To account for the phenomenon of quantum decoherence of a macroscopic object, such as the localization and disappearance of interference, we invoke the adiabatic quantum entanglement between its collective states(such as that of the center-of-mass (C.M)) and its inner states based on our recent investigation. Under the adiabatic limit that motion of C.M dose not excite the transition of inner states, it is shown that the wave function of the macroscopic object can be written as an entangled state with correlation between adiabatic inner states and quasi-classical motion configurations of the C.M. Since the adiabatic inner states are factorized with respect to each parts composing the macroscopic object, this adiabatic separation can induce the quantum decoherence. This observation thus provides us with a possible solution to the Schroedinger cat paradox
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shor's algorithm. We implement the algorithm in an NMR quantum information processor and experimentally factorize the number 21. Numerical simulations indicate that the running time grows only quadratically with the number of qubits.
Quantum processors are potentially superior to their classical counterparts for many computational tasks including factorization. Circuit methods as well as adiabatic methods have already been proposed and implemented for finding the factors of a given composite number. The main challenge in scaling it to larger numbers is the unavailability of large number of qubits. Here we propose a hybrid scheme that involves both classical and quantum computation, which reduces the number of qubits required for factorization. The classical part involves setting up and partially simplifying a set of bit-wi…
In previous implementations of adiabatic quantum algorithms using spin systems, the average Hamiltonian method with Trotter's formula was conventionally adopted to generate an effective instantaneous Hamiltonian that simulates an adiabatic passage. However, this approach had issues with the precision of the effective Hamiltonian and with the adiabaticity of the evolution. In order to address these, we here propose and experimentally demonstrate a novel scheme for adiabatic quantum computation by using the intrinsic Hamiltonian of a realistic spin system to represent the problem Hamiltonian whi…
The adiabatic quantum computation is a universal and robust method of quantum computing. In this architecture, the problem can be solved by adiabatically evolving the quantum processor from the ground state of a simple initial Hamiltonian to that of a final one, which encodes the solution of the problem. By far, there is no experimental realization of adiabatic quantum computation on a single solid spin system under ambient conditions, which has been proved to be a compatible candidate for scalable quantum computation. In this letter, we report on the first experimental realization of an adiab…
We give a simple way to detect the geometric phase shift and the conditional geometric phase shift with Josephson junction system. Comparing with the previous work(Falcl G, Fazio R, Palma G.M., Siewert J and Verdal V, {\it Nature} {\bf 407}, 355(2000)), our scheme has two advantages. We use the non-adiabatic operation, thus the detection is less affected by the decoherence. Also, we take the time evolution on zero dynamic phase loop, we need not take any extra operation to cancel the dynamic phase.
Quantum integer factorization is a potential quantum computing solution that may revolutionize cryptography. Nevertheless, a scalable and efficient quantum algorithm for noisy intermediate-scale quantum computers looks far-fetched. We propose an alternative factorization method, within the digitized-adiabatic quantum computing paradigm, by digitizing an adiabatic quantum factorization algorithm enhanced by shortcuts to adiabaticity techniques. We find that this fast factorization algorithm is suitable for available gate-based quantum computers. We test our quantum algorithm in an IBM quantum c…
Quantum algorithms could be much faster than classical ones in solving the factoring problem. Adiabatic quantum computation for this is an alternative approach other than Shor's algorithm. Here we report an improved adiabatic factoring algorithm and its experimental realization to factor the number 143 on a liquid crystal NMR quantum processor with dipole-dipole couplings. We believe this to be the largest number factored in quantum-computation realizations, which shows the practical importance of adiabatic quantum algorithms.
