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Symbolic Expressions

QuantumSavory supports symbolic expressions for the quantum states and operations being simulated thanks to the QuantumSymbolics library. It serves two purposes:

  • It provides for algebraic manipulation of mathematical expressions related to your models. Particularly helpful when having to automatically generate or simplify expressions of significant complexity.
  • An assortment of "expression translators" are provided that can turn a symbolic expression into a numerical one in any of the representations supported by the simulator (e.g. wavefunctions, tableaux, etc).

Below we list some commonly used expressions. For more detailed documentation consult QuantumSymbolics.jl.

Symbolic ExampleConversion 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)sparse([1, 2, 1, 2], [1, 1, 2, 2], ComplexF64[0.7071067811865475 + 0.0im, 0.7071067811865475 + 0.0im, 0.7071067811865475 + 0.0im, -0.7071067811865475 + 0.0im], 2, 2)
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)sparse([1, 2], [1, 2], ComplexF64[0.5 + 0.0im, 0.5 + 0.0im], 2, 2)
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.