It has previously been shown that probabilistic quantum logic operations can be performed using linear optical elements, additional photons (ancilla), and post-selection based on the output of single-photon detectors. Here we describe the operation of several quantum logic operations of an elementary nature, including a quantum parity check and a quantum encoder, and we show how they can be combined to implement a controlled-NOT (CNOT) gate. All of these gates can be constructed using polarizing beam splitters that completely transmit one state of polarization and totally reflect the orthogonal state of polarization, which allows a simple explanation of each operation. We also describe a polarizing beam splitter implementation of a CNOT gate that is closely analogous to the quantum teleportation technique previously suggested by Gottesman and Chuang [Nature 402, 390 (1999)]. Finally, our approach has the interesting feature that it makes practical use of a quantum-eraser technique. II. CNOT USING FOUR-PHOTON ENTANGLED STATESAs we mentioned earlier, Gottesman and Chuang showed in a pioneering paper [4] that a CNOT operation could be performed using a modified form of quantum teleportation. Although the required Bell-state measure-

We report a proof-of-principle demonstration of a probabilistic controlled-NOT gate for single photons. Single-photon control and target qubits were mixed with a single ancilla photon in a device constructed using only linear optical elements. The successful operation of the controlled-NOT gate relied on post-selected three-photon interference effects which required the detection of the photons in the output modes.There has been considerable interest in a linear optics approach to quantum computing [1,2], in which probabilistic two-qubit logic operations are implemented using linear optical elements and measurements made on a set of n additional (ancilla) photons. Here we report a proofof-principle demonstration of a probabilistic controlled-NOT (CNOT) gate using a single ancilla photon. Two of the required single-photons were produced using parametric down-conversion [3] while a third photon was obtained from an attenuated laser pulse. The use of only one ancilla photon required that all three photons be detected, in which case the device was known to have correctly performed a CNOT logic operation.Logic operations are inherently nonlinear, so it is somewhat surprising that they can be performed using simple linear optical elements [1,4,5,6,7,8,9,10]. The necessary nonlinearity is obtained by mixing the input photons with n ancilla photons using linear elements, and then measuring the state of the ancilla photons after the interaction. The measurement process is nonlinear [11], since a single-photon detector either records a photon or not, and it projects out the desired logical output state provided that certain results are obtained from the measurements. The results of the operation are known to be correct whenever these specific measurement results are obtained, which occurs with a failure rate that scales as

We describe a quantum algorithm that generalizes the quantum linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] to arbitrary problem specifications. We develop a state preparation routine that can initialize generic states, show how simple ancilla measurements can be used to calculate many quantities of interest, and integrate a quantum-compatible preconditioner that greatly expands the number of problems that can achieve exponential speedup over classical linear systems solvers. To demonstrate the algorithm's applicability, we show how it can be used to compute the electromagnetic scattering cross section of an arbitrary target exponentially faster than the best classical algorithm.The potential power of quantum computing was first described by Feynman, who showed that the exponential growth of the Hilbert space of a quantum computer allows efficient simulations of quantum systems, whereas a classical computer would be overwhelmed [1]. Shor extended the applicability of quantum computing when he developed a quantum factorization algorithm that also provides exponential speedup over the best classical algorithm [2].More recently, Harrow et al.[3] demonstrated a quantum algorithm for solving a linear system of equations that, for well-conditioned matrices, gives exponential speedup over the best classical method. In that paper, the authors demonstrated how to invert a sparse matrix to solve the quantum linear system A|x = |b . The requirements for achieving exponential speedup were: 1) the elements of A be efficiently computable via a blackbox oracle; 2) the matrix A must to be sparse, or efficiently decomposable into sparse form; 3) the condition number of A must scale as polylog N where N is the size of the linear system. As presented, the algorithm had three features that made it difficult to apply to generic problem specifications and achieve the promised exponential speedup. These included: State preparation -preparing the generic state |b is an unsolved problem [4][5][6][7][8], and no mention on how one might do this was provided. Solution readoutsince the solution is stored in a quantum state |x , measurement of it is impractical. The authors suggested that it could be used to calculate some expectation values of an arbitrary operator x|R|x . However, no measurement procedure was specified, and estimating x|R|x is not trivial in general. Condition number -in order for the quantum algorithm to achieve exponential speedup, the condition number can scale at most poly logarithmically with the size of the matrix A. This is a very strict condition that greatly limits the class of problems that can achieve exponential speedup.In this letter, we provide solutions to these three problems, greatly expanding the applicability of the Quantum Linear Systems Algorithm (QLSA). In addition, we show how our new techniques enable the first start-to-finish application of the QLSA to a problem of broad interest and importance. Namely, we show how to solve for the scattering cross section of an arbit...

Abstract:We show that the quantum Zeno effect can be used to implement several quantum logic gates for photonic qubits, including a gate that is similar to the squareroot of SWAP operation. The operation of these devices depends on the fact that photons can behave as if they were non-interacting fermions instead of bosons in the presence of a strong Zeno effect. These results are discussed within the context of several no-go theorems for non-interacting fermions or bosons.

Photon number-resolving detectors are needed for a variety of applications including linear-optics quantum computing. Here we describe the use of time-multiplexing techniques that allows ordinary single photon detectors, such as silicon avalanche photodiodes, to be used as photon numberresolving detectors. The ability of such a detector to correctly measure the number of photons for an incident number state is analyzed. The predicted results for an incident coherent state are found to be in good agreement with the results of a proof-of-principle experimental demonstration.