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notes:2026-08-18 [August 18, 2026 at 11:51] – created Ivan Janevskinotes:2026-08-18 [August 18, 2026 at 11:53] (current) – Ivan Janevski
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-# Quantum computing modalities, physical systems, and encodings+# Notes 2026-08-18
  
-## General picture+## (Quantum 1) Quantum computing modalities, physical systems, and encodings 
 + 
 +### General picture
  
 It is useful to distinguish several levels: It is useful to distinguish several levels:
  
-\[+$$
 \text{physical platform} \text{physical platform}
 \rightarrow \rightarrow
Line 15: Line 17:
 \rightarrow \rightarrow
 \text{logical qubit}. \text{logical qubit}.
-\]+$$
  
 A logical qubit always has Hilbert space A logical qubit always has Hilbert space
  
-\[+$$
 \mathcal H_L \cong \mathbb C^2. \mathcal H_L \cong \mathbb C^2.
-\]+$$
  
 An encoding specifies how this abstract two-dimensional system is represented inside a physical Hilbert space: 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}}. V:\mathbb C^2\hookrightarrow\mathcal H_{\mathrm{phys}}.
-\]+$$
  
 The corresponding code/computational subspace is The corresponding code/computational subspace is
  
-\[+$$
 \mathcal C \mathcal C
 = =
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 = =
 \operatorname{span}\{|0_L\rangle,|1_L\rangle\}. \operatorname{span}\{|0_L\rangle,|1_L\rangle\}.
-\]+$$
  
 Thus: Thus:
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 --- ---
  
-## Physical platforms+### Physical platforms
  
 Major physical approaches include: Major physical approaches include:
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 "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: "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} \text{superconducting circuits}
 \rightarrow \rightarrow
 \text{transmon}. \text{transmon}.
-\]+$$
  
 --- ---
  
-# Transmon qubit+## Transmon qubit
  
 The transmon Hamiltonian is The transmon Hamiltonian is
  
-\[+$$
 H H
 = =
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 - -
 E_J\cos\hat\phi. E_J\cos\hat\phi.
-\]+$$
  
 It can first be represented in the charge basis It can first be represented in the charge basis
  
-\[+$$
 \{|n\rangle\}_{n\in\mathbb Z}. \{|n\rangle\}_{n\in\mathbb Z}.
-\]+$$
  
 Diagonalizing the Hamiltonian gives Diagonalizing the Hamiltonian gives
  
-\[+$$
 H|E_j\rangle=E_j|E_j\rangle, H|E_j\rangle=E_j|E_j\rangle,
-\]+$$
  
 where where
  
-\[+$$
 |E_j\rangle |E_j\rangle
 = =
 \sum_n c_n^{(j)}|n\rangle. \sum_n c_n^{(j)}|n\rangle.
-\]+$$
  
 Thus Thus
  
-\[+$$
 |n\rangle |n\rangle
 \quad\longrightarrow\quad \quad\longrightarrow\quad
 |E_j\rangle |E_j\rangle
-\]+$$
  
 is merely a **change of basis**. is merely a **change of basis**.
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 The ordinary transmon encoding then chooses The ordinary transmon encoding then chooses
  
-\[+$$
 |0_L\rangle=|E_0\rangle, |0_L\rangle=|E_0\rangle,
 \qquad \qquad
 |1_L\rangle=|E_1\rangle. |1_L\rangle=|E_1\rangle.
-\]+$$
  
 Hence Hence
  
-\[+$$
 \mathcal C_{\mathrm{transmon}} \mathcal C_{\mathrm{transmon}}
 = =
 \operatorname{span}\{|E_0\rangle,|E_1\rangle\}. \operatorname{span}\{|E_0\rangle,|E_1\rangle\}.
-\]+$$
  
 States such as States such as
  
-\[+$$
 |E_2\rangle,|E_3\rangle,\ldots |E_2\rangle,|E_3\rangle,\ldots
-\]+$$
  
 are outside the computational subspace. Population entering them constitutes **leakage**. are outside the computational subspace. Population entering them constitutes **leakage**.
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 The distinction is therefore The distinction is therefore
  
