# qiskit.opflow.converters.PauliBasisChange¶

class PauliBasisChange(destination_basis=None, traverse=True, replacement_fn=None)[Quellcode]

Converter for changing Paulis into other bases. By default, the diagonal basis composed only of Pauli {Z, I}^n is used as the destination basis to which to convert. Meaning, if a Pauli containing X or Y terms is passed in, which cannot be sampled or evolved natively on some Quantum hardware, the Pauli can be replaced by a composition of a change of basis circuit and a Pauli composed of only Z and I terms (diagonal), which can be evolved or sampled natively on the Quantum hardware.

The replacement function determines how the PauliOps should be replaced by their computed change-of-basis CircuitOps and destination PauliOps. Several convenient out-of-the-box replacement functions have been added as static methods, such as measurement_replacement_fn.

This class uses the typical basis change method found in most Quantum Computing textbooks (such as on page 210 of Nielsen and Chuang’s, „Quantum Computation and Quantum Information“, ISBN: 978-1-107-00217-3), which involves diagonalizing the single-qubit Paulis with H and S† gates, mapping the eigenvectors of the diagonalized origin Pauli to the diagonalized destination Pauli using CNOTS, and then de-diagonalizing any single qubit Paulis to their non-diagonal destination values. Many other methods are possible, as well as variations on this method, such as the placement of the CNOT chains.

Parameter
• destination_basis (Union[Pauli, PauliOp, None]) – The Pauli into the basis of which the operators will be converted. If None is specified, the destination basis will be the diagonal ({I, Z}^n) basis requiring only single qubit rotations.

• traverse (bool) – If true and the operator passed into convert contains sub-Operators, such as ListOp, traverse the Operator and apply the conversion to every applicable sub-operator within it.

• replacement_fn (Optional[Callable]) –

A function specifying what to do with the basis-change CircuitOp and destination PauliOp when converting an Operator and replacing converted values. By default, this will be

1. For StateFns (or Measurements): replacing the StateFn with ComposedOp(StateFn(d), c) where c is the conversion circuit and d is the destination Pauli, so the overall beginning and ending operators are equivalent.

2. For non-StateFn Operators: replacing the origin p with c·d·c†, where c is the conversion circuit and d is the destination, so the overall beginning and ending operators are equivalent.

__init__(destination_basis=None, traverse=True, replacement_fn=None)[Quellcode]
Parameter
• destination_basis (Union[Pauli, PauliOp, None]) – The Pauli into the basis of which the operators will be converted. If None is specified, the destination basis will be the diagonal ({I, Z}^n) basis requiring only single qubit rotations.

• traverse (bool) – If true and the operator passed into convert contains sub-Operators, such as ListOp, traverse the Operator and apply the conversion to every applicable sub-operator within it.

• replacement_fn (Optional[Callable]) –

A function specifying what to do with the basis-change CircuitOp and destination PauliOp when converting an Operator and replacing converted values. By default, this will be

1. For StateFns (or Measurements): replacing the StateFn with ComposedOp(StateFn(d), c) where c is the conversion circuit and d is the destination Pauli, so the overall beginning and ending operators are equivalent.

2. For non-StateFn Operators: replacing the origin p with c·d·c†, where c is the conversion circuit and d is the destination, so the overall beginning and ending operators are equivalent.

