QFT

class QFT(num_qubits=None, approximation_degree=0, do_swaps=True, inverse=False, insert_barriers=False, name='qft')

Quantum Fourier Transform Circuit.

The Quantum Fourier Transform (QFT) on \(n\) qubits is the operation

\[|j\rangle \mapsto \frac{1}{2^{n/2}} \sum_{k=0}^{2^n - 1} e^{2\pi ijk / 2^n} |k\rangle\]

The circuit that implements this transformation can be implemented using Hadamard gates on each qubit, a series of controlled-U1 (or Z, depending on the phase) gates and a layer of Swap gates. The layer of Swap gates can in principle be dropped if the QFT appears at the end of the circuit, since then the re-ordering can be done classically. They can be turned off using the do_swaps attribute.

For 4 qubits, the circuit that implements this transformation is:

The inverse QFT can be obtained by calling the inverse method on this class. The respective circuit diagram is:

One method to reduce circuit depth is to implement the QFT approximately by ignoring controlled-phase rotations where the angle is beneath a threshold. This is discussed in more detail in https://arxiv.org/abs/quant-ph/9601018 or https://arxiv.org/abs/quant-ph/0403071.

Here, this can be adjusted using the approximation_degree attribute: the smallest approximation_degree rotation angles are dropped from the QFT. For instance, a QFT on 5 qubits with approximation degree 2 yields (the barriers are dropped in this example):

Construct a new QFT circuit.

Parameters

• num_qubits (Optional[int]) – The number of qubits on which the QFT acts.

• approximation_degree (int) – The degree of approximation (0 for no approximation).

• do_swaps (bool) – Whether to include the final swaps in the QFT.

• inverse (bool) – If True, the inverse Fourier transform is constructed.

• insert_barriers (bool) – If True, barriers are inserted as visualization improvement.

• name (str) – The name of the circuit.

Attributes

QFT.approximation_degree	The approximation degree of the QFT.
QFT.clbits	Returns a list of classical bits in the order that the registers were added.
QFT.data	Return the circuit data (instructions and context).
QFT.do_swaps	Whether the final swaps of the QFT are applied or not.
QFT.extension_lib
QFT.header
QFT.insert_barriers	Whether barriers are inserted for better visualization or not.
QFT.instances
QFT.n_qubits	Deprecated, use num_qubits instead.
QFT.num_clbits	Return number of classical bits.
QFT.num_parameters	Convenience function to get the number of parameter objects in the circuit.
QFT.num_qubits	The number of qubits in the QFT circuit.
QFT.parameters	Convenience function to get the parameters defined in the parameter table.
QFT.prefix
QFT.qregs	A list of the quantum registers associated with the circuit.
QFT.qubits	Returns a list of quantum bits in the order that the registers were added.

