**Recent publications**

- JY Sim, J Shang, HK Ng, and B-G Englert,
*Proper error bars for self-calibrating quantum tomography*, Phys Rev A 100, 022333 (2019); arXiv:1904.11202.

**Abstract**:Self-calibrating quantum state tomography aims at reconstructing the unknown quantum state and certain properties of the measurement devices from the same data. Since the estimates of the state and device parameters come from the same data, one should employ a joint estimation scheme, including the construction and reporting of joint state-device error regions to quantify uncertainty. We explain how to do this naturally within the framework of optimal error regions. As an illustrative example, we apply our procedure to the double-crosshair measurement of the BB84 scenario in quantum cryptography and so reconstruct the state and estimate the detection efficiencies simultaneously and reliably. We also discuss the practical situation of a satellite-based quantum key distribution scheme, for which self-calibration and proper treatment of the data are necessities.

- Y Gazit, HK Ng, and J Suzuki,
*Quantum process tomography via optimal design of experiments*, Phys Rev A 100, 012350 (2019); arXiv:1904.11849.

**Abstract**:Quantum process tomography — a primitive in many quantum information processing tasks — can be cast within the framework of the theory of design of experiment (DoE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DoE for use in quantum problems. The classical techniques from DoE cannot, however, be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation problems. Here, we properly formulate quantum process tomography as a DoE problem, and examine several examples to illustrate the link and the methods. In particular, we discuss the common issue of nuisance parameters, and point out interesting features in the quantum problem absent in the usual classical setting.

- J Qi and HK Ng,
*Comparing the randomized benchmarking figure with the average infidelity of a quantum gate-set*, Int J Quant Inf 4, 1950031 (2019); arXiv:1805.10622.

**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*, Phys Rev A 98, 022307 (2018); arXiv:1804:09486.

**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.

- Y Zheng, C-Y Lai, and TA 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.

- YL Len, J Dai, B-G Englert, and LA Krivitsky,
*Unambiguous path discrimination in a two-path interferometer*, Phys Rev A 98, 022110 (2018); arXiv:1708:01408.

**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.

Earlier publications (2013-2017)

**Preprints**

- JY Sim, J Suzuki, B-G Englert, and HK Ng,
*User-specified random sampling of quantum channels and its applications*, arXiv:1905.00696 (2019).

**Abstract**:Random samples of quantum channels have many applications in quantum information processing tasks. Due to the Choi–Jamio\l{}kowski isomorphism, there is a well-known correspondence between channels and states, and one can imagine adapting \emph{state} sampling methods to sample quantum channels. Here, we discuss such an adaptation, using the Hamiltonian Monte Carlo method, a well-known classical method capable of producing high quality samples from arbitrary, user-specified distributions. Its implementation requires an exact parameterization of the space of quantum channels, with no superfluous parameters and no constraints. We construct such a parameterization, and demonstrate its use in three common channel sampling applications.

- DJ Nott, M Seah, L Al-Labadi, M Evans, HK Ng, and B-G Englert,
*Using prior expansions for prior-data conflict checking*, arXiv:1902.10393 (2019).

**Abstract**:Any Bayesian analysis involves combining information represented through different model components, and when different sources of information are in conflict it is important to detect this. Here we consider checking for prior-data conflict in Bayesian models by expanding the prior used for the analysis into a larger family of priors, and considering a marginal likelihood score statistic for the expansion parameter. Consideration of different expansions can be informative about the nature of any conflict, and extensions to hierarchically specified priors and connections with other approaches to prior-data conflict checking are discussed. Implementation in complex situations is illustrated with two applications. The first concerns testing for the appropriateness of a LASSO penalty in shrinkage estimation of coefficients in linear regression. Our method is compared with a recent suggestion in the literature designed to be powerful against alternatives in the exponential power family, and we use this family as the prior expansion for constructing our check. A second application concerns a problem in quantum state estimation, where a multinomial model is considered with physical constraints on the model parameters. In this example, the usefulness of different prior expansions is demonstrated for obtaining checks which are sensitive to different aspects of the prior.

- Y Quek, S Fort, and HK Ng,
*Adaptive Quantum State Tomography with Neural Networks*, arXiv:1812.06693 (2018).

**Abstract**:Quantum State Tomography is the task of determining an unknown quantum state by making measurements on identical copies of the state. Current algorithms are costly both on the experimental front — requiring vast numbers of measurements — as well as in terms of the computational time to analyze those measurements. In this paper, we address the problem of analysis speed and flexibility, introducing

*Neural Adaptive Quantum State Tomography*(NA-QST), a machine learning based algorithm for quantum state tomography that adapts measurements and provides orders of magnitude faster processing while retaining state-of-the-art reconstruction accuracy. Our algorithm is inspired by particle swarm optimization and Bayesian particle-filter based adaptive methods, which we extend and enhance using neural networks. The resampling step, in which a bank of candidate solutions — particles — is refined, is in our case learned directly from data, removing the computational bottleneck of standard methods. We successfully replace the Bayesian calculation that requires computational time of O(poly(n)) with a learned heuristic whose time complexity empirically scales as O(log(n)) with the number of copies measured n, while retaining the same reconstruction accuracy. This corresponds to a factor of a million speedup for 10^7 copies measured. We demonstrate that our algorithm learns to work with basis, symmetric informationally complete (SIC), as well as other types of POVMs. We discuss the value of measurement adaptivity for each POVM type, demonstrating that its effect is significant only for basis POVMs. Our algorithm can be retrained within hours on a single laptop for a two-qubit situation, which suggests a feasible time-cost when extended to larger systems. It can also adapt to a subset of possible states, a choice of the type of measurement, and other experimental details.

- B-G Englert, M Evans, GH Jang, HK Ng, DJ Nott, and Y-L 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.