# Diagonal

*class *`qiskit.circuit.library.Diagonal(diag)`

Bases: `QuantumCircuit`

Diagonal circuit.

Circuit symbol:

```
┌───────────┐
q_0: ┤0 ├
│ │
q_1: ┤1 Diagonal ├
│ │
q_2: ┤2 ├
└───────────┘
```

Matrix form:

$\text{DiagonalGate}\ q_0, q_1, .., q_{n-1} = \begin{pmatrix} D[0] & 0 & \dots & 0 \\ 0 & D[1] & \dots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \dots & D[n-1] \end{pmatrix}$Diagonal gates are useful as representations of Boolean functions, as they can map from $\{0,1\}^{2^n}$ to $\{0,1\}^{2^n}$ space. For example a phase oracle can be seen as a diagonal gate with $\{1, -1\}$ on the diagonals. Such an oracle will induce a $+1$ or :math`-1` phase on the amplitude of any corresponding basis state.

Diagonal gates appear in many classically hard oracular problems such as Forrelation or Hidden Shift circuits.

Diagonal gates are represented and simulated more efficiently than a dense $2^n \times 2^n$ unitary matrix.

The reference implementation is via the method described in Theorem 7 of [1]. The code is based on Emanuel Malvetti’s semester thesis at ETH in 2018, supervised by Raban Iten and Prof. Renato Renner.

**Reference:**

[1] Shende et al., Synthesis of Quantum Logic Circuits, 2009 arXiv:0406176

**Parameters**

**diag** (*Sequence[**complex**]*) – List of the $2^k$ diagonal entries (for a diagonal gate on $k$ qubits).

**Raises**

**CircuitError** – if the list of the diagonal entries or the qubit list is in bad format; if the number of diagonal entries is not $2^k$, where $k$ denotes the number of qubits.

## Attributes

### ancillas

A list of `AncillaQubit`

s in the order that they were added. You should not mutate this.

### calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form `{'gate_name': {(qubits, params): schedule}}`

### clbits

A list of `Clbit`

s in the order that they were added. You should not mutate this.

### data

The circuit data (instructions and context).

**Returns**

a list-like object containing the `CircuitInstruction`

s for each instruction.

**Return type**

QuantumCircuitData

### global_phase

The global phase of the current circuit scope in radians.

### instances

Default value: `160`

### layout

Return any associated layout information about the circuit

This attribute contains an optional `TranspileLayout`

object. This is typically set on the output from `transpile()`

or `PassManager.run()`

to retain information about the permutations caused on the input circuit by transpilation.

There are two types of permutations caused by the `transpile()`

function, an initial layout which permutes the qubits based on the selected physical qubits on the `Target`

, and a final layout which is an output permutation caused by `SwapGate`

s inserted during routing.

### metadata

Arbitrary user-defined metadata for the circuit.

Qiskit will not examine the content of this mapping, but it will pass it through the transpiler and reattach it to the output, so you can track your own metadata.

### num_ancillas

Return the number of ancilla qubits.

### num_captured_vars

The number of real-time classical variables in the circuit marked as captured from an enclosing scope.

This is the length of the `iter_captured_vars()`

iterable. If this is non-zero, `num_input_vars`

must be zero.

### num_clbits

Return number of classical bits.

### num_declared_vars

The number of real-time classical variables in the circuit that are declared by this circuit scope, excluding inputs or captures.

This is the length of the `iter_declared_vars()`

iterable.

### num_input_vars

The number of real-time classical variables in the circuit marked as circuit inputs.

This is the length of the `iter_input_vars()`

iterable. If this is non-zero, `num_captured_vars`

must be zero.

### num_parameters

The number of parameter objects in the circuit.

### num_qubits

Return number of qubits.

### num_vars

The number of real-time classical variables in the circuit.

This is the length of the `iter_vars()`

iterable.

### op_start_times

Return a list of operation start times.

This attribute is enabled once one of scheduling analysis passes runs on the quantum circuit.

**Returns**

List of integers representing instruction start times. The index corresponds to the index of instruction in `QuantumCircuit.data`

.

**Raises**

**AttributeError** – When circuit is not scheduled.

### parameters

The parameters defined in the circuit.

This attribute returns the `Parameter`

objects in the circuit sorted alphabetically. Note that parameters instantiated with a `ParameterVector`

are still sorted numerically.

**Examples**

The snippet below shows that insertion order of parameters does not matter.

```
>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
>>> circuit = QuantumCircuit(1)
>>> circuit.rx(b, 0)
>>> circuit.rz(elephant, 0)
>>> circuit.ry(a, 0)
>>> circuit.parameters # sorted alphabetically!
ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])
```

Bear in mind that alphabetical sorting might be unintuitive when it comes to numbers. The literal “10” comes before “2” in strict alphabetical sorting.

```
>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
>>> circuit = QuantumCircuit(1)
>>> circuit.u(*angles, 0)
>>> circuit.draw()
┌─────────────────────────────┐
q: ┤ U(angle_1,angle_2,angle_10) ├
└─────────────────────────────┘
>>> circuit.parameters
ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])
```

To respect numerical sorting, a `ParameterVector`

can be used.

```
>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
>>> x = ParameterVector("x", 12)
>>> circuit = QuantumCircuit(1)
>>> for x_i in x:
... circuit.rx(x_i, 0)
>>> circuit.parameters
ParameterView([
ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
..., ParameterVectorElement(x[11])
])
```

**Returns**

The sorted `Parameter`

objects in the circuit.

### prefix

Default value: `'circuit'`

### qubits

A list of `Qubit`

s in the order that they were added. You should not mutate this.

### name

Type: `str`

A human-readable name for the circuit.

### qregs

Type: `list[QuantumRegister]`

A list of the `QuantumRegister`

s in this circuit. You should not mutate this.

### cregs

Type: `list[ClassicalRegister]`

A list of the `ClassicalRegister`

s in this circuit. You should not mutate this.

### duration

Type: `int | float | None`

The total duration of the circuit, set by a scheduling transpiler pass. Its unit is specified by `unit`

.

### unit

The unit that `duration`

is specified in.