Adaptive Circuit Learning for Quantum Metrology Ziqi Ma*1, Pranav Gokhale1, Tian-Xing Zheng2, Sisi Zhou2, Xiaofei Yu2, Liang Jiang2, Peter Maurer2, and Frederic T. Chong †1 1Department of Computer Science, University of Chicago 2Pritzker School of Molecular Engineering, University of Chicago Abstract Quantum advantage is enabled by entanglement, which al- Quantum metrology is an important application of emerging lows for higher sensitivity than what can be achieved by a clas- quantum technologies. We explore whether a hybrid system of sical parallelization of the sensing qubits, called the classical quantum sensors and quantum circuits can surpass the clas- limit. As shown in Figure1, the classical limit gives a square sical limit of sensing. In particular, we use a circuit learning root scaling of Signal-to-Noise Ratio (SNR) versus the number approach to search for encoder and decoder circuits that scal- of sensing qubits. By contrast, the theoretical limit—called ably improve sensitivity under given application and noise the Heisenberg limit—achieves linear scaling [1, 8, 9, 10]. In characteristics. recent years, there has been exciting experimental progress Our approach uses a variational algorithm that can learn beyond the classical limit in areas ranging from spectroscopy a quantum sensing circuit based on platform-specific control [3] to precision measurement [11, 12] to the famous LIGO ex- capacity, noise, and signal distribution. The quantum circuit periment for gravitational wave detection [13, 14]. While the is composed of an encoder which prepares the optimal sensing state-of-the-art shows great promise in achieving quantum ad- state and a decoder which gives an output distribution con- vantage in sensing, we still expect further improvement—the taining information of the signal. We optimize the full circuit Hilbert space of entangled states is large and existing protocols to maximize the Signal-to-Noise Ratio (SNR). Furthermore, only explore a small set of sensing states. Furthermore, the this learning algorithm can be run on real hardware scalably current state-of-the-art in entanglement-enhanced sensing is by using the “parameter-shift” rule which enables gradient still mostly proof-of-concept experiments, and such protocols evaluation on noisy quantum circuits, avoiding the exponential do not necessarily yield good performance on real hardware— cost of quantum system simulation. We demonstrate a 1.69x each platform is subject to a unique set of noise and control SNR improvement over the classical limit on a 5-qubit IBM constraints. Given the constraints of limited and imperfect con- quantum computer. More notably, our algorithm overcomes trol, device-specific noise, and readout errors of practical hard- the plateauing (or even decreasing) performance of existing ware, theoretically-optimal sensing protocols yield suboptimal entanglement-based protocols with increased system sizes. performance. For example, the Greenberger–Horne–Zeilinger state (GHZ state) is optimal without noise but decoheres easily 1. Introduction in noisy cases. Another theoretically optimal state, called the In recent years, we have seen enormous growth in emerging spin-squeezed state, is hard to create if we do not have global quantum technologies that exploit quantum mechanics for var- interaction or all-to-all qubit connectivity [7]. We believe ious applications, such as computation, communication, and that a circuit learning approach, which optimizes the sensing sensing. The sensitivity of quantum states to changes in the ex- apparatus circuit based on characteristics of the underlying ternal environment, while seen as an obstacle in computation hardware, noise model, and signal can provide a solution. and communication, becomes a valuable advantage in sensing. Our circuit learning algorithm is inspired by recent works Quantum sensing is believed to have the most immediate real- in quantum computation. A class of quantum-classical hybrid arXiv:2010.08702v1 [quant-ph] 17 Oct 2020 world impacts, likely before other quantum technologies [1]. algorithms, called variational algorithms, have been proposed The applications span a wide range of areas, including time- for Noisy Intermediate-Scale Quantum hardware [15, 6, 16]. keeping [2], spectroscopy [3], tests of fundamental physics [4], Variational algorithms operate by alternating between exe- and probing nanoscale systems such as condensed matter and cution on quantum hardware and execution on a classical biological systems [5]. There has been exciting experimental optimizer which guides subsequent iterations of quantum op- progress on various physical platforms, such as atomic vapor, erations. By limiting the number of operations on quantum trapped ions, Rydberg atoms, superconducting