> ## Documentation Index > Fetch the complete documentation index at: https://docs.haiqu.ai/llms.txt > Use this file to discover all available pages before exploring further. # Backpropagate Observables Mcp > Backpropagate observables through a circuit to shorten what must be run. Use this tool to move part of the computation from the circuit into the observable: Haiqu rewrites each observable through the tail of the circuit and stores a correspondingly shortened circuit. Running the shorter circuit with the rewritten observable estimates the same expectation value with less hardware noise. When to Use: - Call this when the user wants expectation values from a circuit that is too deep to run faithfully on hardware. - Call this before ``run_circuits_on_qpu_or_simulator`` for expectation-value workloads, and pass the returned circuit and observable together. Constraints: - Backpropagation is synchronous and runs no quantum job. - Each input observable yields one output circuit, returned under the same key you gave the observable: run ``optimized_circuit_ids[name]`` with ``backpropagated_observables[name]``. - Truncation budgets trade accuracy for depth. Without ``max_error_total`` or ``max_error_per_slice`` the rewritten observable can grow large. Notes: - Pauli coefficients are given and returned as ``(pauli, real, imaginary)`` triples, one per term -- not as the parallel ``(pauli_strings, coefficients)`` lists that ``run_variational_workload`` and ``run_circuits_on_qpu_or_simulator`` take. - Compare the returned circuits against the input with ``get_circuit_by_id`` to show how much depth was removed. Args: user: Authenticated user resolved from the API key. data: Circuit identifier, observables, and optional truncation budgets. db: Active database session. Returns: The generated circuit IDs with their rewritten observables. Raises: HTTPException: Raised with ``404`` when the circuit is not available to the caller, ``422`` when an observable cannot be parsed, or ``501`` when the backpropagation dependency is unavailable. ## OpenAPI ````yaml https://api.haiqu.ai/openapi.json post /ai/backpropagate_observables openapi: 3.1.0 info: title: Haiqu API summary: Haiqu RESTful API service. description: Cloud service providing the access to Haiqu cloud runtime contact: name: Haiqu Inc. url: https://haiqu.ai/ email: info@haiqu.ai version: 1.6.0 servers: [] security: [] paths: /ai/backpropagate_observables: post: summary: Backpropagate Observables Mcp description: >- Backpropagate observables through a circuit to shorten what must be run. Use this tool to move part of the computation from the circuit into the observable: Haiqu rewrites each observable through the tail of the circuit and stores a correspondingly shortened circuit. Running the shorter circuit with the rewritten observable estimates the same expectation value with less hardware noise. When to Use: - Call this when the user wants expectation values from a circuit that is too deep to run faithfully on hardware. - Call this before ``run_circuits_on_qpu_or_simulator`` for expectation-value workloads, and pass the returned circuit and observable together. Constraints: - Backpropagation is synchronous and runs no quantum job. - Each input observable yields one output circuit, returned under the same key you gave the observable: run ``optimized_circuit_ids[name]`` with ``backpropagated_observables[name]``. - Truncation budgets trade accuracy for depth. Without ``max_error_total`` or ``max_error_per_slice`` the rewritten observable can grow large. Notes: - Pauli coefficients are given and returned as ``(pauli, real, imaginary)`` triples, one per term -- not as the parallel ``(pauli_strings, coefficients)`` lists that ``run_variational_workload`` and ``run_circuits_on_qpu_or_simulator`` take. - Compare the returned circuits against the input with ``get_circuit_by_id`` to show how much depth was removed. Args: user: Authenticated user resolved from the API key. data: Circuit identifier, observables, and optional truncation budgets. db: Active database session. Returns: The generated circuit IDs with their rewritten observables. Raises: HTTPException: Raised with ``404`` when the circuit is not available to the caller, ``422`` when an observable cannot be parsed, or ``501`` when the backpropagation dependency is unavailable. operationId: backpropagate_observables requestBody: content: application/json: schema: $ref: '#/components/schemas/ContextBackpropagationSubmitModel' required: true responses: '200': description: Successful Response content: application/json: schema: $ref: '#/components/schemas/ContextBackpropagationResultModel' '422': description: Validation Error content: application/json: schema: $ref: '#/components/schemas/HTTPValidationError' security: - APIKeyQuery: [] - APIKeyHeader: [] components: schemas: ContextBackpropagationSubmitModel: properties: circuit_id: type: string title: Circuit Id observables: additionalProperties: items: prefixItems: - type: string - type: number - type: number type: array maxItems: 3 minItems: 3 type: array type: object minProperties: 1 title: Observables description: >- Observables to backpropagate, keyed by a name you choose. Each observable is a list of Pauli terms given as `[pauli_string, real_coefficient, imaginary_coefficient]`, for example `{"energy": [["ZZ", 1.0, 0.0], ["XX", 0.5, 0.0]]}`. The results come back under the same keys. max_qwc_groups: anyOf: - type: integer - type: 'null' title: Max Qwc Groups description: Optional cap on the number of qubit-wise-commuting groups retained. max_error_total: anyOf: - type: number - type: 'null' title: Max Error Total description: Optional total truncation error budget across the backpropagation. max_error_per_slice: anyOf: - type: number - type: 'null' title: Max Error Per Slice description: Optional truncation error budget per circuit slice. type: object required: - circuit_id - observables title: ContextBackpropagationSubmitModel description: |- Define payload for backpropagating observables through a circuit. Attributes: circuit_id: Circuit to backpropagate through. observables: Observables to rewrite, keyed by caller-chosen names. Each is a list of ``(pauli, real, imaginary)`` terms. max_qwc_groups: Optional cap on qubit-wise-commuting groups. max_error_total: Optional total truncation error budget. max_error_per_slice: Optional per-slice truncation error budget. examples: - circuit_id: circ-123 max_error_total: 0.01 observables: energy: - - ZZ - 1 - 0 ContextBackpropagationResultModel: properties: optimized_circuit_ids: additionalProperties: type: string type: object title: Optimized Circuit Ids backpropagated_observables: additionalProperties: items: prefixItems: - type: string - type: number - type: number type: array maxItems: 3 minItems: 3 type: array type: object title: Backpropagated Observables context: type: string title: Context type: object required: - optimized_circuit_ids - backpropagated_observables - context title: ContextBackpropagationResultModel description: |- Represent the outcome of an observable backpropagation. Attributes: optimized_circuit_ids: Circuit produced for each input observable, keyed by the input's name. backpropagated_observables: Rewritten observables as ``(pauli, real, imaginary)`` term lists, keyed by the input's name. context: Summary of what was produced and the next step. HTTPValidationError: properties: detail: items: $ref: '#/components/schemas/ValidationError' type: array title: Detail type: object title: HTTPValidationError ValidationError: properties: loc: items: anyOf: - type: string - type: integer type: array title: Location msg: type: string title: Message type: type: string title: Error Type type: object required: - loc - msg - type title: ValidationError securitySchemes: APIKeyQuery: type: apiKey in: query name: HAIQU_API_KEY APIKeyHeader: type: apiKey in: header name: authorization ```` This documentation is built and hosted on [Mintlify](https://mintlify.com), a developer documentation platform.