac{a+a^\dagger}{\sqrt2},
\qquad
\hat p
=
\frac{a-a^\dagger}{i\sqrt2}.
$$

Logical states correspond ideally to periodic grid-like states in oscillator phase space.

Thus GKP is another encoding

$$
\mathbb C^2\hookrightarrow\mathcal F,
$$

but with a very different code-space geometry from cat states.

---

## Binomial code

Binomial codes construct logical states from selected finite superpositions of Fock states.

Schematically, an example can look like

$$
|0_L\rangle
=
\frac{|0\rangle+|4\rangle}{\sqrt2},
\qquad
|1_L\rangle
=
|2\rangle.
$$

Thus

$$
\boxed{
\text{cat},\quad
\text{GKP},\quad
\text{binomial}
}
$$

are alternative **bosonic codes/encodings**.

---

## Photonic quantum computing

"Photonic" primarily specifies a **physical platform**: optical electromagnetic modes are used as the physical quantum system.

It does not uniquely determine the encoding.

Possible photonic encodings include:

- dual rail;
- polarization;
- time bin;
- GKP;
- other bosonic encodings.

Therefore

$$
\boxed{\text{photonic}=\text{physical realization}}
$$

while

$$
\boxed{\text{dual rail/GKP/etc.}=\text{encoding}}.
$$

A photonic system can therefore implement a bosonic code, but photonic quantum computing is not synonymous with bosonic coding.

---

## Dual-rail encoding

Dual rail encodes one logical qubit into the **single-excitation subspace of two distinguishable modes or subsystems**.

For two modes $a,b$,

$$
\mathcal H
=
\mathcal F_a\otimes\mathcal F_b.
$$

The encoding is

$$
|0_L\rangle=|1,0\rangle,
\qquad
|1_L\rangle=|0,1\rangle.
$$

Hence

$$
|\psi_L\rangle
=
\alpha|1,0\rangle+\beta|0,1\rangle.
$$

Equivalently,

$$
|\psi_L\rangle
=
(\alpha a^\dagger+\beta b^\dagger)|0,0\rangle.
$$

The essential definition is therefore

$$
\boxed{
\text{dual rail}
=
\text{one excitation coherently shared between two distinguishable rails}.
}
$$

Dual rail does **not** specifically require optical photons.

For two transmons $A,B$, for example,

$$
|0_L\rangle
=
|E_1\rangle_A|E_0\rangle_B,
$$

$$
|1_L\rangle
=
|E_0\rangle_A|E_1\rangle_B.
$$

Using the usual shorthand,

$$
|0_L\rangle=|10\rangle,
\qquad
|1_L\rangle=|01\rangle.
$$

---

## Schwinger-boson representation

Two bosonic modes naturally realize the algebra $\mathfrak{su}(2)$.

Define

$$
J_+
=
a^\dagger b,
\qquad
J_-
=
b^\dagger a,
$$

or equivalently

$$
J_x
=
\frac12(a^\dagger b+b^\dagger a),
$$

$$
J_y
=
\frac{1}{2i}(a^\dagger b-b^\dagger a),
$$

$$
J_z
=
\frac12(a^\dagger a-b^\dagger b).
$$

They satisfy

$$
[J_i,J_j]
=
i\epsilon_{ijk}J_k.
$$

The total occupation

$$
N=a^\dagger a+b^\dagger b
$$

is conserved by these operators.

For fixed $N$, the corresponding subspace has dimension

$$
N+1
$$

and realizes the spin

$$
j=\frac N2
$$

irreducible representation of $SU(2)$.

In particular,

$$
N=1
\quad\Longrightarrow\quad
j=\frac12.
$$

The states

$$
|1,0\rangle,\qquad|0,1\rangle
$$

therefore carry the fundamental two-dimensional representation of $SU(2)$.

This explains mathematically why dual rail naturally behaves as a qubit.

---

## Dual rail versus cat

Both can start from bosonic Hilbert spaces, but they use them very differently.

| | Dual rail | Cat |
|---|---|---|
| Typical number of modes | 2 | 1 |
| Logical basis | $|10\rangle,|01\rangle$ | coherent-state superpositions |
| Occupation | fixed total $N=1$ | distributed across many $n$ |
| Main idea | which rail contains excitation | structured states within oscillator |
| Uses large Fock space | minimally | deliberately |

Thus, schematically,

$$
\boxed{
\text{dual rail}
=
\text{constrain occupation across multiple modes}
}
$$

whereas

$$
\boxed{
\text{cat}
=
\text{construct structured superpositions within a mode}.
}
$$

---

## Dual rail and leakage

For two ideal two-level systems,

$$
\mathcal H_{\mathrm{phys}}
=
\operatorname{span}
\{
|00\rangle,|01\rangle,|10\rangle,|11\rangle
\}.
$$

Dual rail selects only

$$
\mathcal C
=
\operatorname{span}
\{|10\rangle,|01\rangle\}.
$$

Therefore

$$
|00\rangle,\ |11\rangle
$$

are outside the logical code space.