Methods

QFT.AND(qr_variables, qb_target, qr_ancillae)	Build a collective conjunction (AND) circuit in place using mct.
QFT.OR(qr_variables, qb_target, qr_ancillae)	Build a collective disjunction (OR) circuit in place using mct.
QFT.__getitem__(item)	Return indexed operation.
QFT.__len__()	Return number of operations in circuit.
QFT.add_register(*regs)	Add registers.
QFT.append(instruction[, qargs, cargs])	Append one or more instructions to the end of the circuit, modifying the circuit in place.
QFT.assign_parameters(param_dict[, inplace])	Assign parameters to new parameters or values.
QFT.barrier(*qargs)	Apply Barrier.
QFT.bind_parameters(value_dict)	Assign numeric parameters to values yielding a new circuit.
QFT.cast(value, _type)	Best effort to cast value to type.
QFT.cbit_argument_conversion(…)	Converts several classical bit representations (such as indexes, range, etc.) into a list of classical bits.
QFT.ccx(control_qubit1, control_qubit2, …)	Apply CCXGate.
QFT.ch(control_qubit, target_qubit, *[, …])	Apply CHGate.
QFT.cls_instances()	Return the current number of instances of this class, useful for auto naming.
QFT.cls_prefix()	Return the prefix to use for auto naming.
QFT.cnot(control_qubit, target_qubit, *[, …])	Apply CXGate.
QFT.combine(rhs)	Append rhs to self if self contains compatible registers.
QFT.compose(other[, qubits, clbits, front, …])	Compose circuit with other circuit or instruction, optionally permuting wires.
QFT.copy([name])	Copy the circuit.
QFT.count_ops()	Count each operation kind in the circuit.
QFT.crx(theta, control_qubit, target_qubit, *)	Apply CRXGate.
QFT.cry(theta, control_qubit, target_qubit, *)	Apply CRYGate.
QFT.crz(theta, control_qubit, target_qubit, *)	Apply CRZGate.
QFT.cswap(control_qubit, target_qubit1, …)	Apply CSwapGate.
QFT.cu1(theta, control_qubit, target_qubit, *)	Apply CU1Gate.
QFT.cu3(theta, phi, lam, control_qubit, …)	Apply CU3Gate.
QFT.cx(control_qubit, target_qubit, *[, …])	Apply CXGate.
QFT.cy(control_qubit, target_qubit, *[, …])	Apply CYGate.
QFT.cz(control_qubit, target_qubit, *[, …])	Apply CZGate.
QFT.dcx(qubit1, qubit2)	Apply DCXGate.
QFT.decompose()	Call a decomposition pass on this circuit, to decompose one level (shallow decompose).
QFT.depth()	Return circuit depth (i.e., length of critical path).
QFT.diag_gate(diag, qubit)	Deprecated version of QuantumCircuit.diagonal.
QFT.diagonal(diag, qubit)	Attach a diagonal gate to a circuit.
QFT.draw([output, scale, filename, style, …])	Draw the quantum circuit.
QFT.extend(rhs)	Append QuantumCircuit to the right hand side if it contains compatible registers.
QFT.fredkin(control_qubit, target_qubit1, …)	Apply CSwapGate.
QFT.from_qasm_file(path)	Take in a QASM file and generate a QuantumCircuit object.
QFT.from_qasm_str(qasm_str)	Take in a QASM string and generate a QuantumCircuit object.
QFT.h(qubit, *[, q])	Apply HGate.
QFT.hamiltonian(operator, time, qubits[, label])	Apply hamiltonian evolution to to qubits.
QFT.has_register(register)	Test if this circuit has the register r.
QFT.i(qubit, *[, q])	Apply IGate.
QFT.id(qubit, *[, q])	Apply IGate.
QFT.iden(qubit, *[, q])	Deprecated identity gate.
QFT.initialize(params, qubits)	Apply initialize to circuit.
QFT.inverse()	Invert this circuit.
QFT.is_inverse()	Whether the inverse Fourier transform is implemented.
QFT.iso(isometry, q_input, q_ancillas_for_output)	Attach an arbitrary isometry from m to n qubits to a circuit.
QFT.isometry(isometry, q_input, …[, …])	Attach an arbitrary isometry from m to n qubits to a circuit.
QFT.iswap(qubit1, qubit2)	Apply iSwapGate.
QFT.mcmt(gate, control_qubits, target_qubits)	Apply a multi-control, multi-target using a generic gate.
QFT.mcrx(theta, q_controls, q_target[, …])	Apply Multiple-Controlled X rotation gate
QFT.mcry(theta, q_controls, q_target, q_ancillae)	Apply Multiple-Controlled Y rotation gate
QFT.mcrz(lam, q_controls, q_target[, …])	Apply Multiple-Controlled Z rotation gate
QFT.mct(control_qubits, target_qubit[, …])	Apply MCXGate.
QFT.mcu1(lam, control_qubits, target_qubit)	Apply MCU1Gate.
