diff --git a/cirq-ft/cirq_ft/algos/qrom.py b/cirq-ft/cirq_ft/algos/qrom.py
index 85c3eca1976..2bef77aab05 100644
--- a/cirq-ft/cirq_ft/algos/qrom.py
+++ b/cirq-ft/cirq_ft/algos/qrom.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import Callable, Sequence, Tuple
+from typing import Callable, Sequence, Tuple, Set
 
 import attr
 import cirq
@@ -29,7 +29,11 @@ class QROM(unary_iteration_gate.UnaryIterationGate):
     """Gate to load data[l] in the target register when the selection stores an index l.
 
     In the case of multi-dimensional data[p,q,r,...] we use multiple named
-    selection registers [p, q, r, ...] to index and load the data.
+    selection registers [p, q, r, ...] to index and load the data. Here `p, q, r, ...`
+    correspond to registers named `selection0`, `selection1`, `selection2`, ... etc.
+
+    When the input data elements contain consecutive entries of identical data elements to
+    load, the QROM also implements the "variable-spaced" QROM optimization described in Ref[2].
 
     Args:
         data: List of numpy ndarrays specifying the data to load. If the length
@@ -44,6 +48,15 @@ class QROM(unary_iteration_gate.UnaryIterationGate):
             registers. This can be deduced from the maximum element of each of the
             datasets. Should be of length len(data), i.e. the number of datasets.
         num_controls: The number of control registers.
+
+    References:
+        [Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity]
+        (https://arxiv.org/abs/1805.03662).
+            Babbush et. al. (2018). Figure 1.
+
+        [Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization]
+        (https://arxiv.org/abs/2007.07391).
+            Babbush et. al. (2020). Figure 3.
     """
 
     data: Sequence[NDArray]
@@ -152,11 +165,22 @@ def decompose_zero_selection(
             yield cirq.inverse(multi_controlled_and)
             context.qubit_manager.qfree(and_ancilla + [and_target])
 
+    def _break_early(self, selection_index_prefix: Tuple[int, ...], l: int, r: int):
+        global_unique_element: Set[int] = set()
+        for data in self.data:
+            unique_element = np.unique(data[selection_index_prefix][l:r])
+            if len(unique_element) > 1:
+                return False
+            global_unique_element.add(unique_element[0])
+            if len(global_unique_element) > 1:
+                return False
+        return True
+
     def nth_operation(
         self, context: cirq.DecompositionContext, control: cirq.Qid, **kwargs
     ) -> cirq.OP_TREE:
         selection_idx = tuple(kwargs[reg.name] for reg in self.selection_registers)
-        target_regs = {k: v for k, v in kwargs.items() if k in self.target_registers}
+        target_regs = {reg.name: kwargs[reg.name] for reg in self.target_registers}
         yield self._load_nth_data(selection_idx, lambda q: cirq.CNOT(control, q), **target_regs)
 
     def _circuit_diagram_info_(self, _) -> cirq.CircuitDiagramInfo:
@@ -172,4 +196,5 @@ def __pow__(self, power: int):
         return NotImplemented  # pragma: no cover
 
     def _value_equality_values_(self):
-        return (self.selection_registers, self.target_registers, self.control_registers)
+        data_tuple = tuple(tuple(d.flatten()) for d in self.data)
+        return (self.selection_registers, self.target_registers, self.control_registers, data_tuple)
diff --git a/cirq-ft/cirq_ft/algos/qrom_test.py b/cirq-ft/cirq_ft/algos/qrom_test.py
index 7cb67bb84d4..39b89963d8b 100644
--- a/cirq-ft/cirq_ft/algos/qrom_test.py
+++ b/cirq-ft/cirq_ft/algos/qrom_test.py
@@ -116,6 +116,80 @@ def test_t_complexity(data):
     assert cirq_ft.t_complexity(g.gate).t == max(0, 4 * n - 8), n
 
