Complexification of Quantum Signal Processing and its Ramifications

• URL: http://arxiv.org/abs/2407.04780v2
• Date: Thu, 18 Jul 2024 22:51:44 GMT
• Title: Complexification of Quantum Signal Processing and its Ramifications
• Authors: V. M. Bastidas, K. J. Joven,
• Abstract summary: We show a relation between a circuit defining a Floquet operator in a single period and its space-time dual defining QSP sequences for the Lie algebra sl$(2,mathbbC)$. We also show that unitary representations of our QSP sequences exist, although they are infinite-dimensional and are defined for bosonic operators in the Heisenberg picture.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In recent years there has been an increasing interest on the theoretical and experimental investigation of space-time dual quantum circuits. They exhibit unique properties and have applications to diverse fields. Periodic space-time dual quantum circuits are of special interest, due to their iterative structure defined by the Floquet operator. A very similar iterative structure naturally appears in Quantum Signal processing (QSP), which has emerged as a framework that embodies all the known quantum algorithms. However, it is yet unclear whether there is deeper relation between these two apparently different concepts. In this work, we establish a relation between a circuit defining a Floquet operator in a single period and its space-time dual defining QSP sequences for the Lie algebra sl$(2,\mathbb{C})$, which is the complexification of su$(2)$. First, we show that our complexified QSP sequences can be interpreted in terms of action of the Lorentz group on density matrices and that they can be interpreted as hybrid circuits involving unitaries and measurements. We also show that unitary representations of our QSP sequences exist, although they are infinite-dimensional and are defined for bosonic operators in the Heisenberg picture. Finally, we also show the relation between our complexified QSP and the nonlinear Fourier transform for sl$(2,\mathbb{C})$, which is a generalization of the previous results on su$(2)$ QSP.

Related papers

• Relational Quantum Geometry [0.0]
  We identify non-commutative or quantum geometry as a mathematical framework which unifies three objects. We first provide a rigorous account of the extended phase space, and demonstrate that it can be regarded as a classical principal bundle with a Poisson manifold base. We conclude that the quantum orbifold is equivalent to the G-framed algebra proposed in prior work.
  arXiv  Detail & Related papers  (2024-10-14T19:29:27Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• Quantum Signal Processing with the one-dimensional quantum Ising model [0.0]
  Quantum Signal Processing (QSP) has emerged as a promising framework to manipulate and determine properties of quantum systems. We provide examples and applications of our approach in diverse fields ranging from space-time dual quantum circuits and quantum simulation, to quantum control.
  arXiv  Detail & Related papers  (2023-09-08T18:01:37Z)
• A vertical gate-defined double quantum dot in a strained germanium double quantum well [48.7576911714538]
  Gate-defined quantum dots in silicon-germanium heterostructures have become a compelling platform for quantum computation and simulation. We demonstrate the operation of a gate-defined vertical double quantum dot in a strained germanium double quantum well. We discuss challenges and opportunities and outline potential applications in quantum computing and quantum simulation.
  arXiv  Detail & Related papers  (2023-05-23T13:42:36Z)
• Quantum signal processing with continuous variables [0.0]
  Quantum singular value transformation (QSVT) enables the application of functions to singular values of near arbitrary linear operators embedded in unitary transforms. We show that one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ans"atze to other Lie groups.
  arXiv  Detail & Related papers  (2023-04-27T17:50:16Z)
• Quantum walks, limits and transport equations [0.0]
  Quantum Walks (QWs) consist of single and isolated quantum systems evolving in discrete or continuous time steps. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations.
  arXiv  Detail & Related papers  (2021-12-22T12:12:09Z)
• Depth-efficient proofs of quantumness [77.34726150561087]
  A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify quantum advantage of an untrusted prover. In this paper, we give two proof of quantumness constructions in which the prover need only perform constant-depth quantum circuits.
  arXiv  Detail & Related papers  (2021-07-05T17:45:41Z)
• Information Scrambling in Computationally Complex Quantum Circuits [56.22772134614514]
  We experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We show that while operator spreading is captured by an efficient classical model, operator entanglement requires exponentially scaled computational resources to simulate.
  arXiv  Detail & Related papers  (2021-01-21T22:18:49Z)
• The duality-character Solution-Information-Carrying (SIC) unitary propagators [0.0]
  HSSS quantum search process owns the dual character that it obeys both the unitary quantum dynamics and the mathematical-logical principle of the unstructured search problem.
  arXiv  Detail & Related papers  (2020-12-24T13:47:05Z)
• Relevant OTOC operators: footprints of the classical dynamics [68.8204255655161]
  The OTOC-RE theorem relates the OTOCs summed over a complete base of operators to the second Renyi entropy. We show that the sum over a small set of relevant operators, is enough in order to obtain a very good approximation for the entropy. In turn, this provides with an alternative natural indicator of complexity, i.e. the scaling of the number of relevant operators with time.
  arXiv  Detail & Related papers  (2020-07-31T19:23:26Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.