Coherent Superposition States as Quantum Rulers

We explore the sensitivity of an interferometer based on a quantum circuit for coherent states. We show that its sensitivity is at the Heisenberg limit. Moreover we show that this arrangement can measure very small length intervals. There exists a well known isomorphism between interferometers and basic quantum processing circuits. In particular the circuit comprising a Hadamard gate followed by a phase gate and then a second Hadamard gate is equivalent to a single photon, optical interferom-eter with a phase shift in one arm (see Fig.1). Historically this observation has helped to identify candidate quantum circuits. An alternative viewpoint is to consider the efficacy of quantum circuits in performing more traditionally interferometric tasks [1–3]. We have recently proposed an efficient quantum computation scheme based on a coherent state qubit encoding [4]. In this paper we investigate how sensitively distance measurements can be made using the equivalent of the circuit in Fig.1(a) when realized using this new scheme. We find that its sensitivity to small perturbations in length is at the Heisenberg limit. Further more we find that its sensitivity in measuring small length intervals is also at the Heisenberg limit. We refer to this effect as a quantum ruler. Our logical qubits are encoded as follows: the zero state is the vacuum, |0 L = |0, and the one state is the coherent state of amplitude α, |1 L = |α. We assume that the coherent amplitude is real and that α >> 1. We begin by investigating the sensitivity of the idealized circuit of Fig.1(a) using our coherent state qubit encoding and comparing this with the sensitivity of a standard interferometer with a squeezed vacuum input. We then introduce a physical realization of the quantum circuit and consider some more practical issues. Consider the case of the logical zero state, ie the vacuum, entering the first Hadamard gate. The effect of a Hadamard gate is to produce the following transformations in the logical basis: