Nearly Deterministic Linear Optical Controlled-NOT Gate

Abstract

We show how to construct a near deterministic CNOT using several single photons sources, linear optics, photon number resolving quantum non-demolition detectors and feed-forward. This gate does not require the use of massively entangled states common to other implementations and is very efficient on resources with only one ancilla photon required. The key element of this gate are non-demolition detectors that use a weak cross-Kerr nonlinearity effect to conditionally generate a phase shift on a coherent probe, if a photon is present in the signal mode. These potential phase shifts can then be measured using highly efficient homodyne detection.PACS numbers: 03.67. Lx, 03.67.Mn, In the past few years we have seen the emergence of single photon optics with polarisation states as a realistic path for achieving universal quantum computation. This started with the pioneering work of Knill, Laflamme and Milburn [KLM][1] who showed that with only single photon sources and detectors and linear elements such as beam-splitters, a near deterministic CNOT gate could be created, through with the use of significant but polynomial resources. With this architecture for the CNOT gate and trivial single qubit rotations a universal set of gates is hence possible and a route forward for creating large devices can be seen. Since this original work there has been significant progress both theoretically [2,3,4,5,6] and experimentally [7,8,9], with a number of CNOT gates actually demonstrated.Much of the theoretical effort has focused on determining more efficient ways to perform the controlled logic. The standard model for linear logic uses only[1]:• Single photon sources, • Linear optical elements including feed-forward, • Photon number resolving single photon detectors, and it has been shown by Knill[4] that the maximum probability for achieving the CNOT gate is 3/4. While these upper bounds are not thought to be tight, with the best success probabilities for the CNOT gate being 2/27[10], it does indicate that near deterministic gates are not possible using only the above resources and strategy. These gates can be made efficient using the "standard" optical teleportation tricks which require the use of massively entangled resources. Are there other natural ways to increase the efficient of these gate operations? Franson et al. [2] showed that if you can increase your allowed physical resources to include maximally entangled two photon states, then the CNOT gate can have its probability of success boosted to 1/4, though this is still far below the 3/4 maximum. Alternatively it is possible to use single photons for the cluster state method of one way quantum computation [5,6]. This can dramatically decrease the number of single photons sources required to perform a CNOT gate (from up to 10000 for KLM logic to 45 for the cluster approaches). The overhead here in single photon sources is large (but polynomial and hence still efficient in a sense). Can we however build near deterministic (or deterministic) linear optics gates with a low ...

Observation of the nonlinear phase shift due to single post-selected photons

“…Sum-frequency generation of two heralded single photons has also been demonstrated 31 . In parallel, there has been some controversy about the applicability of such strong interactions to quantum logic due to fundamental noise limits [32][33][34] and suggestions that intermediate-strength nonlinearities might be a powerful approach 35,36 . At the present time, the nonlinear phase shift written by a single photon on a probe beam has to our knowledge never been reported, and prior experiments have been performed with classical pulses with average photon numbers on the order of a few hundred 37 in free space, or as low as 16 or 0.1 in a hollow-core fibre filled with atomic vapour 20 or a nonlinear photonic-crystal fibre 19 , respectively.…”

Two-step measurement of the concurrence for hyperentangled state

“…Such measurement can be finished by a general homodyne-heterodyne measurement [39]. The subscript m means the momentum entanglement.…”

“…Here we should point out that we should adopt the different way to measure the coherent state, which makes the phase shift ±θ undistinguished. This measurement can be achieved by choosing the local oscillator phase π/2 offset from the probe phase, which is called an X quadrature measurement [39]. The total concurrence is…”

“…It can be used to perform the quantum nondemolition (QND) measurement, which has been widely used in quantum information processing, such as construction of the CNOT gate [39,40], performing the Bell-state analysis [28,41], entanglement purification and concentration [42][43][44][45], and so on [46][47][48][49][50][51][52]. The Hamiltonian of a cross-Kerr nonlinear interaction is H =hχa † s a s a † p a p [39].…”

Efficient optical quantum information processing

“…On the other hand, much smaller non-linearities can be generated, for example, with electromagnetically induced transparencies (EIT) [10,11,12]. In this paper, we show that with modest additional optical resources these small non-linearities provide the building blocks for efficient optical QIP [13,14,15,16,17]. We present a highly efficient non-absorbing single photon number resolving detector, a nondestructive two qubit parity detector, a nondestructive Bell state measurement, and a near deterministic controlled-not (CNOT) gate.…”