-\[+$$
 \boxed{\text{diagonalization}=\text{change of basis}} \boxed{\text{diagonalization}=\text{change of basis}}
-\]+$$
  
 versus versus
  
-\[+$$
 \boxed{\text{encoding}=\text{choice of physical representation of logical information}}. \boxed{\text{encoding}=\text{choice of physical representation of logical information}}.
-\]+$$
  
 --- ---
  
-# Bosonic systems+## Bosonic systems
  
 A bosonic mode has Fock space A bosonic mode has Fock space
  
-\[+$$
 \mathcal F \mathcal F
 = =
 \operatorname{span} \operatorname{span}
 \{|0\rangle,|1\rangle,|2\rangle,\ldots\} \{|0\rangle,|1\rangle,|2\rangle,\ldots\}
-\]+$$
  
 with ladder operators with ladder operators
  
-\[+$$
 [a,a^\dagger]=1. [a,a^\dagger]=1.
-\]+$$
  
 For a harmonic oscillator, For a harmonic oscillator,
  
-\[+$$
 H H
 = =
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 a^\dagger a+\frac12 a^\dagger a+\frac12
 \right). \right).
-\]+$$
  
 Electromagnetic field modes are bosonic modes, so this description applies both to optical photons and microwave cavity photons. Electromagnetic field modes are bosonic modes, so this description applies both to optical photons and microwave cavity photons.
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 A **bosonic code** uses the larger oscillator Hilbert space to encode a smaller logical system: A **bosonic code** uses the larger oscillator Hilbert space to encode a smaller logical system:
  
-\[+$$
 \mathbb C^2\hookrightarrow\mathcal F. \mathbb C^2\hookrightarrow\mathcal F.
-\]+$$
  
 Important bosonic codes include: Important bosonic codes include:
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 --- ---
  
-# Cat code+## Cat code
  
 A coherent state satisfies A coherent state satisfies
  
-\[+$$
 a|\alpha\rangle=\alpha|\alpha\rangle a|\alpha\rangle=\alpha|\alpha\rangle
-\]+$$
  
 and has Fock expansion and has Fock expansion
  
-\[+$$
 |\alpha\rangle |\alpha\rangle
 = =
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 \sum_{n=0}^{\infty} \sum_{n=0}^{\infty}
 \frac{\alpha^n}{\sqrt{n!}}|n\rangle. \frac{\alpha^n}{\sqrt{n!}}|n\rangle.
-\]+$$
  
 Cat states are superpositions such as Cat states are superpositions such as
  
-\[+$$
 |C_\alpha^\pm\rangle |C_\alpha^\pm\rangle
 = =
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 |\alpha\rangle\pm|-\alpha\rangle |\alpha\rangle\pm|-\alpha\rangle
 \right). \right).
-\]+$$
  
 For example, one possible logical basis is For example, one possible logical basis is
  
-\[+$$
 |0_L\rangle=|C_\alpha^+\rangle, |0_L\rangle=|C_\alpha^+\rangle,
 \qquad \qquad
 |1_L\rangle=|C_\alpha^-\rangle. |1_L\rangle=|C_\alpha^-\rangle.
-\]+$$
  
 Thus a logical state is encoded across many oscillator occupation numbers. Thus a logical state is encoded across many oscillator occupation numbers.
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 --- ---
  
-# GKP code+## GKP code
  
 The GKP code uses oscillator quadratures The GKP code uses oscillator quadratures
  
-\[+$$
 \hat q \hat q
 = =
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 = =
 \frac{a-a^\dagger}{i\sqrt2}. \frac{a-a^\dagger}{i\sqrt2}.
-\]+$$
  
 Logical states correspond ideally to periodic grid-like states in oscillator phase space. Logical states correspond ideally to periodic grid-like states in oscillator phase space.
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 Thus GKP is another encoding Thus GKP is another encoding
  
