Generate SqDRIFT circuits

With sample-based quantum diagonalization (SQD), you must choose an ansatz from which to sample bitstrings. The SqDRIFT variant uses an ensemble of time evolution circuits constructed directly from the target Hamiltonian instead. This is achieved by subsampling smaller time evolution operators from the Hamiltonian based on its coefficients, which is known as the qDRIFT Trotterization method.

This getting started guide shows how to generate an ensemble of such randomized circuits.

1. Hamiltonian setup

For the purposes of this guide, load the electronic structure Hamiltonian of N2 from an FCIDUMP file. There are other means of constructing the FermionOperator. Be sure to consult its documentation, as well as the qiskit_fermions.operators.library.

[x] PYTHON

>>> from qiskit_fermions.operators.library import FCIDump
>>> from qiskit_fermions.operators import FermionOperator
>>>
>>> fcidump = FCIDump.from_file("docs/guides/n2.fcidump")
>>> num_modes = 2 * fcidump.norb
>>> hamil = FermionOperator.from_fcidump(fcidump)

[ ] C

#include <qiskit_fermions.h>

QfFCIDump* fcidump = NULL;
qf_fcidump_from_file("docs/guides/n2.fcidump", &fcidump);
QfFermionOperator* hamil = qf_ferm_op_from_fcidump(fcidump);
uint32_t num_modes = 2 * qf_fcidump_norb(fcidump);

2. Group Hamiltonian terms

Use the many symmetries that are present in the electronic structure Hamiltonian by grouping related terms that have identical coefficients. This action changes the operator coefficient distribution that the qDRIFT protocol samples from, but it does not affect its convergence guarantees. Crucially, grouping terms that are related by symmetry results in a favorable cancellation of Pauli terms, resulting in an overall shorter circuit depth when time evolving a state under their action.

The qiskit_fermions.operators.terms.grouping module provides convenience functions for grouping an operator’s terms. This is explained in more detail in this guide.

Caution

The implementation of the group_terms_by_electronic_structure() assumes the terms of the Hamiltonian to be normal-ordered!

[x] PYTHON

>>> from qiskit_fermions.operators.terms.grouping import group_terms_by_electronic_structure
>>> from qiskit_fermions.operators.terms.ordering import canonical_order
>>>
>>> canon = canonical_order(hamil.normal_ordered().simplify(atol=1e-16))
>>> exit_code = group_terms_by_electronic_structure(canon, num_modes, two_body_physicist_order=False)
>>> assert exit_code is None
>>> print(canon.groups)  # the groups attribute now contains some list of group indices
[0, ...]

[ ] C

QfFermionOperator* normal = qf_ferm_op_normal_ordered(hamil, NULL);
QfFermionOperator* simplified = qf_ferm_op_simplify(normal, 1e-16);
QfFermionOperator* canon = qf_ferm_op_canonical_order(simplified);
QfExitCode exit = qf_ferm_op_group_terms_by_electronic_structure(canon, num_modes, false);
assert(exit == QfExitCode_Success);

qf_ferm_op_free(normal);
qf_ferm_op_free(simplified);

Hint

The full electronic structure Hamiltonian contains certain terms whose inclusion in a time-evolution circuit has no impact on the perceived bitstrings and, thus, only results in an increased sampling overhead. Therefore, it is recommended that such terms be filtered from the Hamiltonian at this point, before constructing the Evolution gate in the next step.

The terms that fit this description are those that are diagonal in the occupation-number basis, that is, the products of number operators ( $a^\dagger_i a_i$ ). This includes the constant energy offset, whose time evolution only introduces a global phase into the circuit, the individual number operators whose time evolution amounts to single-qubit Z rotations, as well as higher-order products such as $n_i n_j$ . None of them can change a mode’s occupation, so dropping them costs no excitation content. Unlike the per-draw rejection in step 5, this filtering happens once, on the Hamiltonian itself, so the coefficient distribution the protocol samples from is rebuilt consistently from the filtered operator rather than renormalized mid-draw.

