The resources required to characterize the dynamics of engineered quantum systems-such as quantum computers and quantum sensors-grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to processes that are nearly sparse in a certain basis and can be implemented using only single-body preparations and measurements. We perform efficient, highfidelity estimation of process matrices of a photonic two-qubit logic gate. The database is obtained under various decoherence strengths. Our technique is both accurate and noise robust, thus removing a key roadblock to the development and scaling of quantum technologies. Understanding and controlling the world at the nanoscale-be it in biological, chemical or physical phenomena-requires quantum mechanics. It is therefore essential to characterize and monitor realistic complex quantum systems that inevitably interact with typically uncontrollable environments. One of the most general descriptions of the dynamics of an open quantum system is a quantum map-typically represented by a process matrix [1]. Methods to identify this matrix are collectively known as quantum process tomography (QPT) [1,2]. For a d-dimensional quantum system, they require Oðd 4 Þ experimental configurations: combinations of input states, on which the process acts, and a set of output observables. For a system of n qubits-two level quantum systemsd ¼ 2 n . The required physical resources hence scale exponentially with system size. Recently, a number of alternative methods have been developed for efficient and selective estimation of quantum processes [3]. However, full characterization of quantum dynamics of comparably small systems, such as an 8-qubit ion trap [4], would still require over a billion experimental configurations, clearly impractical. So far, process tomography has therefore been limited by experimental and off-line computational resources, to systems of 2 and 3 qubits [5][6][7].Here we adapt techniques from compressive sensing to develop an experimentally efficient method for QPT. It requires only Oðs logdÞ configurations if the process matrix is s compressible in some known basis, i.e., it is nearly sparse in that it can be well approximated by an s-sparse process matrix. This is usually the case, because engineered quantum systems aim to implement a unitary process which is maximally sparse in its eigenbasis. In practice, as observed in liquid-state NMR [8], photonics [5,9,10], ion traps [11], and superconducting circuits [6], a near-unitary process will still be nearly sparse in this basis, and still compressible. The near sparsity is due to few dominant system environment interactions. This is more apparent for weakly decohering systems [12].We experimentally demonstrate our algorithm by estimating the 240 real parameters of the process matrix of a canonical photonic two-qubit gate, Fig. 1, from a reduced number of configurations. From...
An optimally controlled quantum system possesses a search landscape defined by the physical objective as a functional of the control field. This paper particularly explores the topological structure of quantum mechanical transition probability landscapes. The quantum system is assumed to be controllable and the analysis is based on the Euler-Lagrange variational equations derived from a cost function only requiring extremizing the transition probability. It is shown that the latter variational equations are automatically satisfied as a mathematical identity for control fields that either produce transition probabilities of zero or unit value. Similarly, the variational equations are shown to be inconsistent ͑i.e., they have no solution͒ for any control field that produces a transition probability different from either of these two extreme values. An upper bound is shown to exist on the norm of the functional derivative of the transition probability with respect to the control field anywhere over the landscape. The trace of the Hessian, evaluated for a control field producing a transition probability of a unit value, is shown to be bounded from below. Furthermore, the Hessian at a transition probability of unit value is shown to have an extensive null space and only a finite number of negative eigenvalues. Collectively, these findings show that ͑a͒ the transition probability landscape extrema consists of values corresponding to no control or full control, ͑b͒ approaching full control involves climbing a gentle slope with no false traps in the control space and ͑c͒ an inherent degree of robustness exists around any full control solution. Although full controllability may not exist in some applications, the analysis provides a basis to understand the evident ease of finding controls that produce excellent yields in simulations and in the laboratory. P i→f , ͑1͒and this paper is concerned with analyzing the structure of the control landscapewhich is a functional of the control field ͓30͔. In the laboratory other factors can enter, but an ultimate common goal is the clean performance of state-to-state transfer in Eq. ͑1͒.Knowledge of the general topology of this landscape, including its extrema values, slopes, and curvature, is fundamental to understanding the ability of finding good quality robust controls in the laboratory and in simulations. This paper will carry out an analysis of the control landscape in Eq. ͑2͒, treating ͑t͒ as an arbitrary continuous temporal function. As a result, the search to maximize P i→f is formally over an infinite dimensional space. However, in PHYSICAL REVIEW A 74, 012721 ͑2006͒
Abstract. Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one-and two-qubit unitary gates in the presence of strong environmentallyinduced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
Characterising complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as 'compressed sensing' has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that compressed sensing tomography of quantum systems is essentially guaranteed by a special property of quantum mechanics itself-that the mathematical objects that describe the system in quantum mechanics are matrices with non-negative eigenvalues. This result has an impact on the way quantum tomography is understood and implemented. In particular, it implies that the information obtained about a quantum system through compressed sensing methods exhibits a new sense of 'informational completeness.' This has important consequences on the efficiency of the data taking for quantum tomography, and enables us to construct informationally complete measurements that are robust to noise and modelling errors. Moreover, our result shows that one can expand the numerical tool-box used in quantum tomography and employ highly efficient algorithms developed to handle large dimensional matrices on a large dimensional Hilbert space. Although we mainly present our results in the context of quantum tomography, they apply to the general case of positive semidefinite matrix recovery.
Manipulation of quantum interference requires that the system under control remains coherent, avoiding (or at least postponing) the phase randomization that can ensue from coupling to an uncontrolled environment. We show that closed-loop coherent control can be used to mitigate the rate of quantum dephasing in a gas-phase ensemble of potassium dimers (K2), which acts as a model system for testing the general concepts of controlling decoherence. Specifically, we adaptively shaped the light pulse used to prepare a vibrational wave packet in electronically excited K2, with the amplitude of quantum beats in the fluorescence signal used as an easily measured surrogate for the purpose of optimizing coherence. The optimal pulse increased the beat amplitude from below the noise level to well above it, and thereby increased the coherence life time as compared with the beats produced by a transform-limited pulse. Closed-loop methods can thus effectively identify states that are robust against dephasing without any previous information about the system-environment interaction.
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