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We generalize the projection-based quantum measurement-driven $k$-SAT algorithm of Benjamin, Zhao, and Fitzsimons (BZF, arxiv:1711.02687) to arbitrary strength quantum measurements, including the limit of continuous monitoring. In doing so, we clarify that this algorithm is a particular case of the measurement-driven quantum control strategy elsewhere referred to as "Zeno dragging". We argue that the algorithm is most efficient with finite time and measurement resources in the continuum limit, where measurements have an infinitesimal strength and duration. Moreover, for solvable $k$-SAT problems, the dynamics generated by the algorithm converge deterministically towards target dynamics in the long-time (Zeno) limit, implying that the algorithm can successfully operate autonomously via Lindblad dissipation, without detection. We subsequently study both the conditional and unconditional dynamics of the algorithm implemented via generalized measurements, quantifying the advantages of detection for heralding errors. These strategies are investigated first in a computationally-trivial $2$-qubit $2$-SAT problem to build intuition, and then we consider the scaling of the algorithm on $3$-SAT problems encoded with $4 - 10$ qubits. The average number of shots needed to obtain a solution scales with qubit number as $\lambda^n$. For vanishing dragging time (with final readout only), we find $\lambda = 2$ (corresponding to a brute-force search over possible solutions). However, the deterministic (autonomous) property of the algorithm in the adiabatic (Zeno) limit implies that we can drive $\lambda$ arbitrarily close to $1$, at the cost of a growing pre-factor. We numerically investigate the tradeoffs in these scalings with respect to algorithmic runtime and assess their implications for using this analog measurement-driven approach to quantum computing in practice., Comment: 23 + 8 pages, 15 figures

The quantum Zeno effect asserts that quantum measurements inhibit simultaneous unitary dynamics when the "collapse" events are sufficiently strong and frequent. This applies in the limit of strong continuous measurement or dissipation. It is possible to implement a dissipative control that is known as "Zeno Dragging", by dynamically varying the monitored observable, and hence also the eigenstates which are attractors under Zeno dynamics. This is similar to adiabatic processes, in that the Zeno dragging fidelity is highest when the rate of eigenstate change is slow compared to the measurement rate. We demonstrate here two theoretical methods for using such dynamics to achieve control of quantum systems. The first, which we shall refer to as "shortcut to Zeno" (STZ), is analogous to the shortcuts to adiabaticity (counterdiabatic driving) that are frequently used to accelerate unitary adiabatic evolution. In the second approach we apply the Chantasri Dressel Jordan (2013, CDJ) stochastic action, and demonstrate that the extremal-probability readout paths derived from this are well suited to setting up a Pontryagin-style optimization of the Zeno dragging schedule. A fundamental contribution of the latter approach is to show that an action suitable for measurement-driven control optimization can be derived quite generally from statistical arguments. Implementing these methods on the Zeno dragging of a qubit, we find that both approaches yield the same solution, namely, that the optimal control is a unitary that matches the motion of the Zeno-monitored eigenstate. We then show that such a solution can be more robust than a unitary-only operation, and comment on solvable generalizations of our qubit example embedded in larger systems. These methods open up new pathways toward systematically developing dynamic control of Zeno subspaces to realize dissipatively-stabilized quantum operations., Comment: 24 + 13 pages, 8 figures. Comments welcome!

The Zeno effect, in which repeated observation freezes the dynamics of a quantum system, stands as an iconic oddity of quantum mechanics. When a measurement is unable to distinguish between states in a subspace, the dynamics within that subspace can be profoundly altered, leading to non-trivial behavior. Here we show that such a measurement can turn a non-interacting system with only single-qubit control into a two- or multi-qubit entangling gate, which we call a Zeno gate. The gate works by imparting a geometric phase on the system, conditioned on it lying within a particular nonlocal subspace. We derive simple closed-form expressions for the gate fidelity under a number of non-idealities and show that the gate is viable for implementation in circuit and cavity QED systems. More specifically, we illustrate the functioning of the gate via dispersive readout in both the Markovian and non-Markovian readout regimes, and derive conditions for longitudinal readout to ideally realize the gate., Comment: 20+12 pages. 13 figures

Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a workaround for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity even in the case of inefficient detection., Comment: 8 + 8 Pages, 7 Figures

The Zeno effect occurs in quantum systems when a very strong measurement is applied, which can alter the dynamics in non-trivial ways. Despite being dissipative, the dynamics stay coherent within any degenerate subspaces of the measurement. Here we show that such a measurement can turn a single-qubit operation into a two- or multi-qubit entangling gate, even in a non-interacting system. We demonstrate this gate between two effectively non-interacting transmon qubits. Our Zeno gate works by imparting a geometric phase on the system, conditioned on it lying within a particular non-local subspace. These results show how universality can be generated not only by coherent interactions as is typically employed in quantum information platforms, but also by Zeno measurements.

