arXiv subjects

Hui Khoon Ng

Publications and source records attributed to Hui Khoon Ng.

At least 55 records · Page 3Linked to original sources

arXiv2016

Random samples of quantum states: Online resources

This is the documentation for generating random samples from the quantum state space in accordance with a specified distribution, associated with this webpage: http://tinyurl.com/QSampling . Ready-made samples (each with at least a million points) from various distributions are available for download, or one can generate one's own samples from a chosen distribution using the provided source codes. The sampling relies on the Hamiltonian Monte Carlo algorithm as described in New J. Phys. 17, 043018 (2015). The random samples are reposited in the hope that they would be useful for a variety of tasks in quantum information and quantum computation. Constructing credible regions for tomographic data, optimizing a function over the quantum state space with a complicated landscape, testing the typicality of entanglement among states from a multipartite quantum system, or computing the average of some quantity of interest over a subset of quantum states are but some exemplary applications among many.

quant-ph↗

arXiv2016

Optimal error intervals for properties of the quantum state

Quantum state estimation aims at determining the quantum state from observed data. Estimating the full state can require considerable efforts, but one is often only interested in a few properties of the state, such as the fidelity with a target state, or the degree of correlation for a specified bipartite structure. Rather than first estimating the state, one can, and should, estimate those quantities of interest directly from the data. We propose the use of optimal error intervals as a meaningful way of stating the accuracy of the estimated property values. Optimal error intervals are analogs of the optimal error regions for state estimation [New J. Phys. 15, 123026 (2013)]. They are optimal in two ways: They have the largest likelihood for the observed data and the pre-chosen size, and are the smallest for the pre-chosen probability of containing the true value. As in the state situation, such optimal error intervals admit a simple description in terms of the marginal likelihood for the data for the properties of interest. Here, we present the concept and construction of optimal error intervals, report on an iterative algorithm for reliable computation of the marginal likelihood (a quantity difficult to calculate reliably), explain how plausible intervals --- a notion of evidence provided by the data --- are related to our optimal error intervals, and illustrate our methods with single-qubit and two-qubit examples.

quant-ph↗

arXiv2016

The initial system-bath state via the maximum-entropy principle

The initial state of a system-bath composite is needed as the input for prediction from any quantum evolution equation to describe subsequent system-only reduced dynamics or the noise on the system from joint evolution of the system and the bath. The conventional wisdom is to write down an uncorrelated state as if the system and the bath were prepared in the absence of each other; yet, such a factorized state cannot be the exact description in the presence of system-bath interactions. Here, we show how to go beyond the simplistic factorized-state prescription using ideas from quantum tomography: We employ the maximum-entropy principle to deduce an initial system- bath state consistent with the available information. For the generic case of weak interactions, we obtain an explicit formula for the correction to the factorized state. Such a state turns out to have little correlation between the system and the bath, which we can quantify using our formula. This has implications, in particular, on the subject of subsequent non-completely-positive dynamics of the system. Deviation from predictions based on such an almost uncorrelated state is indicative of accidental control of hidden degrees of freedom in the bath.

quant-ph↗

arXiv2016

Leading gradient correction to the kinetic energy for two-dimensional fermion gases

Density functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizsäcker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

cond-mat.quant-gas↗

arXiv2016

Implementing a neutral-atom controlled-phase gate with a single Rydberg pulse

One can implement fast two-qubit entangling gates by exploiting the Rydberg blockade. Although various theoretical schemes have been proposed, experimenters have not yet been able to demonstrate two-atom gates of high fidelity due to experimental constraints. We propose a novel scheme, which only uses a single Rydberg pulse illuminating both atoms, for the construction of neutral-atom controlled-phase gates. In contrast to the existing schemes, our approach is simpler to implement and requires neither individual addressing of atoms nor adiabatic procedures. With parameters estimated based on actual experimental scenarios, a gate fidelity higher than 0.99 is achievable.

quant-ph↗

arXiv2015

Monte Carlo sampling from the quantum state space. I

High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the standard strategies of rejection sampling, importance sampling, and Markov-chain sampling can be adapted to this context, where the samples must obey the constraints imposed by the positivity of the statistical operator. For a comparison of these sampling methods, we generate sample points in the probability space for two-qubit states probed with a tomographically incomplete measurement, and then use the sample for the calculation of the size and credibility of the recently-introduced optimal error regions [see New J. Phys. 15 (2013) 123026]. Another illustration is the computation of the fractional volume of separable two-qubit states.

quant-ph↗

arXiv2015

Monte Carlo sampling from the quantum state space. II

High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the Markov-chain Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. The resulting random walk is entirely inside the physical parameter space, and the Hamiltonian dynamics enable us to take big steps, thereby avoiding strong correlations between successive sample points while enjoying a high acceptance rate. We use examples of single and double qubit measurements for illustration.

quant-ph↗

arXiv2014

Quantum state tomography: Mean squared error matters, bias does not

Because of the constraint that the estimators be bona fide physical states, any quantum state tomography scheme - including the widely used maximum likelihood estimation - yields estimators that may have a bias, although they are consistent estimators. Schwemmer et al. (arXiv:1310.8465 [quant-ph]) illustrate this by observing a systematic underestimation of the fidelity and an overestimation of entanglement in estimators obtained from simulated data. Further, these authors argue that the simple method of linear inversion overcomes this (perceived) problem of bias, and there is the suggestion to abandon time-tested estimation procedures in favor of linear inversion. Here, we discuss the pros and cons of using biased and unbiased estimators for quantum state tomography. We conclude that the little occasional benefit from the unbiased linear-inversion estimation does not justify the high price of using unphysical estimators, which are typically the case in that scheme.

