Add base representation of SparseObservable

This adds the base representation of `SparseObservable`, including the
simple constructors from Python space and the ability to view the data
buffers.

This commit does not include the mathematical manipulations of the
operators, nor some of the helper methods that will be used to
manipulate the operators in the context of primitives execution. These
will follow in subsequent patches.

The design and implementation notes of `SparseObservable` are described
in a Qiskit RFC that preceeded this patch series[^1], and it's best to
consult that document for full details on the operator considerations.

[^1]: https://github.com/Qiskit/RFCs/blob/7a74b08793475b7b0142d3a3f7142cabcfd33ab8/0021-sparse-observable.md

crates/accelerate/Cargo.toml

@@ -26,6 +26,7 @@ qiskit-circuit.workspace = true
 thiserror.workspace = true
 ndarray_einsum_beta = "0.7"
 once_cell = "1.20.0"
+bytemuck.workspace = true
 
 [dependencies.smallvec]
 workspace = true

crates/accelerate/src/lib.rs

@@ -39,6 +39,7 @@ pub mod remove_diagonal_gates_before_measure;
 pub mod results;
 pub mod sabre;
 pub mod sampled_exp_val;
+pub mod sparse_observable;
 pub mod sparse_pauli_op;
 pub mod split_2q_unitaries;
 pub mod star_prerouting;

crates/circuit/src/imports.rs

@@ -116,6 +116,8 @@ pub static UNITARY_GATE: ImportOnceCell = ImportOnceCell::new(
     "qiskit.circuit.library.generalized_gates.unitary",
     "UnitaryGate",
 );
+pub static NUMPY_COPY_ONLY_IF_NEEDED: ImportOnceCell =
+    ImportOnceCell::new("qiskit._numpy_compat", "COPY_ONLY_IF_NEEDED");
 
 /// A mapping from the enum variant in crate::operations::StandardGate to the python
 /// module path and class name to import it. This is used to populate the conversion table

crates/pyext/src/lib.rs

@@ -52,6 +52,7 @@ fn _accelerate(m: &Bound<PyModule>) -> PyResult<()> {
     add_submodule(m, ::qiskit_accelerate::results::results, "results")?;
     add_submodule(m, ::qiskit_accelerate::sabre::sabre, "sabre")?;
     add_submodule(m, ::qiskit_accelerate::sampled_exp_val::sampled_exp_val, "sampled_exp_val")?;
+    add_submodule(m, ::qiskit_accelerate::sparse_observable::sparse_observable, "sparse_observable")?;
     add_submodule(m, ::qiskit_accelerate::sparse_pauli_op::sparse_pauli_op, "sparse_pauli_op")?;
     add_submodule(m, ::qiskit_accelerate::split_2q_unitaries::split_2q_unitaries_mod, "split_2q_unitaries")?;
     add_submodule(m, ::qiskit_accelerate::star_prerouting::star_prerouting, "star_prerouting")?;

qiskit/__init__.py

@@ -77,6 +77,7 @@
 sys.modules["qiskit._accelerate.results"] = _accelerate.results
 sys.modules["qiskit._accelerate.sabre"] = _accelerate.sabre
 sys.modules["qiskit._accelerate.sampled_exp_val"] = _accelerate.sampled_exp_val
+sys.modules["qiskit._accelerate.sparse_observable"] = _accelerate.sparse_observable
 sys.modules["qiskit._accelerate.sparse_pauli_op"] = _accelerate.sparse_pauli_op
 sys.modules["qiskit._accelerate.star_prerouting"] = _accelerate.star_prerouting
 sys.modules["qiskit._accelerate.stochastic_swap"] = _accelerate.stochastic_swap

qiskit/quantum_info/__init__.py

@@ -28,6 +28,7 @@
    Pauli
    Clifford
    ScalarOp
+   SparseObservable
    SparsePauliOp
    CNOTDihedral
    PauliList
@@ -113,6 +114,9 @@
 """
 
 from __future__ import annotations
+
+from qiskit._accelerate.sparse_observable import SparseObservable
+
 from .analysis import hellinger_distance, hellinger_fidelity, Z2Symmetries
 from .operators import (
     Clifford,

releasenotes/notes/sparse-observable-7de70dcdf6962a64.yaml

@@ -0,0 +1,32 @@
+---
+features_quantum_info:
+  - |
+    A new observable class has been added.  :class:`.SparseObservable` represents observables as a
+    sum of terms, similar to :class:`.SparsePauliOp`, but with two core differences:
+
+    1. Each complete term is stored as (effectively) a series of ``(qubit, bit_term)`` pairs,
+       without storing qubits that undergo the identity for that term.  This significantly improves
+       the memory usage of observables such as the weighted sum of Paulis :math:`\sum_i c_i Z_i`.
+
+    2. The single-qubit term alphabet is overcomplete for the operator space; it can represent Pauli 
+       operators (like :class:`.SparsePauliOp`), but also projectors onto the eigenstates of the
+       Pauli operators, like :math:`\lvert 0\rangle\langle 0\rangle`.  Such projectors can be
+       measured on hardware equally as efficiently as their corresponding Pauli operator, but
+       :class:`.SparsePauliOp` would require an exponential number of terms to represent
+       :math:`{\lvert0\rangle\langle0\rvert}^{\otimes n}` over :math:`n` qubits, while
+       :class:`.SparseObservable` needs only a single term.
+
+    You can construct and manipulate :class:`.SparseObservable` using an interface familiar to users
+    of :class:`.SparsePauliOp`::
+
+      from qiskit.quantum_info import SparseObservable
+
+      obs = SparseObservable.from_sparse_list([
+        ("XZY", (2, 1, 0), 1.5j),
+        ("+-", (100, 99), 0.5j),
+        ("01", (50, 49), 0.5),
+      ])
+
+    :class:`.SparseObservable` is not currently supported as an input format to the primitives
+    (:mod:`qiskit.primitives`), but we expect to expand these interfaces to include them in the
+    future.