Mathematics and physics at the quantum-classical interface
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general physics↗
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Quantum Resonance for Solving NP-Complete Problems by Simulations
In this paper, the mathematical formalism of quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems.
Quantum computing N-P Complete↗
Resolving Wave-Particle Duality Could Accelerate the Mass Production of Quantum Computers
Quantum computers, hypothesized in 1980s, use concepts of superposition and entanglement phenomena. Although theoretical propositions and associated search algorithms for accurate measurements are being generated, the development of practical quantum computers themselves are advancing very slowly requiring enormous time and investments. The underlying concepts of a quantum computer are not new to the optical domain. However, the crucial enabling concepts of Entanglement and Superposition Principle are remaining clouded under the unresolved postulates, Wave-Particle Duality (WPD), and Wave Packet Reduction (WPR), implicating incompleteness in the interpretations of the mathematical formalism behind Quantum Mechanics. The WPD debate started during late1600 between Newton and Huygens. Young’s resolution of WPD through his double-slit experiment in 1802 was effectively overturned by Einstein’s interpretation of photoelectric effect as due to “indivisible light quanta”. However, Einstein disowned his “light quanta” postulate shortly before his death in1955, even though it had earned him the Nobel Prize. We resolve WPD by synthesizing Newton’s and Maxwell’s concepts and assume atoms do emit quanta but propagate as time-finite exponential pulses. This assumption also resolves WPR for light-matter interaction with the assumption that Schrodinger’s ψ represents atom’s internal dipolar amplitude stimulations. This over-turns Born’s interpretation that ψ only represents the abstract mathematical probability amplitude, rather than the physical “internal amplitude stimulation” of the quantum entity. However, our concept of atomic pulse emission forces us to re-derive the expression for the N-slit grating-spectrometer response since the classical derivation uses CW light, which does not exist. This pulsespectrometric response function strengthens our postulate since the grating response to the exponential pulse appears to be the convolution of a Lorentzian spectrum with the classical CW response function of the grating. The Fourier Transform of an exponential function is Lorentzian and QM predicts spontaneous emission line width to be Lorentzian. Then, conceptually one can extend the grating-expression (with N=2) to get the double-slit pattern. This approach preserves the classical causality that each of the two slits, like the N-signals out of a grating, are physically real and jointly stimulate the quantum detector array at the far field to generate the “Local” cosine fringes. The detector array executes the square modulus operation on its imposed dipolar amplitude stimulation and absorbs the necessary energy to fill up their quantum cups. Hence the double-slit pattern must also be “Local”, just as the N-slit grating spectrum is generated locally at the exit spectral-plane of the spectrometer. This removes the need to believe that “single photons” mysteriously generate the double slit pattern. Quantum computers, hypothesized in 1980s, use concepts of superposition and entanglement phenomena. Although theoretical propositions and associated search algorithms for accurate measurements are being generated, the development of practical quantum computers themselves are advancing very slowly requiring enormous time and investments. The underlying concepts of a quantum computer are not new to the optical domain. However, the crucial enabling concepts of Entanglement and Superposition Principle are remaining clouded under the unresolved postulates, Wave-Particle Duality (WPD), and Wave Packet Reduction (WPR), implicating incompleteness in the interpretations of the mathematical formalism behind Quantum Mechanics. The WPD debate started during late1600 between Newton and Huygens. Young’s resolution of WPD through his double-slit experiment in 1802 was effectively overturned by Einstein’s interpretation of photoelectric effect as due to “indivisible light quanta”. However, Einstein disowned his “light quanta” postulate shortly before his death in1955, even though it had earned him the Nobel Prize. We resolve WPD by synthesizing Newton’s and Maxwell’s concepts and assume atoms do emit quanta but propagate as time-finite exponential pulses. This assumption also resolves WPR for light-matter interaction with the assumption that Schrodinger’s ψ represents atom’s internal dipolar amplitude stimulations. This over-turns Born’s interpretation that ψ only represents the abstract mathematical probability amplitude, rather than the physical “internal amplitude stimulation” of the quantum entity. However, our concept of atomic pulse emission forces us to re-derive the expression for the N-slit grating-spectrometer response since the classical derivation uses CW light, which does not exist. This pulsespectrometric response function strengthens our postulate since the grating response to the exponential pulse appears to be the convolution of a Lorentzian spectrum with the classical CW response function of the grating. The Fourier Transform of an exponential function is Lorentzian and QM predicts spontaneous emission line width to be Lorentzian. Then, conceptually one can extend the grating-expression (with N=2) to get the double-slit pattern. This approach preserves the classical causality that each of the two slits, like the N-signals out of a grating, are physically real and jointly stimulate the quantum detector array at the far field to generate the “Local” cosine fringes. The detector array executes the square modulus operation on its imposed dipolar amplitude stimulation and absorbs the necessary energy to fill up their quantum cups. Hence the double-slit pattern must also be “Local”, just as the N-slit grating spectrum is generated locally at the exit spectral-plane of the spectrometer. This removes the need to believe that “single photons” mysteriously generate the double slit pattern.
