Modular exponentiation (ME) operators are one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. I propose a method for constructing the ME operators that relies upon the simple observation that the work register starts in state $\vert 1 \rangle$. Therefore, we do not have to create an ME operator $U$ that accepts a general input, but rather, one that takes an input from the periodic sequence of states $\vert f(x) \rangle$ for $x \in \{0, 1, \cdots, r-1\}$, where $f(x)$ is the ME function with period $r$. The operator $U$ can be partitioned into $r$ levels, where the gates in level $x \in \{0, 1, \cdots, r-1\}$ increment the state $\vert f(x) \rangle$ to the state $\vert f(x+1) \rangle$. The gates below $x$ do not affect the state $\vert f(x+1) \rangle$. The obvious problem with this method is that it is self-defeating: If we knew the operator $U$, then we would know the period $r$ of the ME function, and there would be no need for Shor's algorithm. I show, however, that the ME operators are very forgiving, and truncated approximate forms in which levels have been omitted are able to extract factors just as well as the exact operators. I demonstrate this by factoring the numbers $N = 21, 33, 35, 143, 247$ by using less than half the requisite number of levels in the ME operators. This procedure works because the method of continued fractions only requires an approximate phase value. This is the basis for a factorization strategy in which we fill the circuits for the ME operators with more and more gates, and the correlations between the various composite operators $U^p$ (where $p$ is a power of two) compensate for the missing levels.

We provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected.Quanta 2023; 12: 41–130.

We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover's algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary {\em steering operators}. We also outline a generalized Grover's algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform sample of computational basis elements when performing the phase selection and diffusion. In contrast, steered operators bias the selection process, thereby providing more flexibility in selecting the target state. Our method is a generalization of the recently proposal pattern matching algorithm of Hiroyuki et al..

This is the third in a series of lectures on the technique of dimensional continuation, employed by Brown, Preston and Singleton (BPS), for calculating Coulomb energy exchange rates in a plasma. Two important examples of such processes are the charged particle stopping power and the temperature equilibration rate between different plasma species. The first lecture was devoted to understanding the machinery of dimensional continuation, and the second concentrated on calculating the electron-ion temperature equilibration rate in the extreme quantum limit. In this lecture, I will examine one of the main theoretical underpinnings of the BPS theory, namely, the dimensional reduction of the BBGKY hierarchy. I will prove that to leading order in the plasma coupling $g$, the BBGKY hierarchy reduces to the BE for dimensions greater then three and to the LBE for dimensions less than three. We must eventually return to three dimensions, and the BPS formalism shows that the simple poles associated with the BE and the LBE exactly cancel, rendering the limit in three dimensions finite. Furthermore, the leading order behavior of the LBE becomes next-to-leading order when the dimensions is analytically continued from less than three to greater than three. This provides the leading and next-to-leading order terms in $g$ exactly, which is equivalent to an exact calculation of the so-called Coulomb logarithm with no use of an integral cut-off. In this way, BPS takes all Coulomb interactions into account to leading and next-to-leading order in $g$.

Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry and statistics. Traditional solution methods such as Gaussian elimination become very time consuming for large matrices, and more efficient computational methods are desired. In the twilight of Moore's Law, quantum computing is perhaps the most direct path out of the darkness. There are two complementary paradigms for quantum computing, namely, gated systems and quantum annealers. In this paper, we express floating point operations such as division and matrix inversion in terms of a quadratic unconstrained binary optimization (QUBO) problem, a formulation that is ideal for a quantum annealer. We first address floating point division, and then move on to matrix inversion. We provide a general algorithm for any number of dimensions, but we provide results from the D-Wave quantum anneler for $2\times 2$ and $3 \times 3$ general matrices. Our algorithm scales to very large numbers of linear equations. We should also mention that our algorithm provides the full solution the the matrix problem, while HHL provides only an expectation value.

