The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.

Measurement underpins all quantitative science. A key example is the measurement of optical phase, used in length metrology and many other applications. Advances in precision measurement have consistently led to important scientific discoveries. At the fundamental level, measurement precision is limited by the number N of quantum resources (such as photons) that are used. Standard measurement schemes, using each resource independently, lead to a phase uncertainty that scales as 1/ √ N -known as the standard quantum limit. However, it has long been conjectured [1,2] that it should be possible to achieve a precision limited only by the Heisenberg uncertainty principle, dramatically improving the scaling to 1/N [3]. It is commonly thought that achieving this improvement requires the use of exotic quantum entangled states, such as the NOON state [4,5]. These states are extremely difficult to generate. Measurement schemes with counted photons or ions have been performed with N ≤ 6 [6, 7, 8, 9, 10, 11, 12, 13, 14, 15], but few have surpassed the standard quantum limit [12,14] and none have shown Heisenberg-limited scaling. Here we demonstrate experimentally a Heisenberg-limited phase estimation procedure. We replace entangled input states with multiple applications of the phase shift on unentangled single-photon states. We generalize Kitaev's phase estimation algorithm [16] using adaptive measurement theory [17,18,19,20] to achieve a standard deviation scaling at the Heisenberg limit. For the largest number of resources used (N = 378), we estimate an unknown phase with a variance more than 10 dB below the standard quantum limit; achieving this variance would require more than 4,000 resources using standard interferometry. Our results represent a drastic reduction in the complexity of achieving quantumenhanced measurement precision.Phase estimation is a ubiquitous measurement primitive, used for precision measurement of length, displacement, speed, optical properties, and much more. Recent work in quantum interferometry has focused on nphoton NOON states [5,6,7,8,9,10,11,12,21], (|n |0 + |0 |n ) / √ 2, expressed in terms of number states of the two arms of the interferometer. With this state, an improved phase sensitivity results from a decrease in the phase period from 2π to 2π/n. We achieve improved phase sensitivity more simply using an insight from quantum computing. We apply Kitaev's phase estimation algorithm [16,22] to quantum interferometry, wherein the entangled input state is replaced by multiple passes through the phase shift. The idea of using multi-pass protocols to gain a quantum advantage was proposed for the problem of aligning spatial reference frames [23], and further developed in relation to clock synchronization [24] and phase estimation [25,26].The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. 1a. The algorithm yields, with K + 1 bits of precision, an estimate φ est of a classical phase parameter φ, where e iφ is an eigenvalue of a unitary operator U . It requires us t...

Quantum computation promises to solve fundamental, yet otherwise intractable, problems across a range of active fields of research. Recently, universal quantum logic-gate sets-the elemental building blocks for a quantum computer-have been demonstrated in several physical architectures. A serious obstacle to a full-scale implementation is the large number of these gates required to build even small quantum circuits. Here, we present and demonstrate a general technique that harnesses multi-level information carriers to significantly reduce this number, enabling the construction of key quantum circuits with existing technology. We present implementations of two key quantum circuits: the three-qubit Toffoli gate and the general two-qubit controlled-unitary gate. Although our experiment is carried out in a photonic architecture, the technique is independent of the particular physical encoding of quantum information, and has the potential for wider application.T he realization of a full-scale quantum computer presents one of the most challenging problems facing modern science. Even implementing small-scale quantum algorithms requires a high level of control over multiple quantum systems. Recently, much progress has been made with demonstrations of universal quantum gate sets in a number of physical architectures including ion traps 1,2 , linear optics 3-6 , superconductors 7,8 and atoms 9,10 . In theory, these gates can now be put together to implement any quantum circuit and build a scalable quantum computer. In practice, there are many significant obstacles that will require both theoretical and technological developments to overcome. One is the sheer number of elemental gates required to build quantum logic circuits.Most approaches to quantum computing use qubits-the quantum version of bits. A qubit is a two-level quantum system that can be represented mathematically by a vector in a two-dimensional Hilbert space. Realizing qubits typically requires enforcing a twolevel structure on systems that are naturally far more complex and which have many readily accessible degrees of freedom, such as atoms, ions or photons. Here, we show how harnessing these extra levels during computation significantly reduces the number of elemental gates required to build key quantum circuits. Because the technique is independent of the physical encoding of quantum information and the way in which the elemental gates are themselves constructed, it has the potential to be used in conjunction with existing gate technology in a wide variety of architectures. Our technique extends a recent proposal 11 , and we use it to demonstrate two key quantum logic circuits: the Toffoli and controlled-unitary 12 gates. We first outline the technique in a general context, then present an experimental realization in a linear optic architecture: without our resource-saving technique, linear optic implementations of these gates are infeasible with current technology. Simplifying the Toffoli gateOne of the most important quantum logic gates is the Toffoli 1...

Entanglement is the defining feature of quantum mechanics, and understanding the phenomenon is essential at the foundational level and for future progress in quantum technology. The concept of steering was introduced in 1935 by Schrödinger [1] as a generalization of the Einstein-Podolsky-Rosen (EPR) paradox [2]. Surprisingly, it has only recently been formalized as a quantum information task with arbitrary bipartite states and measurements [3-5], for which the existence of entanglement is necessary but not sufficient. Previous experiments in this area [6-9] have been restricted to the approach of Reid [10], which followed the original EPR argument in considering only two different measurement settings per side. Here we implement more than two settings so as to be able to demonstrate experimentally, for the first time, that EPR-steering occurs for mixed entangled states that are Bell-local (that is, which cannot possibly demonstrate Bell-nonlocality). Unlike the case of Bell inequalities [11][12][13], increasing the number of measurement settings beyond two-we use up to six-dramatically increases the robustness of the EPR-steering phenomenon to noise.