 
+def _assert_qrom_has_diagram(qrom: cirq_ft.QROM, expected_diagram: str):
+    gh = cirq_ft.testing.GateHelper(qrom)
+    op = gh.operation
+    context = cirq.DecompositionContext(qubit_manager=cirq_ft.GreedyQubitManager(prefix="anc"))
+    circuit = cirq.Circuit(cirq.decompose_once(op, context=context))
+    selection = [
+        *itertools.chain.from_iterable(gh.quregs[reg.name] for reg in qrom.selection_registers)
+    ]
+    selection = [q for q in selection if q in circuit.all_qubits()]
+    anc = sorted(set(circuit.all_qubits()) - set(op.qubits))
+    selection_and_anc = (selection[0],) + sum(zip(selection[1:], anc), ())
+    qubit_order = cirq.QubitOrder.explicit(selection_and_anc, fallback=cirq.QubitOrder.DEFAULT)
+    cirq.testing.assert_has_diagram(circuit, expected_diagram, qubit_order=qubit_order)
+
+
+def test_qrom_variable_spacing():
+    # Tests for variable spacing optimization applied from https://arxiv.org/abs/2007.07391
+    data = [1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8]  # Figure 3a.
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build(data)).t == (8 - 2) * 4
+    data = [1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5]  # Figure 3b.
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build(data)).t == (5 - 2) * 4
+    data = [1, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7]  # Negative test: t count is not (g-2)*4
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build(data)).t == (8 - 2) * 4
+    # Works as expected when multiple data arrays are to be loaded.
+    data = [1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5]
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build(data, data)).t == (5 - 2) * 4
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build(data, 2 * np.array(data))).t == (16 - 2) * 4
+    # Works as expected when multidimensional input data is to be loaded
+    qrom = cirq_ft.QROM.build(
+        np.array(
+            [
+                [1, 1, 1, 1, 1, 1, 1, 1],
+                [1, 1, 1, 1, 1, 1, 1, 1],
+                [2, 2, 2, 2, 2, 2, 2, 2],
+                [2, 2, 2, 2, 2, 2, 2, 2],
+            ]
+        )
+    )
+    # Value to be loaded depends only the on the first bit of outer loop.
+    _assert_qrom_has_diagram(
+        qrom,
+        r'''
+selection00: ───X───@───X───@───
+                    │       │
+target00: ──────────┼───────X───
+                    │
+target01: ──────────X───────────
+    ''',
+    )
+    # When inner loop range is not a power of 2, the inner segment tree cannot be skipped.
+    qrom = cirq_ft.QROM.build(
+        np.array(
+            [[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2]],
+            dtype=int,
+        )
+    )
+    _assert_qrom_has_diagram(
+        qrom,
+        r'''
+selection00: ───X───@─────────@───────@──────X───@─────────@───────@──────
+                    │         │       │          │         │       │
+selection10: ───────(0)───────┼───────@──────────(0)───────┼───────@──────
+                    │         │       │          │         │       │
+anc_1: ─────────────And───@───X───@───And†───────And───@───X───@───And†───
+                          │       │                    │       │
+target00: ────────────────┼───────┼────────────────────X───────X──────────
+                          │       │
+target01: ────────────────X───────X───────────────────────────────────────
+        ''',
+    )
+    # No T-gates needed if all elements to load are identical.
+    assert cirq_ft.t_complexity(cirq_ft.QROM.build([3, 3, 3, 3])).t == 0
+
+
 @pytest.mark.parametrize(
     "data",
     [[np.arange(6).reshape(2, 3), 4 * np.arange(6).reshape(2, 3)], [np.arange(8).reshape(2, 2, 2)]],
diff --git a/cirq-ft/cirq_ft/algos/unary_iteration_gate.py b/cirq-ft/cirq_ft/algos/unary_iteration_gate.py
index 4ff83080100..f37c2718b3b 100644
--- a/cirq-ft/cirq_ft/algos/unary_iteration_gate.py
+++ b/cirq-ft/cirq_ft/algos/unary_iteration_gate.py
@@ -13,7 +13,7 @@
 # limitations under the License.
 
 import abc
-from typing import Dict, Iterator, List, Sequence, Tuple
+from typing import Callable, Dict, Iterator, List, Sequence, Tuple
 from numpy.typing import NDArray
 
 import cirq
@@ -34,6 +34,7 @@ def _unary_iteration_segtree(
     r: int,
     l_iter: int,
     r_iter: int,
+    break_early: Callable[[int, int], bool],
 ) -> Iterator[Tuple[cirq.OP_TREE, cirq.Qid, int]]:
     """Constructs a unary iteration circuit by iterating over nodes of an implicit Segment Tree.
 