The filter_diagonal_terms() function removes such terms from an operator in place:

[x] PYTHON

>>> from qiskit_fermions.operators.terms.filtering import filter_diagonal_terms
>>>
>>> filter_diagonal_terms(canon)

[ ] C

qf_ferm_op_filter_diagonal_terms(canon);

Filtering here, once, is considerably cheaper than filtering repeatedly. QDriftTrotterization runs once per transpiled circuit, so filtering the Hamiltonian initially, rather than on every call, avoids redoing work for every circuit generated from it.

Hint

For the same reason, it pays to order the Hamiltonian’s terms by group index here. Group indices are a per-term tag that says nothing about where those terms sit, so a group’s terms are in general scattered throughout the operator: the grouping applied above assigns indices in order of first encounter, which leaves a group’s second term far from its first.

QDriftTrotterization samples whole groups and looks each drawn group up with split_out_groups(). Finding a group among scattered terms means scanning all of them, so that lookup costs the same whether one group is drawn or forty. group_order() gathers each group into one contiguous run, after which a lookup is a binary search over the group boundaries and costs what the drawn groups cost rather than what the held terms cost. The lookup itself becomes dramatically cheaper; how much of a given transpilation that saves depends on what else the pipeline does, since mapping and synthesis dominate once the scan is gone:

[x] PYTHON

>>> from qiskit_fermions.operators.terms.ordering import group_order
>>>
>>> canon = group_order(canon)
>>> print(canon.groups[:4])
[0, 0, ...]

[ ] C

QfFermionOperator* grouped = qf_ferm_op_group_order(canon);

This reorders the terms without changing the operator’s value, and needs to happen only once no matter how many circuit randomizations are drawn from it.

3. Prepare the time evolution circuit

Prepare the Hamiltonian’s time evolution circuit and the base circuit from which to draw samples. The qiskit_fermions.circuit.library contains all the required components to do so, in compliance with Qiskit conventions.

[x] PYTHON

>>> from qiskit_fermions.circuit import FermionicCircuit
>>> from qiskit_fermions.circuit.library import Evolution
>>>
>>> time = 1.0  # you can choose a desired scaling factor here
>>> evo_gate = Evolution(num_modes, canon, time)
>>>
>>> circ = FermionicCircuit(num_modes)
>>> circ.append(evo_gate, circ.modes)

[ ] C

// WARNING: Qiskit's C API does not yet allow us to implement circuits
// with custom gate definitions.

Note

This example neither initializes the fermionic modes with particles, nor measures their final state.

4. Transpile the circuit with QDrift Trotterization

The qiskit_fermions.transpiler module integrates directly with Qiskit’s transpilation pipeline, allowing the FermionicCircuit constructed above to be directly transpiled to a QuantumCircuit.

Use the jordan_wigner() fermion-to-qubit mapping to convert the Hamiltonian expressed in terms of fermions to be expressed in Pauli strings instead. This can be done directly as part of the transpilation process by using the EvolutionSynthesis transpilation pass plugin. Use generate_preset_jw_pass_manager() to build FermionicStagedPassManager, which ensures that the Jordan-Wigner encoding is used consistently for all circuit instructions.

Crucially, add the QDriftTrotterization transpilation pass to the optimization stage of the transpilation pipeline. This ensures that the circuit does not use the time evolution of the entire Hamiltonian, whose depth would exceed the capabilities of currently available quantum computing hardware.

Instead, it subsamples a fixed number of groups of Hamiltonian terms for each circuit, every time the circuit is transpiled. Through this, you can generate multiple circuit randomizations as required by the SqDRIFT algorithm by repeatedly running the transpilation pipeline.

