gav-quantum 0.0.8

A quantum information utility package.

gav-quantum

gav-quantum is a quantum information utility package developed by Emil Marinov and Gray Alien Ventures.

Installation

pip install gav-quantum

Usage

Sample import statement below:
from gav_quantum import X, Y, Z, operator_from_sparse_pauli

Objects

ZERO - A column vector representation of the 0-state, [1 0]^T.

ONE - A column vector representation of the 1-state, [0 1]^T.

X, Y, Z - Standard X, Y, and Z Pauli operators.

PAULIS - An array of the Pauli operators in standard quantum information convention. Convenient for enumerations.

BELL_STATES[PHI_PLUS | PHI_MINUS | PSI_PLUS | PSI_MINUS] - The four maximally entangled Bell states.

Functions

W(N)

Inputs

• N - The number of qubits in the quantum state.

Output

• A vector corresponding to the maximally entangled N-qubit W-state.

GHZ(N)

Inputs

• N - The number of qubits in the quantum state.

Output

• A vector corresponding to the maximally entangled N-qubit GHZ-state.

operator_from_sparse_pauli(N, nonidentities) - Creates an operator from a list of tuples of qubits and the non-identity Pauli operators acting upon them.

Inputs

• N - The number of qubits in the quantum state.
• nonidentities - List of tuples specifying qubit number (indexed at 0) in ascending order and corresponding Pauli operator. Example: [(0,X),(2,Y)]

Output

• A 2^N by 2^N operator.

Notes

• Make sure to have also imported whichever Pauli operators (X, Y, Z) are used in the nonidentities argument

Paulis_N_k(N, k)

Generates a list of the subset of operator bases over N qubits of weight less than or equal to k; it is a "tensor product of N operators, each of which is in {I, X, Y, Z} such that at most k are not I" [1].

Inputs

• N - The number of qubits in the quantum state.
• k - The maximum locality of Pauli operators acting upon the qubits.

Output

• A 2-tuple containing the set of operators where the first element is a list of the operators in matrix form and the second element is a verbose human-friendly list specifying qubit number (indexed at 0) in ascending order and corresponding Pauli operator. Example: [(0,X),(2,Y)]

Notes

• Returns the full operator basis if k = N

QMaxCutHamiltonian(N, k, mode="Ising")

Generates a Quantum MaxCut Hamiltonian.

Inputs

• N - The number of qubits in the quantum state.
• k - The locality of qubit interations.
• mode - "Ising" | "Heisenberg"

Output

• A 2^N by 2^N matrix corresponding to a QMaxCut Hamiltonian.

moment_constraint_matrix(N, E, k)

Generates the constraint matrix accompanying the moment matrix for a Lasserre semidefinite program assuming an anti-ferromagnetic state. Entries correspond to a matrix with rows and columns indexed by the subset of operator bases (see Paulis_N_k(N, k)). A row with adjacent edges with the same Pauli operator and a column with the same adjacent edges and Pauli operator between them are penalized with a -0.5 weight for their anti-alignment. For example, if the edges in question are 0 and 2, row X_0 X_2 and column X_0 X_2 are penalized for not exhibiting anti-alignment.

Inputs

• N - The number of qubits in the quantum state or nodes in the constraint graph.
• E - A 2-tuple corresponding to the edge of the constraint graph with constraint graph node numbers in ascending order as constraints. Example: (0, 2)
• k - The locality of qubit interactions.

Output

• A $\gamma$ by $\gamma$ matrix where $\gamma$ is the size of the subset of operator bases corresponding to the inputted N and k and the entries are real.

rootsOfUnity(n)

Returns a list of the $n$th roots of unity.

Output

• A list of the $n$th roots of unity i.e. [1, \omega, \omega^2, ..., \omega^{n-1}]

References

[1] Parekh, O., & Thompson, K. (2021). Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms. arXiv. https://doi.org/10.48550/arXiv.2105.05698