qcircuits.state module¶

The state module contains the State class, instances of which represent quantum states of multi-qubit systems, and factory functions for creating specific quantum states.

The State class and each of the factory functions used to create states are aliased at the top-level module, so that, for example, one can call qcircuits.zeros() instead of qcircuits.state.zeros().

class qcircuits.state.State(tensor)¶

Bases: qcircuits.tensors.Tensor

A container class for a tensor representing the state of a quantum system, and associated methods.

Parameters:	tensor (numpy complex128 multidimensional array) – The tensor representing the quantum state, giving the probability amplitudes.

amplitudes¶

Get the state tensor, i.e., the probability amplitudes.

Returns:	The probability amplitudes of the state.
Return type:	numpy complex128 multidimensional array

density_operator()¶

Compute the density operator for this pure state.

Returns:	The density operator of this pure state.
Return type:	DensityOperator

dot(arg)¶

Give the dot product between this and another State.

Parameters:	arg (State) – The state with which we take the dot product.
Returns:	The dot product with State arg.
Return type:	complex

static from_column_vector(v)¶

QCircuits represents d-qubit states by type (d, 0) tensors. This function constructs a state from the more common column vector Kronecker-product representation of the state.

Parameters:	complex128 multidimensional array (numpy) – The column vector representation of the state, of size 2**d.
Returns:	A d-qubit state.
Return type:	State

measure(qubit_indices=None, remove=False)¶

Measure the state with respect to the computational bases of the qubits indicated by qubit_indices. Measuring a state will modify the state in-place. If no indices are indicated, the whole state is measured.

Parameters:	qubit_indices (int or iterable) – An index or indices indicating the qubit(s) whose computational bases the measurement of the state will be made with respect to. If no qubit_indices are given, the whole state is measured. remove (bool) – Indicates whether the measured qubits should be removed from the state vector.
Returns:	The measurement outcomes for the measured qubit(s). If the qubit_indices parameter is supplied as an int, an int is returned, otherwise a tuple.
Return type:	int or tuple of int

permute_qubits(axes, inverse=False)¶

Permute the qubits in the state. Equivalent to crossing wires in a circuit. This method modifies the state in-place, but also returns the resulting state to allow chaining of operations.

Parameters:	axes (list of int) – Permute the qubits according to the values given. inverse (bool) – If true, perform the inverse permutation of the qubits.
Returns:	The resulting state.
Return type:	State

probabilities¶

Get the probability of observing each computational basis vector upon making a measurement.

Returns:	The probability associated with each computational basis vector.
Return type:	numpy float64 multidimensional array

reduced_density_operator(qubit_indices)¶

Compute the reduced density operator of the given qubits by tracing out the qubits not given.

Parameters:	qubit_indices (iterable) – Indices indicating the qubit(s) that we compute the reduced density operator of.
Returns:	The reduced density operator of the sub-system.
Return type:	DensityOperator

renormalize_()¶

Renormalize the state so that the sum of squared amplitude magnitudes is 1.

schmidt_number(indices)¶

For a d-qubit state (d>1) in vector space \(A\otimes B\), get the Schmidt number, a measure of entanglement, for the Schmidt decomposition of that state for the subsystems A and B.

Parameters:	indices (list of int) – The qubit indices for one of the subsystems. This should include at least one qubit index and also exclude at least one index.
Returns:	The Schmidt number.
Return type:	int

swap_qubits(axis1, axis2)¶

Swap two qubits in the state. Equivalent to crossing wires in a circuit. This method modifies the state in-place, but also returns the resulting state to allow chaining of operations.

Parameters:	axis1 (int) – First axis. axis2 (int) – Second axis.
Returns:	The resulting state.
Return type:	State

to_column_vector()¶

QCircuits represents d-qubit states by type (d, 0) tensors. This function returns the more common column vector Kronecker-product representation of the state.

Returns:	The column vector representation of the state.
Return type:	numpy complex128 multidimensional array

qcircuits.state.bell_state(a=0, b=0)¶

Produce one of the four Bell states for a two-qubit system, \(|\beta_{ab}⟩\).

Parameters:	a ({0, 1}) – The computational basis state of the first qubit before entanglement. b ({0, 1}) – The computational basis state of the second qubit before entanglement.
Returns:	A rank 2 tensor describing one of the four two-qubit Bell states. E.g., \(|\beta_{00}⟩ = \frac{1}{\sqrt{2}} (|00⟩ + |11⟩)\)
Return type:	State

See also

qubit(), zeros(), ones(), bitstring(), positive_superposition()

qcircuits.state.bitstring(*bits)¶

Produce a computational basis state from a given bit sequence.

Parameters:	bits – A variable number of arguments each in {0, 1}.
Returns:	If d arguments are supplied, returns a rank d tensor describing the computational basis state given by the sequence of input bits.
Return type:	State

See also

qubit(), zeros(), ones(), positive_superposition(), bell_state()

qcircuits.state.ones(d=1)¶

Produce the all-one computational basis vector for d qubits. I.e., produces \(|1⟩^{\otimes d}\).

Parameters:	d (int) – The number of qubits d for which we produce a computational basis vector, and the rank of the produced tensor.
Returns:	A d-rank tensor describing the all-one d-qubit computational basis vector, \(|1⟩^{\otimes d}\)
Return type:	State

See also

qubit(), zeros(), bitstring(), positive_superposition(), bell_state()

qcircuits.state.positive_superposition(d=1)¶

Produce the positive superposition for a d qubit system, i.e., the state resulting from applying the Hadamard() gate to each of d |0⟩ computational basis states, \(H^{\otimes d}(|0⟩^{\otimes d}) = |+⟩^{\otimes d}.\)

Parameters:	d (int) – The number of qubits d for the state.
Returns:	A d-rank tensor describing the positive superposition for a d-qubit system, \(|+⟩^{\otimes d}.\)
Return type:	State

See also

qubit(), zeros(), ones(), bitstring(), bell_state()

qcircuits.state.qubit(*, alpha=None, beta=None, theta=None, phi=None, global_phase=0.0)¶

Produce a given state for a single qubit.

Parameters:	alpha (float) – The probability amplitude for the |0⟩ basis vector. Should not be specified in conjunction with theta or phi. beta (float) – The probability amplitude for the |1⟩ basis vector. Should not be specified in conjunction with theta or phi. theta (float) – The angle of the state between |0⟩ and |1⟩ on the Bloch sphere. Should not be specified in conjunction with alpha or beta. phi (float) – The phase of the |1⟩ component. Should not be specified in conjunction with alpha or beta. global_phase (float) – A global phase applied to the state.
Returns:	A rank 1 tensor describing the state of a single qubit.
Return type:	State

See also

zeros(), ones(), bitstring(), positive_superposition(), bell_state()

qcircuits.state.zeros(d=1)¶

Produce the all-zero computational basis vector for d qubits. I.e., produces \(|0⟩^{\otimes d}\).

Parameters:	d (int) – The number of qubits d for which we produce a computational basis vector, and the rank of the produced tensor.
Returns:	A d-rank tensor describing the all-zero d-qubit computational basis vector, \(|0⟩^{\otimes d}\)
Return type:	State

See also

qubit(), ones(), bitstring(), positive_superposition(), bell_state()