Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics
Stephen D. Bartlett, Eleni Diamanti, Barry C. Sanders, Yoshihisa Yamamoto
备注：
9 pages, 3 figures
发表信息：
Proceedings of Free-Space Laser Communication and Laser Imaging II, Vol. 4821, SPIE International Symposium on Optical Science and Technology (7-11 July 2002), eds. J C Ricklin and D G Voelz (SPIE, Bellingham, WA, 2002)
The performance of nondeterministic nonlinear gates in linear optics relies on the photon counting scheme being employed and the efficiencies of the detectors in such schemes. We assess the performance of the nonlinear sign gate, which is a critical component of linear optical quantum computing, for two standard photon counting methods: the double detector array and the visible light photon counter. Our analysis shows that the double detector array is insufficient to provide the photon counting capability for effective nondeterministic nonlinear transformations, and we determine the gate fidelity for both photon counting methods as a function of detector efficiencies.
We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC^0[q], where n-ary Mod-q gates are also allowed. We show that it is possible to make a `cat' state on n qubits in constant depth if and only if we can construct a parity or Mod-2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC^0[2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct Mod-q gates in constant depth fo…
Coherent multidimensional optical spectroscopy techniques are broadly applied across the electromagnetic spectrum ranging from NMR to the UV. These reveal properties of matter through correlation plots of signal fields generated in response to sequences of short pulses with variable delays. Here we discuss a new class of multidimensional techniques obtained by time-and-frequency resolved photon coincidence counting measurements of N photons which constitutes a 2N dimensional spectrum. A compact description of these signals is developed based on time ordered superoperators rather than the norma…
We propose definitions of $\QAC^0$, the quantum analog of the classical class $\AC^0$ of constant-depth circuits with AND and OR gates of arbitrary fan-in, and $\QACC[q]$, the analog of the class $\ACC[q]$ where $\Mod_q$ gates are also allowed. We prove that parity or fanout allows us to construct quantum $\MOD_q$ gates in constant depth for any $q$, so $\QACC[2] = \QACC$. More generally, we show that for any $q,p > 1$, $\MOD_q$ is equivalent to $\MOD_p$ (up to constant depth). This implies that $\QAC^0$ with unbounded fanout gates, denoted $\QACwf^0$, is the same as $\QACC[q]$ and $\QACC$ for…
We discuss techniques for producing, manipulating and measureing qubits encoded optically as vacuum and single photon states. We show that a universal set of non-deterministic gates can be constructed using linear optics and photon counting. We investigate the efficacy of a test gate given realistic detector efficiencies.
We investigate the implementation of binary projective measurements with linear optics. This problem can be viewed as a single-shot discrimination of two orthogonal pure quantum states. We show that any two orthogonal states can be perfectly discriminated using only linear optics, photon counting, coherent ancillary states, and feedforward. The statement holds in the asymptotic limit of large number of these physical resources.
We implement a photon-counting Optical Time Domain Reflectometer (OTDR) at 1.55um which exhibits a high 2-point resolution and a high accuracy. It is based on a low temporal-jitter photon-counting module at 1.55um. This detector is composed of a periodically poled Lithium niobate (PPLN) waveguide, which provides a wavelength conversion from near infrared to visible light, and a low jitter silicon photon-counting detector. With this apparatus, we obtain centimetre resolution over a measurement range of tens of kilometres.
Although universal continuous-variable quantum computation cannot be achieved via linear optics (including squeezing), homodyne detection and feed-forward, inclusion of ideal photon counting measurements overcomes this obstacle. These measurements are sometimes described by arrays of beam splitters to distribute the photons across several modes. We show that such a scheme cannot be used to implement ideal photon counting and that such measurements necessarily involve nonlinear evolution. However, this requirement of nonlinearity can be moved "off-line," thereby permitting universal continuous-…
Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, where the number of required SWAP gates is to be minimized. We introduce an Integer Linear Programming model of the problem of which the size scales polynomially in the number of qubits and gates. Further…
Single photon detector (SPD) has a maximum count rate due to its dead time, which results in that the dynamic range of photon counting optical time-domain reflectometry (PC-OTDR) de-creases with the length of monitored fiber. To further improve the dynamic range of PC-OTDR, we propose and demonstrate an externally time-gated scheme. The externally time-gated scheme is realized by using a high-speed optical switch, i.e. a Mach-Zehnder interferometer, to modulate the back-propagation optical signal, and to allow that only a certain segment of the fiber is monitored by the SPD. The feasibility of…
The uses of a silicon-pixel camera with very good time resolution ($\sim$nanosecond) for detecting multiple, bunched optical photons is explored. We present characteristics of the camera and describe experiments proving its counting capabilities. We use a spontaneous parametric down-conversion source to generate correlated photon pairs, and exploit the Hong-Ou-Mandel interference effect in a fiber-coupled beam splitter to bunch the pair onto the same output fiber. It is shown that the time and spatial resolution of the camera enables independent detection of two photons emerging simultaneously…
We propose a scheme for linear optical quantum Fredkin gate based on the combination of recently experimentally demonstrated linear optical partial SWAP gate and controlled-Z gates. Both heralded gate and simplified postselected gate operating in the coincidence basis are designed. The suggested setups have a simple structure and require stabilization of only a single Mach-Zehnder interferometer. A proof-of-principle experimental demonstration of the postselected Fredkin gate appears to be feasible and within the reach of current technology.
