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arXiv:2407.04780

Complexification of Quantum Signal Processing and its Ramifications

V. M. Bastidas, K. J. Joven

Published Jul 5, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2407.04780

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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,C)(2,\mathbb{C})(2,C), which is the complexification of su(2)(2)(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,C)(2,\mathbb{C})(2,C), which is a generalization of the previous results on su(2)(2)(2) QSP.
Quantum signal processing (QSP) has emerged as a promising framework to manipulate and determine properties of quantum systems. QSP not only unifies most existing quantum algorithms but also provides tools to discover new ones. Quantum signal processing is applicable to single-qubit or multiqubit systems that can be ``qubitized'' so one can exploit the SU(2) structure of system evolution within special invariant two-dimensional subspaces. In the context of quantum algorithms, this SU(2) structure is artificially imposed on the system through highly nonlocal evolution operators that are difficult to implement on near-term quantum devices. In this work, we propose QSP protocols for the infinite-dimensional Onsager Lie algebra, which is relevant to the physical dynamics of quantum devices that can simulate the transverse-field Ising model. To this end, we consider QSP sequences in the Heisenberg picture, allowing us to exploit the emergent SU(2) structure in momentum space and ``synthesize'' QSP sequences for the Onsager algebra. Our results demonstrate a concrete connection between QSP techniques and noisy intermediate scale quantum protocols. We provide examples and applications of our approach in diverse fields ranging from space-time dual quantum circuits and quantum simulation to quantum control.
Abstract Many-body quantum dynamics defined on a spatial lattice and in discrete time—either as stroboscopic Floquet systems or quantum circuits—has been an active area of research for several years. Being discrete in space and time, a natural question arises: when can such a model be viewed as evolving unitarily in space as well as in time? Models with this property, which sometimes goes by the name space-time duality, have been shown to have a number of interesting features related to entanglement growth and correlations. One natural way in which the property arises in the context of (brickwork) quantum circuits is by choosing dual unitary gates: two site operators that are unitary in both the space and time directions. We introduce a class of models with q states per site, defined on the square lattice by a complex partition function and paremeterized in terms of q × q Hadamard matrices, that have the property of space-time duality. These may interpreted as particular dual unitary circuits or stroboscopically evolving systems, and generalize the well studied self-dual kicked Ising model. We explore the operator dynamics in the case of Clifford circuits, making connections to Clifford cellular automata (Schlingemann et al 2008 J. Math. Phys. 49 112104) and in the q → ∞ limit to the classical spatiotemporal cat model of many body chaos (Gutkin et al 2021 Nonlinearity 34 2800). We establish integrability and the corresponding conserved charges for a large subfamily and show how the long-range entanglement protocol discussed in the recent paper (Lotkov et al 2022 Phys. Rev. B 105 144306) can be reinterpreted in purely graphical terms and directly applied here.
Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.
Making sense of the world happens through receiving and processing signals, whether it be sound, light, or other waves that interact with us. The fields of nuclear magnetic resonance and digital signal processing each built a rigorous language for manipulating and extracting information from signals of particular types. Quantum signal processing (QSP) connects these two traditions, providing a unified algebraic framework for understanding how quantum systems are transformed by both classical and quantum signals. QSP provides a systematic framework for characterizing possible transformations. QSP was developed for qubit systems, but physical quantum systems are rarely two-level and the full richness of high-dimensional Hilbert spaces has remained largely unexplored. This thesis extends and applies QSP in two directions: from qubits to high-dimensional systems such as finite-level qudits and infinite-dimensional oscillators and from static signals to signals that include continuous-time noise processes. We will also show how QSP for qudits leads to novel experimental applications in trapped-ion systems. One type of high-dimensional system is a qubit coupled to an oscillator, and a qubit-oscillator qudit can be formed and universally controlled using only a limited set of interactions. In this qubit-oscillator space, a bosonic version of QSP allows novel incarnations of quantum algorithms and sensing. The quantum Fourier transform can be performed by clever conversion of information in a many-qubit system into a single oscillator, and an interferometric protocol uses binary decisions from a qubit to sense a signal affecting the oscillator. Another type of high-dimensional system is a qudit made from the electronic levels of an atom. This thesis presents a new construction of Grover’s algorithm using a qudit in an atomic hyperfine manifold, along with a calibration method that finds accurate QSP pulse parameters for the experiment. For decision algorithms, QSP in qudits enables theoretically perfect decision-making for quantum channel discrimination problems with certain symmetries. In the face of noise, temporal QSP is developed to extend the canonical framework to continuous-time stochastic noise. Combined with Kubo linear-response theory, a noise filter function description emerges which formalizes the optimization of atomic clock performance. These results from qudit, bosonic, and temporal QSP establish QSP as a natural language for high-dimensional quantum control and sensing.
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The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on SU(1,1)\mathrm{SU}(1,1)SU(1,1) has been widely studied, its SU(2)\mathrm{SU}(2)SU(2) variant has only recently attracted attention due to emerging applications in quantum signal processing (QSP) and quantum singular value transformation (QSVT). In this paper, we investigate the inverse NLFT on SU(2)\mathrm{SU}(2)SU(2) and establish the numerical stability of the layer stripping algorithm for the first time under suitable conditions. Furthermore, we develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity. This algorithm is applicable to computing phase factors for both QSP and the generalized QSP (GQSP).