**Recent publications**

- Y Zheng, Ching-Yi Lai, and Todd A. Brun,
*Efficient Preparation of Large Block Code Ancilla States for Fault-tolerant Quantum Computation*, Phys Rev A 97, 032331 (2018); arXiv:1710:00389.

**Abstract**:Fault-tolerant quantum computation (FTQC) schemes that use multi-qubit large block codes can potentially reduce the resource overhead to a great extent. A major obstacle is the requirement of a large number of clean ancilla states of different types without correlated errors inside each block. These ancilla states are usually logical stabilizer states of the data code blocks, which are generally difficult to prepare if the code size is large. Previously we have proposed an ancilla distillation protocol for Calderbank-Shor-Steane (CSS) codes by classical error-correcting codes. It was assumed that the quantum gates in the distillation circuit were perfect; however, in reality, noisy quantum gates may introduce correlated errors that are not treatable by the protocol. In this paper, we show that additional postselection by another classical error-detecting code can be applied to remove almost all correlated errors. Consequently, the revised protocol is fully fault-tolerant and capable of preparing a large set of stabilizer states sufficient for FTQC using large block codes. At the same time, the yield rate can be boosted from O(t^{−2}) to O(1) in practice for an [[n,k,d=2t+1]] CSS code. Ancilla preparation for the [[23,1,7]] quantum Golay code is numerically studied in detail through Monte Carlo simulation. The results support the validity of the protocol when the gate failure rate is reasonably low. To the best of our knowledge, this approach is the first attempt to prepare general large block stabilizer states free of correlated errors for FTQC in a fault-tolerant and efficient manner.

- Y Zheng and HK Ng,
*A digital quantum simulator in the presence of a bath*, Phys Rev A 96, 042329 (2017); arXiv:1707:04407

**Abstract**:For a digital quantum simulator (DQS) imitating a target system, we ask the following question: Under what conditions is the simulator dynamics similar to that of the target in the presence of coupling to a bath? In this paper, we derive conditions for close simulation for three different physical regimes, replacing previous heuristic arguments on the subject with rigorous statements. In fact, we find that the conventional wisdom that the simulation cycle time should always be short for good simulation need not always hold up. Numerical simulations of two specific examples strengthen the evidence for our analysis, and go beyond to explore broader regimes.

- M-I Trappe, YL Len, HK Ng, and B-G Englert,
*Airy-averaged gradient corrections for two-dimensional fermion gases*, Ann Phys 385, 136 (2017); arXiv:1612.04048.

**Abstract**:Building on the discussion in PRA 93, 042510 (2016), we present a systematic derivation of gradient corrections to the kinetic-energy functional and the one-particle density, in particular for two-dimensional systems. We derive the leading gradient corrections from a semiclassical expansion based on Wigner’s phase space formalism and demonstrate that the semiclassical kinetic-energy density functional at zero temperature cannot be evaluated unambiguously. In contrast, a density-potential functional description that effectively incorporates interactions provides unambiguous gradient corrections. Employing an averaging procedure that involves Airy functions, thereby partially resumming higher-order gradient corrections, we facilitate a smooth transition of the particle density into the classically forbidden region of arbitrary smooth potentials. We find excellent agreement of the semiclassical Airy-averaged particle densities with the exact densities for very low but finite temperatures, illustrated for a Fermi gas with harmonic potential energy. We furthermore provide criteria for the applicability of the semiclassical expansions at low temperatures. Finally, we derive a well-behaved ground-state kinetic-energy functional, which improves on the Thomas-Fermi approximation.

- B-G Englert, K Horia, J Dai, YL Len, and HK Ng,
*Past of a quantum particle revisited*, Phys Rev A 96, 022126 (2017); arXiv:1704.03722.

**Abstract**:

We analyze Vaidman’s three-path interferometer with weak path marking [Phys. Rev. A 87, 052104 (2013)] and find that common sense yields correct statements about the particle’s path through the interferometer. This disagrees with the original claim that the particles have discontinuous trajectories at odds with common sense. In our analysis, “the particle’s path” has operational meaning as acquired by a path-discriminating measurement. For a quantum-mechanical experimental demonstration of the case, one should perform a single-photon version of the experiment by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] with unambiguous path discrimination. We present a detailed proposal for such an experiment.

- J. Shang, Z. Zhang, and HK Ng,
*Superfast maximum likelihood reconstruction for quantum tomography*, Phys Rev A 95, 062336 (2017); arXiv:1609.07881.