It is well known that a minimum error quantum measurement for arbitrary binary optical coherent states can be realized by a receiver that comprises interfering with a coherent reference light, photon counting, and feedback control. We show that, for ternary optical coherent states, a minimum error measurement cannot always be realized by such a receiver. The problem of finding an upper bound on the maximum success probability of such a receiver can be formulated as a convex programming. We derive its dual problem and numerically find the upper bound. At least for ternary phase-shift keyed cohe…
The behavior of an atomic double lambda system in the presence of a strong off-resonant classical field and a few-photon resonant quantum field is examined. It is shown that the system possesses properties that allow a single-photon state to be distilled from a multi-photon input wave packet. In addition, the system is also capable of functioning as an efficient photodetector discriminating between one- and two-photon wave packets with arbitrarily high efficiency.
Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multi-mode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though non-universal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is s…
We present an InGaAs/InP avalanche photodiode with an active quenching circuit on an ASIC (application specific integrated circuit) that is capable of operating in both gated and free-running modes. The 1.6mm2 ASIC chip is fabricated using CMOS (complementary metal oxide semiconductor) technology guaranteeing long-term stability, reliability and compactness. In the free-running mode we find a single photon detection efficiency of 10% with <2kHz of noise.
We investigate the possibility of implementing a given projection measurement using linear optics and arbitrarily fast feedforward based on the continuous detection of photons. In particular, we systematically derive the so-called Dolinar scheme that achieves the minimum error discrimination of binary coherent states. Moreover, we show that the Dolinar-type approach can also be applied to projection measurements in the regime of photonic-qubit signals. Our results demonstrate that for implementing a projection measurement with linear optics, in principle, unit success probability may be approa…
Universal quantum gates lie at the heart of designing quantum computer. We construct two compact quantum circuits to implement post-selected controlled-phase-flip (CPF) gate and Toffoli gate with linear optics assisted by one and two single photons, respectively. The current existing maximum success probability of 1/4 for linear optical CPF gate is achieved by resorting to an ancillary single photon rather than an entangled photon pair or two single photons. Remarkably, our Toffoli gate is accomplished with current maximum success probability of 1/30 without using additional entangled photon p…
We show that a universal set of gates for quantum computation with optics can be quantum teleported through the use of EPR entangled states, homodyne detection, and linear optics and squeezing operations conditioned on measurement outcomes. This scheme may be used for fault-tolerant quantum computation in any optical scheme (qubit or continuous variable). The teleportation of nondeterministic nonlinear gates employed in linear optics quantum computation is discussed.
We study the class of two qubit gates which can be achieved using only linear optical elements (beam splitters and phase shifters) and post-selection. We are able to exactly characterize this set, and find that it is impossible to implement most two qubit gates in this way. The proof also gives rise to an algorithm for calculating the optimal success probability of those gates which are achievable.
We clarify the connections between the erasure scheme of probabilistic CNOT gate implementation recently proposed by Pittman, Jacobs and Franson [Phys. Rev. A 64, 062311 (2001)] and quantum teleportation.
We propose a deterministic SWAP gate for spatially encoded qubits. The gate is constructed from waveguide crossings, Mach Zender Interferometers and phase shifters providing the gate reconfigurability. Through manipulating the phase of the phase shifters, we can apply either the SWAP or identity gates. As an essential element of nearest neighbor qubit networks, the SWAP gate has a simple structure and presents minimal overhead, making it viable for universal quantum computation.
The performance of three types of InGaAs/InP avalanche photodiodes is investigated for photon counting at 1550 nm in the temperature range of thermoelectric cooling. The best one yields a dark count probability of $% 2.8\cdot 10^{-5}$ per gate (2.4 ns) at a detection efficiency of 10% and a temperature of -60C. The afterpulse probability and the timing jitter are also studied. The results obtained are compared with those of other papers and applied to the simulation of a quantum key distribution system. An error rate of 10% would be obtained after 54 kilometers.
Recently it was realized that linear optics and photo-detectors with feedback can be used for theoretically efficient quantum information processing. The first of three steps toward efficient linear optics quantum computation (eLOQC) was to design a simple non-deterministic gate, which upon post-selection based on a measurement result implements a non-linear phase shift on one mode. Here a computational strategy is given for finding non-deterministic gates for bosonic qubits with helper photons. A more efficient conditional sign flip gate is obtained.
We present a method for computing the action of conditional linear optical transformations, conditioned on photon counting, for arbitrary signal states. The method is based on the Q-function, a quasi probability distribution for anti normally ordered moments. We treat an arbitrary number of signal and ancilla modes. The ancilla modes are prepared in an arbitrary product number state. We construct the conditional, non unitary, signal transformations for an arbitrary photon number count on each of the ancilla modes.
We consider the realization of quantum Fredkin gate with only linear optics and single photons. First we construct a heralded Fredkin gate using four heralded controlled-not (CNOT) gates. Then we simplify this method to a post-selected one utilizing only two CNOT gates. We also give a possible realization of this method which is feasible with current experimental technology. Another post-selected scheme requires time entanglement of the input photons but needs no ancillary photons.
We propose a scheme to perform probabilistic quantum gates on remote trapped atom qubits through interference of optical frequency qubits. The method does not require localization of the atoms to the Lamb-Dicke limit, and is not sensitive to interferometer phase instabilities. Such probabilistic gates can be used for scalable quantum computation.
Aiming the construction of quantum computers and quantum communication systems based on optical devices, in this work we present possible implementations of quantum and classical CNOTs gates, as well an optical setup for generation and distribution of bipartite entangled states, using linear optical devices and photon number quantum non-demolition measurement.
We describe the construction of a conditional quantum control-not (CNOT) gate from linear optical elements following the program of Knill, Laflamme and Milburn [Nature {\bf 409}, 46 (2001)]. We show that the basic operation of this gate can be tested using current technology. We then simplify the scheme significantly.
We present a linear-optical scheme for a controlled-phase gate with tunable phase shift programmed by a qubit state. In contrast to all previous tunable controlled-phase gates, the phase shift is not hard-coded into the optical setup, but can be tuned to any value from 0 to pi by the state of a so-called program qubit. Our setup is feasible with current level of technology using only linear-optical components. We provide an experimental feasibility study to assess the gate's implementability. We also discuss options for increasing the success probability up to 1/12 which approaches the success…
We describe the operation and tolerances of a non-deterministic, coincidence basis, quantum CNOT gate for photonic qubits. It is constructed solely from linear optical elements and requires only a two-photon source for its demonstration.
We describe a simple scheme for implementing the non-linear sign gate of Knill, Laflamme and Milburn (Nature, {\bf 409}, 46-52, Jan. 4 (2001)) which forms the basis of an experiment underway at the University of Vienna.
We present a linear-optical implementation of a class of two-qubit partial SWAP gates for polarization states of photons. Different gate operations, including the SWAP and entangling square root of SWAP, can be obtained by changing a classical control parameter -- namely the path difference in the interferometer. Reconstruction of output states, full process tomography and evaluation of entanglement of formation prove very good performance of the gates.
We have modelled the Zeno effect Control-Sign gate of Franson et al (PRA 70, 062302, 2004) and shown that high two-photon to one-photon absorption ratios, $κ$, are needed for high fidelity free standing operation. Hence we instead employ this gate for cluster state fusion, where the requirement for $κ$ is less restrictive. With the help of partially offline one-photon and two-photon distillations, we can achieve a fusion gate with unity fidelity but non-unit probability of success. We conclude that for $κ> 2200$, the Zeno fusion gate will out perform the equivalent linear optics gate.
Physical realizations of the canonical phase measurement for the optical phase are unknown. Single-shot phase estimation, which aims to determine the phase of an optical field in a single shot, is critical in quantum information processing and metrology. Here we present a family of strategies for single-shot phase estimation of coherent states based on adaptive non-Gaussian, photon counting, measurements with coherent displacements that maximize information gain as the measurement progresses, which have higher sensitivities over the best known adaptive Gaussian strategies. To gain understandin…
We report the first experimental demonstration of an optical controlled-NOT gate constructed entirely in fibre. We operate the gate using two heralded optical fibre single photon sources and find an average logical fidelity of 90% and an average process fidelity of 0.83<F<0.91. On the basis of a simple model we are able to conclude that imperfections are primarily due to the photon sources, meaning that the gate itself works with very high fidelity.