-\[+$$
 \mathbb C^2\hookrightarrow\mathcal F, \mathbb C^2\hookrightarrow\mathcal F,
-\]+$$
  
 but with a very different code-space geometry from cat states. but with a very different code-space geometry from cat states.
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 --- ---
  
-# Binomial code+## Binomial code
  
 Binomial codes construct logical states from selected finite superpositions of Fock states. Binomial codes construct logical states from selected finite superpositions of Fock states.
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 Schematically, an example can look like Schematically, an example can look like
  
-\[+$$
 |0_L\rangle |0_L\rangle
 = =
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 = =
 |2\rangle. |2\rangle.
-\]+$$
  
 Thus Thus
  
-\[+$$
 \boxed{ \boxed{
 \text{cat},\quad \text{cat},\quad
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 \text{binomial} \text{binomial}
 } }
-\]+$$
  
 are alternative **bosonic codes/encodings**. are alternative **bosonic codes/encodings**.
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 --- ---
  
-# Photonic quantum computing+## Photonic quantum computing
  
 "Photonic" primarily specifies a **physical platform**: optical electromagnetic modes are used as the physical quantum system. "Photonic" primarily specifies a **physical platform**: optical electromagnetic modes are used as the physical quantum system.
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 Therefore Therefore
  
-\[+$$
 \boxed{\text{photonic}=\text{physical realization}} \boxed{\text{photonic}=\text{physical realization}}
-\]+$$
  
 while while
  
-\[+$$
 \boxed{\text{dual rail/GKP/etc.}=\text{encoding}}. \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. A photonic system can therefore implement a bosonic code, but photonic quantum computing is not synonymous with bosonic coding.
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 --- ---
  
-# Dual-rail encoding+## Dual-rail encoding
  
 Dual rail encodes one logical qubit into the **single-excitation subspace of two distinguishable modes or subsystems**. Dual rail encodes one logical qubit into the **single-excitation subspace of two distinguishable modes or subsystems**.
  
-For two modes \(a,b\),+For two modes $a,b$,
  
-\[+$$
 \mathcal H \mathcal H
 = =
 \mathcal F_a\otimes\mathcal F_b. \mathcal F_a\otimes\mathcal F_b.
-\]+$$
  
 The encoding is The encoding is
  
-\[+$$
 |0_L\rangle=|1,0\rangle, |0_L\rangle=|1,0\rangle,
 \qquad \qquad
 |1_L\rangle=|0,1\rangle. |1_L\rangle=|0,1\rangle.
-\]+$$
  
 Hence Hence
  
-\[+$$
 |\psi_L\rangle |\psi_L\rangle
 = =
 \alpha|1,0\rangle+\beta|0,1\rangle. \alpha|1,0\rangle+\beta|0,1\rangle.
-\]+$$
  
 Equivalently, Equivalently,
  
-\[+$$
 |\psi_L\rangle |\psi_L\rangle
 = =
 (\alpha a^\dagger+\beta b^\dagger)|0,0\rangle. (\alpha a^\dagger+\beta b^\dagger)|0,0\rangle.
-\]+$$
  
 The essential definition is therefore The essential definition is therefore
  
-\[+$$
 \boxed{ \boxed{
 \text{dual rail} \text{dual rail}
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 \text{one excitation coherently shared between two distinguishable rails}. \text{one excitation coherently shared between two distinguishable rails}.
 } }
-\]+$$
  
 Dual rail does **not** specifically require optical photons. Dual rail does **not** specifically require optical photons.
  
-For two transmons \(A,B\), for example,+For two transmons $A,B$, for example,
  
-\[+$$
 |0_L\rangle |0_L\rangle
 = =
 |E_1\rangle_A|E_0\rangle_B, |E_1\rangle_A|E_0\rangle_B,
-\]+$$
  
-\[+$$
 |1_L\rangle |1_L\rangle
 = =
 |E_0\rangle_A|E_1\rangle_B. |E_0\rangle_A|E_1\rangle_B.
-\]+$$
  
 Using the usual shorthand, Using the usual shorthand,
  
-\[+$$
 |0_L\rangle=|10\rangle, |0_L\rangle=|10\rangle,
 \qquad \qquad
 |1_L\rangle=|01\rangle. |1_L\rangle=|01\rangle.
-\]+$$
  
 --- ---
  
-# Schwinger-boson representation+## Schwinger-boson representation
  
-Two bosonic modes naturally realize the algebra \(\mathfrak{su}(2)\).+Two bosonic modes naturally realize the algebra $\mathfrak{su}(2)$.
  
