Quantum Signal Processing Research Articles

Quantum parameter estimation on coherently superposed noisy channels

A generic qubit unitary operator affected by quantum noise is duplicated and inserted in a coherently superposed channel, superposing two paths offered to a probe qubit across the noisy unitary, and driven by a control qubit. A characterization is performed of the transformation realized by the superposed channel on the joint state of the probe-control qubit pair. The superposed channel is then specifically analyzed for the fundamental metrological task of phase estimation on the noisy unitary, with the performance assessed by the Fisher information, classical or quantum. A comparison is made with conventional estimation techniques and also with a quantum switched channel with indefinite causal order recently investigated for a similar task of phase estimation. In the analysis here, a first important observation is that the control qubit of the superposed channel, although it never directly interacts with the unitary being estimated, can nevertheless be measured alone for effective estimation, while discarding the probe qubit that interacts with the unitary. This property is also present with the switched channel but is inaccessible with conventional techniques. The optimal measurement of the control qubit here is characterized in general conditions. A second important observation is that the noise plays an essential role in coupling the control qubit to the unitary, and that the control qubit remains operative for phase estimation at very strong noise, even with a fully depolarizing noise, whereas conventional estimation and the switched channel become inoperative in these conditions. The results extend the analysis of the capabilities of coherently controlled channels which represent novel devices exploitable for quantum signal and information processing.

Quantum Windowed Fourier Transform and its Application to Quantum Signal Processing

Quantum Windowed Fourier Transform and its Application to Quantum Signal Processing

Quantum simulation of gauge theory via orbifold lattice

We propose a new framework for simulating U(k) Yang-Mills theory on a universal quantum computer. This construction uses the orbifold lattice formulation proposed by Kaplan, Katz, and Unsal, who originally applied it to supersymmetric gauge theories. Our proposed approach yields a novel perspective on quantum simulation of quantum field theories, carrying certain advantages over the usual Kogut-Susskind formulation. We discuss the application of our constructions to computing static properties and real-time dynamics of Yang-Mills theories, from glueball measurements to AdS/CFT, making use of a variety of quantum information techniques including qubitization, quantum signal processing, Jordan-Lee-Preskill bounds, and shadow tomography. The generalizations to certain supersymmetric Yang-Mills theories appear to be straightforward, providing a path towards the quantum simulation of quantum gravity via holographic duality.

Nonlinearity-Induced Nonreciprocity—Part II

This is the second part of our study of the properties and limitations of nonlinearity-induced nonreciprocal devices. In the first part, we discussed the general working principle of nonreciprocal devices based on resonators filled with Kerr-like nonlinearities, and we showed that strong nonreciprocal responses can be obtained when an asymmetric resonator is excited from only one port with a quasi-monochromatic signal. In this second part, we further investigate their behavior under more complex input signals. Specifically, we discuss simultaneous two-port excitations, which are expected to be detrimental to isolation purposes due to the lack of linearity of these devices. We show, however, that this regime can be used to achieve novel functionalities, such as optically controlled transmission switching. In order to assess the device performance in realistic scenarios, we also investigate how noise and random inputs affect the nonreciprocal response. Finally, we show how increasing the system complexity can help to overcome some device limitations, and lead to novel applications such as bandpass nonreciprocal filters. Overall, this work shows that nonlinearity-induced nonreciprocal devices, with their appealing passive, bias-free operation, offer interesting opportunities for various practical application scenarios and may form the basis for a plethora of new technologies of great relevance for microwave engineering systems, from radars and lidars to classical and quantum signal processing and computing.