Knill, Laflamme, and Milburn (KLM) proved that it is possible to build a scalable universal quantum computer using only linear-optics elements and conditional dynamics [Nature (London) {\bf 409}, 46 (2001)\cite{Knill}]. However, the practical realization of the quantum logic gate for the scheme is still technically difficult. A major difficulty is the requirement for sub-wavelength level stabilization of the interlocking interferometers. Following our recent experimental work[Phys. Rev. Lett.{\bf 92}, 017902 (2004)\cite{Sanaka2}], we describe a more feasible scheme to implement the gate that g…
We suggest a scheme of using two-mode squeezed vacuum for conditional teleportation of quantum states of optical field. Alice mixes the input state with one of the squeezed modes on another squeezing device and detects the output photon numbers. The result is then communicated to Bob who shifts the photon number of his part accordingly. This is a principally realizable modification of the recent scheme [G.J. Milburn and S.L. Braunstein, Phys. Rev. A 60, 937 (1999)] where measurements of photon number difference and phase sum are considered. We show that for some classes of states this method c…
We present and experimentally demonstrate a novel optical nondestructive controlled-NOT gate without using entangled ancilla. With much fewer measurements compared with quantum process tomography, we get a good estimation of the gate fidelity. The result shows a great improvement compared with previous experiments. Moreover, we also show that quantum parallelism is achieved in our gate and the performance of the gate can not be reproduced by local operations and classical communications.
We propose a scheme for conditional implementation of a quantum phase gate by using distant atoms trapped in different optical cavities. Instead of direct interaction between atoms, the present scheme makes use of quantum interference of polarized photons decaying from the optical cavities to conditionally create the desired quantum phase gate between two distant atoms. The proposed scheme only needs linear optical elements and a two-fold coincidence detection, and are insensitive to the quantum noise. The scheme can be directly used to prepare any quantum state of many distant atoms.
We report on the first experimental realization of optimal linear-optical controlled phase gates for arbitrary phases. The realized scheme is entirely flexible in that the phase shift can be tuned to any given value. All such controlled phase gates are optimal in the sense that they operate at the maximum possible success probabilities that are achievable within the framework of any postselected linear-optical implementation. The quantum gate is implemented using bulk optical elements and polarization encoding of qubit states. We have experimentally explored the remarkable observation that the…
Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such number-path entanglement has thought to have been resource intensive to create in the first place --- typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feed-forward, which are difficult to implement. Recently in [Phys. Rev. Lett. 114, 170802 (2015)] we showed that number-path entanglement from a BosonSampling inspired interferometer can be used to beat the shot-no…
Knill, Laflamme, and Milburn [Nature 409, 46 (2001)] showed that linear optics techniques could be used to implement a nonlinear sign gate. They also showed that two of their nonlinear sign gates could be combined to implement a controlled-phase gate, which has a number of practical applications. Here we describe an alternative implementation of a controlled-phase gate that only requires the use of a single nonlinear sign gate. This gives a much higher average probability of success when the required ancilla photons are generated using heralding techniques. This implementation of a controlled-…
We introduce an elementary optical gate for expanding polarization entangled W states, in which every pair of photons are entangled alike. The gate is composed of a pair of 50:50 beamsplitters and ancillary photons in the two-photon Fock state. By seeding one of the photons in an $n$-photon W state into this gate, we obtain an $(n+2)$-photon W state after post-selection. This gate gives a better efficiency and a simpler implementation than previous proposals for $\rm W$-state preparation.
