diff --git a/.azure/lint-linux.yml b/.azure/lint-linux.yml
deleted file mode 100644
index a1809bd829bd..000000000000
--- a/.azure/lint-linux.yml
+++ /dev/null
@@ -1,62 +0,0 @@
-parameters:
-  - name: "pythonVersion"
-    type: string
-    displayName: "Version of Python to use"
-
-jobs:
-  - job: 'Lint'
-    pool: {vmImage: 'ubuntu-latest'}
-
-    variables:
-      PIP_CACHE_DIR: $(Pipeline.Workspace)/.pip
-
-    steps:
-      - task: UsePythonVersion@0
-        inputs:
-          versionSpec: '${{ parameters.pythonVersion }}'
-        displayName: 'Use Python ${{ parameters.pythonVersion }}'
-
-      - bash: |
-          set -e
-          python -m pip install --upgrade pip setuptools wheel virtualenv
-          virtualenv test-job
-          source test-job/bin/activate
-          python -m pip install -U pip setuptools wheel
-          python -m pip install -U \
-            -c constraints.txt \
-            -r requirements.txt \
-            -r requirements-dev.txt \
-            -e .
-          # Build and install both qiskit and qiskit-terra so that any optionals
-          # depending on `qiskit` will resolve correctly.
-        displayName: 'Install dependencies'
-        env:
-          SETUPTOOLS_ENABLE_FEATURES: "legacy-editable"
-
-      - bash: |
-          set -e
-          source test-job/bin/activate
-          echo "Running black, any errors reported can be fixed with 'tox -eblack'"
-          black --check qiskit test tools examples setup.py
-          echo "Running rustfmt check, any errors reported can be fixed with 'cargo fmt'"
-          cargo fmt --check
-        displayName: "Formatting"
-
-      - bash: |
-          set -e
-          source test-job/bin/activate
-          echo "Running ruff"
-          ruff qiskit test tools examples setup.py
-          echo "Running pylint"
-          pylint -rn qiskit test tools
-          echo "Running Cargo Clippy"
-          cargo clippy -- -D warnings
-          echo "Running license header check"
-          tools/verify_headers.py qiskit test
-          echo "Running check for optional imports on bare 'import qiskit'"
-          python tools/find_optional_imports.py
-          echo "Running check for release notes in incorrect directories"
-          tools/find_stray_release_notes.py
-          echo "Running reno lint"
-          reno lint
-        displayName: 'Lint'
diff --git a/.azure/docs-linux.yml b/.azure/lint_docs_qpy-linux.yml
similarity index 51%
rename from .azure/docs-linux.yml
rename to .azure/lint_docs_qpy-linux.yml
index a44722bd0c38..99d2b7dc41c9 100644
--- a/.azure/docs-linux.yml
+++ b/.azure/lint_docs_qpy-linux.yml
@@ -4,7 +4,8 @@ parameters:
     displayName: "Version of Python to use"
 
 jobs:
-  - job: 'Docs'
+  - job: 'Lint_Docs_QPY'
+    displayName: 'Lint, documentation and QPY'
     pool: {vmImage: 'ubuntu-latest'}
 
     variables:
@@ -20,14 +21,13 @@ jobs:
         displayName: 'Use Python ${{ parameters.pythonVersion }}'
 
       - bash: tools/install_ubuntu_docs_dependencies.sh
-        displayName: 'Install dependencies'
+        displayName: 'Install docs dependencies'
 
-      - bash: |
-          set -e
-          tox -e docs
-          # Clean up Sphinx detritus.
-          rm -rf docs/_build/html/{.doctrees,.buildinfo}
-        displayName: 'Run Docs build'
+      - bash: tox run -e docs,lint
+        displayName: 'Lint and docs'
+
+      - bash: rm -rf docs/_build/html/{.doctrees,.buildinfo}
+        displayName: 'Clean up docs detritus'
 
       - task: ArchiveFiles@2
         inputs:
@@ -43,3 +43,24 @@ jobs:
           artifactName: 'html_docs'
           Parallel: true
           ParallelCount: 8
+
+      - task: Cache@2
+        inputs:
+          key: 'qpy | test/qpy_compat/test_qpy.py | "$(Build.BuildNumber)"'
+          restoreKeys: |
+            qpy | test/qpy_compat/test_qpy.py
+          path: qpy_files
+        displayName: Cache old QPY files
+
+      - bash: |
+          set -e
+          # Reuse the docs environment to avoid needing to rebuild another
+          # version of Qiskit.
+          source .tox/docs/bin/activate
+          mv qpy_files/* test/qpy_compat || :
+          pushd test/qpy_compat
+          ./run_tests.sh
+          popd
+          mkdir qpy_files || :
+          mv test/qpy_compat/qpy_* qpy_files/.
+        displayName: 'Run QPY backwards compat tests'
diff --git a/.azure/test-linux.yml b/.azure/test-linux.yml
index 512e1d773c35..4200b03e887b 100644
--- a/.azure/test-linux.yml
+++ b/.azure/test-linux.yml
@@ -6,9 +6,6 @@ parameters:
   - name: "testRust"
     type: boolean
 
