On Qubits as Discrete Signals

Chur Chin

Applying the Poincaré theorems and Euler integral in the qubit based signal and sensing framework can be as Qubit State Space. Defined by rational amplitudes or unit-fraction-based norms. Signal as Path: A time-evolving qubit state (signal) as a path through this discrete manifold.Use Euler integration or Poincaré recurrence to describe or detect repeating patterns, signal stability and minimal sensing requirements.Signal values across this grid can be analyzed via Euler calculus. Signal coverage, gaps, and redundancy are reflected in topological counts and we can apply Euler integrals to compress, summarize, or filter the signal.

Osama M. Raisuddin, Suvranu De

Meisheng Zhao, Tao Qin, Yongde Zhang

Artyom M. Grigoryan, Alexis A. Gomez, Sos С. Agaian

The article presents a new approach to the problem of transforming one quantum state into another. It is shown that a r-qubit superposition |x can be obtained from another r-qubit superposition |y, by using only (2r-1) rotations, each presented by one controlled rotation gate. The quantum superpositions with real amplitudes are considered. The traditional two-stage approach Uy-1Ux:|x→|0⊕r→|y requires twice as many rotations. Here, both transformations to the conventual basis state, Ux: |x→ |0⊕r and Uy: |y→ |0⊕r, use (2r-1) rotations each on two binary planes and many of these rotations require additional sets of CNOTs to be represented as 1- or 2-qubit controlled gates. The proposed method is based on the concept of the discrete signal-induced heap transform (DsiHT) which is unitary and generated by a vector and a set of angular equations with given parameters. The quantum analogue of this transform is described. The main characteristic of the DsiHT is the path of processing the data. It is shown that exist such fast paths that allow for effective computing of the DsiHT, which leads to the simple quantum circuits for state preparation and transformation. Examples of such paths are given and quantum circuits for preparation and transformation of 2-, 3-, and 4-qubits are described in detail. CNOT gates are not used, but only controlled gates of elementary rotations around the y-axis. It is shown that the transformation and, in particular, only rotation gates with control qubits are required for initialization of 4-qubits. The quantum circuits are simple and have a recursive form, which makes them easy to implement for arbitrary r-qubit superposition, with r≥2. This approach significantly reduces the complexity of quantum state transformations, paving the way for more efficient quantum algorithms and practical implementations on near-term quantum devices.

Andre Schneider, Jochen Braumüller, Lingzhen Guo +5 more

Analyzing weak microwave signals in the GHz regime is a challenging task if the signal level is very low and the photon energy widely undefined. A superconducting qubit can detect signals in the low photon regime, but due to its discrete level structure, it is only sensitive to photons of certain energies. With a multilevel quantum system (qudit) in contrast, the unknown signal frequency and amplitude can be deduced from the higher level ac Stark shift. The measurement accuracy is given by the signal amplitude, its detuning from the discrete qudit energy level structure, and the anharmonicity. We demonstrate an energy sensitivity in the order of ${10}^{\ensuremath{-}3}$ with a measurement range of more than $1\phantom{\rule{0.28em}{0ex}}\mathrm{GHz}$. Here, using a transmon qubit, we experimentally observe shifts in the transition frequencies involving up to three excited levels. These shifts are in good agreement with an analytic circuit model and master equation simulations. For large detunings, we find the shifts to scale linearly with the power of the applied microwave drive. Exploiting the effect, we demonstrated a power meter which makes it possible to characterize the microwave transmission from source to sample.