This step also introduces the few parameters you can use to customize the circuits to generate:

The number of circuits to generate: num_sqdrift_randomizations
The length of each circuit in terms of excitation groups: num_groups
Optionally, precomputed sampling weights, if the transpilation itself becomes a bottleneck: weights (see (Optional) Precompute the sampling weights below)

[x] PYTHON

>>> from qiskit_fermions.transpiler import FermionicPassManager
>>> from qiskit_fermions.transpiler.presets import generate_preset_jw_pass_manager
>>> from qiskit_fermions.transpiler.passes import QDriftTrotterization
>>>
>>> num_groups = 10
>>> qdrift = QDriftTrotterization(num_groups, rng=19)
>>>
>>> pm = generate_preset_jw_pass_manager()
>>> pm.optimization = FermionicPassManager([qdrift])
>>>
>>> num_sqdrift_randomizations = 10
>>> sqdrift_circuits = [
...     pm.run(circ) for _ in range(num_sqdrift_randomizations)
... ]

[ ] C

// WARNING: Qiskit's C API does not yet allow us to implement circuits
// with custom gate definitions, which we therefore also cannot transpile
// via this API.

Note

The preceding example fixes the seed for the random number generator used inside of the QDriftTrotterization transpilation pass.

5. Filter drawn excitations by occupation

Beyond the diagonal terms filtered out in the previous step, a drawn excitation can act entirely within a set of modes whose occupation is already fixed (all occupied or all unoccupied), where it cannot move a particle from one side to the other. Such an excitation tells you nothing about the sampled bitstrings, so a slot spent on it is a slot wasted. Setting filter_trivial=True on the QDriftTrotterization pass rejects such excitations as they are drawn and draws a replacement, so that each of the num_groups slots of the resulting circuit carries an excitation that couples the two sides.

Warning

This filtering is not free, and it is off by default for that reason. What it conserves is the budget of num_groups sampled slots, not the circuit depth: since that budget is fixed, a rejected excitation is replaced rather than dropped, and a cheap one gives way to a coupling excitation that costs more to synthesize. Expect a filtered circuit to be deeper than an unfiltered one drawn from the same seed.

Renormalizing the draw over the accepted excitations changes the distribution the qDRIFT protocol samples from, and unlike the grouping in step 2 it also breaks the protocol’s convergence guarantee: the sampled product no longer averages to the evolution under the Hamiltonian you passed in. Every retained excitation ends up weighted by the reciprocal of the acceptance probability, the rejected ones contribute nothing, and neither the total coefficient magnitude nor the per-gate evolution time is adjusted to compensate.

The acceptance rule is also an over-approximation. It compares mode supports only, so it carries no amplitude information, and the tracked sets only ever grow: an excitation confined to the occupied set is genuinely inert on a product state, but the same reasoning gets weaker for later draws, once earlier accepted excitations have left the state entangled and marked most modes “uncertain”.

Use filter_trivial=True for the bitstring-sampling workflow this guide describes, where the circuits feed SQD post-processing and only the sampled bitstrings matter. Do not use it if you intend to estimate an expectation value from these circuits, or to rely on the qDRIFT error bound in any other way: those results carry a bias that the pass does not correct for.

This filtering needs to know which modes start out occupied. It therefore requires an InitializeModes gate preceding the Evolution gates in the circuit; InitializeModes.from_hartree_fock() is a convenient way to construct one. Add one here for the N2 Hartree-Fock reference (seven alpha and seven beta electrons in 14 spatial orbitals) and compare the sampled excitations with and without filter_trivial=True:

[x] PYTHON

>>> from qiskit_fermions.circuit.library import InitializeModes
>>>
>>> init = InitializeModes.from_hartree_fock(fcidump.norb, (7, 7))
>>>
>>> hf_circ = FermionicCircuit(num_modes)
>>> hf_circ.append(init, hf_circ.modes)
>>> hf_circ.append(Evolution(num_modes, canon, time), hf_circ.modes)
>>>
>>> num_groups = 5
>>> qdrift_unfiltered = QDriftTrotterization(num_groups, rng=3480)
>>> qdrift_trivial = QDriftTrotterization(
...     num_groups, filter_trivial=True, rng=3480
... )
>>>
>>> for instruction in FermionicPassManager(qdrift_unfiltered).run(hf_circ)._inner.data:
...     if instruction.operation.name == "Evolution":
...         print(sorted(instruction.operation.operator.get_support()))
[2, 4]
[41, 45, 52]
[15, 45, 55]
[41, 52, 53]
[10, 16, 37, 38]
>>>
>>> for instruction in FermionicPassManager(qdrift_trivial).run(hf_circ)._inner.data:
...     if instruction.operation.name == "Evolution":
...         print(sorted(instruction.operation.operator.get_support()))
[0, 1, 6, 7]
[0, 1, 28, 29]
[4, 13, 55]
[13, 20, 40, 41]
[0, 1, 28, 29]

[ ] C

// WARNING: Qiskit's C API does not yet allow us to implement circuits
// with custom gate definitions, which we therefore also cannot transpile
// via this API.

None of the excitations drawn without filtering touch the occupied set (0-6 and 28-34) at all, so none of them can move a particle between an occupied and an unoccupied mode. With filter_trivial=True, all five are rejected and replaced by excitations that do couple an occupied mode with an unoccupied one. For example, the first accepted excitation [0, 1, 6, 7] moves a particle between occupied modes 0, 1, and 6 and unoccupied mode 7.

Once an excitation is accepted, every mode in its support becomes “uncertain” and, thus, eligible to play either role for later samples, so the occupied and unoccupied mode sets keep growing as more excitations get accepted. This is what makes the second excitation, [0, 1, 28, 29], acceptable. All four of its modes are among the originally occupied ones, so it does not couple to any originally unoccupied mode. It is only accepted because modes 0 and 1 became uncertain (and thus eligible as the “unoccupied” side of the coupling) once the first excitation touched them.

This growth is what makes the rule an over-approximation rather than a test: as more modes turn uncertain, fewer draws are rejected, and the filtering does the most to the sampling distribution on the earliest draws.

To judge how far the filtering moved the distribution, read the rejection counts the pass records in the circuit metadata. One entry is stored per filtered Evolution gate, and the accepted-to-total ratio estimates the acceptance probability, which is the factor by which the retained excitations were over-weighted:

[x] PYTHON

>>> # a freshly seeded pass, so the counts do not depend on earlier draws
>>> qdrift_counted = QDriftTrotterization(
...     num_groups, filter_trivial=True, rng=3480
... )
>>> filtered = FermionicPassManager(qdrift_counted).run(hf_circ)
>>> discarded = filtered.metadata.get("filter_trivial.discarded")
>>> emitted = filtered.metadata.get("filter_trivial.emitted")
>>> print(f"{emitted[0]} kept, {discarded[0]} discarded")
5 kept, 7 discarded
>>> print(f"acceptance probability: {emitted[0] / (emitted[0] + discarded[0]):.2f}")
acceptance probability: 0.42

[ ] C

// WARNING: Qiskit's C API does not yet allow us to implement circuits
// with custom gate definitions, which we therefore also cannot transpile
// via this API.

Note

Filtering needs occupation information to filter against. A PrepareSlaterDeterminant gate seeds it just as InitializeModes does, and any OrbitalRotation marks every mode it acts on as “uncertain”. Without any of them, or when the seeded modes all land on one side (every mode occupied, or every mode unoccupied), filter_trivial=True emits a UserWarning and leaves the sampling unfiltered for that Evolution gate. On a Hamiltonian whose remaining terms can no longer couple the two tracked sets, the pass instead raises RuntimeError after QDriftTrotterization.MAX_SAMPLE_RETRIES consecutive rejections.

(Optional) Precompute the sampling weights

Reach for this only if the transpilation itself is a bottleneck. It changes nothing about the circuits you get.

