From Linear Optical Quantum Computing to Heisenberg-Limited Interferometry

The working principles of linear optical quantum computing are based on photodetection, namely, projective measurements. The use of photodetection can provide efficient nonlinear interactions between photons at the single-photon level, which is technically problematic otherwise. We report an application of such a technique to prepare quantum correlations as an important resource for Heisenberglimited optical interferometry, where the sensitivity of phase measurements can be improved beyond the usual shot-noise limit. Furthermore, using such nonlinearities, optical quantum nondemolition measurements can now be carried out at the singlephoton level. § To whom correspondence should be addressed ([email protected]) From Linear Optical Quantum Computing ... 2 1. Effective nonlinearities by projective measurements Looking back, scalable quantum computation with linear optics was considered to be impossible due to the lack of efficient two-qubit logic gates, despite the ease of implementation of one-qubit gates. Two-qubit gates necessarily need a nonlinear interaction between the two photons, and the efficiency of this nonlinear interaction is typically very tiny in bulk materials [1]. However, Knill, Laflamme, and Milburn recently showed that this barrier can be circumvented with effective nonlinearities produced by projective measurements [2], and with this work scalable linear optical quantum computation (LOQC) becomes a reality. Let us consider the Kerr nonlinearity, which can be described by a Hamiltonian [3] HKerr = h̄κâ†âb̂†b̂, (1) where κ is a coupling constant depending on the third-order nonlinear susceptibility, and â†, b̂† and â, b̂ are the creation and annihilation operators for two optical modes. One convenient choice of the logical qubit can then be represented by the two modes containing a single photon, denoted as |0〉L = |0〉l |1〉k |1〉L = |1〉l |0〉k, (2) where l, k represent the relevant modes, and we have used the notation |·〉L for a logical qubit, in order to distinguish it from the photon-number states |·〉k. For a two-qubit gate, let us assign mode 1,2 for the control qubit, and 3,4 for the target qubit. Suppose now only the modes 2,4 are coupled under the interaction given by Eq.(1). For a given interaction time τ , the transformation can be written as |0〉L|0〉L → |0〉L|0〉L |0〉L|1〉L → |0〉L|1〉L |1〉L|0〉L → |1〉L|0〉L |1〉L|1〉L → e|1〉L|1〉L, (3) where φ ≡ κnanbτ and na = 〈â†â〉, nb = 〈b̂†b̂〉. This operation yields a conditional phase shift [4]. When φ = π, we have the two two-qubit gate called the conditional sign-flip gate. A typical two-qubit gate, controlled-NOT (CNOT), is then simply constructed by using the conditional sign flip and two one-qubit gates (e.g., Hadamard on the target, followed by the conditional sign flip and another Hadamard on the target). In order to have φ ∼ π at the single-photon level, however, a huge third-order nonlinear coupling is required [5]. Instead, Knill, Laflamme, and Milburn devised a nondeterministic conditional sign flip gate using nonlinear sign gate defined by α|0〉+ β|1〉+ γ|2〉 −→ α|0〉+ β|1〉 − γ|2〉. (4) The nonlinear sign gate can be implemented non-deterministically by three beam splitters, two photo-detectors, and one ancilla photon [6] (see Fig. 1). The implementation of conditional sign flip gate is then made by the combination of the From Linear Optical Quantum Computing ... 3