circuits, and hardware per iteration, variational methods reduce errors while nitrogen-vacancy centers in diamond. Even today, practical maintaining the quantum advantage. Such algorithms also al- quantum sensors such as SQUID magnetometers [6], atomic low us to leverage classical computing power to aid in quantum vapors, and atomic clocks [2] have already become the state- computation. of-the-art in magnetometry and timekeeping [7]. There have been recent proposals on extending the vari- *[email protected] ational approach to sensing [17, 18, 19]. However, these †[email protected] proposals: Signal-to-Noise Ratio Scaling the exponentially large Hilbert space of all possible entangled states, we propose circuit templates based on both physics principles and hardware capacities. The circuit template pa- Heisenberg Limit Entanglement rameterizes a circuit into a real-valued vector, framing circuit enhanced design into a multivariate optimization problem. We use an SNR objective function based on estimation theory, as well as prior Optimized results on noisy gradient estimation [21], which allows for Existing efficient and scalable search evaluated on real hardware. Classical Limit We achieve 1.75x SNR improvement over the best baseline Number of Qubits protocol under realistic simulation. We experimentally vali- date our technique on a 5-qubit IBM quantum computer and Figure 1: SNR scaling. The classical parallel scheme (Classi- measure a 1.69x improvement from the classical limit, with cal Limit) gives square root scaling of SNR versus the num- larger gains expected for increased system size. More impor- ber of sensing qubits. The theoretical upper bound of quan- tantly, we are able to overcome the plateauing/decreasing tum sensing is the Heisenberg Limit, which gives linear scal- performance of known entanglement-based protocols, ob- ing. Shaded region corresponds to entanglement-enhanced taining consistent sensitivity gain from additional qubits. sensing. Existing and optimized curves are based on exper- Compared to the classical limit, our result is equivalent to iments on IBM hardware (dotted lines are extrapolations)— 3x savings in the number of qubits needed, or 3x savings in circuit learning is able to overcome the plateauing/decreasing total sensing time to achieve the same SNR. Considering the performance of existing protocols. increasing difficulty of engineering large quantum systems, a • have fixed circuit structures; 3x reduction in system size has significant practical benefits. • require deep circuits or global interactions which are not Furthermore, a smaller system size also allows for smaller practical on many platforms; sensing volume, which is key in invasive applications such as • consider limited noise sources; biological sensing. Reduction of the sensing tim, in addition, • assume noise-free or limited “decoding” procedures for ex- allows for improved precision in sensing any time-dependent traction of quantum information into classical information; signal. • are not evaluated on hardware, which often varies consider- We enable quantum sensing to be implemented with high ably from simulation, e.g. in the difficulty of convergence. fidelity on real near-term noise-prone quantum hardware. Our We address these problems by proposing a flexible, solution considerably extends the state-of-the-art by incorpo- architecture-aware circuit structure design and a hardware- rating noise-awareness, realistic classical-quantum interfacing, based learning procedure that could be run under device noise. and adaptivity to diverse physical platforms and applications. The algorithm schematic is shown in Figure2. We divide The main contributions of this work are as follows: the sensing circuit into four components: encoder, signal ac- • Enabling a guided exploration of a larger space of entangled cumulation, decoder, and measurement. Given the available states than existing protocols; control gateset, platform-specific noise, and gate/readout er- • Leveraging platform-specific information via on-device rors, our approach aims to find the optimal encoder/decoder training under realistic hardware and noise constraints; pair. The encoder prepares the sensing state prior to exposure • Optimizing full sensing circuit including information ex- to the signal by applying superposition and entanglement to traction (rather than just state preparation), which we show the initial state. to have important effects on performance, especially given Then the sensing qubit(s) will be exposed to the signal gate noise and readout errors; for a certain amount of time, which results in changes to the • Considering a holistic noise model including gate noise, in- state due to the signal as well as interrogation noise (noise terrogation