The classic Noh verification test problem is extended beyond the traditional ideal gas and applied to shock compression of condensed matter. Using the stiff-gas equation of state (EOS), which admits an exact analytical solution for the planar Noh problem, we examine the shock compression of Al, Fe, Cu, and W. Analytical EOS predictions for the jump in density and the location of the shock are compared to numerical results obtained using the same EOS within Los Alamos compressible-flow codes Flag and xRage. Excellent agreement between the numerical and exact results is observed. Both codes exhibit first-order spatial convergence with increasing mesh resolution.

The alpha-particle energy deposition mechanism modifies the ignition conditions of the thermonuclear Deuterium-Tritium fusion reactions, and constitutes a key issue in achieving high gain in Inertial Confinement Fusion implosions. One-dimensional hydrodynamic calculations have been performed with the code Multi-IFE [R. Ramis, J. Meyer-ter-Vehn, Comput. Phys. Commun. 203, 226 (2016)] to simulate the implosion of a capsule directly irradiated by a laser beam. The diffusion approximation for the alpha energy deposition has been used to optimize three laser profiles corresponding to different implosion velocities. A Monte-Carlo package has been included in Multi-IFE to calculate the alpha energy transport, and in this case the energy deposition uses both the LP [C.K. Li, R.D. Petrasso, Phys. Rev. Lett. 70, 3059 (1993)] and the BPS [L.S. Brown, D.L. Preston, R.L. Singleton Jr., Phys. Rep. 410, 237 (2005)] stopping power models. Homothetic transformations that maintain a constant implosion velocity have been used to map out the transition region between marginally-igniting and high-gain configurations. The results provided by the two models have been compared and it is found that – close to the ignition threshold – in order to produce the same fusion energy, the calculations performed with the BPS model require about 10% more invested energy with respect to the LP model.

The Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan-Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which in combination with the Rankine-Hugoniot jump conditions provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case the solution can be applied to shocks of arbitrary strength, but for cylindrical and spherical geometries it is required that the analysis be restricted to strong shocks. The exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode FLAG.

The preceding paper [Brown, Preston, and Singleton Jr., Phys. Rev. E 86, 016406 (2012)] presented precise results for the partition of the initial energy E(0) of a fast particle into the ions and electrons--E(I)/E(0) and E(e)/E(0)--when the fast particle slows in a plasma whose ion and electron temperatures may differ. As emphasized in that paper, this is an important problem because nuclear fusion reactions, such as those that occur in an inertial confinement fusion capsule, involve ion temperatures that run away from the electron temperatures. As also noted in the preceding paper, a precise evaluation entails the use of a well-defined Fokker-Planck equation for the phase-space evolution of initially fast projectile particles. When the plasma has differing ion and electron temperatures, the projectiles must slow into a "schizophrenic" final ensemble of particles that has neither the electron nor the ion temperature. This is not a simple Maxwell-Boltzmann distribution since the electrons are not in thermal equilibrium with the ions. Thus, detailed calculations are required for the solution of the problem. These we provide here for a weakly to moderately coupled plasma. The Fokker-Planck equation holds to first subleading order in the dimensionless plasma coupling constant, which translates to computing to order n ln n (leading) and n (subleading) in the plasma density n. The energy partitions for a background plasma in thermal equilibrium have been previously computed, but the order n terms have not been calculated, only estimated. The "schizophrenic" final ensemble of slowed particles gives a new mechanism to bring the electron and ion temperatures together. The rate at which this new mechanism brings the electrons and ions in the plasma into thermal equilibrium will be computed.

Brown, Preston, and Singleton (BPS) produced an analytic calculation for energy exchange processes for a weakly to moderately coupled plasma: the electron-ion temperature equilibration rate and the charged particle stopping power. These precise calculations are accurate to leading and next-to-leading order in the plasma coupling parameter and to all orders for two-body quantum scattering within the plasma. Classical molecular dynamics can provide another approach that can be rigorously implemented. It is therefore useful to compare the predictions from these two methods, particularly since the former is theoretically based and the latter numerically. An agreement would provide both confidence in our theoretical machinery and in the reliability of the computer simulations. The comparisons can be made cleanly in the purely classical regime, thereby avoiding the arbitrariness associated with constructing effective potentials to mock up quantum effects. We present here the classical limit of the general result for the temperature equilibration rate presented in BPS. In particular, we examine the validity of the m(electron)/m(ion)-> 0 limit used in BPS to obtain a very simple analytic evaluation of the long-distance collective effects in the background plasma.