Quantum hypothesis testing with group structure

The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., $C_n, D_{2n}, A_4, S_4, A_5$) the recently-developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing (G-QHT), intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in $n$, the size of the channel set and group, compared to na\"ive repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

Analysis and design of secure quantum communication systems utilizing electromagnetic Schrodinger coherent states

We develop a general mathematical formalism and computational method suitable for analyzing the performance of quantum communication systems utilizing coherent radiation states (Schrodinger states). The key motivation is to demonstrate how the analysis and design of quantum communication systems may benefit from the integration of the physical structure (here, the optical quantum coherent state) with the quantum signal processing aspects (here, quantum estimation and prediction theory). As an application, we show how the quantum-electromagnetic signature of optical radiation emitted by quantum antennas, the Schrodinger's coherent state structure, may be exploited to jointly design the transmitter and receiver of a K-ary digital communication link. We present a concrete example comprised of a quantum quadrature-amplitude modulation (QAM) system (with ), including a complete description of the required general design principles developed from quantum mechanics and the quantum antenna's radiated coherent state structure. The simulation results illustrate improvement in spectral efficiency due to the use of over smaller values of K. We also compare the quantum antenna-based link performance with classical QAM utilizing classical antenna-based communication systems and report superiority of the quantum link over the classical version. To provide for security protection, our system is equipped with a quantum encryption protocol for quantum key distribution, and a demonstration of the complete quantum QAM system with encryption is presented. The main message of the article is the fruitfulness of incorporating a multidisciplinary approach to our thinking about electromagnetic wireless systems through the joint deployment of electromagnetic, quantum, and communication theories in the design and development process of current and future advanced technologies.

Toward simulating superstring/M-theory on a quantum computer

We present a novel framework for simulating matrix models on a quantum computer. Supersymmetric matrix models have natural applications to superstring/M-theory and gravitational physics, in an appropriate limit of parameters. Furthermore, for certain states in the Berenstein-Maldacena-Nastase (BMN) matrix model, several supersymmetric quantum field theories dual to superstring/M-theory can be realized on a quantum device. Our prescription consists of four steps: regularization of the Hilbert space, adiabatic state preparation, simulation of real-time dynamics, and measurements. Regularization is performed for the BMN matrix model with the introduction of energy cut-off via the truncation in the Fock space. We use the Wan-Kim algorithm for fast digital adiabatic state preparation to prepare the low-energy eigenstates of this model as well as thermofield double state. Then, we provide an explicit construction for simulating real-time dynamics utilizing techniques of block-encoding, qubitization, and quantum signal processing. Lastly, we present a set of measurements and experiments that can be carried out on a quantum computer to further our understanding of superstring/M-theory beyond analytic results.

Quantum advantage for computations with limited space

Quantum computations promise the ability to solve problems intractable in the classical setting. Restricting the types of computations considered often allows to establish a provable theoretical advantage by quantum computations, and later demonstrate it experimentally. In this paper, we consider space-restricted computations, where input is a read-only memory and only one (qu)bit can be computed on. We show that $n$-bit symmetric Boolean functions can be implemented exactly through the use of quantum signal processing as restricted space quantum computations using $O(n^2)$ gates, but some of them may only be evaluated with probability $1/2 + O(n/\sqrt{2}^n)$ by analogously defined classical computations. We experimentally demonstrate computations of $3$-, $4$-, $5$-, and $6$-bit symmetric Boolean functions by quantum circuits, leveraging custom two-qubit gates, with algorithmic success probability exceeding the best possible classically. This establishes and experimentally verifies a different kind of quantum advantage -- one where quantum scrap space is more valuable than analogous classical space -- and calls for an in-depth exploration of space-time tradeoffs in quantum circuits.

Microwave-driven generation and group delay control of optical pulses from an ultra-dilute atomic ensemble.

A cyclic atomic level scheme interacting with an optical and a microwave field is proposed for the generation and group-delay control of few-photon optical pulses. Our analysis exploits a hybrid second order-nonlinearity under conditions of electromagnetically induced transparency to generate an optical pulse. The generated pulse can be delayed or advanced through microwave intensity control of the absolute phase of the second-order-nonlinearity. Importantly, this handle on group delay of the generated pulse is number density-independent. Our scheme is thus ideally suited for the generation and control of few-photon optical pulses using ultra-dilute atomic samples. Our results will enable microscopic atomic interface systems that serve as controllable delay channels for both classical and quantum signal processing.

Efficient phase-factor evaluation in quantum signal processing

Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than $10\phantom{\rule{0.16em}{0ex}}000$ with errors below ${10}^{\ensuremath{-}12}$.