This means encoding creates additional opportunities for leakage, but that additional Hilbert-space structure can also make errors detectable.

For an ordinary transmon,

$$
|0_L\rangle=|0\rangle,
\qquad
|1_L\rangle=|1\rangle,
$$

and relaxation gives

$$
|1_L\rangle\rightarrow|0_L\rangle.
$$

The error has mapped one valid logical state onto another.

For dual rail,

$$
|0_L\rangle=|10\rangle,
\qquad
|1_L\rangle=|01\rangle,
$$

and single-excitation loss gives

$$
|10\rangle\rightarrow|00\rangle,
$$

or

$$
|01\rangle\rightarrow|00\rangle.
$$

Since

$$
|00\rangle\notin\mathcal C,
$$

the loss can be recognized as leaving the code space.

However, the original amplitudes in

$$
\alpha|10\rangle+\beta|01\rangle
$$

are lost after complete excitation loss. Dual rail therefore does not by itself correct this error; it can turn it into a **detectable erasure/leakage event**.

---

## Dual rail versus repetition code

Dual rail:

$$
|0_L\rangle=|10\rangle,
\qquad
|1_L\rangle=|01\rangle.
$$

Two-qubit repetition encoding:

$$
|0_L\rangle=|00\rangle,
\qquad
|1_L\rangle=|11\rangle.
$$

Three-qubit bit-flip repetition code:

$$
|0_L\rangle=|000\rangle,
\qquad
|1_L\rangle=|111\rangle.
$$

These should not be confused.

The three-qubit repetition code is designed to correct one bit-flip error:

$$
X_i.
$$

Dual rail instead has useful properties against excitation-loss errors because valid states have fixed total excitation number

$$
N=1.
$$

Thus they solve different problems:

$$
\boxed{
\text{dual rail}
\rightarrow
\text{encoding with useful error-detection/erasure properties}
}
$$

while

$$
\boxed{
\text{3-qubit repetition}
\rightarrow
\text{error-correcting code for bit flips}.
}
$$

---

## Why use a larger Hilbert space?

Encoding often increases the physical Hilbert-space dimension.

For example,

$$
\dim\mathcal C=2
$$

for one logical qubit, while two physical two-level systems provide

$$
\dim\mathcal H_{\mathrm{phys}}=4.
$$

This creates more states into which the system can leak.

However, the unused dimensions also provide somewhere for physical errors to go **without becoming another valid logical state**.

This is one of the fundamental ideas behind quantum error detection and correction:

$$
\boxed{
\mathbb C^2
\hookrightarrow
\mathcal H_{\mathrm{phys}},
\qquad
\dim\mathcal H_{\mathrm{phys}}>2.
}
$$

The additional degrees of freedom provide redundancy that can allow physical errors to become detectable syndromes, leakage events, or correctable transformations.

---

## Overall taxonomy

The most useful classification separates **hardware** from **encoding**:

$$
\boxed{
\underbrace{\text{What physical system do I have?}}_{\text{platform/device}}
\quad\times\quad
\underbrace{\text{How do I represent }\mathbb C^2\text{ in it?}}_{\text{encoding/code}}
}
$$

Examples:

| Physical system | Possible encoding |
|---|---|
| Transmon | $|E_0\rangle,|E_1\rangle$ |
| Two transmons | dual rail |
| Two optical modes | dual rail |
| Optical polarization | polarization encoding |
| Optical temporal modes | time-bin encoding |
| Microwave cavity | cat, GKP, binomial |
| Optical bosonic mode | GKP or other bosonic code |

Encodings can themselves subsequently become the physical building blocks of higher-level error-correcting codes:

$$
\text{physical oscillator}
\rightarrow
\text{bosonic encoded qubit}
\rightarrow
\text{higher-level QEC code}
\rightarrow
\text{fault-tolerant logical qubit}.
$$

The overarching idea is

$$
\boxed{
\text{find a controllable and robust representation of }
\mathbb C^2
\text{ inside a physical Hilbert space}.
}
$$

Different quantum-computing modalities and encodings are different solutions to this same physical and information-theoretic problem.