**Abstract**:Conventional methods for computing maximum-likelihood estimators (MLE) often converge slowly in practical situations, leading to a search for simplifying methods that rely on additional assumptions for their validity. In this work, we provide a fast and reliable algorithm for maximum-likelihood reconstruction that avoids this slow convergence. Our method utilizes the state-of-the-art convex optimization scheme, an accelerated projected-gradient method, that allows one to accommodate the quantum nature of the problem in a different way than in the standard methods. We demonstrate the power of our approach by comparing its performance with other algorithms for n-qubit state tomography. In particular, an eight-qubit situation that purportedly took weeks of computation time in 2005 can now be completed in under a minute for a single set of data, with far higher accuracy than previously possible. This refutes the common claim that MLE reconstruction is slow and reduces the need for alternative methods that often come with difficult-to-verify assumptions. In fact, recent methods assuming Gaussian statistics or relying on compressed sensing ideas are demonstrably inapplicable for the situation under consideration here. Our algorithm can be applied to general optimization problems over the quantum state space; the philosophy of projected gradients can further be utilized for optimization contexts with general constraints.

Earlier publications (2013-2016)

**Preprints**

- J Qi and HK Ng,
*Randomized benchmarking does not measure average infidelity of gates*, arXiv:1805.10622 (2018). (Under review at Phys Rev Lett.)

**Abstract**:Randomized benchmarking (RB) is a popular procedure used to gauge the

performance of a set of gates useful for quantum information processing (QIP).

Recently, Proctor et al. [Phys. Rev. Lett. 119, 130502 (2017)] demonstrated a

practically relevant example where the RB measurements give a number $r$ very

different from the actual average gate-set infidelity $\epsilon$, despite past

theoretical assurances that the two should be equal. Here, we derive formulas

for $\epsilon$, and for $r$ from the RB protocol, in a manner permitting easy

comparison of the two. We show that $r\neq \epsilon$, i.e., RB does not measure

average infidelity, and, in fact, neither one bounds the other. We give several

examples, all plausible in real experiments, to illustrate the differences in

$\epsilon$ and $r$. Many recent papers on experimental implementations of QIP

have claimed the ability to perform high-fidelity gates because they

demonstrated small $r$ values using RB. Our analysis shows that such a

conclusion cannot be drawn from RB alone.

- YL Len and HK Ng,
*Open-system quantum error correction*, arXiv:1804:09486 (2018). (Under review at Phys Rev A.)

**Abstract**:We study the performance of quantum error correction (QEC) on a system undergoing open-system (OS) dynamics. The noise on the system originates from a joint quantum channel on the system-bath composite, a framework that includes and interpolates between the commonly used system-only quantum noise channel model and the system-bath Hamiltonian noise model. We derive the perfect OSQEC conditions, with QEC recovery only on the system and not the inaccessible bath. When the noise is only approximately correctable, the generic case of interest, we quantify the performance of OSQEC using worst-case fidelity. We find that the leading deviation from unit fidelity after recovery is quadratic in the uncorrectable part, a result reminiscent of past work on approximate QEC for system-only noise, although the approach here requires the use of different techniques than in past work.

- Berthold-Georg Englert, Michael Evans, Gun Ho Jang, HK Ng, David Nott, and Yi-Lin Seah,
*Checking the Model and the Prior for the Constrained Multinomial*, arXiv:1804:06906 (2018).

**Abstract**:The multinomial model is one of the simplest statistical models. When constraints are placed on the possible values for the probabilities, however, it becomes much more difficult to deal with. Model checking and checking for prior-data conflict is considered here for such models. A theorem is proved that establishes the consistency of the check on the prior. Applications are presented to models that arise in quantum state estimation as well as the Bayesian analysis of models for ordered probabilities.

- YL Len, Jibo Dai, Berthold-Georg Englert, and Leonid A. Krivitsky,
*Unambiguous path discrimination in a two-path interferometer*, arXiv:1708:01408 (2017).

**Abstract**:When a photon is detected after passing through an interferometer one might wonder which path it took, and a meaningful answer can only be given if one has the means of monitoring the photon’s whereabouts. We report the realization of a single-photon experiment for a two-path interferometer with path marking. In this experiment, the path of a photon (“signal”) through a Mach–Zehnder interferometer becomes known by unambiguous discrimination between the two paths. We encode the signal path in the polarization state of a partner photon (“idler”) whose polarization is examined by a three-outcome measurement: one outcome each for the two signal paths plus an inconclusive outcome. Our results agree fully with the theoretical predictions from a common-sense analysis of what can be said about the past of a quantum particle: The signals for which we get the inconclusive result have full interference strength, as their paths through the interferometer cannot be known; and every photon that emerges from the dark output port of the balanced interferometer has a known path.