Symbolic Computation for Quantum States

Compute measurements, transformations and more symbolically

In the Wolfram Quantum framework, a quantum state can be defined symbolically. Therefore, its transformation due to, e.g., measurement, quantum channel, unitary transformation and more can be computed symbolically, too.

First, install and load the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]Needs["Wolfram`QuantumFramework`"]

In the following, we will show examples of different symbolic computations on a quantum state.

Hadamard on a symbolic state:

In[2]:=

QuantumOperator["H"]@QuantumState[{α,β}]//TraditionalForm

Out[2]//TraditionalForm=

|0〉+

|1〉

Generalized Pauli-X in 3D transforms a symbolic 3D quantum state:

In[3]:=

QuantumOperator[{"X",3}][QuantumState[{α,β,γ},3]]//TraditionalForm

Out[3]//TraditionalForm=

β|0〉+

α+

γ|1〉+

β|2〉

Symbolic basis transformation:

In[4]:=

ψ0=QuantumState[{x,y},QuantumBasis[0{a,b},1{c,d}]];QuantumState[ψ0,"X"]//TraditionalForm

Out[4]//TraditionalForm=

ax+cy

bx+dy



+

ax+cy

bx+dy





Symbolic computation of quantum measurement:

In[5]:=

mea=QuantumMeasurementOperator[]QuantumStateCos[θ/2],

ϕ



Sin[θ/2]

Out[5]=

QuantumMeasurement

Target: {1}

Measurement Outcomes: 2



Measurement probabilities:

In[6]:=

FullSimplify[#,Assumptions{ϕ∈Reals,θ∈Reals}]&/@mea["Probabilities"]

Out[6]=

0

Cos



,1

Sin





One can do symbolic computation, using quantum circuits or channels on a quantum state.

Define a symbolic multiplexer circuit:

In[7]:=

qc=QuantumCircuitOperator[{"Multiplexer",{"RX",α},{"RX",β}}];qc["Diagram"]

Out[7]=

Find the final state vector, after acting the above circuit on a symbolic 2-qubit state:

In[8]:=

FullSimplify/@Normal[qc[QuantumState[{c1,c2,c3,c4}]]["StateVector"]]

Out[8]=

c1Cos

-c2Sin

,c2Cos

-c1Sin

,c3Cos

-c4Sin

,c4Cos

-c3Sin



Given a symbolic quantum channel, transform a symbolic quantum state:

In[9]:=

ρ=QuantumChannel[{"BitFlip",p}]QuantumStateCos[θ/2],

ϕ



Sin[θ/2]

Out[9]=

QuantumState

Unknown state	Qudits: 1
Type: Matrix	Dimension: 2
Picture: Schrödinger



Represent the final density matrix:

In[10]:=

FullSimplify[#,Assumptions{0≤p≤1,ϕ∈Reals,θ∈Reals}]&/@Normal[ρ["DensityMatrix"]]//MatrixForm

Out[10]//MatrixForm=

1 2 (1+Cos[θ]-2pCos[θ])	1 2 Sin[θ](Cos[ϕ]+(-1+2p)Sin[ϕ])
1 2 Sin[θ](Cos[ϕ]+(1-2p)Sin[ϕ])	1 2 (1+(-1+2p)Cos[θ])

Evolve a symbolic quantum state by a symbolic Hamiltonian in time:

In[11]:=

QuantumEvolve[QuantumOperator["X"],QuantumState[{α,β}],t]["StateVector"]//MatrixForm

Out[11]//MatrixForm=

αCos[t]-βSin[t]

βCos[t]-αSin[t]

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Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

Symbolic Computation for Quantum States

See Also

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