Fig. 4

The quantum transducer for the perturbed coin can generate appropriate future statistics via the quantum circuit in a. Suppose a transducer, in state |τ _i 〉, receives input x at time t. To generate output y, it transforms |τ _i 〉 to a 4-qubit quantum state |s _i 〉 by application of an appropriate unitary U on |τ _i 〉 and three ancilla qubits in state |000〉. The transducer then discards qubits one and two if x = 0, or qubits three and four if x = 1. The two remaining qubits, labeled B ₁ and B ₂, are subsequently transformed by a unitary V that maps |00〉 to \(\left|0{\tau }_{0}\right\rangle\) and |11〉 to \(\left|1{\tau }_{1}\right\rangle\) (this is always possible as \(\langle 0{\tau }_{0}\mathrm{|1}{\tau }_{1}\rangle \mathrm{=0}\)). B ₁ is emitted as output while B ₂ is retained by the transducer as the causal state for the subsequent time-step. Measurement of B ₁ in the computational basis yields y. Iteration of this procedure replicates correct future input–output statistics. The resulting improved efficiency is highlighted in b, which depicts the maximum memory required by a quantum transducer \(\bar{Q}\) (orange surface) to simulated the actively perturbed coin vs. its classical counterpart, the structural complexity \(\bar{C}\) (blue surface) for various p and q. While the ε-transducer generally requires 1 bit of memory, the quantum transducer requires less, and becomes increasingly more efficient as p,q→0.5