Symbolic Expressions Reference

This page is the example-oriented reference for symbolic states and operators in QuantumSavory.

For the conceptual role of the symbolic frontend, start with Symbolic Frontend. For the full symbolic API, see the QuantumSymbolics.jl documentation.

Below are the most commonly used expression families and examples of converting them to numerical representations with express.

Common Symbolic Objects

Symbolic Example Conversion Example

	Symbolic Example	Conversion Example
Qubit Basis States `X1`, `X2`, `Y1`, `Y2`, `Z1`, `Z2`	`Z1` \[\left\|Z_1\right\rangle\]	`express(Z1)` `Ket(dim=2) basis: Spin(1/2) 1.0 + 0.0im 0.0 + 0.0im` `express(Y2, CliffordRepr())` `𝒟ℯ𝓈𝓉𝒶𝒷 + Z 𝒮𝓉𝒶𝒷 - Y`
Common gates: `CNOT`, `H`, etc	`CNOT` \hat CNOT	`express(H)` `Operator(dim=2x2) basis: Spin(1/2) 0.7071067811865475 + 0.0im 0.7071067811865475 + 0.0im 0.7071067811865475 + 0.0im -0.7071067811865475 + 0.0im` `express(CNOT, CliffordRepr(), UseAsOperation())` `QuantumClifford.sCNOT`
Tensor products `⊗` and sums `+`	`(X1⊗Z2 + Y1⊗Y2 ) / √3` \[0.5773502691896258 \left\|Y_1\right\rangle\otimes\left\|Y_2\right\rangle + \left\|X_1\right\rangle\otimes\left\|Z_2\right\rangle\]	`express(X1⊗Z1)` `Ket(dim=4) basis: [Spin(1/2) ⊗ Spin(1/2)] 0.7071067811865475 + 0.0im 0.7071067811865475 + 0.0im 0.0 + 0.0im 0.0 + 0.0im` `express(X1⊗Y2, CliffordRepr())` `𝒟ℯ𝓈𝓉𝒶𝒷 + Z_ + _Z 𝒮𝓉𝒶𝒷 + X_ - _Y`
Projectors, pure density matrices	`SProjector(X1⊗Z2)` \[\mathrm{projector}\left( \left\|X_1\right\rangle\otimes\left\|Z_2\right\rangle \right)\]	`express(SProjector(X1⊗Z1))` `Operator(dim=4x4) basis: [Spin(1/2) ⊗ Spin(1/2)] 0.5+0.0im 0.5+0.0im 0.0+0.0im 0.0+0.0im 0.5+0.0im 0.5+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im` `express(SProjector(X1⊗Z1), CliffordRepr())` `𝒟ℯ𝓈𝓉𝒶𝒷 + Z_ + _X 𝒮𝓉𝒶𝒷 + X_ + _Z`
Completely depolarized (mixed) state	`MixedState(X1)` \mathbb{M}	`express(MixedState(X1))` `Operator(dim=2x2) basis: Spin(1/2) 0.5 + 0.0im ⋅ ⋅ 0.5 + 0.0im` `express(MixedState(X1), CliffordRepr())` `𝒟ℯ𝓈𝓉𝒶𝒷 𝒳ₗ━ + X 𝒮𝓉𝒶𝒷 𝒵ₗ━ + Z`
Impure states, represented as sum of density matrices	`(MixedState(X1)+SProjector(Z1)) / 2` \[0.5 \mathbb{M} + \mathrm{projector}\left( \left\|Z_1\right\rangle \right)\]	`express((MixedState(X1)+SProjector(Z1)) / 2)` `Operator(dim=2x2) basis: Spin(1/2) 0.75+0.0im 0.0+0.0im 0.0+0.0im 0.25+0.0im` When a Clifford representation is used, an efficient sampler is generated, and stabilizer states are randomly sampled from the correct distribution: `express(MixedState(X1)/2+SProjector(Z1)/2, CliffordRepr())` `𝒟ℯ𝓈𝓉𝒶𝒷 𝒳ₗ━ + X 𝒮𝓉𝒶𝒷 𝒵ₗ━ + Z`

