FermionicSuzukiTrotter

class FermionicSuzukiTrotter(order=2, reps=1)

Bases: FermionicEvolutionSynthesis

The higher-order Suzuki-Trotter product formula, applied in fermionic space.

A product formula composes the factors of an operator symmetrically, so that the leading error terms cancel. For \(\texttt{order} = 2\) and factors \(H_j\), the approximation is the palindrome

\[e^{-i t \sum_j H_j} \approx \left( \prod_{j

where \(\mathrm{R}\) denotes the reversed order. Higher even orders are built from this by Suzuki’s recursion, each level composing five sweeps of the level below.

The evolved operator is split by its groups when it has them (see Group operator terms: use the operator structure), and term-by-term otherwise. Every order splits it the same way: what a higher order changes is the ordering of the factors and their time scales, not the partition.

Note

A higher order buys accuracy with depth: an order-\(k\) formula emits roughly \(5^{(k-2)/2}\) times as many factors as an order-2 one. Increasing reps instead divides the evolution time into more, shorter steps, which reduces the error at a different rate for the same kind of cost. Which is the better trade depends on the operator.

Note

Applying this in fermionic space at all is optional, and it composes with the product formula chosen for the fermion-to-qubit stage. Where both approximate, the accuracy of the result is governed by the weaker of the two – so raising the order here while the qubit-side formula stays first-order buys little. See synthesis.

Caution

Each factor must be Hermitian for its exponential to be unitary, which does not follow from their sum being Hermitian: splitting a Hermitian operator can produce non-Hermitian groups (for example, separating \(a^\dagger_0 a_1\) from its conjugate partner \(a^\dagger_1 a_0\)). It is the caller’s responsibility to group accordingly; this is not verified.

Note that the symmetrization partially cancels the error of a non-Hermitian factor, so an incorrectly grouped operator can look markedly better at an even order than at first order while still being wrong.

Initializing an instance of this synthesis method can be done with the arguments below.

Parameters:

• order (int) – the order of the product formula. Must be 1 (which reduces to FermionicLieTrotter) or even, since the Suzuki formulas are symmetric.

• reps (int) – the number of times to repeat the formula, each repetition evolving for time / reps.

Raises:

ValueError – if order is neither 1 nor even, or if reps is not positive.

Attributes

order

The order of the product formula.

reps

The number of times the product formula is repeated.

Methods

synthesize(gate)

Synthesizes the gate into the ordered factors of the product formula.

See the class documentation for the formula this implements.

Parameters:

gate (Evolution) – the gate to synthesize.

Returns:

A FermionicCircuit holding one narrowed Evolution gate per factor of the formula. A factor may appear more than once, at different evolution times.

Return type:

FermionicCircuit