Yuxiang Yang

We present a systematic framework addressing the challenge of identifying optimal sequential strategies for noisy quantum metrology under resource constraints, with a focus on restricted ancillae. While achieving the optimal metrological precision generally requires quantum error correction, we derive rigorous sufficient conditions for attaining the Heisenberg limit using ancilla-free sequential strategies, either without control or with identical unitary controls, based on a spectral analysis of the quantum channel. Complementing this asymptotic analysis, we introduce an efficient tensor network algorithm for optimizing ancilla-constrained metrological strategies in the finite-query regime, adaptable to a wide variety of noise models and experimental control capabilities.

One of the main quests in quantum metrology is to attain the ultimate precision limit with given resources, where the resources are not only of the number of queries, but more importantly of the allowed strategies. With the same number of queries, the restrictions on the strategies constrain the achievable precision. In this work, we establish a systematic framework to identify the ultimate precision limit of different families of strategies, including the parallel, the sequential, and the indefinite-causal-order strategies, and provide an efficient algorithm that determines an optimal strategy within the family of strategies under consideration. With our framework, we show there exists a strict hierarchy of the precision limits for different families of strategies.

Abstract A universal quantum processor is a device that takes as input a (quantum) program, containing an encoding of an arbitrary unitary gate, and a (quantum) data register, on which the encoded gate is applied. While no perfect universal quantum processor can exist, approximate processors have been proposed in the past two decades. A fundamental open question is how the size of the smallest quantum program scales with the approximation error. Here we answer the question, by proving a bound on the size of the program and designing a concrete protocol that attains the bound in the asymptotic limit. Our result is based on a connection between optimal programming and the Heisenberg limit of quantum metrology, and establishes an asymptotic equivalence between the tasks of programming, learning, and estimating unitary gates.

Noisy intermediate-scale quantum (NISQ) devices pave the way to implement quantum algorithms that exhibit supremacy over their classical counterparts. Due to the intrinsic noise and decoherence in the physical system, NISQ computations are naturally modeled as large-size but low-depth quantum circuits. Practically, to execute such quantum circuits, we need to pass commands to a programmable quantum computer. Existing programming approaches, dedicated to generic unitary transformations, are inefficient in terms of the computational resources under the low-depth assumption and remain far from satisfactory. As such, to realize NISQ algorithms, it is crucial to find an efficient way to program low-depth circuits as the qubit number $N$ increases. Here, we investigate the gate complexity and the size of quantum memory (known as the program cost) required to program low-depth brickwork circuits. We unveil a $\sim N \poly \log N$ worst-case program cost of universal programming of low-depth brickwork circuits in the large $N$ regime, which is a tight characterization. Moreover, we analyze the trade-off between the cost of describing the layout of local gates and the cost of programming them to the targeted unitaries via the light-cone argument. Our findings suggest that faithful gate-wise programming is optimal in the low-depth regime.

The traditional framework of quantum metrology commonly assumes unlimited access to resources, overlooking resource constraints in realistic scenarios. As such, the optimal strategies therein can be infeasible in practice. Here, we investigate quantum metrology where the total energy consumption of the probe state preparation, intermediate control operations, and the final measurement is subject to a constraint. We establish a comprehensive theoretical framework for characterizing energy-constrained multi-step quantum processes, based on which we develop a general optimization method for energy-constrained quantum metrology that determines both the optimal precision and the corresponding strategy. Using the method, we determine the ultimate precision limit of energy-constrained phase estimation and identify a novel advantage of quantum superpositions of causal orders in enhancing the energy efficiency of adaptive quantum estimation.