Brown, Preston, and Singleton (BPS) produced an analytic calculation for energy exchange pro- cesses for a weakly to moderately coupled plasma: the electron-ion temperature equilibration rate and the charged particle stopping power. These precise calculations are accurate to leading and next-to-leading order in the plasma coupling parameter, and to all orders for two-body quantum scattering within the plasma. Classical molecular dynamics can provide another approach that can be rigorously implemented. It is therefore useful to compare the predictions from these two meth- ods, particularly since the former is theoretically based and the latter numerically. An agreement would provide both confidence in our theoretical machinery and in the reliability of the computer simulations. The comparisons can be made cleanly in the purely classical regime, thereby avoiding the arbitrariness associated with constructing effective potentials to mock up quantum effects. We present here the classical limit of the general result for the temperature equilibration rate pre- sented in BPS. We examine the validity of the melectron/mion → 0 limit used in BPS to obtain a very simple analytic evaluation of the long-distance, collective effects in the background plasma.

The charged particle stopping power in a highly ionized and weakly to moderately coupled plasma has been calculated exactly to leading and next-to-leading accuracy in the plasma density by Brown, Preston and Singleton (BPS). Since the calculational techniques of BPS might be unfamiliar to some, and since the same methodology can also be used for other energy transport phenomena, we will review the main ideas behind the calculation. BPS used their stopping power calculation to derive a Fokker-Planck equation, also accurate to leading and next-to-leading orders, and we will also review this. We use this Fokker-Planck equation to compute the electron-ion energy partitioning of a charged particle traversing a plasma. The motivation for this application is ignition for inertial confinement fusion-more energy delivered to the ions means a better chance of ignition, and conversely. It is therefore important to calculate the fractional energy loss to electrons and ions as accurately as possible. One method by which one calculates the electron-ion energy splitting of a charged particle traversing a plasma involves integrating the stopping power dE/dx. However, as the charged particle slows down and becomes thermalized into the background plasma, this method of calculating the electron-ion energy splitting breaks down. As a result, it suffers a systematic error that may be as large as T/E-0, where T is the plasma temperature and E-0 is the initial energy of the charged particle. The formalism presented here is designed to account for the thermalization process and it provides results that are near-exact.

I will discuss a new method for calculating transport quantities, such as the charged particle stopping power, in a weakly to moderately coupled plasma. This method, called dimensional continuation, lies within the framework of convergent kinetic equations, and it is powerful enough to allow for systematic perturbative expansions in the plasma coupling constant. Inparticular, it provides an exact evaluation of the stopping power to leading and next-to-leading order in the plasma coupling, with the systematic error being of cubic order. Consequently, the calculation is near-exact for a weakly coupled plasma, and quite accurate for a moderately coupled plasma. The leading order term in this expansion has been known since the classic work of Spitzer. In contrast, the next-to-leading order term has been calculated only recently by Brown,Preston, and Singleton (BPS), using the aforementioned method, to account for all short-and long-distance physics accurate to second order in the plasma coupling, including an exact treatment of the quantum-to-classical scattering transition. Under conditions relevant for inertial confinement fusion, BPS find the alpha particle range in the DT plasma to be about 30% longer than typical model predictions in the literature. Preliminary numerical studies suggest that this renders the ignition threshold proportionally higher, thereby having potential adverse implications for upcoming high energy density facilities. Since the key ideas behind the BPS calculation are possibly unfamiliar to plasma physicists, and the implications might be important, I will use this opportunity to explain the method in a pedagogical fashion.

