diff --git a/.azure/lint_docs_qpy-linux.yml b/.azure/lint_docs_qpy-linux.yml
index 99d2b7dc41c9..c08ea702ede5 100644
--- a/.azure/lint_docs_qpy-linux.yml
+++ b/.azure/lint_docs_qpy-linux.yml
@@ -61,6 +61,6 @@ jobs:
           pushd test/qpy_compat
           ./run_tests.sh
           popd
-          mkdir qpy_files || :
+          mkdir -p qpy_files
           mv test/qpy_compat/qpy_* qpy_files/.
         displayName: 'Run QPY backwards compat tests'
diff --git a/test/qpy_compat/compare_versions.py b/test/qpy_compat/compare_versions.py
deleted file mode 100755
index 5deec1b9ca5b..000000000000
--- a/test/qpy_compat/compare_versions.py
+++ /dev/null
@@ -1,40 +0,0 @@
-#!/usr/bin/env python3
-# This code is part of Qiskit.
-#
-# (C) Copyright IBM 2023.
-#
-# This code is licensed under the Apache License, Version 2.0. You may
-# obtain a copy of this license in the LICENSE.txt file in the root directory
-# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
-#
-# Any modifications or derivative works of this code must retain this
-# copyright notice, and modified files need to carry a notice indicating
-# that they have been altered from the originals.
-
-"""Compare Qiskit versions to determine if we should run qpy compat tests."""
-
-import argparse
-import sys
-
-from qiskit.qpy.interface import VERSION_PATTERN_REGEX
-
-
-def main():
-    """Main function."""
-    parser = argparse.ArgumentParser(prog="compare_version", description="Compare version strings")
-    parser.add_argument(
-        "source_version", help="Source version of Qiskit that is generating the payload"
-    )
-    parser.add_argument("test_version", help="Version under test that will load the QPY payload")
-    args = parser.parse_args()
-    source_match = VERSION_PATTERN_REGEX.search(args.source_version)
-    target_match = VERSION_PATTERN_REGEX.search(args.test_version)
-    source_version = tuple(int(x) for x in source_match.group("release").split("."))
-    target_version = tuple(int(x) for x in target_match.group("release").split("."))
-    if source_version > target_version:
-        sys.exit(1)
-    sys.exit(0)
-
-
-if __name__ == "__main__":
-    main()
diff --git a/test/qpy_compat/get_versions.py b/test/qpy_compat/get_versions.py
new file mode 100644
index 000000000000..a89e3b8508de
--- /dev/null
+++ b/test/qpy_compat/get_versions.py
@@ -0,0 +1,99 @@
+# This code is part of Qiskit.
+#
+# (C) Copyright IBM 2024.
+#
+# This code is licensed under the Apache License, Version 2.0. You may
+# obtain a copy of this license in the LICENSE.txt file in the root directory
+# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
+#
+# Any modifications or derivative works of this code must retain this
+# copyright notice, and modified files need to carry a notice indicating
+# that they have been altered from the originals.
+
+"""List the versions of Qiskit and Terra that should be tested for QPY compatibility."""
+
+import json
+import sys
+import urllib.request
+
+import packaging.version
+import packaging.tags
+
+import qiskit
+
+
+def tags_from_wheel_name(wheel: str):
+    """Extract the wheel tag from its filename."""
+    assert wheel.lower().endswith(".whl")
+    # For more information, see:
+    # - https://packaging.python.org/en/latest/specifications/binary-distribution-format/
+    # - https://packaging.python.org/en/latest/specifications/platform-compatibility-tags/
+    #
+    # In particular, take note that a wheel's filename can include "compressed tag sets" (and our
+    # Linux wheels generally do this), which is why this function returns an iterable of tags.
+    _prefix, interpreters, abis, platforms = wheel[:-4].rsplit("-", 3)
+    yield from (
+        packaging.tags.Tag(interpreter, abi, platform)
+        for interpreter in interpreters.split(".")
+        for abi in abis.split(".")
+        for platform in platforms.split(".")
+    )
+
+
+def available_versions():
+    """Get all the versions of Qiskit that support exporting QPY, and are installable with the
+    active version of Python on this platform."""
+    our_version = packaging.version.parse(qiskit.__version__)
+    supported_tags = set(packaging.tags.sys_tags())
+
+    def available_versions_for_package(package, min_version=None, max_version=None):
+        with urllib.request.urlopen(f"https://pypi.org/pypi/{package}/json") as fd:
+            data = json.load(fd)
+        min_version = min_version and packaging.version.parse(min_version)
+        max_version = max_version and packaging.version.parse(max_version)
+        for other_version, payload in data["releases"].items():
+            other_version = packaging.version.parse(other_version)
+            if min_version is not None and other_version < min_version:
+                continue
+            if max_version is not None and other_version >= max_version:
+                continue
+            if other_version > our_version:
+                continue
+            if other_version.pre is not None and other_version.pre[0] in ("a", "b"):
+                # We skip alpha and beta prereleases, but we currently want to test for
+                # compatibility with release candidates.
+                continue
+            # Note: this ignores versions that are uninstallable because we're using a Python
+            # version that's too new, which can be a problem for the oldest Terras, especially from
+            # before we built for abi3.  We're not counting sdists, since if we didn't release a
+            # compatible wheel for the current Python version, there's no guarantee it'll install.
+            if not any(
+                tag in supported_tags
+                for release in payload
+                if release["packagetype"] == "bdist_wheel"
+                for tag in tags_from_wheel_name(release["filename"])
+            ):
+                print(
+                    f"skipping '{other_version}', which has no installable binary artifacts",
+                    file=sys.stderr,
+                )
+                continue
+            yield other_version
+
+    yield from (
+        ("qiskit-terra", version)
+        for version in available_versions_for_package("qiskit-terra", "0.18.0", "1.0.0")
+    )
+    yield from (
+        ("qiskit", version) for version in available_versions_for_package("qiskit", "1.0.0")
+    )
+
+
+def main():
+    """main"""
+    for package, version in available_versions():
+        print(package, version)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/test/qpy_compat/process_version.sh b/test/qpy_compat/process_version.sh
index f5ea685394a3..33faf4225535 100755
--- a/test/qpy_compat/process_version.sh
+++ b/test/qpy_compat/process_version.sh
@@ -14,46 +14,50 @@
 set -e
 set -x
 