QDriftTrotterization draws each group with a probability proportional to the magnitude of its coefficient, and derives those magnitudes from the Hamiltonian on every call: one value per Hamiltonian term, reduced down to one per group. Because the pass is stateless it repeats that reduction for every randomization, even though the result is identical every time. The weights argument lets you compute it once with group_coeff_means() and hand the result over, moving the cost out of the loop while leaving the pass free of any cached, Hamiltonian-dependent state.

This is the counterpart of the term ordering in step 2. That one hoists the group lookup out of the loop, this one the weight reduction, and each is only as noticeable as the other is absent, so it is worth applying both together. Since canon was already reordered above, the two compose directly:

[x] PYTHON

>>> from qiskit_fermions.operators.terms.grouping import group_coeff_means
>>>
>>> weights = group_coeff_means(canon)  # `canon` is group-ordered, from step 2
>>> qdrift_weighted = QDriftTrotterization(num_groups, rng=19, weights=weights)
>>>
>>> pm.optimization = FermionicPassManager([qdrift_weighted])
>>> weighted_circuits = [
...     pm.run(circ) for _ in range(num_sqdrift_randomizations)
... ]
>>> len(weighted_circuits)
10

[ ] C

// WARNING: Qiskit's C API does not yet allow us to implement circuits
// with custom gate definitions, which we therefore also cannot transpile
// via this API.

Since those are exactly the weights the pass would have computed, the circuits are the same as before; only the repeated reduction is gone.

One entry is expected per group, not per term, whenever the operator carries groups, since a grouped operator is sampled group-wise. Entries must be non-negative: a weight is a magnitude, because the qDRIFT protocol decomposes $H = \sum_j h_j H_j$ with positive $h_j$ , and a coefficient’s sign belongs to $H_j$ , where the pass reads it off the operator directly.

Caution

The weights are absolute magnitudes, not relative preferences. Their sum also fixes the evolution time, so multiplying every entry by $\gamma$ samples identically but evolves for $\gamma t$ instead of $t$ . Passing anything other than the Hamiltonian’s own coefficient magnitudes therefore changes which evolution the ensemble approximates, which is why this is a performance tool rather than a way to tune the sampling.

Because a weights array describes the groups of one particular operator, it is validated against each Evolution gate it is applied to, and a circuit holding more than one such gate is rejected. Leaving weights unset keeps such a circuit supported, since each gate then derives its own.

(Optional) Optimize the fermionic mode indexing

You can add an additional optimization step to the transpilation pipeline that minimizes the distance of the fermionic excitation spans by relabeling the fermionic mode indices. This optimization was introduced in the SqDRIFT paper and is implemented by build_excitation_span_minimization_model(). It can be easily inserted into the transpiler pipeline by using the RelabelModes pass:

[x] PYTHON

>>> from pyomo.environ import SolverFactory
>>> from qiskit_fermions.transpiler.passes import RelabelModes
>>>
>>> solver = SolverFactory("appsi_highs")
>>> solver.options["time_limit"] = 10
>>>
>>> qdrift = QDriftTrotterization(5, rng=19)
>>> relabel = RelabelModes(solver=solver)
>>>
>>> pm.optimization = FermionicPassManager([qdrift, relabel])
>>>
>>> relabeled_circ = pm.run(circ)
>>> # if the automatic mode relabeling was successful, the circuit's
>>> # metadata will contain the mode `permutation` information

[ ] C

// WARNING: This feature is not available via the C API.

Note

Using the automatic optimization inside RelabelModes (which leverages build_excitation_span_minimization_model()) requires the optional dependency managed by HAS_PYOMO.

Important

In order to perform the correct subspace diagonalization, the bitstrings sampled from circuits that were transpiled with the RelabelModes optimization pass must be post-processed based on the permutation information contained in the circuits’ metadata!

Next steps

Now that you have successfully generated an ensemble of circuits, you can sample bitstrings from them. To do so, the circuits must be executed on hardware. Refer to the Qiskit documentation for detailed instructions.

Once the bitstring samples have been obtained, these can be used in combination with the Hamiltonian coefficients to perform SQD post-processing, as explained in the SQD addon tutorials.

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