# Quantum Metrology

# Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum amplification. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework.

# Calibrating the actuated quantum control signal is essential to the quantum system engineering: unlike classical computers, quantum computers need to be calibrated whenever its been rebooted. I’ve been working on two directions in regard to quantum metrology:

# (1) **physics modeling** of imperfections in realistic quantum systems

**physics modeling**of imperfections in realistic quantum systems

# (2) **quantum metrology algorithm** designs to learn parameters of physics models in (1).

**quantum metrology algorithm**designs to learn parameters of physics models in (1).

## Physics Modeling

Experimental measurements of exponential decay of excited state for four qubits from the same quantum processor: probabilities of remaining in the excited state after a swap-spectroscopy gate as a function of frequency and hold time. Accelerated decay caused by the coupling of qubits to environmental defects manifest as deep blue lines.

# Two-level-system (TLS) defects in amorphous dielectrics are a major source of noise and decoherence in solid-state qubits. Gate-dependent non-Markovian errors caused by TLS-qubit coupling are detrimental to fault-tolerant quantum computation and have not been rigorously treated in the existing literature. In Learning Non-Markovian Quantum Noise from Moire-Enhanced Swap Spectroscopy with Deep Evolutionary Algorithm, we derive the non-Markovian dynamics between TLS and qubits during a SWAP-like two-qubit gate and the associated average gate fidelity for frequency-tunable Transmon qubits. This gate-dependent error model facilitates using qubits as sensors to simultaneously learn practical imperfections in both the qubit’s environment and control waveforms.

## Quantum Metrology Algorithm

## Evolutionary Algorithm with Deep Learning

# We combine the-state-of-art machine learning algorithm with Moir´e enhanced swap spectroscopy to achieve robust learning using noisy experimental data. Deep neural networks are used to represent the functional map from experimental data to TLS parameters, and are trained through an evolutionary algorithm. Our method achieves the highest learning efficiency and robustness against experimental imperfections to-date, representing an important step towards in-situ quantum control optimization over environmental and control defects.

Fig. 1. Diagram of the DNN learning architecture: each section of the two-dimensional data (represented by the black frame) from different frequency regions are input to a four layer fully connected DNN with hidden layer dimensions 20, 30, 12 and 4. By using an evolutionary algorithm, the last layer of the DNN is trained to output the TLS parameters: λ, Γ2, ωTLS, and tr that best reproduce the experimental data given our theoretical model.

## Periodic Calibration

# Calibration of quantum gates is one of the most crucial steps in achieving high-fidelity quantum computation and its large-scale deployment. Temporal instabilities, including drifts and fluctuations in the control fields and qubit frequencies, can propagate and accumulate coherently in large quantum circuits. Therefore, it is crucial to develop fast and accurate calibration methods to characterize and mitigate these errors. However, common calibration tools, such as randomized benchmarking, compressed sensing, gate set tomography, and cross-entropy benchmarking [65] are often too slow to capture drifts and fluctuations in the hardware. Quantum metrology offers a route to this goal. The Heisenberg limit O(1/n) sets a fundamental lower error bound in phase estimation with n photons, whereas the standard quantum limit O(1/ √ n) refers to the minimum uncertainty allowed by using semiclassical states. Modified versions of the quantum phase estimation algorithm have been shown to reach Heisenberg scaling theoretically and experimentally. Based on this idea, Kimmel et al. proposed a protocol to characterize the axis and angle of a single-qubit rotation. It achieves uncertainty O(1/n) by repeating identical operations O(n) times.

Fig. 2, Two-qubit gate calibration: strategy overview. a General representation of a photon-conserving two-qubit gate, truncated to the single-particle subspace. This model has four parameters. The parameter θ describes how much the particle hops between qubits. The parameter ζ is the phase the particle accumulates when it stays on the same site (corresponding to a local field). The parameter χ is the phase the particle accumulates when it hops (corresponding to a complex hopping). The parameter γ is a global phase. b Two methods for extracting parameters in the Fourier domain by repeated application of the two-qubit gate separated by single-qubit z-rotations. The z-rotation provides a probe which can be varied to determine parameters. c Table showing the Fourier frequencies that each method resolves. d Calibration procedure for determining θ, ζ and γ from the measured Fourier frequencies. The remaining parameter χ cannot be determined from frequencies at two qubits as it corresponds to a flux and thus requires a ring of qubits.