-# version is the source version, this is the release with which to generate
-# qpy files with to load with the version under test
-version=$1
-parts=( ${version//./ } )
-# qiskit_version is the version under test, We're testing that we can correctly
-# read the qpy files generated with source version with this version.
-qiskit_version=`./qiskit_venv/bin/python -c "import qiskit;print(qiskit.__version__)"`
-qiskit_parts=( ${qiskit_version//./ } )
+function usage {
+    echo "usage: ${BASH_SOURCE[0]} -p /path/to/qiskit/python <package> <version>" 1>&2
+    exit 1
+}
 
-
-# If source version is less than 0.18 QPY didn't exist yet so exit fast
-if [[ ${parts[0]} -eq 0 && ${parts[1]} -lt 18 ]] ; then
-    exit 0
+python="python"
+while getopts "p:" opt; do
+    case "$opt" in
+        p)
+            python="$OPTARG"
+            ;;
+        *)
+            usage
+            ;;
+    esac
+done
+shift "$((OPTIND-1))"
+if [[ $# != 2 ]]; then
+    usage
 fi
 
-# Exclude any non-rc pre-releases as they don't have stable API guarantees
-if [[ $version == *"b"* || $version == *"a"* ]] ; then
-    exit 0
-fi
+# `package` is the name of the Python distribution to install (qiskit or qiskit-terra). `version` is
+# the source version: the release with which to generate qpy files with to load with the version
+# under test.
+package="$1"
+version="$2"
 