We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements. We deduce the stochastic evolution equations for position and momentum expectation values and the covariance matrix elements from the system's characteristic function. By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 2015) to this continuous variable system, we construct its stochastic Hamiltonian and action. Action extremization gives us the equations for the most-likely readout paths and quantum trajectories. For steady states of the covariance matrix elements, the analytical solutions for these most-likely paths are obtained. Using the CDJ formalism we calculate final state probability densities exactly starting from any initial state. We also demonstrate the agreement between the optimal path solutions and the averages of simulated clustered stochastic trajectories. Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement engine/refrigerator experiments., Comment: 15 pages, 7 figures

The Wigner's friend paradox concerns one of the most puzzling problems of quantum mechanics: the consistent description of multiple nested observers. Recently, a variation of Wigner's gedankenexperiment, introduced by Frauchiger and Renner, has lead to new debates about the self-consistency of quantum mechanics. At the core of the paradox lies the description of an observer and the object it measures as a closed system obeying the Schr\"odinger equation. We revisit this assumption to derive a necessary condition on a quantum system to behave as an observer. We then propose a simple single-photon interferometric setup implementing Frauchiger and Renner's scenario, and use the derived condition to shed a new light on the assumptions leading to their paradox. From our description, we argue that the three apparently incompatible properties used to question the consistency of quantum mechanics correspond to two logically distinct contexts: either one assumes that Wigner has full control over his friends' lab, or conversely that some parts of the labs remain unaffected by Wigner's subsequent measurements. The first context may be seen as the quantum erasure of the memory of Wigner's friend. We further show these properties are associated with observables which do not commute, and therefore cannot take well-defined values simultaneously. Consequently, the three contradictory properties never hold simultaneously., Comment: 11 pages, 3 figures. Accepted in Quantum (24 June 2021)

Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local pi-pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e. using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around a Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e. using diffusive quantum trajectories), [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. The method we describe here combines proven techniques in a novel way, complementing existing protocols, and offering a pathway towards generating and protecting entangled states so that they may be used in various applications on demand., Comment: 10 + 5 pages, 7 + 2 figures

We consider protocols to generate quantum entanglement between two remote qubits, through joint time-continuous detection of their spontaneous emission. We demonstrate that schemes based on homodyne detection, leading to diffusive quantum trajectories, lead to identical average entanglement yield as comparable photodetection strategies; this is despite substantial differences in the two-qubit state dynamics between these schemes, which we explore in detail. The ability to use different measurements to achieve the same ends may be of practical significance; the less-well-known diffusive scheme appears far more feasible on superconducting qubit platforms in the near term., Comment: 6 pages, 3 figures. Supplementary information can be found in the longer version at arXiv:1910.01206

We discuss recent developments in measurement protocols that generate quantum entanglement between two remote qubits, focusing on the theory of joint continuous detection of their spontaneous emission. We consider a device geometry similar to that used in well-known Bell-state measurements, which we analyze using a conceptually transparent model of stochastic quantum trajectories; we use this to review photodetection, the most straightforward case, and then generalize to the diffusive trajectories from homodyne and heterodyne detection as well. Such quadrature measurement schemes are a realistic two-qubit extension of existing circuit-QED experiments which obtain quantum trajectories by homodyning or heterodyning a superconducting qubit's spontaneous emission, or an adaptation of existing optical measurement schemes to obtain jump trajectories from emitters. We mention key results, presented from within a single theoretical framework, and draw connections to concepts in the wider literature on entanglement generation by measurement (such as path information erasure and entanglement swapping). The photon which-path information acquisition, and therefore the two-qubit entanglement yield, is tunable under the homodyne detection scheme we discuss, at best generating equivalent average entanglement dynamics as in the comparable photodetection case. In addition to deriving this known equivalence, we extend past analyses in our characterization of the measurement dynamics: We include derivations of bounds on the fastest possible evolution towards a Bell state under joint homodyne measurement dynamics, and characterize the maximal entanglement yield possible using inefficient (lossy) measurements.