 Define Define
  
-\[+$$
 J_+ J_+
 = =
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 = =
 b^\dagger a, b^\dagger a,
-\]+$$
  
 or equivalently or equivalently
  
-\[+$$
 J_x J_x
 = =
 \frac12(a^\dagger b+b^\dagger a), \frac12(a^\dagger b+b^\dagger a),
-\]+$$
  
-\[+$$
 J_y J_y
 = =
 \frac{1}{2i}(a^\dagger b-b^\dagger a), \frac{1}{2i}(a^\dagger b-b^\dagger a),
-\]+$$
  
-\[+$$
 J_z J_z
 = =
 \frac12(a^\dagger a-b^\dagger b). \frac12(a^\dagger a-b^\dagger b).
-\]+$$
  
 They satisfy They satisfy
  
-\[+$$
 [J_i,J_j] [J_i,J_j]
 = =
 i\epsilon_{ijk}J_k. i\epsilon_{ijk}J_k.
-\]+$$
  
 The total occupation The total occupation
  
-\[+$$
 N=a^\dagger a+b^\dagger b N=a^\dagger a+b^\dagger b
-\]+$$
  
 is conserved by these operators. is conserved by these operators.
  
-For fixed \(N\), the corresponding subspace has dimension+For fixed $N$, the corresponding subspace has dimension
  
-\[+$$
 N+1 N+1
-\]+$$
  
 and realizes the spin and realizes the spin
  
-\[+$$
 j=\frac N2 j=\frac N2
-\]+$$
  
-irreducible representation of \(SU(2)\).+irreducible representation of $SU(2)$.
  
 In particular, In particular,
  
-\[+$$
 N=1 N=1
 \quad\Longrightarrow\quad \quad\Longrightarrow\quad
 j=\frac12. j=\frac12.
-\]+$$
  
 The states The states
  
-\[+$$
 |1,0\rangle,\qquad|0,1\rangle |1,0\rangle,\qquad|0,1\rangle
-\]+$$
  
-therefore carry the fundamental two-dimensional representation of \(SU(2)\).+therefore carry the fundamental two-dimensional representation of $SU(2)$.
  
 This explains mathematically why dual rail naturally behaves as a qubit. This explains mathematically why dual rail naturally behaves as a qubit.
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 --- ---
  
-# Dual rail versus cat+## Dual rail versus cat
  
 Both can start from bosonic Hilbert spaces, but they use them very differently. Both can start from bosonic Hilbert spaces, but they use them very differently.
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 |---|---|---| |---|---|---|
 | Typical number of modes | 2 | 1 | | Typical number of modes | 2 | 1 |
-| Logical basis | \(|10\rangle,|01\rangle\) | coherent-state superpositions | +| Logical basis | $|10\rangle,|01\rangle$ | coherent-state superpositions | 
-| Occupation | fixed total \(N=1\) | distributed across many \(n\) |+| Occupation | fixed total $N=1$ | distributed across many $n$ |
 | Main idea | which rail contains excitation | structured states within oscillator | | Main idea | which rail contains excitation | structured states within oscillator |
 | Uses large Fock space | minimally | deliberately | | Uses large Fock space | minimally | deliberately |
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 Thus, schematically, Thus, schematically,
  
-\[+$$
 \boxed{ \boxed{
 \text{dual rail} \text{dual rail}
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 \text{constrain occupation across multiple modes} \text{constrain occupation across multiple modes}
 } }
-\]+$$
  
 whereas whereas
  
-\[+$$
 \boxed{ \boxed{
 \text{cat} \text{cat}
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 \text{construct structured superpositions within a mode}. \text{construct structured superpositions within a mode}.
 } }
-\]+$$
  