Qubit Basis States X1, X2, Y1, Y2, Z1, Z2

Z1

\[\left|Z_1\right\rangle\]

express(Z1)

Ket(dim=2)
  basis: Spin(1/2)
 1.0 + 0.0im
 0.0 + 0.0im

express(Y2, CliffordRepr())

𝒟ℯ𝓈𝓉𝒶𝒷
+ Z
𝒮𝓉𝒶𝒷
- Y

Common gates: CNOT, H, etc

CNOT

\hat CNOT

express(H)

Operator(dim=2x2)
  basis: Spin(1/2)
 0.7071067811865475 + 0.0im   0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im  -0.7071067811865475 + 0.0im

express(CNOT, CliffordRepr(), UseAsOperation())

QuantumClifford.sCNOT

Tensor products ⊗ and sums +

(X1⊗Z2 + Y1⊗Y2 ) / √3

\[0.5773502691896258 \left|Y_1\right\rangle\otimes\left|Y_2\right\rangle + \left|X_1\right\rangle\otimes\left|Z_2\right\rangle\]

express(X1⊗Z1)

Ket(dim=4)
  basis: [Spin(1/2) ⊗ Spin(1/2)]
 0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im

express(X1⊗Y2, CliffordRepr())

𝒟ℯ𝓈𝓉𝒶𝒷
+ Z_
+ _Z
𝒮𝓉𝒶𝒷
+ X_
- _Y

Projectors, pure density matrices

SProjector(X1⊗Z2)

\[\mathrm{projector}\left( \left|X_1\right\rangle\otimes\left|Z_2\right\rangle \right)\]

express(SProjector(X1⊗Z1))

Operator(dim=4x4)
  basis: [Spin(1/2) ⊗ Spin(1/2)]
 0.5+0.0im  0.5+0.0im  0.0+0.0im  0.0+0.0im
 0.5+0.0im  0.5+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im

express(SProjector(X1⊗Z1), CliffordRepr())

𝒟ℯ𝓈𝓉𝒶𝒷
+ Z_
+ _X
𝒮𝓉𝒶𝒷
+ X_
+ _Z

Completely depolarized (mixed) state

MixedState(X1)

\mathbb{M}

express(MixedState(X1))

Operator(dim=2x2)
  basis: Spin(1/2)
 0.5 + 0.0im       ⋅     
      ⋅       0.5 + 0.0im

express(MixedState(X1), CliffordRepr())

𝒟ℯ𝓈𝓉𝒶𝒷

𝒳ₗ━
+ X
𝒮𝓉𝒶𝒷

𝒵ₗ━
+ Z

Impure states, represented as sum of density matrices

(MixedState(X1)+SProjector(Z1)) / 2

\[0.5 \mathbb{M} + \mathrm{projector}\left( \left|Z_1\right\rangle \right)\]

express((MixedState(X1)+SProjector(Z1)) / 2)

Operator(dim=2x2)
  basis: Spin(1/2)
 0.75+0.0im   0.0+0.0im
  0.0+0.0im  0.25+0.0im

When a Clifford representation is used, an efficient sampler is generated, and stabilizer states are randomly sampled from the correct distribution:

express(MixedState(X1)/2+SProjector(Z1)/2, CliffordRepr())

𝒟ℯ𝓈𝓉𝒶𝒷

𝒳ₗ━
+ X
𝒮𝓉𝒶𝒷

𝒵ₗ━
+ Z

Stabilizer state expressions

The state written as $\frac{|Z₁⟩⊗|Z₁⟩+|Z₂⟩⊗|Z₂⟩}{√2}$ is a well known stabilizer state, namely a Bell state. However, automatically expressing it as a stabilizer is a prohibitively expensive computational operation in general. We do not perform that computation automatically. If you want to ensure that states you define can be automatically converted to tableaux for Clifford simulations, avoid using summation of kets. On the other hand, in all of our Clifford Monte-Carlo simulations, ⊗ is fully supported, as well as SProjector, MixedState, StabilizerState, and summation of density matrices.