TernaryTreeEncoding

class TernaryTreeEncoding(tree, mode_map=None)

Bases: object

A ternary tree compiled into the Pauli images of its Majorana operators.

Pairing the tree’s legs into Majorana operators and building their Pauli strings is done once, here, rather than per mapped operator. Pass one of these to the ternary-tree mappers.

The pairing follows Algorithm 1 of the Bonsai paper [1]: from each node, descend its X link and then Z links until reaching a leg, which carries $\gamma$ of that node’s mode; the Y link likewise gives $\gamma'$ . This is what makes $|0\cdots0\rangle$ the fermionic vacuum, so it is not an adjustable convention.

[1]

A. Miller, Z. Zimborás, S. Knecht, S. Maniscalco and G. García-Pérez, Bonsai algorithm: grow your own fermion-to-qubit mappings, PRX Quantum 4, 030314 (2023), arXiv:2212.09731.

Usage

>>> from qiskit_fermions.mappers import TernaryTree, TernaryTreeEncoding
>>> enc = TernaryTreeEncoding(TernaryTree.breadth_first(4, 3))
>>> enc.num_modes, enc.max_weight
(4, 2)

Two encodings compare equal when they assign the same Pauli string to every Majorana operator, which is what lets a set or dictionary deduplicate them. Leaving out the mode map is the same as giving the identity, whereas any other one changes which Majorana operator each string belongs to:

>>> tree = TernaryTree.chain(3, "Z")
>>> TernaryTreeEncoding(tree) == TernaryTreeEncoding(tree, [0, 1, 2])
True
>>> TernaryTreeEncoding(tree) == TernaryTreeEncoding(tree, [2, 1, 0])
False

The encoding is built once and reused across every operator mapped through it, so the pairing and string construction are not repeated per call. The qubit count of a mapped operator is given to the mapper rather than fixed here, so one encoding serves registers of any width from num_modes upwards.

An encoding cannot be pickled. It is a derived artifact: compiling discards the tree and the mode map it was built from, so supporting pickle would mean retaining them purely to serialize them. Pickle the TernaryTree (and the mode map) and compile again instead.

Attributes

max_weight

The largest Pauli weight of any single Majorana operator.

Always equal to the TernaryTree.max_weight of the tree it was compiled from: the leg left unpaired is never deeper than the deepest paired one.

num_modes

The number of fermionic modes this encoding covers.

Methods

majorana_image

majorana_image(majorana)

Return the Pauli image of Majorana operator majorana, as (qubit, label) pairs.

The index is 2 * mode + is_prime, matching gamma(). The string is ordered by ascending qubit.

>>> from qiskit_fermions.mappers import TernaryTree, TernaryTreeEncoding
>>> enc = TernaryTreeEncoding(TernaryTree.chain(3, "Z"))
>>> enc.majorana_image(4)
[(0, 'Z'), (1, 'Z'), (2, 'X')]

The coefficient is always exactly $1$ : a leg’s string is a product of Paulis on distinct qubits, so it is Hermitian and squares to the identity with no phase to correct.

Parameters

majorana – the Majorana index, 2 * mode + is_prime.

Raises

ValueError – if majorana is not below 2 * num_modes.
OverflowError – if majorana is negative or does not fit in 32 bits.

total_parity

total_parity(num_qubits)

Build the total fermionic parity observable of this encoding.

A tree’s $2N+1$ legs pair into $2N$ Majorana operators, leaving exactly one over, and that leftover is the total parity $\prod_j (1 - 2 n_j)$ .

Note

This is an observable to measure, not a constraint to impose. Both of its eigenvalues are physical, so do not project a state onto the $+1$ sector: a ternary-tree encoding spends one qubit per mode and is a unitary isomorphism onto the whole $2^N$ -dimensional space, with both parity sectors represented at multiplicity $2^{N-1}$ and no unphysical subspace to project away.

The string is not in general the all- $Z$ string. It is the leg reached by the all- $Z$ path, which carries the identity on every node not on that path. On the Jordan-Wigner chain it is the full $Z$ string:

>>> from qiskit_fermions.mappers import TernaryTree, TernaryTreeEncoding
>>> TernaryTreeEncoding(TernaryTree.chain(4, "Z")).total_parity(4)
<SparseObservable with 1 term on 4 qubits: (1+0j)(Z_3 Z_2 Z_1 Z_0)>

On the balanced tree at four modes it is not:

>>> TernaryTreeEncoding(TernaryTree.breadth_first(4, 3)).total_parity(4)
<SparseObservable with 1 term on 4 qubits: (1+0j)(Z_3 Z_0)>

Only the support depends on the tree: the coefficient is $+1$ on every tree and under every mode map.

Parameters

num_qubits – the number of qubits for the resulting observable. Must be at least the encoding’s mode count; a larger value pads with the identity.

Returns

a single $Z$ string with coefficient $+1$ .

Return type

The total-parity observable

Raises

ValueError – if num_qubits is below the encoding’s mode count.
OverflowError – if num_qubits is negative or does not fit in 32 bits.

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