00 : Conceptual overview

The most commonly used notation for representing quantum algorithms is the quantum circuit model introduced by Deutsch. The model describes the computation as a sequence of elementary quantum gates acting on a collection of qubits.

In tweedledum, the representation of quantum programs is a quantum circuit, hereinafter called circuit (see ). A circuit is a collection of gates. An operation is a gate that operates on a specific subset of wires.

Let’s analyze a small quantum circuit to illustrate these concepts:

../_images/concept_back.svg

Wires

At the very bottom of the layers of abstraction rest concepts of quantum bit, qubit, and classical bit, cbit. I expect the reader to be already familiar with those. So I will start describing the wires:

../_images/concept_wires.svg

A wire can be either a quantum or classical. A quantum wire holds the state of a qubit, and it is represented by a line in quantum circuit diagrams. In tweedledum, a quantum wire is equivalent to a qubit. Similarly, a classical wire holds the state of a cbit, and it is represented by a double line in quantum circuit diagrams.

In a quantum circuit, each wire has a wire::id. The wire::id is used to uniquely identify a wire, and to it indicate whether the wire is quantum or classical. Wires created are by calling one of the create_qubit() or create_cbit() methods from a circuit. We can also directly create wire using wire::make_qubit or wire::make_cbit. A wire created using these functions, however, won’t be part of a quantum circuit.

#include <tweedledum/tweedledum.hpp>

int main(int argc, char** argv)
{
    wire::id q0 = wire::make_qubit(0);
    wire::id q1 = wire::make_qubit(1);
    wire::id c0 = wire::make_cbit(2);
}

Gates

A gate is an effect that can be applied to a subset of wires. Most often this effect is an unitary evolution, hence the gate is a quantum gate. In our small example, we have two ‘pure’ quantum gates: the Hadamard gate \(\mathrm{H}\), and the \(\mathrm{CNOT}\) gate \(\mathrm{CX}\).

../_images/concept_gates.svg

The weird looking ‘meter gate’ is actually a measurement gate. As measurement is irreversible, it is not a quantum gate. Finaly, the last gate is NOT gate \(\mathrm{\oplus}\) that is applied whenever the state of the cbit is true.

Operations

An operation is a gate that operates on a specific subset wires:

../_images/concept_ops.svg 
#include <tweedledum/tweedledum.hpp>

int main(int argc, char** argv)
{
    wire::id q0 = wire::make_qubit(0);
    wire::id q1 = wire::make_qubit(1);
    wire::id c0 = wire::make_cbit(2);
    operation h_op(gate_lib::h, q1);
    operation cx_op(gate_lib::cx, q1, q0);
    operation m_op(gate_lib::measure_z, q0, c0);
}

Networks

  • A netlist represents the circuit as a list of gates to be applied sequentially. It is convenient because each range in the array represents a valid sub-circuit.

  • Directed acyclic graph (DAG) representation, op_dag. The vertices of the DAG are the operations of the circuit and the edges encode their relationships. The DAG representation has the advantage of making adjacency between gates easy to access.

  • Mapped DAG representation, mapped_dag. The same as op_dag but mapped to a particular device architecture.

  • Unitary representation. A unitary matrix representation of the circuit. Not scalable at all, the unitary is literally represented as a \(2^n \times 2^n\) matrix, where \(n\) is the number of qubits.

  • Phase polynomials representation. (work-in-progress)

  • Path polynomials representation. (work-in-progress)

  • Exponents of Pauli representation. (work-in-progress)