Quantum Computer↗
A retrospective review of von Neumann’s analysis of hidden variables in quantum mechanics
This article reviews the history of J. von Neumann’s analysis of hidden variables in quantum mechanics and the subsequent analysis by others. In his book The Mathematical Foundations of Quantum Mechanics , published in 1932, von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He arrived at two principal conclusions: first, hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and second, if they did exist, the theory would have already failed in situations where it has been successfully applied. This analysis has been taken as an “incorrect proof” against the existence of hidden variables, possibly due to a mistranslation of the German word prufen . von Neumann’s so-called proof isn’t even wrong as such a proof does not exist, but it is an examination of the limitations imposed by internal consistency of the Hilbert space formulation of the theory. One of the earliest attempts to eliminate uncertainty, by D. Bohm, requires a major modification of quantum mechanics (observables are not represented by Hermitian operators), which supports von Neumann’s first principal conclusion. However, testing the Bohm theory requires constructing a physically impossible initial state. As such, the theory has no experimental consequences, so W. Pauli referred to it as an “uncashable check”. As there are no observable consequences, the Bohm theory is possibly a counterexample to von Neumann’s second conclusion that hidden variables in particular would have already led to a failure of the theory.
density matrix↗
Using Complex Probability Amplitudes to Simulate Solute Transport in Composite Porous Media
Probability amplitudes are fundamental to quantum mechanics and offer robust descriptions of complicated systems, which have allowed physicists to explain behaviors inaccessible to classical physics. This article ponders how some of the same conceptual underpinnings of the mathematics used for modeling quantum systems might be applied to subsurface water resources problems and speculates how these tools could facilitate applications on quantum computers. A probability amplitude-based model for describing advective-dispersive transport in porous media using linear operators is investigated. The proposed complex valued model decomposes spreading into two “sub-continuum partial dispersion” coefficients, and this recovers classical spreading when the sum of these coefficients is the Fickian dispersion coefficient. However, the probability amplitudes have a manyto-one relationship to a probability distribution, so it embeds a level of heterogeneity into seemingly equivalent functions. Two propagators with different sub-continuum coefficients may have the same macroscopic behavior when either is considered in isolation, but when they act on the other the system’s behavior changes. Additionally, differences in the amplitudes cause a reduction in spreading as velocity correlations are disrupted, despite both propagators having identical dispersion coefficients, and this cannot be achieved using classical methods without changing the dispersion coefficient. The main point is that these amplitude-based models offer a way to embed information about the system into the propagators, instead of just “averaging it out” when making an upscaled model.
54 ENVIRONMENTAL SCIENCES↗
Quantum-inspired weight-constrained neural network: Reducing variable numbers by 100× compared to standard neural networks
Although quantum machine learning has shown great promise, the practical application of quantum computers remains constrained in the noisy intermediate-scale quantum era. To take advantage of quantum machine learning, we investigate the underlying mathematical principles of these quantum models and find that the quantum neural network with amplitude encoding is equivalent to a weight-constrained neural network. Motivated by this discovery, we develop a classical weight-constrained neural network. We find that this approach can reduce the number of variables in a classical neural network by a factor of 135 while preserving its accuracy. In addition, we develop a dropout method to enhance the robustness of quantum machine learning models, which are highly susceptible to adversarial attacks. This technique can also be applied to improve the adversarial robustness of the classical weight-constrained neural network, which is essential for industry applications, such as self-driving vehicles. Our work offers an approach to reduce the complexity of large classical neural networks, addressing a critical challenge in machine learning.
quantum algorithms & computation↗
Quantum mechanical closure of partial differential equations with symmetries
We develop a statistical framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. Employing the mathematical framework of quantum mechanics to embed the original classical dynamics into a quantum mechanical representation, we use the space of quantum density operators to model the unresolved degrees of freedom of the original dynamics in a statistical sense, and the framework of quantum measurement to predict their contributions to the resolved dynamics. The embedded dynamics is discretized by a positivity preserving process, leading to a compressed representation that is invariant under the dynamical symmetries of the resolved dynamics. We present a data based formulation of the closure scheme and apply it to a closure problem for the shallow water equations. The numerical results demonstrate that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.
