# Qudits¶

Most of the time in quantum computation, we work with qubits, which are 2-level quantum systems. A qu-d-it is a generalization of a qubit to a d-level or d-dimension system.

Qudits with known values for d have specific names. A qubit has dimension 2, a qutrit has dimension 3, a ququart has dimension 4, and so on. In Cirq, qudits work exactly like qubits except they have a dimension attribute other than 2, and they can only be used with gates specific to that dimension.

Both qubits and qudits are represented by a Qid object.

To apply a gate to some qudits, the dimensions of the qudits must match the dimensions it works on. For example, if a gate represents a unitary evolution on three qudits, a qubit, a qutrit, and another qutrit, the gate’s “qid shape” is (2, 3, 3) and its on method will accept exactly 3 Qids with dimension 2, 3, and 3.

This is an example single qutrit gate used in a circuit:

[1]:

import cirq

class QutritPlusGate(cirq.SingleQubitGate):
def _qid_shape_(self):
return (3,)

def _unitary_(self):
return np.array([[0, 0, 1],
[1, 0, 0],
[0, 1, 0]])

def _circuit_diagram_info_(self, args):
return '[+1]'

q0 = cirq.LineQid(0, dimension=3)
circuit = cirq.Circuit(
QutritPlusGate().on(q0)
)
print(circuit)

0 (d=3): ───[+1]───


## Qids¶

Qid is the type that represents both qubits and qudits. By default, a qid like cirq.NamedQubit('a') is a qubit.

While Cirq has the built-in qubit types, it also provides the corresponding Qid types:

• cirq.NamedQid: To create a qutrit named ‘a’, specify the dimension with cirq.NamedQid('a', dimension=3).

• cirq.GridQid: To create a 2x2 grid of ququarts, use cirq.GridQid.rect(2, 2, dimension=4).

• cirq.LineQid: In addition, the LineQid constructor also supports a dimension argument directly cirq.LineQid(0, dimension=4).

## cirq.qid_shape and def _qid_shape_¶

Quantum gates, operations, and other types that act on a sequence of qudits can specify the dimension of each qudit they act on by implementing the _qid_shape_ magic method. This method returns a tuple of integers corresponding to the required dimension of each qudit it operates on, e.g. (2, 3, 3) means an object that acts on a qubit, a qutrit, and another qutrit.

When Qids are used with Gates, Operations, and Circuits, the dimension of each qid must match the corresponding entry in the qid shape. An error is raised otherwise.

Callers can query the qid shape of an object or a list of Qids by calling cirq.qid_shape on it. By default, cirq.qid_shape will return the equivalent qid shape for qubits if _qid_shape_ is not defined.

For a qubit-only gate the qid shape is a tuple of 2s containing one 2 for each qubit e.g. (2,) * cirq.num_qubits(gate).

## Unitaries, Mixtures, and Channels¶

The magic methods _unitary_, _apply_unitary_, _mixture_, and _channel_ used to define unitary operations, mixtures, and channels can be used with qudits with one caveat.

The matrix dimensions for qudits will be larger than for qubits based on the values of the qudit dimensions (the object’s qid shape). The size of the matrix is determined from the product of the qudit dimensions.

For example, a single qubit unitary is a 2x2 matrix, whereas a single qutrit unitary is a 3x3 matrix. A two qutrit unitary is a 9x9 matrix (3 * 3 = 9) and a qubit-ququart unitary is an 8x8 matrix (2 * 4 = 8). The size of the matrices for mixtures and channels follow the same rule.

## Simulators and Samplers¶

Simulators like cirq.Simulator and cirq.DensityMatrixSimulator will return simulation results with larger matrices than the same size qubit circuit when simulating qudit circuits.

The size of the matrix is determined by the product of the dimensions of the qudits being simulated. The state vector output of cirq.Simulator after simulating a circuit on a qubit, a qutrit, and a qutrit will have 2 * 3 * 3 = 18 elements.

Call cirq.qid_shape(simulation_result) to check the qudit dimensions.

Measurement results from running a qudit circuit are integers in the range 0 to qid.dimension-1.