Quantum state discrimination circuits inspired by Deutschian closed timelike curves

It is known that a party with access to a Deutschian closed timelike curve (DCTC) can perfectly distinguish multiple nonorthogonal quantum states. In this paper we propose a practical method for discriminating multiple nonorthogonal states, by using a previously known quantum circuit designed to simulate DCTCs. This method relies on multiple copies of an input state, multiple iterations of the circuit, and a fixed set of unitary operations. We first characterize the performance of this circuit and study its asymptotic behavior. We also show how it can be equivalently recast as a local adaptive circuit that may be implemented simply in an experiment. Finally, we prove that our state discrimination strategy achieves the multiple Chernoff bound when discriminating an arbitrary set of pure qubit states.

Figure 1

The upper system is the CR system and the lower system is the CV system. The past and future mouths of the wormhole are represented by the double bars on the left and right. The CR and CV systems interact according to the unitary $V_{SC}$, defined in (7), before the CV system enters the future mouth of the wormhole.

Figure 2

This circuit simulates a DCTC with interaction unitary $V_{SC}$. This figure shows three rounds of interaction. As the number of rounds $n$ increases, the simulation approaches the behavior of a DCTC.

Figure 3

Series of simplifications of the original state discrimination circuit in Fig. 2 that ultimately leads to the simple iterative circuit in Fig. 3. First, we explicitly expand $V_{SC}$ and set the state of the $G_n$ registers to $|1\rangle \langle 1|$, as this is optimal for performing our state discrimination scheme. This leads to the circuit in (a). We also note that the controlled-$U_i$ gate is short for the gate ${\sum }_i|i\rangle \langle i|\otimes U_i$. Further, as the controlling registers are effectively classical, the circuit can be simplified to that in (b). Finally, the iterative circuit is rewritten in the compact form of (c). The arrows do not depict a time-travel loop, but instead depict that the output of the unitary $U_i$ at one time interval is fed as input to the $C$ system at the next time interval.