QFT.mcx(control_qubits, target_qubit[, …])	Apply MCXGate.
QFT.measure(qubit, cbit)	Measure quantum bit into classical bit (tuples).
QFT.measure_active([inplace])	Adds measurement to all non-idle qubits.
QFT.measure_all([inplace])	Adds measurement to all qubits.
QFT.mirror()	Mirror the circuit by reversing the instructions.
QFT.ms(theta, qubits)	Apply MSGate.
QFT.num_connected_components([unitary_only])	How many non-entangled subcircuits can the circuit be factored to.
QFT.num_nonlocal_gates()	Return number of non-local gates (i.e.
QFT.num_tensor_factors()	Computes the number of tensor factors in the unitary (quantum) part of the circuit only.
QFT.num_unitary_factors()	Computes the number of tensor factors in the unitary (quantum) part of the circuit only.
QFT.qasm([formatted, filename])	Return OpenQASM string.
QFT.qbit_argument_conversion(…)	Converts several qubit representations (such as indexes, range, etc.) into a list of qubits.
QFT.r(theta, phi, qubit, *[, q])	Apply RGate.
QFT.rcccx(control_qubit1, control_qubit2, …)	Apply RC3XGate.
QFT.rccx(control_qubit1, control_qubit2, …)	Apply RCCXGate.
QFT.remove_final_measurements([inplace])	Removes final measurement on all qubits if they are present.
QFT.reset(qubit)	Reset q.
QFT.rx(theta, qubit, *[, label, q])	Apply RXGate.
QFT.rxx(theta, qubit1, qubit2)	Apply RXXGate.
QFT.ry(theta, qubit, *[, label, q])	Apply RYGate.
QFT.ryy(theta, qubit1, qubit2)	Apply RYYGate.
QFT.rz(phi, qubit, *[, q])	Apply RZGate.
QFT.rzx(theta, qubit1, qubit2)	Apply RZXGate.
QFT.rzz(theta, qubit1, qubit2)	Apply RZZGate.
QFT.s(qubit, *[, q])	Apply SGate.
QFT.sdg(qubit, *[, q])	Apply SdgGate.
QFT.size()	Returns total number of gate operations in circuit.
QFT.snapshot(label[, snapshot_type, qubits, …])	Take a statevector snapshot of the internal simulator representation.
QFT.snapshot_density_matrix(label[, qubits])	Take a density matrix snapshot of simulator state.
QFT.snapshot_expectation_value(label, op, qubits)	Take a snapshot of expectation value <O> of an Operator.
QFT.snapshot_probabilities(label, qubits[, …])	Take a probability snapshot of the simulator state.
QFT.snapshot_stabilizer(label)	Take a stabilizer snapshot of the simulator state.
QFT.snapshot_statevector(label)	Take a statevector snapshot of the simulator state.
QFT.squ(unitary_matrix, qubit[, mode, …])	Decompose an arbitrary 2*2 unitary into three rotation gates.
QFT.swap(qubit1, qubit2)	Apply SwapGate.
QFT.t(qubit, *[, q])	Apply TGate.
QFT.tdg(qubit, *[, q])	Apply TdgGate.
QFT.to_gate([parameter_map])	Create a Gate out of this circuit.
QFT.to_instruction([parameter_map])	Create an Instruction out of this circuit.
QFT.toffoli(control_qubit1, control_qubit2, …)	Apply CCXGate.
QFT.u1(theta, qubit, *[, q])	Apply U1Gate.
QFT.u2(phi, lam, qubit, *[, q])	Apply U2Gate.
QFT.u3(theta, phi, lam, qubit, *[, q])	Apply U3Gate.
QFT.uc(gate_list, q_controls, q_target[, …])	Attach a uniformly controlled gates (also called multiplexed gates) to a circuit.
QFT.ucg(angle_list, q_controls, q_target[, …])	Deprecated version of uc.
QFT.ucrx(angle_list, q_controls, q_target)	Attach a uniformly controlled (also called multiplexed) Rx rotation gate to a circuit.
QFT.ucry(angle_list, q_controls, q_target)	Attach a uniformly controlled (also called multiplexed) Ry rotation gate to a circuit.
QFT.ucrz(angle_list, q_controls, q_target)	Attach a uniformly controlled (also called multiplexed gates) Rz rotation gate to a circuit.
QFT.ucx(angle_list, q_controls, q_target)	Deprecated version of ucrx.
QFT.ucy(angle_list, q_controls, q_target)	Deprecated version of ucry.
QFT.ucz(angle_list, q_controls, q_target)	Deprecated version of ucrz.
QFT.unitary(obj, qubits[, label])	Apply unitary gate to q.
QFT.width()	Return number of qubits plus clbits in circuit.
QFT.x(qubit, *[, label, ctrl_state, q])	Apply XGate.
QFT.y(qubit, *[, q])	Apply YGate.
QFT.z(qubit, *[, q])	Apply ZGate.
QFT.__getitem__(item)	Return indexed operation.
QFT.__len__()	Return number of operations in circuit.