 --- ---
  
-# Dual rail and leakage+## Dual rail and leakage
  
 For two ideal two-level systems, For two ideal two-level systems,
  
-\[+$$
 \mathcal H_{\mathrm{phys}} \mathcal H_{\mathrm{phys}}
 = =
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 |00\rangle,|01\rangle,|10\rangle,|11\rangle |00\rangle,|01\rangle,|10\rangle,|11\rangle
 \}. \}.
-\]+$$
  
 Dual rail selects only Dual rail selects only
  
-\[+$$
 \mathcal C \mathcal C
 = =
 \operatorname{span} \operatorname{span}
 \{|10\rangle,|01\rangle\}. \{|10\rangle,|01\rangle\}.
-\]+$$
  
 Therefore Therefore
  
-\[+$$
 |00\rangle,\ |11\rangle |00\rangle,\ |11\rangle
-\]+$$
  
 are outside the logical code space. are outside the logical code space.
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 For an ordinary transmon, For an ordinary transmon,
  
-\[+$$
 |0_L\rangle=|0\rangle, |0_L\rangle=|0\rangle,
 \qquad \qquad
 |1_L\rangle=|1\rangle, |1_L\rangle=|1\rangle,
-\]+$$
  
 and relaxation gives and relaxation gives
  
-\[+$$
 |1_L\rangle\rightarrow|0_L\rangle. |1_L\rangle\rightarrow|0_L\rangle.
-\]+$$
  
 The error has mapped one valid logical state onto another. The error has mapped one valid logical state onto another.
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 For dual rail, For dual rail,
  
-\[+$$
 |0_L\rangle=|10\rangle, |0_L\rangle=|10\rangle,
 \qquad \qquad
 |1_L\rangle=|01\rangle, |1_L\rangle=|01\rangle,
-\]+$$
  
 and single-excitation loss gives and single-excitation loss gives
  
-\[+$$
 |10\rangle\rightarrow|00\rangle, |10\rangle\rightarrow|00\rangle,
-\]+$$
  
 or or
  
-\[+$$
 |01\rangle\rightarrow|00\rangle. |01\rangle\rightarrow|00\rangle.
-\]+$$
  
 Since Since
  
-\[+$$
 |00\rangle\notin\mathcal C, |00\rangle\notin\mathcal C,
-\]+$$
  
 the loss can be recognized as leaving the code space. the loss can be recognized as leaving the code space.
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 However, the original amplitudes in However, the original amplitudes in
  
-\[+$$
 \alpha|10\rangle+\beta|01\rangle \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**. 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**.
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 --- ---
  
-# Dual rail versus repetition code+## Dual rail versus repetition code
  
 Dual rail: Dual rail:
  
-\[+$$
 |0_L\rangle=|10\rangle, |0_L\rangle=|10\rangle,
 \qquad \qquad
 |1_L\rangle=|01\rangle. |1_L\rangle=|01\rangle.
-\]+$$
  
 Two-qubit repetition encoding: Two-qubit repetition encoding:
  
-\[+$$
 |0_L\rangle=|00\rangle, |0_L\rangle=|00\rangle,
 \qquad \qquad
 |1_L\rangle=|11\rangle. |1_L\rangle=|11\rangle.
-\]+$$
  
 Three-qubit bit-flip repetition code: Three-qubit bit-flip repetition code:
  
-\[+$$
 |0_L\rangle=|000\rangle, |0_L\rangle=|000\rangle,
 \qquad \qquad
 |1_L\rangle=|111\rangle. |1_L\rangle=|111\rangle.
-\]+$$
  
 These should not be confused. These should not be confused.
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 The three-qubit repetition code is designed to correct one bit-flip error: The three-qubit repetition code is designed to correct one bit-flip error:
  
-\[+$$
 X_i. X_i.
-\]+$$
  
 Dual rail instead has useful properties against excitation-loss errors because valid states have fixed total excitation number Dual rail instead has useful properties against excitation-loss errors because valid states have fixed total excitation number
  
