**Fault-tolerant quantum computing – from theory to practice**

We work on various aspects of quantum error correction and fault tolerance. Our recent direction has been on reducing the gap between theoretical quantum error correction and fault tolerance ideas and their implementation in experiments. With the rapid development of quantum computing devices, we are beginning to have an inkling of what a quantum computer might look like, and the practical obstacles, to do with noise and scalability, are taking on more concrete shapes. This is thus the right time to re-examine the often generic and abstract theoretical proposals for noise removal, in the light of recent experiments, for progress towards large-scale, useful quantum devices.

Below, we highlight some recent projects within the group.

**Fault-tolerant embedding of circuits viaÂ swap gates**

HK Ng, with Entropica Labs (arXiv:2406.17044)

In near-term quantum devices, qubit connectivity remains limited by architectural constraints. A computational circuit with given connectivity requirements for multi-qubit gates has to be embedded in physical hardware with fixed connectivity. Long-distance gates have to be done by first routing the information together.

The simplest routing strategy uses swap gates to swap information carried by two unconnected qubits to connected ones. Ideal SWAPs just permute qubits; real SWAPs, however, can cause simultaneous errors on the qubits involved and spread errors across the circuit. General swap schemes can thus destroy fault-tolerant features carefully designed into the original circuit.

Here, we show that, by a simple restriction of allowed swap moves, we can embed an arbitrary circuit in a fault-tolerant manner.The embedded circuit will be noisier, but we show, in the examples of surface codes on heavy-hexagonal and hexagonal lattices, that the noise deterioration is not severe.

Our approach is easily incorporated into existing circuit compilation algorithms, and offers an immediate solution to implementing circuits on current hardware in a fault-tolerant manner.

**Circuit-level fault tolerance of cat codes**

LDH My, S Qin, and HK Ng (arXiv:2406.04157)

Bosonic codes, which encode quantum information in the infinite Hilbert space of a harmonic oscillator, are viable alternatives to conventional qubit codes. The family of rotationally symmetric bosonic (RSB) codesis capable of correcting for both photon loss and phase (i.e., rotation) errors, offering robustness against arbitrary physical errors at the base layer of encoding.

We extend the formalism of fault tolerance to RSB codes, and assess the performance of previously proposed teleportation-based error correction (EC) circuits [Grimsmo et al., 2020] for cat codes (a type of RSB codes), accounting for circuit-level noise, i.e., where every physical component of the circuit can be faulty. We find that the noise threshold is significantly worse than found in previous more idealised studies. Through our analysis, we identify crucial circuit settings, such as the choice of code order, the optimal waiting time between EC cycles, and the addition of squeezing to the code states, that improve the noise threshold by an order of magnitude, restoring the noise requirement to a level achievable with near-term quantum hardware.

**Bosonic codes in quantum-dot–resonator systems**

M Ma, HK Ng, with the group of TS Koh in NTU

Recent advancements in coupling quantum-dots to superconducting (SC) resonatorsenable long-range gates between quantum-dot qubits, and present the intriguing possibility of implementing circuit-QED ideas, originally for SC qubits and cavities, in quantum-dotâ€“resonator systems. We study how arbitrary bosonic resonator states can be prepared using a double-quantum-dot system as control, and further investigate how computational operations can be performed on information carried by bosonic codes.

We explore how to prepare arbitrary bosonic resonator states through interaction with a double-quantum-dot system, for different coupling regimes. In the coherent regime, the dual-channel Law-Eberly protocol creates arbitrary superpositions of Fock states. In the dispersive regime, the qcMAP protocol commonly used in SC contexts allows for preparation of superpositions of coherent states. More generally, GRAPE optimization gives flexible state preparation through simultaneous resonator and quantum-dot drives.

While we follow well-understood schemes from standard SC c-QED contexts, the main research thrust here is to examine how well the effective Hamiltonians assumed in those schemes describe the exact dynamics of the double-quantum-dotâ€“resonator system. Our system also offers different tuning knobs than those in SC systems, presenting further opportunities for improved control under noise.

**Surface codes: numerical studies**

M Myers II, M Lavialle, ET Duong, O Valette, and HK Ng

We study the performance of surface codes, one of the most popular routes to error-corrected quantum devices, under realistic noise scenarios.

Quasi-probability methods for simulating general noise. The stabilizer formalism allows for efficient classical simulation of error correction circuits with only Clifford gates and Pauli measurements subjected to Pauli errors. Surface codes, however, can correct general noise, but studies of its experimentally relevant properties have been limited to Pauli noise only because of the stabilizer restrictions. We make use of quasi-probability methods that circumvent the stabilizer restrictions to study the performance of surface codes under non-Pauli noise. While computational costs grow rapidly as one steps away from stabilizer noise, relying on the specific features of error correction circuits, we can access noise types never before simulated for surface codes. Unitary noise (i.e., coherent errors), however, remains difficult and requires novel ideas for feasible simulation.

Logical operations. Another direction of interest is the investigation of fault-tolerance properties of current methods of lattice surgery for implementing logical operations, like the Hadamard and CNOT gates. The significantly higher simulation resource demands for lattice surgery as well as the question of proper decoding in such contexts form our current research focus.

**Noise-adapted fault tolerance**

LDH My and HK Ng, with P Mandayam (IIT Madras) and A Jayashankar (TCG CREST)

Standard fault-tolerant (FT) schemes are designed with codes that correct arbitrary errors and assume no knowledge of the physical noise. Noise-adapted FT schemes, tailor-made to deal with the dominant noise in the device, may have lower resource overheads and less stringent thresholds. Here, we develop a full fault-tolerant quantum computing protocol for amplitude-damping (AD) noise, using Bacon-Shor codes. We describe a universal set of fault-tolerant encoded gadgets and estimate the noise thresholds below which our scheme leads to more accurate computation. This is the first example of a full FT scheme adapted to non-Pauli-type noise.

Our published article [PR Research 4, 023034 (2022)] details the protocol for the smallest instance of the 4-qubit code; a manuscript in preparation gives the generalization to higher-distance Bacon-Shor codes.

**Reinforcement learning for context-aware dynamical gate calibration**

A Strauss, L Voss, and HK Ng

Quantum control techniques have enabled significant improvements in gate fidelities. However, most methods do not provide dynamical and contextual error robustness, likely important for near-term devices. Here, we present (1) a gate calibration procedure based on reinforcement learning (RL) to suppress errors arising from a specific circuit context; (2) a concrete use case showing how contextual and dynamical gate calibrations can successfully increase quantum circuit fidelity.â€‹

Context-aware calibration. Each gate instance carries a unique calibration, adapted to its location in the circuit, i.e., its context.

**Resource costs of quantum metrology**

YL Len, T Acharya, T Lim, and HK Ng, with A AuffĂ¨ves from MajuLab

Quantum metrology promises the possibility of beyond-classical precision in estimation problems, through nonclassical features in the probe states. More exotic states often offer greater advantages, but are more difficult to prepare in practice. Here, we compare the efficiencies of quantum metrology with different probe-state choices by taking into proper account the experimental costs and constraints in preparing the quantum states. Through such a resource-based figure of merit, we arrive at significantly different conclusions about the quantum advantages of different metrology protocols, compared to the oversimplified conventional benchmarks based only on counting the number of probe states consumed.