We demonstrate complete characterization of a two-qubit entangling process -a linear optics controlled-not gate operating with coincident detection -by quantum process tomography. We use a maximum-likelihood estimation to convert the experimental data into a physical process matrix. The process matrix allows accurate prediction of the operation of the gate for arbitrary input states and calculation of gate performance measures such as the average gate fidelity, average purity and entangling capability of our gate, which are 0.90, 0.83 and 0.73 respectively.PACS numbers: 03.67. Lx, 03.65.Wj, 03.67.Mn, Quantum information science offers the potential for major advances such as quantum computing [1] and quantum communication [2], as well as many other quantum technologies [3]. Two-qubit entangling gates, such as the controlled-not (cnot), are fundamental elements in the archetypal quantum computer [1]. A promising proposal for achieving scalable quantum computing is that of Knill, Laflamme and Milburn (KLM), in which linear optics and a measurement-induced Kerr-like nonlinearity can be used to construct cnot gates [4]. Gates such as these can also be used to prepare the required entangled resource for optical cluster state quantum computation [5]. The nonlinearity upon which the KLM and related [6,7] cnot schemes are built can be used for other important quantum information tasks, such as quantum nondemolition measurements [8,9] and preparation of novel quantum states (for example, [10]). An essential step in realizing such advances is the complete characterization of quantum processes.A complete characterization in a particular input/output state space requires determination of the mapping from one to the other. In discrete-variable quantum information, this map can be represented as a state transfer function, expressed in terms of a process matrix χ. Experimentally, χ is obtained by performing quantum process tomography (QPT) [11,12]. QPT has been performed in a limited number of systems. A one-qubit teleportation circuit [13], and a controlled-NOT process acting on a highly mixed two-qubit state [14] have been investigated in liquid-state NMR. In optical systems, where pure qubit states are readily prepared, one-qubit processes have been investigated by both ancilla-assisted [15,16] and standard [17] QPT. Two-qubit optical QPT has been performed on a beamsplitter acting as a Bellstate filter [18].We fully characterize a two-qubit entangling gatea cnot gate acting on pure input states -by QPT, maximum-likelihood reconstruction, and analysis of the resulting process matrix. The maximum likelihood technique overcomes the problem that the naïve matrix inversion procedure in QPT, when performed on real (i.e., inherently noisy) experimental data, typically leads to an unphysical process matrix. In a previous maximumlikelihood QPT experiment [18], a reduced set of fitting constraints was used. Here we present a fully-constrained fitting technique that can be applied to any physical process. After obtaining our physical...

The idea of signal amplification is ubiquitous in the control of physical systems, and the ultimate performance limit of amplifiers is set by quantum physics. Increasing the amplitude of an unknown quantum optical field, or more generally any harmonic oscillator state, must introduce noise 1 . This linear amplification noise prevents the perfect copying of the quantum state 2 , enforces quantum limits on communications and metrology 3 , and is the physical mechanism that prevents the increase of entanglement via local operations. It is known that non-deterministic versions of ideal cloning 4 and local entanglement increase (distillation) 5 are allowed, suggesting the possibility of non-deterministic noiseless linear amplification. Here we introduce, and experimentally demonstrate, such a noiseless linear amplifier for continuous-variables states of the optical field, and use it to demonstrate entanglement distillation of field-mode entanglement. This simple but powerful circuit can form the basis of practical devices for enhancing quantum technologies. The idea of noiseless amplification unifies approaches to cloning and distillation, and will find applications in quantum metrology and communications.A quantum-noise-free amplifier, if it could be constructed, could aid a wide variety of quantum-enhanced information protocols, primarily through its ability to distill and purify continuous-variable entanglement. This

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Photons have been a flagship system for studying quantum mechanics, advancing quantum information science, and developing quantum technologies. Quantum entanglement, teleportation, quantum key distribution and early quantum computing demonstrations were pioneered in this technology because photons represent a naturally mobile and low-noise system with quantum-limited detection readily available. The quantum states of individual photons can be manipulated with very high precision using interferometry, an experimental staple that has been under continuous development since the 19th century. The complexity of photonic quantum computing device and protocol realizations has raced ahead as both underlying technologies and theoretical schemes have continued to develop. Today, photonic quantum computing represents an exciting path to medium-and large-scale processing. It promises to out aside its reputation for requiring excessive resource overheads due to inefficient two-qubit gates. Instead, the ability to generate large numbers of photons-and the development of integrated platforms, improved sources and detectors, novel noise-tolerant theoretical approaches, and more-have solidified it as a leading contender for both quantum information processing and quantum networking. Our concise review provides a flyover of some key aspects of the field, with a focus on experiment. Apart from being a short and accessible introduction, its many references to in-depth articles and longer specialist reviews serve as a launching point for deeper study of the field. CONTENTS arXiv:1907.06331v1 [quant-ph]