A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2(n)). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Cluster-state quantum computation

This article is a short introduction to and review of the cluster-state model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of single-qubit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel properties of the model, including a proof that the cluster state cannot occur as the exact ground state of any naturally occurring physical system, and a proof that measurements on any quantum state which is linearly prepared in one dimension can be efficiently simulated on a classical computer, and thus are not candidates for use as a substrate for quantum computation.

Entangled-state cycles from conditional quantum evolution

A system of cascaded qubits interacting via the one-way exchange of photons is studied. While for general operating conditions the system evolves to a superposition of Bell states (a dark state) in the long-time limit, under a particular resonance condition no steady state is reached within a finite time. We analyze the conditional quantum evolution (quantum trajectories) to characterize the asymptotic behavior under this resonance condition. A distinct bimodality is observed: for perfect qubit coupling, the system either evolves to a maximally entangled Bell state without emitting photons (the dark state) or executes a sustained entangled-state cycle-random switching between a pair of Bell states while emitting a continuous photon stream; for imperfect coupling, two entangled-state cycles coexist, between which a random selection is made from one quantum trajectory to another.

Interconvertibility of single-rail optical qubits

We show how to convert between partially coherent superpositions of a single photon with the vacuum by using linear optics and postselection based on homodyne measurements. We introduce a generalized quantum efficiency for such states and show that any conversion that decreases this quantity is possible. We also prove that our scheme is optimal by showing that no linear optical scheme with generalized conditional measurements, and with one single-rail qubit input, can improve the generalized efficiency. (c) 2006 Optical Society of America.

Quantum computation by communication

We present here a new approach to scalable quantum computing - a 'qubus computer' - which realizes qubit measurement and quantum gates through interacting qubits with a quantum communication bus mode. The qubits could be 'static' matter qubits or 'flying' optical qubits, but the scheme we focus on here is particularly suited to matter qubits. There is no requirement for direct interaction between the qubits. Universal two-qubit quantum gates may be effected by schemes which involve measurement of the bus mode, or by schemes where the bus disentangles automatically and no measurement is needed. In effect, the approach integrates together qubit degrees of freedom for computation with quantum continuous variables for communication and interaction.

Quantum memory scheme based on optical fibers and cavities

An optical quantum memory scheme using two narrow-linewidth cavities and some optical fibers is proposed. The cavities are connected via an optical fiber, and the gap of each cavity can be adjusted to allow photons with a certain bandwidth to transmit through or reflect back. Hence, each cavity acts as a shutter and the photons can be stored in the optical fiber between the cavities at will. We investigate the feasibility of using this device in storing a single photon. We estimate that with current technology storage of a photon qubit for up to 50 clock cycles (round trips) could be achieved with a probability of success of 85%. We discuss how this figure could be improved.

Quantum nondemolition measurements for quantum information

We discuss the characterization and properties of quantum nondemolition (QND) measurements on qubit systems. We introduce figures of merit which can be applied to systems of any Hilbert space dimension, thus providing universal criteria for characterizing QND measurements. The controlled-NOT gate and an optical implementation are examined as examples of QND devices for qubits. We also consider the QND measurement of weak values.

Quantum Information Processing with spin systems: from modeling to possible implementations

The physical implementation of quantum information processing is one of the major challenges of current research. In the last few years, several theoretical proposals and experimental demonstrations on a small number of qubits have been carried out, but a quantum computing architecture that is straightforwardly scalable, universal, and realizable with state-of-the-art technology is still lacking. In particular, a major ultimate objective is the construction of quantum simulators, yielding massively increased computational power in simulating quantum systems. Here we investigate promising routes towards the actual realization of a quantum computer, based on spin systems. The first one employs molecular nanomagnets with a doublet ground state to encode each qubit and exploits the wide chemical tunability of these systems to obtain the proper topology of inter-qubit interactions. Indeed, recent advances in coordination chemistry allow us to arrange these qubits in chains, with tailored interactions mediated by magnetic linkers. These act as switches of the effective qubit-qubit coupling, thus enabling the implementation of one- and two-qubit gates. Molecular qubits can be controlled either by uniform magnetic pulses, either by local electric fields. We introduce here two different schemes for quantum information processing with either global or local control of the inter-qubit interaction and demonstrate the high performance of these platforms by simulating the system time evolution with state-of-the-art parameters. The second architecture we propose is based on a hybrid spin-photon qubit encoding, which exploits the best characteristic of photons, whose mobility is exploited to efficiently establish long-range entanglement, and spin systems, which ensure long coherence times. The setup consists of spin ensembles coherently coupled to single photons within superconducting coplanar waveguide resonators. The tunability of the resonators frequency is exploited as the only manipulation tool to implement a universal set of quantum gates, by bringing the photons into/out of resonance with the spin transition. The time evolution of the system subject to the pulse sequence used to implement complex quantum algorithms has been simulated by numerically integrating the master equation for the system density matrix, thus including the harmful effects of decoherence. Finally a scheme to overcome the leakage of information due to inhomogeneous broadening of the spin ensemble is pointed out. Both the proposed setups are based on state-of-the-art technological achievements. By extensive numerical experiments we show that their performance is remarkably good, even for the implementation of long sequences of gates used to simulate interesting physical models. Therefore, the here examined systems are really promising buildingblocks of future scalable architectures and can be used for proof-of-principle experiments of quantum information processing and quantum simulation.