@@ -53,6 +54,11 @@ def _unary_iteration_segtree(
         r: Right index of the range represented by current node of the segment tree.
         l_iter: Left index of iteration range over which the segment tree should be constructed.
         r_iter: Right index of iteration range over which the segment tree should be constructed.
+        break_early: For each internal node of the segment tree, `break_early(l, r)` is called to
+            evaluate whether the unary iteration should terminate early and not recurse in the
+            subtree of the node representing range `[l, r)`. If True, the internal node is
+            considered equivalent to a leaf node and the method yields only one tuple
+            `(OP_TREE, control_qubit, l)` for all integers in the range `[l, r)`.
 
     Yields:
         One `Tuple[cirq.OP_TREE, cirq.Qid, int]` for each leaf node in the segment tree. The i'th
@@ -68,8 +74,8 @@ def _unary_iteration_segtree(
     if l >= r_iter or l_iter >= r:
         # Range corresponding to this node is completely outside of iteration range.
         return
-    if l == (r - 1):
-        # Reached a leaf node; yield the operations.
+    if l_iter <= l < r <= r_iter and (l == (r - 1) or break_early(l, r)):
+        # Reached a leaf node or a "special" internal node; yield the operations.
         yield tuple(ops), control, l
         ops.clear()
         return
@@ -78,20 +84,24 @@ def _unary_iteration_segtree(
     if r_iter <= m:
         # Yield only left sub-tree.
         yield from _unary_iteration_segtree(
-            ops, control, selection, ancilla, sl + 1, l, m, l_iter, r_iter
+            ops, control, selection, ancilla, sl + 1, l, m, l_iter, r_iter, break_early
         )
         return
     if l_iter >= m:
         # Yield only right sub-tree
         yield from _unary_iteration_segtree(
-            ops, control, selection, ancilla, sl + 1, m, r, l_iter, r_iter
+            ops, control, selection, ancilla, sl + 1, m, r, l_iter, r_iter, break_early
         )
         return
     anc, sq = ancilla[sl], selection[sl]
     ops.append(and_gate.And((1, 0)).on(control, sq, anc))
-    yield from _unary_iteration_segtree(ops, anc, selection, ancilla, sl + 1, l, m, l_iter, r_iter)
+    yield from _unary_iteration_segtree(
+        ops, anc, selection, ancilla, sl + 1, l, m, l_iter, r_iter, break_early
+    )
     ops.append(cirq.CNOT(control, anc))
-    yield from _unary_iteration_segtree(ops, anc, selection, ancilla, sl + 1, m, r, l_iter, r_iter)
+    yield from _unary_iteration_segtree(
+        ops, anc, selection, ancilla, sl + 1, m, r, l_iter, r_iter, break_early
+    )
     ops.append(and_gate.And(adjoint=True).on(control, sq, anc))
 
 
@@ -101,16 +111,17 @@ def _unary_iteration_zero_control(
     ancilla: Sequence[cirq.Qid],
     l_iter: int,
     r_iter: int,
+    break_early: Callable[[int, int], bool],
 ) -> Iterator[Tuple[cirq.OP_TREE, cirq.Qid, int]]:
     sl, l, r = 0, 0, 2 ** len(selection)
     m = (l + r) >> 1
     ops.append(cirq.X(selection[0]))
     yield from _unary_iteration_segtree(
-        ops, selection[0], selection[1:], ancilla, sl, l, m, l_iter, r_iter
+        ops, selection[0], selection[1:], ancilla, sl, l, m, l_iter, r_iter, break_early
     )
     ops.append(cirq.X(selection[0]))
     yield from _unary_iteration_segtree(
-        ops, selection[0], selection[1:], ancilla, sl, m, r, l_iter, r_iter
+        ops, selection[0], selection[1:], ancilla, sl, m, r, l_iter, r_iter, break_early
     )
 
 
@@ -121,9 +132,12 @@ def _unary_iteration_single_control(
     ancilla: Sequence[cirq.Qid],
     l_iter: int,
     r_iter: int,
+    break_early: Callable[[int, int], bool],
 ) -> Iterator[Tuple[cirq.OP_TREE, cirq.Qid, int]]:
     sl, l, r = 0, 0, 2 ** len(selection)
-    yield from _unary_iteration_segtree(ops, control, selection, ancilla, sl, l, r, l_iter, r_iter)
+    yield from _unary_iteration_segtree(
+        ops, control, selection, ancilla, sl, l, r, l_iter, r_iter, break_early
+    )
 