-# If the source version is newer than the version under test exit fast because
-# there is no QPY compatibility for loading a qpy file generated from a newer
-# release with an older release of Qiskit.
-if ! ./qiskit_venv/bin/python ./compare_versions.py "$version" "$qiskit_version" ; then
-    exit 0
-fi
+our_dir="$(realpath -- "$(dirname -- "${BASH_SOURCE[0]}")")"
+cache_dir="$(pwd -P)/qpy_$version"
+venv_dir="$(pwd -P)/${version}"
 
-if [[ ! -d qpy_$version ]] ; then
-    echo "Building venv for qiskit-terra $version"
-    python -m venv $version
-    ./$version/bin/pip install "qiskit-terra==$version"
-    mkdir qpy_$version
-    pushd qpy_$version
-    echo "Generating qpy files with qiskit-terra $version"
-    ../$version/bin/python ../test_qpy.py generate --version=$version
+if [[ ! -d $cache_dir ]] ; then
+    echo "Building venv for $package==$version"
+    "$python" -m venv "$venv_dir"
+    "$venv_dir/bin/pip" install -c "${our_dir}/qpy_test_constraints.txt" "${package}==${version}"
+    mkdir "$cache_dir"
+    pushd "$cache_dir"
+    echo "Generating QPY files with $package==$version"
+    "$venv_dir/bin/python" "${our_dir}/test_qpy.py" generate --version="$version"
 else
     echo "Using cached QPY files for $version"
-    pushd qpy_$version
+    pushd "${cache_dir}"
 fi
-echo "Loading qpy files from $version with dev qiskit-terra"
-../qiskit_venv/bin/python ../test_qpy.py load --version=$version
+echo "Loading qpy files from $version with dev Qiskit"
+"$python" "${our_dir}/test_qpy.py" load --version="$version"
 popd
-rm -rf ./$version
+rm -rf "$venv_dir"
diff --git a/test/qpy_compat/run_tests.sh b/test/qpy_compat/run_tests.sh
index 4fc6bc5b91fd..437585bac581 100755
--- a/test/qpy_compat/run_tests.sh
+++ b/test/qpy_compat/run_tests.sh
@@ -12,6 +12,7 @@
 # copyright notice, and modified files need to carry a notice indicating
 # that they have been altered from the originals.
 set -e
+set -o pipefail
 set -x
 
 # Set fixed hash seed to ensure set orders are identical between saving and
@@ -19,12 +20,17 @@ set -x
 export PYTHONHASHSEED=$(python -S -c "import random; print(random.randint(1, 4294967295))")
 echo "PYTHONHASHSEED=$PYTHONHASHSEED"
 
-python -m venv qiskit_venv
-qiskit_venv/bin/pip install ../..
+our_dir="$(realpath -- "$(dirname -- "${BASH_SOURCE[0]}")")"
 
-parallel bash ./process_version.sh ::: `git tag --sort=-creatordate`
+qiskit_venv="$(pwd -P)/qiskit_venv"
+qiskit_python="$qiskit_venv/bin/python"
+python -m venv "$qiskit_venv"
+# `packaging` is needed for the `get_versions.py` script.
+"$qiskit_venv/bin/pip" install -c "$our_dir/../../constraints.txt" "$our_dir/../.." packaging
+
+"$qiskit_python" "$our_dir/get_versions.py" | parallel --colsep=" " bash "$our_dir/process_version.sh" -p "$qiskit_python"
 
 # Test dev compatibility
-dev_version=`qiskit_venv/bin/python -c 'import qiskit;print(qiskit.__version__)'`
-qiskit_venv/bin/python test_qpy.py generate --version=$dev_version
-qiskit_venv/bin/python test_qpy.py load --version=$dev_version
+dev_version="$("$qiskit_python" -c 'import qiskit; print(qiskit.__version__)')"
+"$qiskit_python" "$our_dir/test_qpy.py" generate --version="$dev_version"
+"$qiskit_python" "$our_dir/test_qpy.py" load --version="$dev_version"