Error tolerance of topological codes with independent bit-flip and measurement errors

Topological quantum error correction codes are currently among the most promising candidates for efficiently dealing with the decoherence effects inherently present in quantum devices. Numerically, their theoretical error threshold can be calculated by mapping the underlying quantum problem to a related classical statistical-mechanical spin system with quenched disorder. Here, we present results for the general fault-tolerant regime, where we consider both qubit and measurement errors. However, unlike in previous studies, here we vary the strength of the different error sources independently. Our results highlight peculiar differences between toric and color codes. This study complements previous results published in New J. Phys. 13, 083006 (2011).

Iterative Phase Optimization of Elementary Quantum Error Correcting Codes

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.

Utilização de pulsos de radiofrequência fortemente modulados no processamento de informação quântica via ressonância magnética nuclear

Nesta dissertação apresentamos um trabalho de desenvolvimento e utilização de pulsos de radiofreqüência modulados simultaneamente em freqüência, amplitude e fase (pulsos fortemente modulados, SMP, do inglês Strongly Modulated Pulses) para criar estados iniciais e executar operações unitárias que servem como blocos básicos para processamento da informação quântica utilizando Ressonância Magnética Nuclear (RMN). As implementações experimentais foram realizas em um sistema de 3 q-bits constituído por spins nucleares de Césio 133 (spin nuclear 7/2) em uma amostra de cristal líquido em fase nemática. Os pulsos SMP´s foram construídos teoricamente utilizando um programa especialmente desenvolvido para esse fim, sendo o mesmo baseado no processo de otimização numérica Simplex Nelder-Mead. Através deste programa, os pulsos SMP foram otimizados de modo a executarem as operações lógicas desejadas com durações consideravelmente menores que aquelas realizadas usando o procedimento usual de RMN, ou seja, seqüências de pulsos e evoluções livres. Isso tem a vantagem de reduzir os efeitos de descoerência decorrentes da relaxação do sistema. Os conceitos teóricos envolvidos na criação dos SMPs são apresentados e as principais dificuldades (experimentais e teóricas) que podem surgir devido ao uso desses procedimentos são discutidas. Como exemplos de aplicação, foram produzidos os estados pseudo-puros usados como estados iniciais de operações lógicas em RMN, bem como operações lógicas que foram posteriormente aplicadas aos mesmos. Utilizando os SMP\'s também foi possível realizar experimentalmente os algoritmos quânticos de Grover e Deutsch-Jozsa para 3 q-bits. A fidelidade das implementações experimentais foi determinadas utilizando as matrizes densidade experimentais obtidas utilizando um método de tomografia da matriz densidade previamente desenvolvido.

