Quantum signal processing (QSP)

Quantum technology (QT) applies quantum mechanical properties such as quantum entanglement, quantum superposition, and No-cloning theorem to quantum systems such as atoms, ions, electrons, photons, or molecules. Quantum bit is the basic unit of quantum information.  Whereas in a classical system, a bit is either in one state or the another. However, quantum qubits can exist in large number of states simultaneously,  property called  Superposition. Quantum entanglement is a phenomenon where entangled particles can stay connected in the sense that the actions performed on one of the particles affects the other no matter what’s the distance between them. No-cloning theorem tells us that quantum information (qubit) cannot be copied.

Quantum technology has many Quantum applications, one of the major class is Quantum computation and simulation. Quantum computers shall bring the power of massive parallel processing, the equivalent of a supercomputer to a single chip. They can simultaneously consider different possible solutions to a problem and quickly converge on the correct solution without checking each possibility individually. This dramatically speeds up certain calculations, such as number factoring.

Since the quantum computers works differently than classical computers therefore, they require different software approach and different quantum algorithms. In quantum computing, a quantum algorithm is an algorithm that runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each step can be performed on a quantum computer.

Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a quantum circuit which acts on some input qubits and terminates with a measurement. A quantum circuit consists of simple quantum gates which act on at most a fixed number of qubits. The number of qubits has to be fixed because a changing number of qubits implies non-unitary evolution. Quantum algorithms may also be stated in other models of quantum computation, such as the Hamiltonian oracle model.

Processing information by manipulating quantum mechanical degrees of freedom has been a holy grail of
quantum science since the design of the first quantum algorithms that utilize superposition and entanglement. (QSP)  is aimed at developing new or modifying existing signal processing algorithms by borrowing from the principles of quantum mechanics and some of its interesting axioms and constraints.

Digital Signal Processing (DSP) take real-world signals like voice, audio, video, temperature, pressure, or position that have been digitized and then mathematically manipulate them. Signals need to be processed so that the information that they contain can be displayed, analyzed, or converted to another type of signal that may be of use. Although real-world signals can be processed in their analog form, processing signals digitally provides the advantages of high speed and accuracy. DSP technology uses specially designed programs and algorithms to manipulate analog signals and produce a signal that is higher-quality, less prone to degradation or easier to transmit. This typically requires the DSP to perform a large number of simple mathematical functions (addition, subtraction, multiplication, division, and the like) within a fixed or constrained time frame.

Many new classes of signal processing algorithms have been developed by emulating the behavior of physical
systems. There are also many examples in the signal processing literature in which new classes of algorithms have
been developed by artificially imposing physical constraints on implementations that are not inherently subject to these constraints.

Among the many well-known examples of digital signal processing algorithms that are derived by using physical phenomena and constraints as metaphors are wave digital filters. This class of filters relies on emulating the physical constraints of passivity and energy conservation associated with analog filters to achieve low sensitivity in coefficient variations in digital filters. As another class of examples, the fractal-like aspects of nature  and related modeling have inspired interesting signal processing paradigms that are not constrained by the associated physics. These include fractal modulation, which emulates the fractal characteristic of nature for communicating over a particular class of unreliable channels, and the various approaches to image compression based on synthetic
generation of fractals.

Likewise, the chaotic behavior of certain features of nature have inspired new classes of signals for secure
communications, remote sensing, and a variety of other signal processing applications. Other examples of algorithms using  physical systems as a simile include solitons, genetic algorithms, simulated annealing, and
neural networks.

Three fundamental interrelated underlying principles of quantum mechanics that play a major role in QSP as
presented here are the concept of measurement, the principle of measurement consistency, and the principle
of quantization of the measurement output. In addition, when using quantum systems in a communication context, other principles arise such as inner product constraints. QSP is based on exploiting these various principles and constraints in the context of signal processing algorithms.

