End-to-end channel and output error contract

A finite $L$-stage composition contract consists of nonzero finite-dimensional complex Hilbert spaces $H_0,\dots ,H_L$ and, for each $j=1,\dots ,L$, an ideal channel ${\mathrm{\Phi }}_j:\mathcal{L}(H_{j-1})\to \mathcal{L}(H_j)$ and an implemented channel ${\tilde{\mathrm{\Phi }}}_j$ with the same domain and codomain. Channels are completely positive and trace-preserving on the full operator space, as in Finite-dimensional quantum channel. Products act right to left, so ${\mathrm{\Phi }}_L\cdots {\mathrm{\Phi }}_1$ applies stage $1$ first, following Denotational semantics of a finite layered circuit.

For channels on the same nonzero finite-dimensional complex Hilbert input and output spaces define the normalized diamond distance $d_{\diamond }(\mathrm{\Psi },\mathrm{\Phi }):=\frac{1}{2}\mathrm{\parallel }\mathrm{\Psi }-\mathrm{\Phi }{\mathrm{\parallel }}_{\diamond },$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\diamond }$ is the unhalved supremum of induced trace norms over all finite reference spaces, as in Diamond Norm. For density operators $\rho ,\sigma$ define $D(\rho ,\sigma )=\frac{1}{2}\mathrm{\parallel }\rho -\sigma {\mathrm{\parallel }}_1$, as in Trace Distance. For probability distributions $p,q$ on a common finite output alphabet define $d_{\mathrm{T}\mathrm{V}}(p,q)=\frac{1}{2}{\sum }_a\mathrm{\mid }p(a)-q(a)\mathrm{\mid }$.

The local channel error at stage $j$ is $d_{\diamond }({\tilde{\mathrm{\Phi }}}_j,{\mathrm{\Phi }}_j)$. The final channel error is $d_{\diamond }({\tilde{\mathrm{\Phi }}}_L\cdots {\tilde{\mathrm{\Phi }}}_1,{\mathrm{\Phi }}_L\cdots {\mathrm{\Phi }}_1)$. A final output-distribution error is total variation distance between the finite laws obtained by applying a declared final POVM to the two output states. A failure probability is the probability of a declared failure event in the finite execution history. These four quantities have different definitions and must be reported separately.

When an input is entangled with a finite reference register, include that reference in the density operator of Density operator and retain it through the channel call; the diamond distance already takes the supremum over such references. Channel, state, and classical output contracts are distinct. Cost coordinates for input description and preprocessing, state preparation, gates, queries including controlled and inverse access, workspace, measurement, postprocessing, and classical output are specified separately by Classical output, quantum-state output, and access costs and Circuit-family and computation cost model.