Delay embedding↗
Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons
Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that—when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)—these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer’s records. However, our “let quantum be quantum” approach poses questions absent in Bohr’s Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply “too relative”, and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born’s rule, p k =|ψ k | 2 , relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection—environment—induced superselection)—are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the ‘quantum theory of the classical’ one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability—hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability—the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states—(ii) entanglement—assisted invariance or envariance—implies Born’s rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool—reduced density matrices. A simple and manifestly noncircular derivation of p k =|ψ k | 2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness—to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons—composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Interligand Coupling Drives Fast Triplet Energy Transfer Routes in PbS/Tetracene Quantum Dot Hybrids
The binding of photoactive organic ligands to inorganic quantum dots (QDs) creates a versatile hybrid architecture that allows access to photophysical processes such as efficient triplet exciton generation with near-infrared radiation. Here we report the subnanosecond generation of a hybrid triplet state with mixed ligand-QD character by replacing native oleate ligands on small PbS QDs with 5,12-tetracenepropiolic acid, a bifunctional ligand with two carboxylic acids that tends to lie face-on with the QD surface at low loadings. The face-on geometry engenders a regime of strong electronic coupling that is evident in steady-state absorption and hastens triplet energy flow by several orders of magnitude compared with more typical tetracene-based ligands exhibiting weak coupling. We further determined via Fourier transform infrared (FTIR) and supported by density functional theory (DFT)-based geometry optimizations that high ligand loading causes a shift in QD-ligand mutual disposition toward an edge-on geometry that instigates the formation of intermolecular excited states characterized by triplet excimer-like features in photoluminescence and transient absorption. Our results demonstrate the ability to control strongly coupled ligand-QD systems toward ultrafast generation of photophysically relevant species such as triplets that are valuable for photon upconversion and catalysis.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Theory of ultrasound absorption by free electrons in a metal in a quantizing magnetic field. 1
Mathematical model of quantum oscillations of ultrasonic absorption coefficients in quantizing magnetic fields in approximation of relaxation time
Filippov, B. N.↗
Variational fast forwarding for quantum simulation beyond the coherence time
Abstract Trotterization-based, iterative approaches to quantum simulation (QS) are restricted to simulation times less than the coherence time of the quantum computer (QC), which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called variational fast forwarding (VFF), for decreasing the quantum circuit depth of QSs. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function’s operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. In addition, we implement VFF on Rigetti’s QC to demonstrate simulation beyond the coherence time. Finally, we show how to estimate energy eigenvalues using VFF.
97 MATHEMATICS AND COMPUTING↗
New formalism for two-photon quantum optics. I - Quadrature phases and squeezed states. II - Mathematical foundation and compact notation
A new formalism for analyzing two-photon devices, such as parametric amplifiers and phase-conjugate mirrors, is proposed in part I, focusing on the properties and the significance of the quadrature-phase amplitudes and two-mode squeezed states. Time-stationary quasi-probability noise is also detailed for the case of Gaussian noise, and uncertainty principles for the quadrature-phase amplitudes are outlined, as well as some important properties of the two-mode states. Part II establishes a mathematical foundation for the formalism, with introduction of a vector notation for compact representation of two-mode properties. Fundamental unitary operators and special quantum states are also examined with an emphasis on the two-mode squeezed states. The results are applied to a previously studied degenerate limit (epsilon = 0).
Caves, C. M.↗
Quantum electron transport in degenerate donor–acceptor systems
In this work, we develop a mathematically rigorous theory for the quantum transfer processes in degenerate donor–acceptor dimers in contact with a thermal environment. We explicitly calculate the transfer rates and the acceptor population efficiency. The latter depends critically on the initial donor state. Additionally, we show that quantum coherence in the initial state enhances the transfer process. If the electron is initially shared coherently by the donor levels, then the efficiency can reach values close to 100%, while an incoherent initial donor state will significantly suppress the efficiency. The results are useful for a better understanding of the quantum electron transport in many chemical, solid state, and biological systems with complex degenerate and quasi-degenerate energy landscapes.
36 MATERIALS SCIENCE↗
Quantum simulation algorithms that optimally exploit problem structure [Slides]
The overarching goal is to improve quantum algorithms for the simulation of quantum systems.
97 MATHEMATICS AND COMPUTING↗
Wavelets and the squeezed states of quantum optics
Wavelets are new mathematical objects which act as 'designer trigonometric functions.' To obtain a wavelet, the original function space of finite energy signals is generalized to a phase-space, and the translation operator in the original space has a scale change in the new variable adjoined to the translation. Localization properties in the phase-space can be improved and unconditional bases are obtained for a broad class of function and distribution spaces. Operators in phase space are 'almost diagonal' instead of the traditional condition of being diagonal in the original function space. These wavelets are applied to the squeezed states of quantum optics. The scale change required for a quantum wavelet is shown to be a Yuen squeeze operator acting on an arbitrary density operator.
Defacio, B.↗
Unitary Coupled Cluster: Seizing the Quantum Moment
Not Available
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Theoretical guarantees for permutation-equivariant quantum neural networks
Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., symmetry group $S_n$), and show how to build $S_n$-equivariant QNNs We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees for equivariant QNNs, thus indicating the power and potential of GQML.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