## `eigenkit.algorithms.hhl` ### `HHL` Bases: `BaseQuantumSolver` Quantum linear solver based on the algorithm of Harrow, Hassidim and Lloyd (HHL). Solves `Ax = b` for any invertible square matrix `A`. If `A` is Hermitian, it is used directly, even when it is not positive semidefinite. Any other matrix is converted to the normal equations `A†Ax = A†b`, which have the same solution and a Hermitian, positive semidefinite system matrix. The cost of this conversion is that the condition number `κ` is squared. The circuit itself inverts the eigenvalues. It applies controlled `RY` rotations to the ancilla, with one fixed angle for each state of the clock register, so the reciprocal `1/λ` of each eigenvalue is encoded in the amplitudes without any classical division by the eigenvalues. The readout is different: it uses the simulator's full statevector and keeps only the part where the ancilla is in `|1⟩` and the clock register is in `|0⟩`. HHL produces the quantum state `|x⟩`, so reading a classical vector from it is only possible on a simulator. See HHLResult. Warning The backend must provide `statevector()` on its compiled circuit. The built-in numpy simulator, PennyLane `default.qubit` and `lightning.qubit`, and Qiskit Aer all do. A custom backend registered with register() must implement it, otherwise `solve()` raises `NotImplementedError`. **Scope and scaling.** HHL is efficient only when three conditions hold: `A` is a sum of a few short Pauli terms, `A` is well conditioned, and the state `|b⟩` can be prepared efficiently. - *Hamiltonian simulation.* `e^{iAt}` is Trotterised from the Pauli decomposition of `A`, so its cost follows the number of Pauli terms. This means sparsity in the Pauli basis, which is different from sparsity of the matrix. The module `algorithms/_hamiltonian_sim.py` explains what this means in practice. In short, a sparse matrix is not necessarily cheap to simulate here. - *State preparation.* `|b⟩` is prepared with AmplitudeEncoding, which is treated as an oracle of unit cost. Loading general amplitudes costs `O(N)` gates, so a real speedup assumes that `b` comes from a formula, from an earlier quantum routine, or from a quantum random-access memory (qRAM). - *Conditioning.* The condition number `κ` is never an input. It only bounds the achievable success probability and the size of the clock register. - *Inversion angles.* The gates are elementary (see `_uniformly_controlled_ry()`), so the circuit built in `"honest"` mode contains no unitary-matrix gate. However, the `2^n_l` angles, where `n_l` is the number of clock qubits, are listed one by one instead of being computed by the circuit. The classical description of the circuit therefore has size `Θ(2^n_l)`. The circuit can be built from elementary gates, but its description is not compact. Solvers for dense matrices, for example the one of Wossnig, Zhao and Prakash, reach a quadratic speedup when the spectral norm is bounded. They need a specialised memory architecture that this package does not provide. **Real and complex inputs.** For real `A` and `b`, the readout discards the imaginary part. The leading error of the Trotter eigenvectors enters the raw vector as a purely imaginary component, so discarding the imaginary part removes it. This matters at `trotter_order=1`, where it improves the convergence by one order: the measured error falls as `steps⁻²` for real inputs, and as `steps⁻¹` for a complex Hermitian `A` or a complex `b`, which keep that error. At `trotter_order=2`, the default, the first-order commutator already cancels by construction, so there is nothing left to remove, and real and complex inputs both converge as `steps⁻²`. This advantage was not designed. It depends on the test for real inputs in solve(), so a change to that test changes the convergence rate at order 1. Parameters: | Name | Type | Description | Default | | --------------------- | -------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | `n_eigenvalue_qubits` | `int` | The number of qubits in the clock register. Eigenvalues are resolved into 2\*\*n_eigenvalue_qubits buckets. The value must be at least 1, and at least 2 in "honest" mode. More qubits give a finer resolution of the spectrum and a larger circuit. The default is 4. | `4` | | `t` | \`float | None\` | The evolution time for U = e^{iAt}. If given, it must be positive and finite. If None, it is chosen automatically: from the spectrum in "reference" mode, and from a Gershgorin bound