In developing the QSP framework, it is convenient to discuss both signal processing and quantum mechanics in a
vector space setting. Specifically, we consider an arbitrary Hilbert (inner product) space H with inner product 〈x y 〉 for any vectors x y, in H. Typically we will refer to elements of H as vectors or signals interchangeably, explain Yonina C. Eldar and Alan V. Oppenheim.

Measurement consistency is a fundamental postulate of quantum mechanics, i.e., repeated measurements on a
system must yield the same outcomes; otherwise, we would not be able to confirm the output of a measurement. Therefore the state of the system after a measurement must be such that if we instantaneously remeasure
the system in this state, then the final state after this second measurement will be identical to the state after the
first measurement.

Quantization of the measurement outcome is a direct consequence of the consistency requirement. Specifically,
the consistency requirement leads to a class of states referred to as determinate states of the measurement.
These are states of the quantum system for which the measurement yields a known outcome with probability
one and are the states that lie completely in one of the measurement subspaces S. Furthermore, even when the
state of the system is not one of the determinate states, after performing the measurement the system is quantized
to one of these states, i.e., is certain to be in one of these states, where the probability of being in a particular determinate state is a function of the inner products between the state of the system and the determinate states.

Quantum Detection

The constraints imposed by physics on a quantum measurement lead to some interesting problems within
the framework of quantum mechanics. In particular, an interesting problem that arises when using quantum
states for communication is a quantum detection problem.

In a quantum detection problem a sender conveys classical information to a receiver using a quantum-mechanical channel. The sender represents messages by preparing the quantum channel in a pure quantum state drawn from
a collection of known states φ i . The receiver detects the information by subjecting the channel to a quantum measurement with measurement vectors µi that are constrained by the physics to be orthogonal. If the states are
not orthogonal, then no measurement can distinguish perfectly between them. In the general context of quantum mechanics, a fundamental problem is to construct measurements optimized to distinguish between a set of
nonorthogonal pure quantum states. In the context of a communications system, we would like to choose the
measurement vectors to minimize the probability of detection error. In this context, this problem is commonly
referred to as the quantum detection problem.

Specifically, the measurement vectors µi are chosen to be orthogonal and closest in a least-squares (LS) sense to the given set of state vectors φi so that the vectors µi are chosen to minimize the sum of the squared norms of the error vectors. The optimal measurement is referred to as the LS measurement.

Quantum Signal Processing (QSP)

In quantum mechanics, systems are “processed” by performing measurements on them. In signal processing,
signals are processed by applying an algorithm to them.

These algorithms promise a profound transformation of our current computational capabilities, by reducing the
space and time required to accomplish important computational tasks, such as phase estimation, unstructured
search, and Hamiltonian simulation.

Quantum signal processing is a framework for quantum algorithms including Hamiltonian simulation, quantum linear system solving, amplitude amplification, etc. Quantum signal processing (QSP) was originally developed as a technique to design real-variable functions from single-qubit rotations. It has attracted a lot of attention
in the last few years after the discovery that it can be extended to realize also functions of operators, initially only
for Hermitian operators  and culminating in the general formalism of quantum singular value transformations
(QSVT)

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood.

Quantum signal processing performs spectral transformation of any unitary U, given access to an ancilla qubit, a controlled version of U and single-qubit rotations on the ancilla qubit. It first truncates an arbitrary spectral transformation function into a Laurent polynomial, then finds a way to decompose the Laurent polynomial into a sequence of products of controlled-U and single qubit rotations (by certain “QSP phase angles”) on the ancilla. Such routines achieve optimal gate complexity for many of the quantum algorithmic tasks mentioned above. The task achieved is essentially entirely defined by the QSP phase angles employed in the QSP operation sequence, and as such a central part is finding these QSP phase angles, given the desired Laurent polynomial.

References and Resources also include:

https://www.weizmann.ac.il/math/yonina/sites/math.yonina/files/quantum%20signal%20processing.pdf

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