Wolfram/ QuantumFramework

Perform analytic and numeric quantum computations

Contributed by: Wolfram Research, Quantum Computation Framework team

The Wolfram Quantum Framework brings a broad, coherent design for quantum computation, together with a host of leading-edge capabilities and full integration into Mathematica and Wolfram Language. Starting from discrete quantum mechanics, the Framework provides a high-level symbolic representation of quantum bases, states and operators. The Framework can perform measurements and is equipped with various well-known states and operators, such as Bell states and Pauli operators. Using such simulation capabilities as a foundation, one can use the Framework to model and simulate quantum circuits and algorithms, interoperate with many external quantum platforms, and send queries to quantum processing units from a Wolfram notebook.

Within the Framework, all quantum-related functions and objects are natively integrated with Wolfram Language, providing immediate access to a wide array of tools for exploring quantum computational queries and serving as a potent resource for education in the field.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["Wolfram/QuantumFramework"]

To load the code after installation, evaluate this code:
Needs["Wolfram`QuantumFramework`"]

Details

This paclet works with Version 13.1 and higher of Wolfram Language.

Paclet Guide

The Wolfram Quantum Framework offers a novel and convenient approach to quantum computing research with Wolfram Notebooks. It provides a comprehensive set of modeling capabilities for simulating quantum computation, with full integration into Mathematica and Wolfram Language, and also interoperability with many external quantum platforms. Within Wolfram quantum framework, there exists a diverse collection of quantum objects, that fully describe quantum entities, such as quantum basis, state, operations, and processes. These objects can undergo transformations through some operations (for example, quantum state into a quantum measurement, by a measurement operator). Furthermore, their properties can be calculated using other functions (for example, quantum entanglement monotones for a quantum state). Computations can be done either symbolically, or numerically.

Basic Objects

QuditName

— representation of a qudit name with formatting

QuditBasis

— representation of a qudit basis

QuantumBasis

— representation of a quantum basis

QuantumState

— representation of a quantum state

QuantumOperator

— representation of a quantum operator

QuantumMeasurementOperator

— representation of a quantum measurement operator

QuantumMeasurement

— representation of quantum measurement

Quantum Circuits

QuantumChannel

— representation of a quantum channel to describe noise

QuantumCircuitOperator

— representation of a quantum circuit

QuantumCircuitMultiwayGraph

— multiway graph of a quantum circuit

QuantumShortcut

— shorthand representation of operators or circuits

Quantum Functions

QuantumDistance

— distance between quantum states

QuantumEvolve

— time evolution

QuantumMeasurementSimulation

— simulating one or many measurement results, given a quantum state

QuantumMPS

— returns the MPS or MPO of a state or operator/circuit

QuantumPartialTrace

— partial trace of a quantum state, operator or circuit with respect to subsystems

QuantumPartialTranspose

— partial transpose of a quantum state or quantum operator with respect to subsystems

QuantumStateEstimate

— estimating quantum state based on measurement results

QuantumTensorProduct

— tensor product of quantum states, quantum bases, quantum operators, or quantum measurements

QuantumWignerTransform

— discrete Wigner transform of a quantum state, operator or circuit

Entanglement

QuantumEntangledQ

— determination of whether a part of quantum state is entangled with another part

QuantumEntanglementMonotone

— measure entanglement of a quantum state

Examples

The Wolfram quantum framework handles many different quantum objects, including states, operators, channels, measurements, circuits, and more. It offers specialized functions for various computations like quantum evolution, entanglement monotones, partial tracing, Wigner or Weyl transformations, stabilizer formalism, and additional capabilities. Each functionality incorporates common named operations, such as Schwinger basis, GHZ state, Fourier operator, Grover circuit, and others. Perform computations seamlessly with the Wolfram quantum framework using the standard Wolfram kernel, such as the usual evaluation of codes in Mathematica. Alternatively, leverage the framework to send jobs to quantum processing units via service connections.

Create a quantum circuit composed of Pauli-X on qubits 1 and 2, Hadamard on qubit 1, CNOT with qubit 1 as the controlled and qubit 2 as the target, rotation around the z-axis by a symbolic angle ϕ on qubit 1, rotation around the z-axis by a symbolic angle θ on qubit 2, Hadamard on qubits 1 and 2, and finally, measurement in the computational basis on qubits 1 and 2.

In[1]:=

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Out[2]=

Generate the multivariate distribution of measurement results based on the assumption that the angles are real:

In[3]:=

$$Assumptions = {\[Phi], \[Theta]} \[Element] Reals; \[ScriptCapitalD] = qc[]["MultivariateDistribution"]$

Out[4]=

Calculate the quantum correlation P₀₀-P₁₀-P₀₁+P₁₁

In[5]:=

$Sum[(-1)^(i + j) PDF[\[ScriptCapitalD], {i, j}], {i, 0, 1}, {j, 0, 1}] // FullSimplify$

Out[5]=

Calculate the state before measurements and show it in the Dirac notation:

In[6]:=

Out[6]=

Calculate the entanglement monotone of that state:

In[7]:=

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Out[7]=

Add a bit-flip channel with a probability p only on the qubit-2:

In[8]:=

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Out[9]=

Calculate the corresponding density matrix of the final quantum state of this circuit:

In[10]:=

Out[10]=

Create the quantum phase estimation circuit for a phase operator with a given angle and a given number of qubits:

In[11]:=

Out[12]=

Calculate the outcome of circuit:

In[13]:=

Out[13]=

Calculate the corresponding probabilities:

In[14]:=

Out[14]=

Set time and other variables of a Hamiltonian (Rabi drive and detuning):

In[15]:=

$\[CapitalOmega] = 15. Sin[t]^2; \[CapitalDelta] = 10. Sin[t - 1]; Plot[{\[CapitalOmega], \[CapitalDelta]}, {t, 0, 3}, PlotLegends -> {"\[CapitalOmega] (Rabi)", "\[CapitalDelta] (Detuning)"}, AspectRatio -> 1/2, AxesLabel -> {"Time"}]$

Out[16]=

Create a Hamiltonian operator as

In[17]:=

Evolve an initial quantum state using the Hamiltonian:

In[18]:=

Plot the measurement probabilities in the different basis:

In[19]:=

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Out[19]=

Plot the Bloch vector evolution:

In[20]:=

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Out[20]=

Define Lindblad operators (inducing jump from 1 to 0, and vice versa) with given rates:

In[21]:=

Evolve the same initial state using the Lindblad equation:

In[22]:=

Plot the measurement probabilities in the computational basis:

In[23]:=

$Plot[Evaluate@Normal@\[Rho]["ProbabilitiesList"], {t, 0, 3.}, Sequence[PlotLabel -> "Population vs time", PlotLegends -> { Ket[{"0"}], Ket[{"1"}]}, AspectRatio -> 1/2, PlotRange -> {0, 1}, GridLines -> Automatic]]$