Universal Superreplication of Unitary Gates

Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to states. In addition to gate replication, we show that $N$ parallel uses of a completely unknown unitary gate can be compressed into a single gate acting on $O({\log }_{2}N)$ qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely.

Figure 1

Quantum network for gate replication. The network simulates $M$ parallel uses of an unknown unitary gate ${\mathcal{U}}_g$, while querying it only $N$ times. The simulation is obtained by transforming the input state of $M$ systems into the joint state of $N$ systems plus an ancilla (via quantum channel ${\mathcal{C}}_1$), applying the unknown gate on the $N$ systems, and then recombining them with the ancilla via a quantum channel ${\mathcal{C}}_2$, which finally produces $M$ output systems.

Figure 2

Gate compression. Alice sandwiches the given gate ${\mathcal{U}}_x$ (in green) between two quantum channels ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ (in red), thus, compressing it into a gate ${{\mathcal{U}}_x}^{\prime }$ acting on a smaller quantum system. The action of ${\mathcal{U}}_x$ is retrieved through Bob’s operations ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ (in blue), which involve the system processed by Alice and a quantum memory kept in Bob’s laboratory. The protocol reduces the amount of quantum communication needed to simulate the application of Alice’s gate to Bob’s input.