in "honest" mode. The default is None. | | `epsilon` | `float` | The relative eigenvalue cutoff, in \[0, 1). For a Hermitian A, buckets whose estimated eigenvalue is below epsilon * λ_max are treated as zero. For the normal equations, the cutoff applies to the singular values instead, and it is implemented as epsilon² on the squared spectrum. If no eigenvalue is above the cutoff, solve() raises an error. The default is 0.01. | `0.01` | | `provider` | `ProviderSpec` | The backend to use: a registered name ("numpy", "pennylane", "qiskit" and so on), a config instance, or a backend object. It must provide statevector(). The default is "numpy". | `'numpy'` | | `mode` | `str` | How the circuit is built. This argument is keyword-only. "reference" (the default): the eigenvalues are computed classically with eigvalsh, e^{iAt} is a dense QubitUnitary computed with expm, and t is tuned to the known spectrum. This mode is fast and exact, but the O(N³) precomputation and the dense unitary mean that its cost does not follow the scaling of HHL. "honest": the circuit is built as it would be on hardware. No eigendecomposition sets any circuit parameter: t comes from a Gershgorin bound, the scaling constant C of the inversion comes from the resolution of the clock register (see \_honest_qpe_params()), and e^{iAt} is Trotterised into elementary gates. This mode requires n_eigenvalue_qubits >= 2. Input validation still runs an O(N³) rank check. This check makes the solver safer to use and is not part of the algorithm; see solve(). | `'reference'` | | `trotter_steps` | \`int | Literal['auto']\` | The number of Trotter steps for the base evolution time in "honest" mode. It is ignored in "reference" mode. With "auto", the default, the solver picks the smallest number of steps for which a computable bound on the Trotter drift ‖A_eff − A‖ stays below trotter_tolerance × C, which is below the eigenvalue resolution of the clock register. This matters because, with a fixed step size, the circuit is exact HHL for an effective matrix A_eff whose eigenvectors are rotated even when its eigenvalues are exact. That changes the direction of the answer while every check based on the spectrum reports success. An integer turns off the automatic choice, and the solver warns if the drift bound then exceeds C/2. Above that value, an extreme eigenvalue can be captured by the Nyquist bucket or wrap around the branch cut. The automatic choice is limited by a gate budget, and the solver says so when the limit is reached. The default is "auto". | | `trotter_order` | `int` | The Trotter order, 1 or 2. The default is 2. The symmetric product of order 2 is a palindrome, so the first-order commutator cancels, the error falls from O(τ) to O(τ²), and A_eff stays real for a real A. At order 1, A_eff picks up Y-type terms, and a real system returns a complex state. Order 1 is kept for cost comparisons. | `2` | | `trotter_tolerance` | `float` | The target for trotter_steps="auto". This argument is keyword-only. The drift bound must not exceed trotter_tolerance × C. The default, 0.25, keeps the drift at a quarter of a bucket, below the C/2 threshold at which the Nyquist bucket can capture an eigenvalue. At order 2, the cost grows as steps ∝ 1/√tolerance. | `_AUTO_TROTTER_TOLERANCE` | Raises: | Type | Description | | ------------ | ---------------------------------------------------------------------------------------------------------- | | `ValueError` | If n_eigenvalue_qubits is below 1, if t is not a positive finite number, or if epsilon is outside \[0, 1). | Example This example solves a `2 × 2` Hermitian system: ```python import numpy as np from eigenkit import HHL A = np.array([[2, 1], [1, 2]], dtype=float) b = np.array([1, 0], dtype=float) result = HHL(n_eigenvalue_qubits=6).solve(A, b) print(result.solution) print(result.residual) ``` This example solves the same system on a faster simulator: ```python from eigenkit import HHL, PennyLaneConfig solver = HHL( n_eigenvalue_qubits=6, provider=PennyLaneConfig(device_name="lightning.qubit"), ) result = solver.solve(A, b) ``` #### `solve(A, b)` Solve the linear system `Ax = b`. Parameters: | Name | Type | Description | Default | | ---- | --------- | ----------------------------------------------------------------------------------------------------------------------- | ---------- | | `A` | `ndarray` | A real or complex square matrix, which must be invertible. A non-Hermitian matrix is converted to