A completely rigorous first-principles calculation of the charged particle stopping power has recently been performed by Brown, Preston, and Singleton (BPS). This calculation is exact to leading and next-to-leading order in the plasma number density, including an exact treatment of two-body quantum scattering. The BPS calculation is therefore extremely accurate in the plasma regime realized during the ignition and burn of an inertial confinement fusion capsule. For deuterium-tritium fusion, the 3.5MeV alpha particle range tends to be 20%–30% longer than most models in the literature have predicted, and the energy deposition into the ions tends to be smaller. Preliminary numerical simulations indicate that this increases the ρR required to achieve ignition.

This is the first of two lectures on a new powerful technique employed by Brown, Preston, and Singleton (BPS) to calculate transport quantities in a plasma. This exposition will be self-contained and intended for those who are not specialists in quantum field theory, where the technique of dimensional continuation exploited by BPS first arose. The method is highlighted by the interesting analogy between the Coulomb logarithm of Lyman Spitzer on the one hand, and the Lamb shift as calculated by Hans Bethe on the other. BPS employed the method in a novel way that provides the leading and subleading behavior for processes with competing disparate length or energy scales. They calculated the temperature equilibration rate to leading and next-to-leading order in the plasma number density for any two species in a plasma that are in thermal equilibrium with themselves, but not necessarily with each other. A special case of this calculation is the electron-ion temperature equilibration rate. It should be emphasized that the BPS result is not a model, but rather it is an exact calculation of the leading terms in a well-defined perturbation theory. This exact result differs from approximations and models given in the literature.

We calculate analytically the electron-ion temperature equilibration rate in a fully ionized, weakly to moderately coupled plasma, using an exact treatment of the Fermi-Dirac electrons. The temperature is sufficiently high so that the quantum-mechanical Born approximation to the scattering is valid. It should be emphasized that we do not build a model of the energy exchange mechanism, but rather, we perform a systematic first principles calculation of the energy exchange. At the heart of this calculation lies the method of dimensional continuation, a technique that we borrow from quantum field theory and use in a different fashion to regulate the kinetic equations in a consistent manner. We can then perform a systematic perturbation expansion and thereby obtain a finite first-principles result to leading and next-to-leading order. Unlike model building, this systematic calculation yields an estimate of its own error and thus prescribes its domain of applicability. The calculational error is small for a weakly to moderately coupled plasma, for which our result is nearly exact. It should also be emphasized that our calculation becomes unreliable for a strongly coupled plasma, where the perturbative expansion that we employ breaks down, and one must then utilize model building and computer simulations. Besides providing different and potentially useful results, we use this calculation as an opportunity to explain the method of dimensional continuation in a pedagogical fashion. Interestingly, in the regime of relevance for many inertial confinement fusion experiments, the degeneracy corrections are comparable in size to the subleading quantum correction below the Born approximation. For consistency, we therefore present this subleading quantum-to-classical transition correction in addition to the degeneracy correction.

This is the second in a series of two lectures on the technique of dimensional continuation, a new method for analytically calculating certain energy transport quantities in a weakly to moderately coupled plasma. Recently, this method was employed by Brown, Preston, and Singleton (BPS) to calculate the electron-ion temperature equilibration rate and the charged particle stopping power to leading and next-to-leading order in the plasma coupling. In this lecture, I develop the framework further, and then explicitly calculate the electron-ion temperature equilibration rate in the high temperature limit. This method captures all short and long distance physics to second order in the plasma coupling. This analytic perturbative technique is applicable for ignition in inertial confinement fusion and for other processes in hot a weakly coupled plasma.

Dimensional continuation is employed to compute the energy loss rate for a non-relativistic particle moving through a highly ionized plasma. No restriction is made on the charge, mass, or speed of this particle, but it is assumed that the plasma is not strongly coupled in that the dimensionless plasma coupling parameter g = e(2)kappa(D)/4 pi T is small, where kappa(D) is the Debye wave number. To leading order in this coupling, dE/dx is of the generic form g(2) ln[g(2)C]. The prefactor of the logarithm is well known. We compute the constant C under the logarithm exactly. Our result differs from approximations given in the literature, with differences in the range of about 20% for cases relevant to inertial confinement fusion experiments.