Fig. 3
From: Using quantum theory to simplify input–output processes
The quantum circuit that illustrates how a quantum transducer initialized in|s i 〉 at time t−1, simulates the future behavior, when supplied with an input sequence x (t) x (t + 1)…. Upon receiving x at time-step t, it applies a selection operator \({S}_{x}:\left|{s}_{i}\right\rangle \to \left|{s}_{i}^{x}\right\rangle\), followed by the quantum operation A that takes each \(\left|{s}_{i}^{x}\right\rangle \left\langle {s}_{i}^{x}\right|\) to some bipartite state \({\sum }_{y,k}{T}_{ik}^{y|x}\left|y\right\rangle \left|{s}_{k}\right\rangle \left\langle y\right|\left\langle {s}_{k}\right|\) with bipartitions Σ, spanned by |y〉, and \({\mathcal{W}}\), spanned by |s k 〉 (This is always possible with suitable ancilla, see methods). Σ is emitted as output, while \({\mathcal{W}}\) is retained as the quantum causal state at the next time-step. Measurement of Σ in the {|y〉} basis by any outside observer yields outcome y (t). Iterating this procedure then generates correct outputs at each future time-step