Methods

 __init__([destination_basis, traverse, …]) type destination_basis Union[Pauli, PauliOp, None] construct_cnot_chain(diag_pauli_op1, …) Construct a CircuitOp (or PauliOp if equal to the identity) which takes the eigenvectors of diag_pauli_op1 to the eigenvectors of diag_pauli_op2, assuming both are diagonal (or performing this operation on their diagonalized Paulis implicitly if not). convert(operator) Given a PauliOp, or an Operator containing PauliOps if _traverse is True, converts each Pauli into the basis specified by self._destination and a basis-change-circuit, calls replacement_fn with these two Operators, and replaces the PauliOps with the output of replacement_fn. get_cob_circuit(origin) Construct an Operator which maps the +1 and -1 eigenvectors of the origin Pauli to the +1 and -1 eigenvectors of the destination Pauli. get_diagonal_pauli_op(pauli_op) Get the diagonal PualiOp to which pauli_op could be rotated with only single-qubit operations. Construct a CircuitOp with only single-qubit gates which takes the eigenvectors of pauli to eigenvectors composed only of |0⟩ and |1⟩ tensor products. get_tpb_pauli(list_op) Gets the Pauli (not PauliOp!) whose diagonalizing single-qubit rotations is a superset of the diagonalizing single-qubit rotations for each of the Paulis in list_op. measurement_replacement_fn(cob_instr_op, …) A built-in convenience replacement function which produces measurements isomorphic to an OperatorStateFn measurement holding the origin PauliOp. operator_replacement_fn(cob_instr_op, …) A built-in convenience replacement function which produces Operators isomorphic to the origin PauliOp. pad_paulis_to_equal_length(pauli_op1, pauli_op2) If pauli_op1 and pauli_op2 do not act over the same number of qubits, pad identities to the end of the shorter of the two so they are of equal length. statefn_replacement_fn(cob_instr_op, …) A built-in convenience replacement function which produces state functions isomorphic to an OperatorStateFn state function holding the origin PauliOp.

Attributes

 destination The destination PauliOp, or None if using the default destination, the diagonal basis.
construct_cnot_chain(diag_pauli_op1, diag_pauli_op2)[Quellcode]

Construct a CircuitOp (or PauliOp if equal to the identity) which takes the eigenvectors of diag_pauli_op1 to the eigenvectors of diag_pauli_op2, assuming both are diagonal (or performing this operation on their diagonalized Paulis implicitly if not). This works by the insight that the eigenvalue of a diagonal Pauli’s eigenvector is equal to or -1 if the parity is 1 and 1 if the parity is 0, or 1 - (2 * parity). Therefore, using CNOTs, we can write the parity of diag_pauli_op1’s significant bits onto some qubit, and then write out that parity onto diag_pauli_op2’s significant bits.

Parameter
• diag_pauli_op1 (PauliOp) – The origin PauliOp.

• diag_pauli_op2 (PauliOp) – The destination PauliOp.

Rückgabetyp

PrimitiveOp

Rückgabe

The PrimitiveOp performs the mapping.

convert(operator)[Quellcode]

Given a PauliOp, or an Operator containing PauliOps if _traverse is True, converts each Pauli into the basis specified by self._destination and a basis-change-circuit, calls replacement_fn with these two Operators, and replaces the PauliOps with the output of replacement_fn. For example, for the built-in operator_replacement_fn below, each PauliOp p will be replaced by the composition of the basis-change Clifford CircuitOp c with the destination PauliOp d and c†, such that p = c·d·c†, up to global phase.

Parameter

operator (OperatorBase) – The Operator to convert.

Rückgabetyp

OperatorBase

Rückgabe

The converted Operator.

property destination

The destination PauliOp, or None if using the default destination, the diagonal basis.

Rückgabetyp

Optional[PauliOp]

get_cob_circuit(origin)[Quellcode]

Construct an Operator which maps the +1 and -1 eigenvectors of the origin Pauli to the +1 and -1 eigenvectors of the destination Pauli. It does so by

1. converting any |i+⟩ or |i+⟩ eigenvector bits in the origin to |+⟩ and |-⟩ with S†s, then

2. converting any |+⟩ or |+⟩ eigenvector bits in the converted origin to |0⟩ and |1⟩ with Hs, then

3. writing the parity of the significant (Z-measured, rather than I) bits in the origin to a single „origin anchor bit,“ using cnots, which will hold the parity of these bits,

4. swapping the parity of the pauli anchor bit into a destination anchor bit using a swap gate (only if they are different, if there are any bits which are significant in both origin and dest, we set both anchors to one of these bits to avoid a swap).

5. writing the parity of the destination anchor bit into the other significant bits of the destination,

6. converting the |0⟩ and |1⟩ significant eigenvector bits to |+⟩ and |-⟩ eigenvector bits in the destination where the destination demands it (e.g. pauli.x == true for a bit), using Hs 8) converting the |+⟩ and |-⟩ significant eigenvector bits to |i+⟩ and |i-⟩ eigenvector bits in the destination where the destination demands it (e.g. pauli.x == true and pauli.z == true for a bit), using Ss

Parameter

origin (Union[Pauli, PauliOp]) – The Pauli or PauliOp to map.