Optical coherence and energy-level properties of a Tm3+ -doped LiNbO3 waveguide at subkelvin temperatures

We characterize the optical coherence and energy-level properties of the 795 nm $^3$H$_6$ to $^3$H$_4$ transition of Tm$^{3+}$ in a Ti$^{4+}$:LiNbO$_{3}$ waveguide at temperatures as low as 0.65 K. Coherence properties are measured with varied temperature, magnetic field, optical excitation power and wavelength, and measurement time-scale. We also investigate nuclear spin-induced hyperfine structure and population dynamics with varying magnetic field and laser excitation power. Except for accountable differences due to difference Ti$^{4+}$ and Tm$^{3+}$-doping concentrations, we find that the properties of Tm$^{3+}$:Ti$^{4+}$:LiNbO$_{3}$ produced by indiffusion doping are consistent with those of a bulk-doped Tm$^{3+}$:LiNbO$_{3}$ crystal measured under similar conditions. Our results, which complement previous work in a narrower parameter space, support using rare-earth-ions for integrated optical and quantum signal processing.

Computing Spectral Measures and Spectral Types

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on L^2({mathbb {R}}^d). Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

ML Criterion Based Signal Detection of a MIMO-OFDM System Using Quantum and Semi-Quantum Assisted Modified DHA/BBHT Search Algorithm

The quantum search algorithm (QSA) is one of the most recent developments in the era of quantum signal processing (QSP) and quantum information theory (QIT). It shows a faster performance in finding the solutions of system such as multiple-input multiple-output (MIMO) communication system, quantum error correction and others with a polynomial order over the classical one. Grover's algorithm is one of the first proposed QSA in identifying the index number of the solution of a system, whose solutions are known to the oracle. A further improved QSA called Boyer-Brassard-Høyer-Tapp (BBHT) is followed by the Durr-Hoyer algorithm (DHA), which proposes to find the solutions, while the exact number of the solutions is not known. In this treatise, we propose two modifications to the existing DHA and BBHT algorithms. In DHA, we propose a semi-quantum framework, where after each iteration in DHA, the non-solution points are removed in classical framework. This makes the data set iteratively shorter. In the case of BBHT algorithm, we propose to choose the Grover search number based on the Gamma distribution instead of the uniform one, which gives more chances to the new larger values to improve probability of success. The performance of the proposed QSAs demonstrates plausible improvement in search success performance as compared to their existing BBHT and DHA counterparts. Numerical simulations are conducted with the proposed QSA algorithms on a MIMO orthogonal frequency division multiplexing (OFDM) system to demonstrate the performance improvement.

Complexity Measure for Engineering Systems Incorporating System States and Behavior

Complex engineering systems have structures characterized by a large number of components, interconnections, and elaborated configurations. Time-dependent functionality and multiple system states defy accurate predictions of complex systems behavior. Although many measures exist to quantify complexity, most of them consider either structure or function only. States and behavior are critical aspects to consider simultaneously with structure and function when assessing the complexity of modern systems as all of them have a profound impact on system design and future system performance. A more complete understanding of system complexity could improve the accuracy of scheduling and cost estimates during system development. This article proposes an objective, dynamic, and comprehensive complexity measure derived from molecular quantum mechanics and signal processing principles to quantify engineering system complexity. The proposed measure formalizes complexity as a single quantifiable system property that incorporates structure, functionality, and behavior, enabling unbiased comparisons of different conceptual designs, identification of complex elements, and analysis of complexity management. It provides systems architects and systems engineers with a strong basis for evaluating and communicating complexity during system design and architectural modeling. The formulation of the measure and methodology to calculate system complexity is illustrated through test cases using model-based systems engineering.

A Pearl Spectrometer.

