Concepts

LPDO

Site j holds a Kraus tensor V_j with axes (*bonds, kraus, out, in). The full network joins V_j with its complex conjugate through the shared Kraus index, so the channel is Λ(ρ) = Σ_kr V ρ V† and completely positive for any parameter values. Bonds v{i}_{j} (Kraus half) and w{i}_{j} (conjugate half) follow the topology edges.

      kn0      kn1      kn2          <- output, ket side
       |        |        |
   [V_0]==v0_1==[V_1]==v1_2==[V_2]    kraus kr_j shared vertically
       |\       |\       |\
       k0 kr0   k1 kr1   k2 kr2      <- input, ket side
       b0 |     b1 |     b2 |        <- input, bra side
       |/       |/       |/
   [V_0*]=w0_1=[V_1*]=w1_2=[V_2*]
       |        |        |
      bn0      bn1      bn2          <- output, bra side

LPDO.random scales complex Gaussian tensors so that the expected trace is 2**n; LPDO.trace equals 2**n for a trace-preserving channel. LPDO.distance is ‖Λ − Λ'‖_F² / 4**n.

Probability network

For a batch of shots the prepared single-qubit states form tensors (exp, k_j, b_j) and the measured effects tensors (exp, bn_j, kn_j), all sharing the batch hyper-index exp. The ideal circuit sits between the states and the channel as one tensor per site: every two-qubit gate is split by an operator-Schmidt decomposition into two site tensors joined by a bond, so the construction works on any connectivity without compression. Contracting everything with exp left open gives all Tr[E Λ(U ρ U†)] of the batch in one cotengra-optimised contraction.

Loss

L(θ) = −mean log p_i(θ)  +  w · ‖Tr_out Λ(θ) − 𝟙‖_F² / 2**n

The first term is the negative log-likelihood of the shots. The second keeps the channel trace preserving; it is squared so that its gradient vanishes on the manifold and large weights act as a smooth constraint. Because the likelihood alone can be lowered by inflating the channel's trace, the trained trace sits at 2**n (1 + 1/(2w)); the default w = 100 leaves a 0.5 % bias. Custom terms with the same (lpdo_tn, probs) -> scalar signature can be added through TrainingConfig.extra_loss_terms.

Simulated shots

sample_shots draws random (preparation, basis) settings, contracts the exact joint distribution over the 2**n outcomes of each setting once, and draws all its shots from a single multinomial. The default preparations and effects are the six Pauli eigenstates; effect 2 * basis + bit is the projector onto outcome bit of basis Z, X or Y.