 
 def _unary_iteration_multi_controls(
@@ -133,6 +147,7 @@ def _unary_iteration_multi_controls(
     ancilla: Sequence[cirq.Qid],
     l_iter: int,
     r_iter: int,
+    break_early: Callable[[int, int], bool],
 ) -> Iterator[Tuple[cirq.OP_TREE, cirq.Qid, int]]:
     num_controls = len(controls)
     and_ancilla = ancilla[: num_controls - 2]
@@ -142,7 +157,7 @@ def _unary_iteration_multi_controls(
     )
     ops.append(multi_controlled_and)
     yield from _unary_iteration_single_control(
-        ops, and_target, selection, ancilla[num_controls - 1 :], l_iter, r_iter
+        ops, and_target, selection, ancilla[num_controls - 1 :], l_iter, r_iter, break_early
     )
     ops.append(cirq.inverse(multi_controlled_and))
 
@@ -154,6 +169,7 @@ def unary_iteration(
     controls: Sequence[cirq.Qid],
     selection: Sequence[cirq.Qid],
     qubit_manager: cirq.QubitManager,
+    break_early: Callable[[int, int], bool] = lambda l, r: False,
 ) -> Iterator[Tuple[cirq.OP_TREE, cirq.Qid, int]]:
     """The method performs unary iteration on `selection` integer in `range(l_iter, r_iter)`.
 
@@ -181,6 +197,9 @@ def unary_iteration(
     ...     circuit.append(j_ops)
     >>> circuit.append(i_ops)
 
+    Note: Unary iteration circuits assume that the selection register stores integers only in the
+    range `[l, r)` for which the corresponding unary iteration circuit should be built.
+
     Args:
         l_iter: Starting index of the iteration range.
         r_iter: Ending index of the iteration range.
@@ -192,6 +211,11 @@ def unary_iteration(
         controls: Control register of qubits.
         selection: Selection register of qubits.
         qubit_manager: A `cirq.QubitManager` to allocate new qubits.
+        break_early: For each internal node of the segment tree, `break_early(l, r)` is called to
+            evaluate whether the unary iteration should terminate early and not recurse in the
+            subtree of the node representing range `[l, r)`. If True, the internal node is
+            considered equivalent to a leaf node and the method yields only one tuple
+            `(OP_TREE, control_qubit, l)` for all integers in the range `[l, r)`.
 
     Yields:
         (r_iter - l_iter) different tuples, each corresponding to an integer in range
@@ -207,14 +231,16 @@ def unary_iteration(
     assert len(selection) > 0
     ancilla = qubit_manager.qalloc(max(0, len(controls) + len(selection) - 1))
     if len(controls) == 0:
-        yield from _unary_iteration_zero_control(flanking_ops, selection, ancilla, l_iter, r_iter)
+        yield from _unary_iteration_zero_control(
+            flanking_ops, selection, ancilla, l_iter, r_iter, break_early
+        )
     elif len(controls) == 1:
         yield from _unary_iteration_single_control(
-            flanking_ops, controls[0], selection, ancilla, l_iter, r_iter
+            flanking_ops, controls[0], selection, ancilla, l_iter, r_iter, break_early
         )
     else:
         yield from _unary_iteration_multi_controls(
-            flanking_ops, controls, selection, ancilla, l_iter, r_iter
+            flanking_ops, controls, selection, ancilla, l_iter, r_iter, break_early
         )
     qubit_manager.qfree(ancilla)
 
@@ -231,6 +257,9 @@ class UnaryIterationGate(infra.GateWithRegisters):
     indexed operations on a target register depending on the index value stored in a selection
     register.
 
+    Note: Unary iteration circuits assume that the selection register stores integers only in the
+    range `[l, r)` for which the corresponding unary iteration circuit should be built.
+
     References:
             [Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity]
         (https://arxiv.org/abs/1805.03662).
@@ -308,10 +337,38 @@ def decompose_zero_selection(
         """
         raise NotImplementedError("Selection register must not be empty.")
 