-  - name: "testQPY"
-    type: boolean
-
   - name: "testImages"
     type: boolean
 
@@ -166,24 +163,6 @@ jobs:
         displayName: 'Publish images on test failure'
         condition: failed()
 
-      - ${{ if eq(parameters.testQPY, true) }}:
-        - task: Cache@2
-          inputs:
-            key: 'qpy | test/qpy_compat/test_qpy.py | "$(Build.BuildNumber)"'
-            restoreKeys: |
-              qpy | test/qpy_compat/test_qpy.py
-            path: qpy_files
-          displayName: cache old qpy
-        - bash: |
-            set -e
-            mv qpy_files/* test/qpy_compat || :
-            pushd test/qpy_compat
-            ./run_tests.sh
-            popd
-            mkdir qpy_files || :
-            mv test/qpy_compat/qpy_* qpy_files/.
-          displayName: 'Run QPY backwards compat tests'
-
       - ${{ if eq(parameters.testImages, true) }}:
         - bash: |
             set -e
diff --git a/azure-pipelines.yml b/azure-pipelines.yml
index 58370138160a..c3154abf412c 100644
--- a/azure-pipelines.yml
+++ b/azure-pipelines.yml
@@ -16,6 +16,12 @@ pr:
     include:
       - '*'
 
+# A schedule only runs on branches that match include rules from _both_ `main`
+# and the branch itself.  On `main`, we blanket include all branches that might
+# want to be enabled, then particular branches can override it to exclude
+# themselves by removing themselves from the trigger list.  For example, old
+# stable branches can remove `stable/*` from their copy of this file once they
+# reach their end-of-life.
 schedules:
   - cron: "20 6 * * *"
     displayName: "Complete matrix test"
@@ -82,9 +88,13 @@ parameters:
 #   - push to a pull request (`PullRequest`)
 # For `IndividualCI` (push/merge to a branch/tag on Qiskit/qiskit-terra), you
 # need to examine `variables['Build.SourceBranch']` to determine whether it's a
-# branch or a tag.
+# branch or a tag, and if a branch, then whether it's in the merge queue or a
+# push to a "real" branch.
 stages:
   # Nightly cron job.
+  #
+  # For this to run on a branch, the `schedules` trigger  up at the top of this
+  # file needs to match on _both_ `main` and the branch itself.
   - ${{ if eq(variables['Build.Reason'], 'Schedule') }}:
     - stage: "Nightly"
       displayName: "Nightly complete matrix tests"
@@ -93,7 +103,6 @@ stages:
           - template: ".azure/test-linux.yml"
             parameters:
               pythonVersion: ${{ version }}
-              testQPY: false
               testImages: false
               testRust: false
               installOptionals: true
@@ -123,10 +132,10 @@ stages:
                 id: 7864
                 comment: Nightly test job failed at commit $(Build.SourceVersion).  View the logs at $(System.TeamFoundationCollectionUri)$(System.TeamProject)/_build/results?buildId=$(Build.BuildId).
 
-  # Full PR suite.  This also needs to apply to the merge queue, which appears
-  # as a push event.  The queue won't get cron triggered because the schedule
-  # trigger doesn't have it as a target.
-  - ${{ if or(eq(variables['Build.Reason'], 'PullRequest'), contains(variables['Build.SourceBranch'], 'gh-readonly-queue')) }}:
+  # Full PR suite.  PRs need to pass this pipeline in order to be moved to the
+  # merge queue, where they'll use the next rule as the branch-protection rule
+  # for the final merge to the base branch.
+  - ${{ if eq(variables['Build.Reason'], 'PullRequest') }}:
     # The preliminary stage should be small in both total runtime (including
     # provisioning) and resources required.  About half of PR commits result in
     # a CI failure, and over 90% of these are in linting, documention or a test
@@ -135,14 +144,10 @@ stages:
     - stage: "Lint_Docs_Prelim_Tests"
       displayName: "Preliminary tests"
       jobs:
-        - template: ".azure/lint-linux.yml"
+        - template: ".azure/lint_docs_qpy-linux.yml"
           parameters:
             pythonVersion: ${{ parameters.minimumPythonVersion }}
 