quant-ph↗

arXiv2013

One-dimensional transport revisited: A simple and exact solution for phase disorder

Disordered systems have grown in importance in the past decades, with similar phenomena manifesting themselves in many different physical systems. Because of the difficulty of the topic, theoretical progress has mostly emerged from numerical studies or analytical approximations. Here, we provide an exact, analytical solution to the problem of uniform phase disorder in a system of identical scatterers arranged with varying separations along a line. Relying on a relationship with Legendre functions, we demonstrate a simple approach to computing statistics of the transmission probability (or the conductance, in the language of electronic transport), and its reciprocal (or the resistance). Our formalism also gives the probability distribution of the conductance, which reveals features missing from previous approaches to the problem.

cond-mat.dis-nn↗

arXiv2013

Beyond adiabatic elimination: A hierarchy of approximations for multi-photon processes

In multi-level systems, the commonly used adiabatic elimination is a method for approximating the dynamics of the system by eliminating irrelevant, non-resonantly coupled levels. This procedure is, however, somewhat ambiguous and it is not clear how to improve on it systematically. We use an integro-differential equation for the probability amplitudes of the levels of interest, which is equivalent to the original Schrodinger equation for all probability amplitudes. In conjunction with a Markov approximation, the integro-differential equation is then used to generate a hierarchy of approximations, in which the zeroth order is the adiabatic-elimination approximation. It works well with a proper choice of interaction picture; the procedure suggests criteria for optimizing this choice. The first-order approximation in the hierarchy provides significant improvements over standard adiabatic elimination, without much increase in complexity, and is furthermore not so sensitive to the choice of interaction picture. We illustrate these points with several examples.

quant-ph↗

arXiv2013

Optimal error regions for quantum state estimation

Rather than point estimators, states of a quantum system that represent one's best guess for the given data, we consider optimal regions of estimators. As the natural counterpart of the popular maximum-likelihood point estimator, we introduce the maximum-likelihood region---the region of largest likelihood among all regions of the same size. Here, the size of a region is its prior probability. Another concept is the smallest credible region---the smallest region with pre-chosen posterior probability. For both optimization problems, the optimal region has constant likelihood on its boundary. We discuss criteria for assigning prior probabilities to regions, and illustrate the concepts and methods with several examples.

quant-ph↗

arXiv2012

Raman transitions without adiabatic elimination: A simple and accurate treatment

Driven Raman processes --- nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population --- are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann-Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.

quant-ph↗

arXiv2012

Towards a Unified Framework for Approximate Quantum Error Correction

Recent work on approximate quantum error correction (QEC) has opened up the possibility of constructing subspace codes that protect information with high fidelity in scenarios where perfect error correction is impossible. Motivated by this, we investigate the problem of approximate subsystem codes. Subsystem codes extend the standard formalism of subspace QEC to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Here, we demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, for certain classes of subsystem codes and noise processes, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel. This work generalizes our earlier approach [H.K. Ng and P. Mandayam, Phys. Rev. A 81 062342 (2010)] of using the transpose channel for approximate correction of subspace codes to the case of subsystem codes, and brings us closer to a unifying framework for approximate QEC.

quant-ph↗

arXiv2012

Minimax mean estimator for the trine

We explore the question of state estimation for a qubit restricted to the $x$-$z$ plane of the Bloch sphere, with the trine measurement. In our earlier work [H. K. Ng and B.-G. Englert, eprint arXiv:1202.5136[quant-ph] (2012)], similarities between quantum tomography and the tomography of a classical die motivated us to apply a simple modification of the classical estimator for use in the quantum problem. This worked very well. In this article, we adapt a different aspect of the classical estimator to the quantum problem. In particular, we investigate the mean estimator, where the mean is taken with a weight function identical to that in the classical estimator but now with quantum constraints imposed. Among such mean estimators, we choose an optimal one with the smallest worst-case error-the minimax mean estimator-and compare its performance with that of other estimators. Despite the natural generalization of the classical approach, this minimax mean estimator does not work as well as one might expect from the analogous performance in the classical problem. While it outperforms the often-used maximum-likelihood estimator in having a smaller worst-case error, the advantage is not significant enough to justify the more complicated procedure required to construct it. The much simpler adapted estimator introduced in our earlier work is still more effective. Our previous work emphasized the similarities between classical and quantum state estimation; in contrast, this paper highlights how intuition gained from classical problems can sometimes fail in the quantum arena.

quant-ph↗

arXiv2012

A simple minimax estimator for quantum states

Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. In this article, we review the concept of minimax state estimation, and use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.

quant-ph↗

arXiv2011

Combining dynamical decoupling with fault-tolerant quantum computation

We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise, and have a lower overhead cost, than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer, and can be expressed either in terms of the operator norm of the bath's Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.

quant-ph↗

arXiv2010

Information preserving structures: A general framework for quantum zero-error information

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., "noiseless", "unitarily correctible", etc.) and introduce two new and natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational critera, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.

quant-ph↗

arXiv2010

A simple approach to approximate quantum error correction based on the transpose channel

We demonstrate that there exists a universal, near-optimal recovery map---the transpose channel---for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions, and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement.

quant-ph↗