The accuracies of modern quantum logic clocks have surpassed those of standard atomic fountain clocks. These clocks also provide a greater degree of control, because before and after clock queries, we are able to apply chosen unitary operations and measurements. Here, we take advantage of these choices and present a numerical technique designed to increase the accuracy of these clocks. We use a greedy approach, minimizing the phase variance of a noisy classical oscillator with respect to a perfect frequency standard after an interrogation step; we do not optimize over successive interrogations or the probe times. We consider arbitrary prior frequency knowledge and compare clocks with varying numbers of ions and queries interlaced with unitary control. Our technique is based on the semidefinite programming formulation of quantum query complexity, a method first developed in the context of deriving algorithmic lower bounds. The application of semidefinite programming to an inherently continuous problem like that considered here requires discretization; we derive bounds on the error introduced and show that it can be made suitably small.
The accuracies of modern quantum logic clocks have surpassed those of standard atomic fountain clocks. These clocks also provide a greater degree of control, because before and after clock queries, we are able to apply chosen unitary operations and measurements. Here, we take advantage of these choices and present a numerical technique designed to increase the accuracy of these clocks. We use a greedy approach, minimizing the phase variance of a noisy classical oscillator with respect to a perfect frequency standard after an interrogation step; we do not optimize over successive interrogations or the probe times. We consider arbitrary prior frequency knowledge and compare clocks with varying numbers of ions and queries interlaced with unitary control. Our technique is based on the semidefinite programming formulation of quantum query complexity, a method first developed in the context of deriving algorithmic lower bounds. The application of semidefinite programming to an inherently continuous problem like that considered here requires discretization; we derive bounds on the error introduced and show that it can be made suitably small.
The accuracies of modern quantum logic clocks have surpassed those of standard atomic fountain clocks. These clocks also provide a greater degree of control, because before and after clock queries, we are able to apply chosen unitary operations and measurements. Here, we take advantage of these choices and present a numerical technique designed to increase the accuracy of these clocks. We use a greedy approach, minimizing the phase variance of a noisy classical oscillator with respect to a perfect frequency standard after an interrogation step; we do not optimize over successive interrogations or the probe times. We consider arbitrary prior frequency knowledge and compare clocks with varying numbers of ions and queries interlaced with unitary control. Our technique is based on the semidefinite programming formulation of quantum query complexity, a method first developed in the context of deriving algorithmic lower bounds. The application of semidefinite programming to an inherently continuous problem like that considered here requires discretization; we derive bounds on the error introduced and show that it can be made suitably small.
We describe quantum interrogation schemes for passive atomic clocks. During any given interrogation period, the optimal interrogation algorithm depends on the state of the clock -- specifically on the frequency deviation of the flywheel (classical oscillator) from the atomic standard. As a clock runs, it is possible to estimate this deviation. Nonetheless, traditional schemes use the same, fixed algorithm for each interrogation period, which is necessarily independent of this prior knowledge. Here we present a dynamic scheme, tailoring our algorithms to the clock's state before each interrogation. These strategies are derived by constructing a complete model of a passive clock -- specifically, a probability distribution describing the estimated average offset frequency of the flywheel during both the upcoming interrogation period and interrogation periods in the past is updated via appropriate noise models and by measurements of the atomic standard.\nTo reduce the deviation from an ideal clock we optimize the next interrogation algorithm by means of a semidefinite program for atomic state preparation and measurement whose objective function depends on the updated state. This program is based on the semidefinite programming formulation of quantum query complexity, a method first developed in the context of deriving algorithmic lower bounds. The application of semidefinite programming to an inherently continuous problem like that considered here requires discretization; we derive bounds on the error introduced and show that it can be made suitably small.\nFinally, we implement a full simulation of a passive clock with power-law noise models and find significant improvements by applying our techniques.
We describe protocols for passive atomic clocks based on quantum interrogation of the atoms. Unlike previous techniques, our protocols are adaptive and take advantage of prior information about the clock's state. To reduce deviations from an ideal clock, each interrogation is optimized by means of a semidefinite program for atomic state preparation and measurement whose objective function depends on the prior information. Our knowledge of the clock's state is maintained according to a Bayesian model that accounts for noise and measurement results. We implement a full simulation of a running clock with power-law noise models and find significant improvements by applying our techniques.
Abstract Quantum query complexity is pivotal in the analysis of quantum algorithms, encompassing well-known examples like search and period-finding algorithms. These algorithms typically involve a sequence of unitary operations and oracle calls dependent on an input variable. In this study, we introduce a variational learning approach to explore quantum query complexity. Our method employs an efficient parameterization of the unitary operations and utilizes a loss function derived from the algorithm’s error probability. We apply this technique to various quantum query complexities, notably devising a new algorithm that resolves the 5-bit Hamming modulo problem with four queries, addressing an open question from Cornelissen et al (2021 arXiv: 2112.14682 ). This finding is corroborated by a semidefinite programming (SDP) approach. Our numerical method exhibits superior memory efficiency compared to SDP and can identify quantum query algorithms (QQAs) that require a smaller workspace register dimension, an aspect not optimized by SDP. These advancements present a significant step forward in the practical application and understanding of QQAs.
The Quantum Adversary Method has proven to be a successful technique for deriving lower bounds on a wide variety of problems. However, it assumes perfect quantum computation, which in most modern devices, is unrealistic. Here, we develop a generalization of this technique without this assumption, which can be applied to arbitrary small problems automatically. To do this, we start by reformulating the objective value of the semidefinite program of the spectral adversary method. By relating the final measurement stage of a quantum computation to remote state preparation, we prove that the optimal value of the new objective corresponds to the probability that the quantum computer will output the correct value after a specified number of queries. Once in this framework, the addition of decoherence is natural. In particular, the optimum probability of success can be determined for any probability of phase error. In the limit of complete phase decoherence, we recover the optimal p! robability of success for a classical computation. Our semidefinite programming formulation is suitably general, and so has application outside that of algorithms. In particular, we apply it to the optimization of quantum clocks.
Transducers (Belovs, Jeffery and Yolcu, 2024) are a quantum computing framework describing a quantum algorithm as a unitary converting an input state into a target state using a catalyst, an auxiliary vector that is left unchanged. They are a powerful tool in quantum algorithm design, especially in the context of quantum query complexity: feasible points of the (dual) adversary semidefinite program directly translate into transducers and the optimal transduction complexity is equal to the adversary bound, i.e. the Las Vegas complexity, which is known to characterize bounded-error quantum query complexity. Moreover, contrary to bounded-error algorithms, transducers compose exactly, which limits overheads due to controlling errors in algorithms constructed by composition. Constructing efficient, let alone optimal, transducers in terms of quantum query complexity nevertheless remains a hard task since it still requires solving the adversary SDP and constructing the unitary to obtain an explicit algorithm. In this paper, we show how using the symmetry group of state-conversion problems simplifies both steps. First, using a symmetrization argument, we prove an optimal catalyst can always be chosen covariant under a representation of the symmetry group. Second, we prove that the transducer intertwines two different representations of the group and can thus be chosen block diagonal in the isotypic decomposition of the Hilbert space. Using those methods, we then derive optimal transducers, with optimal constants, for different widely used quantum algorithmic primitives, such as unstructured search, amplitude amplification and amplitude estimation. Our approach extends previous work on the use of representation theory to compute adversary lower bounds (Høyer, Lee, and {\v S}palek, 2007; Ambainis, Magnin, Roetteler and Roland, 2011) to the systematic construction of optimal algorithms.