Operator Library

Schrodinger.Sigma1 — Method

Sigma1(N)

Generate the N-d shift matrix $Σ₁$.

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Schrodinger.Sigma3 — Method

Sigma3(N)

Generate the N-d clock matrix $Σ₃$.

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Schrodinger.create — Method

create(N)

Generate a quantum harmonic oscillator raising (creation) operator $\hat{a}^†$ in a truncated Hilbert space of size N. Returns a sparse matrix.

Example

julia> create(4)
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 4
  ⋅    ⋅        ⋅        ⋅
 1.0   ⋅        ⋅        ⋅
  ⋅   1.41421   ⋅        ⋅
  ⋅    ⋅       1.73205   ⋅

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Schrodinger.destroy — Method

destroy(N)

Generate a quantum harmonic oscillator lowering (annihilation) operator $\hat{a}$ in a truncated Hilbert space of size N. Returns a sparse matrix.

Example

julia> destroy(4)
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 4
  ⋅   1.0   ⋅        ⋅
  ⋅    ⋅   1.41421   ⋅
  ⋅    ⋅    ⋅       1.73205
  ⋅    ⋅    ⋅        ⋅

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Schrodinger.displacementop — Method

displacementop(N, α)

Generate a quantum harmonic oscillator displacement operator $\hat{D}(α)$ in a truncated Hilbert space of size N. Returns a dense matrix.

\[\hat{D}(α) = \exp\left(α\hat{a}^† - α^*\hat{a}\right)\]

Example

julia> displacementop(3,0.5im)
3×3 Operator{Matrix{ComplexF64}, 1} with dimensions 3
   0.88262+0.0im            0.0+0.439802im  -0.166001+0.0im
       0.0+0.439802im  0.647859+0.0im             0.0+0.621974im
 -0.166001+0.0im            0.0+0.621974im    0.76524+0.0im

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Schrodinger.numberop — Method

numberop(N)

Generate a number operator $\hat{n}$ in a Hilbert space of size N. Returns a sparse matrix.

Example

julia> numberop(4)
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 4
 0.0   ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅
  ⋅    ⋅   2.0   ⋅
  ⋅    ⋅    ⋅   3.0

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Schrodinger.projectorop — Method

projectorop(N,S)

Generate a projector on the subspaces defined by an integer or a vector/range of integers S:

\[P = ∑_{i∈S} |i⟩⟨i|.\]

Example

julia> projectorop(5,[1,3])
5×5 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 5
  ⋅    ⋅    ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅   1.0   ⋅
  ⋅    ⋅    ⋅    ⋅    ⋅

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Schrodinger.qeye — Method

qeye(N, dims=(N,))

Generate an identity operator for a Hilbert space of size N. It is possible to specify the subspace dimensions with the dims argument. Returns a sparse matrix.

Example

julia> qeye(4,(2,2))
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 2} with dimensions 2⊗2
 1.0   ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅
  ⋅    ⋅   1.0   ⋅
  ⋅    ⋅    ⋅   1.0

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Schrodinger.qzero — Method

qzero(N, dims=(N,))

Generate a zero operator for a Hilbert space of size N. It is possible to specify the subspace dimensions with the dims argument. Returns a sparse matrix.

Example

julia> qzero(4,(2,2))
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 2} with dimensions 2⊗2
  ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅

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Schrodinger.squeezeop — Method

squeezeop(N, z)

Generate a quantum harmonic oscillator squeeze operator $\hat{S}(z)$ in a truncated Hilbert space of size N. Returns a dense matrix.

\[\hat{S}(z) = \exp\left(\frac{1}{2}\left(z^*\hat{a}^2 - z\hat{a}^{†2}\right)\right)\]

Example

julia> squeezeop(3,0.5im)
3×3 Operator{Matrix{ComplexF64}, 1} with dimensions 3
 0.938148-0.0im       0.0-0.0im       0.0-0.346234im
      0.0+0.0im       1.0+0.0im       0.0+0.0im
      0.0-0.346234im  0.0-0.0im  0.938148-0.0im

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Schrodinger.sylvesterop — Method

sylvesterop(N,k,l)

Generate the $(k,j)^\text{th}$ Sylvester generalized Pauli matrix in N-d. https://en.wikipedia.org/wiki/GeneralizationsofPauli_matrices

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Schrodinger.rand_unitary — Function

rand_unitary(N, dims=(N,))

Generate a Haar distributed random unitary operator for a Hilbert space of size N. It is possible to specify the subspace dimensions with the dims argument. Returns a dense matrix.

Example

julia> U = rand_unitary(4,(2,2));

julia> U'*U ≈ qeye(4,(2,2))
true

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Schrodinger.Gate.rotation — Function

Gate.rotation(θ, n=(1,0,0))

Generate a qubit rotation operator giving a θ rad rotation about an axis defined by the vector $\vec{n}$. Note that n will be normalized, allowing for inputs like (1,1,0). The rotation operator is defined as

\[\hat{R}_{\vec{n}}(θ) = \exp\left(−iθ\vec{n}⋅\vec{σ}/2\right) = \cos\frac{θ}{2}𝕀 - i\sin\frac{θ}{2}(n_x σ_x + n_y σ_y + n_z σ_z).\]

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Schrodinger.Gate.H — Constant