Integrated System Technologies for Modular Trapped Ion Quantum Information Processing

Although trapped ion technology is well-suited for quantum information science, scalability of the system remains one of the main challenges. One of the challenges associated with scaling the ion trap quantum computer is the ability to individually manipulate the increasing number of qubits. Using micro-mirrors fabricated with micro-electromechanical systems (MEMS) technology, laser beams are focused on individual ions in a linear chain and steer the focal point in two dimensions. Multiple single qubit gates are demonstrated on trapped 171Yb+ qubits and the gate performance is characterized using quantum state tomography. The system features negligible crosstalk to neighboring ions (< 3e-4), and switching speeds comparable to typical single qubit gate times (< 2 us). In a separate experiment, photons scattered from the 171Yb+ ion are coupled into an optical fiber with 63% efficiency using a high numerical aperture lens (0.6 NA). The coupled photons are directed to superconducting nanowire single photon detectors (SNSPD), which provide a higher detector efficiency (69%) compared to traditional photomultiplier tubes (35%). The total system photon collection efficiency is increased from 2.2% to 3.4%, which allows for fast state detection of the qubit. For a detection beam intensity of 11 mW/cm2, the average detection time is 23.7 us with 99.885(7)% detection fidelity. The technologies demonstrated in this thesis can be integrated to form a single quantum register with all of the necessary resources to perform local gates as well as high fidelity readout and provide a photon link to other systems.

Micro-Fabricated Surface Trap and Cavity Integration for Trapped Ion Quantum Computing

Atomic ions trapped in micro-fabricated surface traps can be utilized as a physical platform with which to build a quantum computer. They possess many of the desirable qualities of such a device, including high fidelity state preparation and readout, universal logic gates, long coherence times, and can be readily entangled with each other through photonic interconnects. The use of optical cavities integrated with trapped ion qubits as a photonic interface presents the possibility for order of magnitude improvements in performance in several key areas of their use in quantum computation. The first part of this thesis describes the design and fabrication of a novel surface trap for integration with an optical cavity. The trap is custom made on a highly reflective mirror surface and includes the capability of moving the ion trap location along all three trap axes with nanometer scale precision. The second part of this thesis demonstrates the suitability of small micro-cavities formed from laser ablated fused silica substrates with radii of curvature in the 300-500 micron range for use with the mirror trap as part of an integrated ion trap cavity system. Quantum computing applications for such a system include dramatic improvements in the photonic entanglement rate up to 10 kHz, the qubit measurement time down to 1 microsecond, and the measurement error rates down to the 10e-5 range. The final part of this thesis details a performance simulator for exploring the physical resource requirements and performance demands to scale such a quantum computer to sizes capable of performing quantum algorithms beyond the limits of classical computation.

Seawater carbonate chemistry and biological processes during experiments with coral Oculina arbuscula, 2010

Anthropogenic elevation of atmospheric pCO2 is predicted to cause the pH of surface seawater to decline by 0.3-0.4 units by 2100 AD, causing a 50% reduction in seawater [CO3] and undersaturation with respect to aragonite in high-latitude surface waters. We investigated the impact of CO2-induced ocean acidification on the temperate scleractinian coral Oculina arbuscula by rearing colonies for 60 days in experimental seawaters bubbled with air-CO2 gas mixtures of 409, 606, 903, and 2,856 ppm pCO2, yielding average aragonite saturation states (Omega aragonite) of 2.6, 2.3, 1.6, and 0.8. Measurement of calcification (via buoyant weighing) and linear extension (relative to a 137Ba/138Ba spike) revealed that skeletal accretion was only minimally impaired by reductions in Omega aragonite from 2.6 to 1.6, although major reductions were observed at 0.8 (undersaturation). Notably, the corals continued accreting new skeletal material even in undersaturated conditions, although at reduced rates. Correlation between rates of linear extension and calcification suggests that reduced calcification under Omega aragonite = 0.8 resulted from reduced aragonite accretion, rather than from localized dissolution. Accretion of pure aragonite under each Omega aragonite discounts the possibility that these corals will begin producing calcite, a less soluble form of CaCO3, as the oceans acidify. The corals' nonlinear response to reduced Omega aragonite and their ability to accrete new skeletal material in undersaturated conditions suggest that they strongly control the biomineralization process. However, our data suggest that a threshold seawater [CO3] exists, below which calcification within this species (and possibly others) becomes impaired. Indeed, the strong negative response of O. arbuscula to Omega aragonite= 0.8 indicates that their response to future pCO2-induced ocean acidification could be both abrupt and severe once the critical Omega aragoniteis reached.

Divisible quantum dynamics satisfies temporal Tsirelson's bound

We show that divisibility of qubit quantum processes implies temporal Tsirelson's bound. We also prove that the classical bound of the temporal Bell's inequality holds for dynamics that can be described by entanglement-breaking channels---a more general class of dynamics than that allowed by classical physics.