the normal equations. | *required* | | `b` | `ndarray` | The right-hand side, of length A.shape[0]. | *required* | Returns: | Type | Description | | ----------- | ------------------------------------------------------------------------------------------------------------------------------------------------------- | | `HHLResult` | An HHLResult. Its solution is read from the simulator's statevector and completed with a classical least-squares rescaling. See the notes on HHLResult. | Raises: | Type | Description | | --------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------- | | `ValueError` | If A is not square or is singular, if b has the wrong length, is zero or is not finite, or if t causes aliasing in quantum phase estimation (QPE). | | `NotImplementedError` | If the backend does not implement statevector(), or if it returns something else, such as a density matrix. | Note Validation runs one `O(N³)` singular value decomposition (SVD) to reject a singular `A` and to compute a condition number for the warnings. No quantity derived from it reaches `t`, `C`, the rotation angles or the number of Trotter steps, and changing it would leave the circuit exactly the same. `mode="honest"` therefore keeps its defining property, but `solve()` itself still does `O(N³)` work. ### `HHLResult` Bases: `ValueEquality` Output of HHL.solve(). The result is read-only. Assigning to a field raises `FrozenInstanceError`, and writing into an array field raises `ValueError`. This prevents a mistyped `=` (instead of `==`) or an in-place edit from leaving a result whose scalar fields describe a different run from the arrays it holds. To get a writable copy of an array, use `np.array(result.field)`. Results compare by value, including their arrays. They are not hashable, because a result is a collection of numbers and not a key, and a correct hash of a float array is expensive to compute. Attributes: | Name | Type | Description | | --------------------- | --------- | ------------------------------------------------------------------------------------------------------------------------------------------------------- | | `solution` | `ndarray` | The best-fit x with Ax ≈ b. It is float64 for real inputs, and complex128 when A or b has a non-zero imaginary part. | | `euclidean_norm` | `float` | The norm ‖solution‖ of the returned solution. It is computed after the classical least-squares rescaling, so it is not a norm that HHL itself measures. | | `residual` | `float` | The relative error ‖Ax − b‖ / ‖b‖, computed with the original A and b. | | `success_probability` | `float` | The probability that the ancilla is in | | `readout_probability` | `float` | The probability of the joint event from which the solution is read: the ancilla in | | `raw_solution` | \`ndarray | None\` | Note HHL produces the quantum state `|x⟩`, not a vector of numbers. The full array in this result is read from the simulator's statevector, which only a simulator can provide. On hardware, only expectation values can be measured, and extracting all `N` components of `|x⟩` costs `O(N)` measurements, which cancels the speedup. In practice, `|x⟩` is passed to another quantum routine, or used to estimate one observable. Recovering the sign of `1/λ` needs a separate protocol. All of this applies to both values of `mode`. Warning `solution` is completed by two classical steps that hardware cannot perform. For real inputs, the imaginary part is discarded. The magnitude and the sign then come from a least-squares projection, `x = raw · ⟨A·raw, b⟩ / ‖A·raw‖²`, which uses the full classical `A` and `b`. Only the direction of `raw` comes from the quantum computation. The rescaling can change `residual` by an order of magnitude, and for an indefinite `A` its sign correction is not reliable, because `⟨Ax, b⟩ ≥ 0` does not imply `⟨x, A⁻¹b⟩ ≥ 0`. #### `__post_init__()` Make the array fields read-only. Freezing the dataclass prevents a field from being reassigned, but it does not prevent writing into the array that the field holds, and a result that has been written into still reports the scalar values of the original run. The result therefore keeps its own copy, so the array it was built from can still be changed by the caller. To get a writable copy, use `np.array(result.solution)`. Both array fields are frozen. `raw_solution` is optional, and `freeze_arrays` would turn a `None` into a zero-dimensional object array, so it is only frozen when the run recorded one.