-        - template: ".azure/docs-linux.yml"
-          parameters:
-            pythonVersion: ${{ parameters.documentationPythonVersion }}
-
         - template: ".azure/test-linux.yml"
           parameters:
             pythonVersion: ${{ parameters.minimumPythonVersion }}
@@ -153,13 +158,12 @@ stages:
             # to fail fast in the first CI stage.
             installOptionals: true
             testRust: true
-            testQPY: true
             testImages: true
 
     # The rest of the PR pipeline is to test the oldest and newest supported
-    # versions of Python. It's very rare for a failure to be specific to an intermediate version of
-    # Python, so we just catch those in the cron-job pipeline to reduce the
-    # amount of resources used.
+    # versions of Python. It's very rare for a failure to be specific to an
+    # intermediate version of Python, so we just catch those in the cron-job
+    # pipeline to reduce the amount of resources used.
     - stage: "Tests"
       displayName: "Main tests"
       dependsOn: "Lint_Docs_Prelim_Tests"
@@ -168,7 +172,6 @@ stages:
           parameters:
             pythonVersion: ${{ parameters.maximumPythonVersion }}
             testRust: false
-            testQPY: false
             testImages: false
             installFromSdist: true
             installOptionals: false
@@ -193,6 +196,46 @@ stages:
             pythonVersion: ${{ parameters.maximumPythonVersion }}
             installOptionals: false
 
+  # Merge queue.  A PR that reaches here has already passed the more rigorous PR
+  # suite, so is very likely to pass.  The main reasons for failures here are
+  # flaky VMs timing out (which we can't do much about), or a merge conflict
+  # with another PR that is also in the merge queue.
+  #
+  # There's no reason to have multiple stages in this case, because we're
+  # expecting it to pass.  Having more than one stage frustrates parallel
+  # throughput in low-contention cases, and guarantees a longer critical path.
+  - ${{ if and(eq(variables['Build.Reason'], 'IndividualCI'), contains(variables['Build.SourceBranch'], 'gh-readonly-queue')) }}:
+    - stage: "Merge_Queue"
+      displayName: "Merge queue"
+      jobs:
+        - template: ".azure/lint_docs_qpy-linux.yml"
+          parameters:
+            pythonVersion: ${{ parameters.minimumPythonVersion }}
+
+        - template: ".azure/test-linux.yml"
+          parameters:
+            pythonVersion: ${{ parameters.minimumPythonVersion }}
+            installOptionals: true
+            testRust: true
+            testImages: true
+
+        - template: ".azure/test-linux.yml"
+          parameters:
+            pythonVersion: ${{ parameters.maximumPythonVersion }}
+            installOptionals: false
+            testRust: false
+            testImages: false
+
+        - template: ".azure/test-macos.yml"
+          parameters:
+            pythonVersion: ${{ parameters.maximumPythonVersion }}
+            installOptionals: false
+
+        - template: ".azure/test-windows.yml"
+          parameters:
+            pythonVersion: ${{ parameters.maximumPythonVersion }}
+            installOptionals: false
+
   # Push to main or the stable branches.  The triggering branches also need to
   # be in the triggers at the top of this file.
   - ${{ if and(eq(variables['Build.Reason'], 'IndividualCI'), or(startsWith(variables['Build.SourceBranch'], 'refs/heads/main'), startsWith(variables['Build.SourceBranch'], 'refs/heads/stable/'))) }}:
@@ -202,6 +245,5 @@ stages:
           parameters:
             pythonVersion: ${{ parameters.branchPushPythonVersion }}
             testRust: true
-            testQPY: true
             testImages: true
             installOptionals: false
diff --git a/tox.ini b/tox.ini
index 74436293597b..71953ac239ba 100644
--- a/tox.ini
+++ b/tox.ini
@@ -1,5 +1,5 @@
 [tox]
-minversion = 3.3.0
+minversion = 4.0
 envlist = py38, py39, py310, py311, py312, lint-incr
 isolated_build = true
 
@@ -23,9 +23,16 @@ commands =
 
 [testenv:lint]
 basepython = python3
+# `pylint` will examine the source code, not the version that would otherwise be
+# installed in `site-packages`, so we use an editable install to make sure the
+# compiled modules are built into a valid place for it to find them.
+package = editable
+allowlist_externals = cargo
 commands =
-  ruff check qiskit test tools examples setup.py
   black --check {posargs} qiskit test tools examples setup.py
+  cargo fmt --check
+  ruff check qiskit test tools examples setup.py
+  cargo clippy -- -D warnings
   pylint -rn qiskit test tools
   # This line is commented out until #6649 merges. We can't run this currently
   # via tox because tox doesn't support globbing