We design linear optics multiqubit quantum logic gates. We assume the traditional encoding of a qubit onto state of a single photon in two modes (e.g. spatial or polarization). We suggest schemes allowing direct probabilistic realization of the fundamental Toffoli and Fredkin gates without resorting to a sequence of single- and two-qubit gates. This yields more compact schemes and potentially reduces the number of ancilla photons. The proposed setups involve passive linear optics, sources of auxiliary single photons or maximally entangled pairs of photons, and single-photon detectors. In parti…
Linear optics quantum logic gates are the best tool to generate multi-photon entanglement. Simplifying a recent approach [Phys. Rev. A 65, 062324; Phys. Rev. A 66, 024308] we were able to implement the conditional phase gate with only one second order interference at a polarization dependent beam splitter, thereby significantly increasing its stability. The improved quality of the gate is evaluated by analysing its entangling capability and by performing full process tomography. The achieved results ensure that this device is well suited for implementation in various multi photon quantum infor…
Recent progress in integrated-optics technology has made photonics a promising platform for quantum networks and quantum computation protocols. Integrated optical circuits are characterized by small device footprints and unrivalled intrinsic interferometric stability. Here, we take advantage of femtosecond-laser-written waveguides' ability to process polarization-encoded qubits and present the first implementation of a heralded controlled-NOT gate on chip. We evaluate the gate performance in the computational basis and a superposition basis, showing that the gate can create polarization entang…
We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to two times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this paper for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed inst…
In principle the Zeno effect controlled-sign gate of Franson et al's (PRA 70, 062302, 2004) is a deterministic two-qubit optical gate. However, when realistic values of photon loss are considered its fidelity is significantly reduced. Here we consider the use of measurement based quantum processing techniques to enhance the operation of the Zeno gate. With the help of quantum teleportation, we show that it is possible to achieve a Zeno CNOT gate (GC-Zeno gate) that gives (near) unit fidelity and moderate probability of success of 0.76 with a one-photon to two-photon transmission ratio $κ=10^4$…
We report a 1.036 GHz gated Geiger mode InGaAs avalanche photodiode with a detection dead time of just 1.93 ns. This is demonstrated by full recovery of the detection efficiency two gate cycles after a detection event, as well as a measured maximum detection rate of 497 MHz. As an application, we measure the second order correlation function $g^{(2)}$ of the emission from a diode laser with a single detector which works reliably at high speed owing to the extremely short dead time of the detector. The device is ideal for high bit rate fiber wavelength quantum key distribution and photonic quan…
We study optical gate array of KLM model of quantum computer for C-Sign operator, which contains linear elements and two nonlinear phase shifts. Linear elements and photon counting are taken to be ideal, whereas nonlinear phase shifts are subject of realistic and unavoidable factors of decoherence: the immediate photon leak from cavities and mediated leakage with the atom, which flies out of the cavity and plays the role of ancilla. We use JC model with Lindblad master equation to estimate the dependence of the validity of C-Sign operator from the rate of photon leak and estimate the limitatio…
A two qubit quantum gate, namely the C-Phase, has been realized by exploiting the longitudinal momentum (i.e. the optical path) degree of freedom of a single photon. The experimental setup used to engineer this quantum gate represents an advanced version of the high stability closed-loop interferometric setup adopted to generate and characterize 2-photon 4-qubit Phased Dicke states. Some experimental results, dealing with the characterization of multipartite entanglement of the Phased Dicke states are also discussed in detail.
Assisted with linear optical manipulation, single photon, entangled photon pairs, photon measurement, and classical communication, a scheme for two-spin qubits phase gate and teleportation of a CNOT gate between two electron spins from acting on local qubits to acting on remote qubits using quantum dots in optical microcavities is proposed. The scheme is based on spin selective photon reflection from the cavity and is achieved in a deterministic way by the sequential detection of photons and the single-qubit rotations of a single electron spin in a self-assembled GaAs/InAs quantum dot. The fea…
The concept of directionally unbiased optical multiports is introduced, in which photons may reflect back out the input direction. A linear-optical implementation is described, and the simplest three-port version studied. Symmetry arguments demonstrate potential for unusual quantum information processing applications. The devices impose group structures on two-photon entangled Bell states and act as universal Bell-state processors to implement probabilistic quantum gates acting on state symmetries. These multiports allow optical scattering experiments to be carried out on arbitrary undirected …
In quantum computation with continous-variable systems, quantum advantage can only be achieved if some non-Gaussian resource is available. Yet, non-Gaussian unitary evolutions and measurements suited for computation are challenging to realize in the lab. We propose and analyze two methods to apply a polynomial approximation of any unitary operator diagonal in the amplitude quadrature representation, including non-Gaussian operators, to an unknown input state. Our protocols use as a primary non-Gaussian resource a single-photon counter. We use the fidelity of the transformation with the target …
We present some deterministic schemes to construct universal quantum gates, that is, controlled- NOT, three-qubit Toffoli, and Fredkin gates, between flying photon qubits and stationary electron-spin qubits assisted by quantum dots inside double-sided optical microcavities. The control qubit of our gates is encoded on the polarization of the moving single photon and the target qubits are encoded on the confined electron spins in quantum dots inside optical microcavities. Our schemes for these universal quantum gates on a hybrid system have some advantages. First, all the gates are accomplished…
One of the main advantages of an optical approach to quantum computing is the fact that optical fibers can be used to connect the logic and memory devices to form useful circuits, in analogy with the wires of a conventional computer. Here we describe an experimental demonstration of a simple quantum circuit of that kind in which two probabilistic exclusive-OR (XOR) logic gates were combined to calculate the parity of three input qubits.