+    def _break_early(self, selection_index_prefix: Tuple[int, ...], l: int, r: int) -> bool:
+        """Derived classes should override this method to specify an early termination condition.
+
+        For each internal node of the unary iteration segment tree, `break_early(l, r)` is called
+        to evaluate whether the unary iteration should not recurse in the subtree of the node
+        representing range `[l, r)`. If True, the internal node is considered equivalent to a leaf
+        node and thus, `self.nth_operation` will be called for only integer `l` in the range [l, r).
+
+        When the `UnaryIteration` class is constructed using multiple selection registers, i.e. we
+        wish to perform nested coherent for-loops, a unary iteration segment tree is constructed
+        corresponding to each nested coherent for-loop. For every such unary iteration segment tree,
+        the `_break_early` condition is checked by passing the `selection_index_prefix` tuple.
+
+        Args:
+            selection_index_prefix: To evaluate the early breaking condition for the i'th nested
+                for-loop, the `selection_index_prefix` contains `i-1` integers corresponding to
+                the loop variable values for the first `i-1` nested loops.
+            l: Beginning of range `[l, r)` for internal node of unary iteration segment tree.
+            r: End (exclusive) of range `[l, r)` for internal node of unary iteration segment tree.
+
+        Returns:
+            True of the `len(selection_index_prefix)`'th unary iteration should terminate early for
+            the given parameters.
+        """
+        return False
+
     def decompose_from_registers(
         self, *, context: cirq.DecompositionContext, **quregs: NDArray[cirq.Qid]
     ) -> cirq.OP_TREE:
-        if self.selection_registers.total_bits() == 0:
+        if self.selection_registers.total_bits() == 0 or self._break_early(
+            (), 0, self.selection_registers[0].iteration_length
+        ):
             return self.decompose_zero_selection(context=context, **quregs)
 
         num_loops = len(self.selection_registers)
@@ -354,6 +411,7 @@ def unary_iteration_loops(
                 return
             # Use recursion to write `num_loops` nested loops using unary_iteration().
             ops: List[cirq.Operation] = []
+            selection_index_prefix = tuple(selection_reg_name_to_val.values())
             ith_for_loop = unary_iteration(
                 l_iter=0,
                 r_iter=self.selection_registers[nested_depth].iteration_length,
@@ -361,6 +419,7 @@ def unary_iteration_loops(
                 controls=controls,
                 selection=[*quregs[self.selection_registers[nested_depth].name]],
                 qubit_manager=context.qubit_manager,
+                break_early=lambda l, r: self._break_early(selection_index_prefix, l, r),
             )
             for op_tree, control_qid, n in ith_for_loop:
                 yield op_tree
@@ -368,6 +427,7 @@ def unary_iteration_loops(
                 yield from unary_iteration_loops(
                     nested_depth + 1, selection_reg_name_to_val, (control_qid,)
                 )
+                selection_reg_name_to_val.pop(self.selection_registers[nested_depth].name)
             yield ops
 
         return unary_iteration_loops(0, {}, self.control_registers.merge_qubits(**quregs))
diff --git a/cirq-ft/cirq_ft/infra/qubit_manager.py b/cirq-ft/cirq_ft/infra/qubit_manager.py
index 098e6231c35..59e4bbded13 100644
--- a/cirq-ft/cirq_ft/infra/qubit_manager.py
+++ b/cirq-ft/cirq_ft/infra/qubit_manager.py
@@ -74,9 +74,9 @@ def qalloc(self, n: int, dim: int = 2) -> List[cirq.Qid]:
         return ret_qubits
 
     def qfree(self, qubits: Iterable[cirq.Qid]) -> None:
-        qs = set(qubits)
+        qs = list(dict(zip(qubits, qubits)).keys())
         assert self._used_qubits.issuperset(qs), "Only managed qubits currently in-use can be freed"
-        self._used_qubits -= qs
+        self._used_qubits = self._used_qubits.difference(qs)
         self._free_qubits.extend(qs)
 
     def qborrow(self, n: int, dim: int = 2) -> List[cirq.Qid]:
diff --git a/cirq-ft/cirq_ft/infra/qubit_manager_test.py b/cirq-ft/cirq_ft/infra/qubit_manager_test.py
index bbfc6e10dd2..5eda575958a 100644
--- a/cirq-ft/cirq_ft/infra/qubit_manager_test.py
+++ b/cirq-ft/cirq_ft/infra/qubit_manager_test.py
@@ -92,3 +92,11 @@ def make_circuit(qm: cirq.QubitManager):
 ancilla_1: ───X───X───
      """,
     )
+
+
+def test_greedy_qubit_manager_preserves_order():
+    qm = cirq_ft.GreedyQubitManager(prefix="anc")
+    ancillae = [cirq.q(f"anc_{i}") for i in range(100)]
+    assert qm.qalloc(100) == ancillae
+    qm.qfree(ancillae)
+    assert qm.qalloc(100) == ancillae