We review the continuous monitoring of a qubit through its spontaneous emission, at an introductory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring, and illustrate its equivalence to the stochastic master equation formalism throughout. Special emphasis is given to homodyne (phase-sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results., Comment: 20+6 pages, 8 figures

Using weak values amplification angular resolution limits, we theoretically investigate the gravitational sensing of objects. By inserting a force-sensing pendulum into a weak values interferometer, the optical response can sense accelerations to a few 10's of $\mathrm{zepto}\text{-}\mathrm{g}/\sqrt{\mathrm{Hz}}$, with optical powers of $1~\mathrm{mW}$. We convert this precision into range and mass sensitivity, focusing in detail on simple and torsion pendula. Various noise sources present are discussed, as well as the necessary cooling that should be applied to reach the desired levels of precision., Comment: 9 pages, 4 figures, Quantum Stud.: Math. Found. (2018)

We predict that continuously monitored quantum dynamics can be chaotic. The optimal paths between past and future boundary conditions can diverge exponentially in time when there is time-dependent evolution and continuous weak monitoring. Optimal paths are defined by extremizing the global probability density to move between two boundary conditions. We investigate the onset of chaos in pure-state qubit systems with optimal paths generated by a periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated in a scheme where two non-commuting observables of a qubit are continuously monitored, and one measurement strength is periodically modulated. The optimal quantum paths in this example bear similarities to the trajectories of the kicked rotor, or standard map, which is a paradigmatic example of classical chaos. We emphasize connections with the concept of resonance between integrable optimal paths and weak periodic perturbations, as well as our previous work on "multipaths", and connect the optimal path chaos to instabilities in the underlying quantum trajectories., Comment: 12+11 pages, 11 figures. Supplemental Animations can be found at << https://drive.google.com/file/d/1cx__Aggt40s3r8ueTe8LlqZyAWLb5NV1/view?usp=sharing >> (all animations in .pdf format), or << https://drive.google.com/drive/folders/12LgI0dCiSjRYoHO9oiWDaXm7S0PzM7Y1?usp=sharing >> (as individual .mp4 files)

We examine most-likely paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of "multipaths" in the dynamics of continuously-monitored qubit systems, wherein multiple most-likely paths travel between the same pre- and post-selected states over the same time interval. Most-likely paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrange manifold in this phase space, and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states, or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously-monitored quantum states. We apply our methods in two systems: a qubit with two non-commuting observables measured simultaneously, and a qubit measured in one observable while subject to Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van-Vleck determinant, and their onset time. We also find multipaths generated by paths with different "winding numbers" around the Bloch sphere in both systems., Comment: 20 pages, 8 figures. Compared with version 1, version 2 contains an additional appendix, as well as minor revisions / clarifications based partly on referee feedback at Phys. Rev. A

We employ phase-sensitive amplification to perform homodyne detection of the resonance fluorescence from a driven superconducting artificial atom. Entanglement between the emitter and its fluorescence allows us to track the individual quantum state trajectories of the emitter conditioned on the outcomes of the field measurements. We analyze the ensemble properties of these trajectories by considering trajectories that connect specific initial and final states. By applying the stochastic path integral formalism, we calculate equations-of-motion for the most likely path between two quantum states and compare these predicted paths to experimental data. Drawing on the mathematical similarity between the action formalism of the most likely quantum paths and ray optics we study the emergence of caustics in quantum trajectories---places where multiple extrema in the stochastic action occur. We observe such multiple most likely paths in experimental data and find these paths to be in reasonable quantitative agreement with theoretical calculations., Comment: 12 pages, 10 figures

The effect of substrate bias and surface gate voltage on the low temperature resistivity of a Si-MOSFET is studied for electron concentrations where the resistivity increases with increasing temperature. This technique offers two degrees of freedom for controlling the electron concentration and the device mobility, thereby providing a means to evaluate the relative importance of electron-electron interactions and disorder in this so-called ``metallic'' regime. For temperatures well below the Fermi temperature, the data obey a scaling law where the disorder parameter ($k_{\rm{F}}l$), and not the concentration, appears explicitly. This suggests that interactions, although present, do not alter the Fermi-liquid properties of the system fundamentally. Furthermore, this experimental observation is reproduced in results of calculations based on temperature-dependent screening, in the context of Drude-Boltzmann theory., Comment: 5 pages, 6 figures