Information recovery from incomplete measurements, typically performed by a numerical means, is beneficial in a variety of classical and quantum signal processing. Random and sparse sampling with nanophotonic and light scattering approaches has received attention to overcome the hardware limitations of conventional spectrometers and hyperspectral imagers but requires high-precision nanofabrications and bulky media. We report a simple spectral information processing scheme in which light transport through an Anderson-localized medium serves as an entropy source for compressive sampling directly in the frequency domain. As implied by the "lustrous" reflection originating from the exquisite multilayered nanostructures, a pearl (or mother-of-pearl) allows us to exploit the spatial and spectral intensity fluctuations originating from strong light localization for extracting salient spectral information with a compact and thin form factor. Pearl-inspired light localization in low-dimensional structures can offer an alternative of spectral information processing by hybridizing digital and physical properties at a material level.

Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimalO~(dκlog⁡(1/ϵ))query complexity for ad-sparse matrix, whereκis the condition number, andϵis the desired precision. Neither algorithm uses phase estimation or amplitude amplification.

Secure optical communication using a quantum alarm

Optical fibre networks are advancing rapidly to meet growing traffic demands. Security issues, including attack management, have become increasingly important for optical communication networks because of the vulnerabilities associated with tapping light from optical fibre links. Physical layer security often requires restricting access to channels and periodic inspections of link performance. In this paper, we report how quantum communication techniques can be utilized to detect a physical layer attack. We present an efficient method for monitoring the physical layer security of a high-data-rate classical optical communication network using a modulated continuous-variable quantum signal. We describe the theoretical and experimental underpinnings of this monitoring system and the monitoring accuracy for different monitored parameters. We analyse its performance for both unamplified and amplified optical links. The technique represents a novel approach for applying quantum signal processing to practical optical communication networks and compares well with classical monitoring methods. We conclude by discussing the challenges facing its practical application, its differences with respect to existing quantum key distribution methods, and its usage in future secure optical transport network planning.

Quantum signal processing for quantum phase estimation: Fourier transform versus maximum likelihood approaches

The phase in quantum states is an essential information carrier for quantum telecommunications, signal processing, and computation. Quantum phase estimation is therefore a fundamental operation to extract and control useful information at the quantum level. Here, we analyze various approaches to quantum phase estimation, when a phase parameter characterizing a quantum process gets imprinted in a relative phase attached to a quantum state serving as a probe signal. The estimation approaches are based on standard concepts of signal processing (Fourier transform, maximum likelihood), yet operated in the quantum realm. We also exploit the Fisher information, both in its classical and its quantum forms, in order to assess the performance of each approach to quantum phase estimation. We demonstrate a possibility of enhanced estimation performance, inaccessible classically, which is obtained via optimized quantum entanglement. Beyond their significance to quantum phase estimation, the results illustrate how standard concepts of signal processing can contribute to the ongoing developments in quantum information and quantum technologies.

Nonreciprocal Transport Based on Cavity Floquet Modes in Optomechanics

Directional transport is obtained in various multimode systems by driving multiple, nonreciprocally interfering interactions between individual bosonic modes. However, systems sustaining the required number of modes become physically complex. In our microwave-optomechanical experiment, we show how to configure nonreciprocal transport between frequency components of a single superconducting cavity coupled to two drumhead oscillators. The frequency components are promoted to Floquet modes and generate the missing dimension to realize an isolator and a directional amplifier. A second cavity left free by this arrangement is used to cool the mechanical oscillators and bring the transduction noise close to the quantum limit. We furthermore uncover a new type of instability specific to nonreciprocal coupling. Our approach is generic and can greatly simplify quantum signal processing and the design of topological lattices from low-dimensional systems.

Signal processing techniques for efficient compilation of controlled rotations in trapped ions

Quantum logic gates with many control qubits are essential in many quantum algorithms, but remain challenging to perform in current experiments. Trapped ion quantum computers natively feature the Mølmer–Sørensen (MS) entangling operation, which effectively applies an Ising interaction to all pairs of qubits at the same time. We consider a sequence of equal all-to-all MS operations, interleaved with single-qubit gates that act only on one special qubit. Using a connection with quantum signal processing techniques, we find that it is possible to perform an arbitray SU(2) rotation on the special qubit if and only if all other qubits are in the state . Such controlled rotation gates with N − 1 control qubits require 2N applications of the MS gate, and can be mapped to a conventional Toffoli gate by demoting a single qubit to ancilla.