-\[+$$
 N=1. N=1.
-\]+$$
  
 Thus they solve different problems: Thus they solve different problems:
  
-\[+$$
 \boxed{ \boxed{
 \text{dual rail} \text{dual rail}
Line 654: Line 656:
 \text{encoding with useful error-detection/erasure properties} \text{encoding with useful error-detection/erasure properties}
 } }
-\]+$$
  
 while while
  
-\[+$$
 \boxed{ \boxed{
 \text{3-qubit repetition} \text{3-qubit repetition}
Line 664: Line 666:
 \text{error-correcting code for bit flips}. \text{error-correcting code for bit flips}.
 } }
-\]+$$
  
 --- ---
  
-# Why use a larger Hilbert space?+## Why use a larger Hilbert space?
  
 Encoding often increases the physical Hilbert-space dimension. Encoding often increases the physical Hilbert-space dimension.
Line 674: Line 676:
 For example, For example,
  
-\[+$$
 \dim\mathcal C=2 \dim\mathcal C=2
-\]+$$
  
 for one logical qubit, while two physical two-level systems provide for one logical qubit, while two physical two-level systems provide
  
-\[+$$
 \dim\mathcal H_{\mathrm{phys}}=4. \dim\mathcal H_{\mathrm{phys}}=4.
-\]+$$
  
 This creates more states into which the system can leak. This creates more states into which the system can leak.
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 This is one of the fundamental ideas behind quantum error detection and correction: This is one of the fundamental ideas behind quantum error detection and correction:
  
-\[+$$
 \boxed{ \boxed{
 \mathbb C^2 \mathbb C^2
Line 698: Line 700:
 \dim\mathcal H_{\mathrm{phys}}>2. \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. The additional degrees of freedom provide redundancy that can allow physical errors to become detectable syndromes, leakage events, or correctable transformations.
Line 704: Line 706:
 --- ---
  
-# Overall taxonomy+## Overall taxonomy
  
 The most useful classification separates **hardware** from **encoding**: The most useful classification separates **hardware** from **encoding**:
  
-\[+$$
 \boxed{ \boxed{
 \underbrace{\text{What physical system do I have?}}_{\text{platform/device}} \underbrace{\text{What physical system do I have?}}_{\text{platform/device}}
Line 714: Line 716:
 \underbrace{\text{How do I represent }\mathbb C^2\text{ in it?}}_{\text{encoding/code}} \underbrace{\text{How do I represent }\mathbb C^2\text{ in it?}}_{\text{encoding/code}}
 } }
-\]+$$
  
 Examples: Examples:
Line 720: Line 722:
 | Physical system | Possible encoding | | Physical system | Possible encoding |
 |---|---| |---|---|
-| Transmon | \(|E_0\rangle,|E_1\rangle\) |+| Transmon | $|E_0\rangle,|E_1\rangle$ |
 | Two transmons | dual rail | | Two transmons | dual rail |
 | Two optical modes | dual rail | | Two optical modes | dual rail |
Line 730: Line 732:
 Encodings can themselves subsequently become the physical building blocks of higher-level error-correcting codes: Encodings can themselves subsequently become the physical building blocks of higher-level error-correcting codes:
  
-\[+$$
 \text{physical oscillator} \text{physical oscillator}
 \rightarrow \rightarrow
Line 738: Line 740:
 \rightarrow \rightarrow
 \text{fault-tolerant logical qubit}. \text{fault-tolerant logical qubit}.
-\]+$$
  
 The overarching idea is The overarching idea is
  
-\[+$$
 \boxed{ \boxed{
 \text{find a controllable and robust representation of } \text{find a controllable and robust representation of }
Line 748: Line 750:
 \text{ inside a physical Hilbert space}. \text{ inside a physical Hilbert space}.
 } }
-\]+$$
  
 Different quantum-computing modalities and encodings are different solutions to this same physical and information-theoretic problem. Different quantum-computing modalities and encodings are different solutions to this same physical and information-theoretic problem.
notes/2026-08-18.1787053881.md.gz · Last modified: by Ivan Janevski