Rückgabetyp

Tuple[PrimitiveOp, PauliOp]

Rückgabe

A tuple of a PrimitiveOp which equals the basis change mapping and a PauliOp which equals the destination basis.

Verursacht
• TypeError – Attempting to convert from non-Pauli origin.

• ValueError – Attempting to change a non-identity Pauli to an identity Pauli, or vice versa.

get_diagonal_pauli_op(pauli_op)[Quellcode]

Get the diagonal PualiOp to which pauli_op could be rotated with only single-qubit operations.

Parameter

pauli_op (PauliOp) – The PauliOp whose diagonal to compute.

Rückgabetyp

PauliOp

Rückgabe

The diagonal PauliOp.

get_diagonalizing_clifford(pauli)[Quellcode]

Construct a CircuitOp with only single-qubit gates which takes the eigenvectors of pauli to eigenvectors composed only of |0⟩ and |1⟩ tensor products. Equivalently, finds the basis-change circuit to take pauli to a diagonal PauliOp composed only of Z and I tensor products.

Note, underlying Pauli bits are in Qiskit endianness, so we need to reverse before we begin composing with Operator flow.

Parameter

pauli (Union[Pauli, PauliOp]) – the Pauli or PauliOp to whose diagonalizing circuit to compute.

Rückgabetyp

OperatorBase

Rückgabe

The diagonalizing CircuitOp.

get_tpb_pauli(list_op)[Quellcode]

Gets the Pauli (not PauliOp!) whose diagonalizing single-qubit rotations is a superset of the diagonalizing single-qubit rotations for each of the Paulis in list_op. TPB stands for Tensor Product Basis.

Parameter

list_op (ListOp) – the ListOp whose TPB Pauli to return.

Rückgabetyp

Pauli

Rückgabe

The TBP Pauli.

static measurement_replacement_fn(cob_instr_op, dest_pauli_op)[Quellcode]

A built-in convenience replacement function which produces measurements isomorphic to an OperatorStateFn measurement holding the origin PauliOp.

Parameter
• cob_instr_op (PrimitiveOp) – The basis-change CircuitOp.

• dest_pauli_op (Union[PauliOp, PauliSumOp, ListOp]) – The destination Pauli type operator.

Rückgabetyp

OperatorBase

Rückgabe

The ~StateFn @ CircuitOp composition equivalent to a measurement by the original PauliOp.

static operator_replacement_fn(cob_instr_op, dest_pauli_op)[Quellcode]

A built-in convenience replacement function which produces Operators isomorphic to the origin PauliOp.

Parameter
• cob_instr_op (PrimitiveOp) – The basis-change CircuitOp.

• dest_pauli_op (Union[PauliOp, PauliSumOp, ListOp]) – The destination PauliOp.

Rückgabetyp

OperatorBase

Rückgabe

The ~CircuitOp @ PauliOp @ CircuitOp composition isomorphic to the original PauliOp.

pad_paulis_to_equal_length(pauli_op1, pauli_op2)[Quellcode]

If pauli_op1 and pauli_op2 do not act over the same number of qubits, pad identities to the end of the shorter of the two so they are of equal length. Padding is applied to the end of the Paulis. Note that the Terra represents Paulis in big-endian order, so this will appear as padding to the beginning of the Pauli x and z bit arrays.

Parameter
• pauli_op1 (PauliOp) – A pauli_op to possibly pad.

• pauli_op2 (PauliOp) – A pauli_op to possibly pad.

Rückgabetyp

Tuple[PauliOp, PauliOp]

Rückgabe

A tuple containing the padded PauliOps.

static statefn_replacement_fn(cob_instr_op, dest_pauli_op)[Quellcode]

A built-in convenience replacement function which produces state functions isomorphic to an OperatorStateFn state function holding the origin PauliOp.

Parameter
• cob_instr_op (PrimitiveOp) – The basis-change CircuitOp.

• dest_pauli_op (Union[PauliOp, PauliSumOp, ListOp]) – The destination Pauli type operator.

Rückgabetyp

OperatorBase

Rückgabe

The ~CircuitOp @ StateFn composition equivalent to a state function defined by the original PauliOp.