What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index $p$ that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index…
How the thermodynamic entropy $S_{TD}$ is related to the Boltzmann entropy $S_{B}$ has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy $S_{TD}$ is equal to a function $\tilde{S}_{B}$ that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether $\tilde{S}_{B}$ completely determines the possibility of a macroscopic adiabatic transformation in the same way as $S_{TD}$ does. In this paper, by formulating the possibility of a macroscopic adiabatic transformation…
An extension of the adiabatic factorization of the time evolution operator is studied for spin in a general time varying magnetic field $B(t)$. When $B(t)$ changes adiabatically, such a factorization reduces to the product of the geometric operator which embodies the Berry phase phenomenon and a usual dynamical operator. For a general time variation of $B(t)$, there should be another operator $N(t)$ in the factorization that is related to non-adiabatic transitions. A simple and explicit expression for the instantaneous angular velocity of this operator is derived. This is done in a way that is…
We investigate the exact wavefunction as a single product of electronic and nuclear wavefunction for a model conical intersection system. Exact factorized spiky potentials and nodeless nuclear wavefunctions are found. The exact factorized potential preserves the symmetry breaking effect when the coupling mode is present. Additionally the nodeless wavefunctions are found to be closely related to the adiabatic nuclear eigenfunctions. This phenomenon holds even for the regime where the non-adiabatic coupling is relevant, and sheds light on the relation between the exact wavefunction factorization…
Quantum decoherence is of primary importance for relaxation to an equilibrium distribution and, accordingly, for equilibrium processes. We demonstrate how coherence breaking implies evolution to a microcanonical distribution (``microcanonical postulate'') and, on that ground, consider an adiabatic process, in which there is no thermostat. We stress its difference from a zero-polytropic process, i.e., a process with zero heat capacity but involving a thermostat. We find the distribution for the adiabatic process and show that (i) in the classical limit this distribution is canonical, (ii) for m…
Prime factorization is a difficult problem with classical computing, whose exponential hardness is the foundation of Rivest-Shamir-Adleman (RSA) cryptography. With programmable quantum devices, adiabatic quantum computing has been proposed as a plausible approach to solve prime factorization, having promising advantage over classical computing. Here, we find there are certain hard instances that are consistently intractable for both classical simulated annealing and un-configured adiabatic quantum computing (AQC). Aiming at an automated architecture for optimal configuration of quantum adiabat…
In two companion papers it was shown how to separate out from a scattering function in quantum electrodynamics a distinguished part that meets the correspondence-principle and pole-factorization requirements. The integrals that define the terms of the remainder are here shown to have singularities on the pertinent Landau singularity surface that are weaker than those of the distinguished part. These remainder terms therefore vanish, relative to the distinguished term, in the appropriate macroscopic limits. This shows, in each order of the perturbative expansion, that quantum electrodynamics do…
We suggest a procedure for demonstrating quantum coherence and measuring decoherence times between different fluxoid states of a SQUID by using ``adiabatic inversion'', where one macroscopic fluxoid state is smoothly transferred into the other, like a spin reversing direction by following a slowly moving magnetic field. This is accomplished by sweeping an external applied flux, and depends on a well-defined quantum phase between the two macroscopic states. Varying the speed of the sweep relative to the decoherence time permits one to move from the quantum regime, where such a well-defined phas…
A procedure for demonstrating quantum coherence and measuring decoherence times between different fluxoid states of a SQUID by using ``adiabatic inversion'' is discussed. One fluxoid state is smoothly transferred into the other, like a spin reversing direction by following a slowly moving magnetic field. This is accomplished by sweeping an external applied flux, and depends on a well-defined quantum phase between the two macroscopic states. Varying the speed of the sweep relative to the decoherence time permits one to move from the quantum regime, where such a well-defined phase exists, to the…
The quantum adiabatic theorem incorporating the Berry phase phenomenon can be characterized as a factorization of the time evolution operator into a path-dependent geometric factor, a usual dynamical factor and a non-adiabatic factor that approaches the identity in the adiabatic limit. We study a case where all these three factors can be constructed explicitly and where the instantaneous Hamiltonian has infinitely degenerate energy eigenstates associated with magnetic translation symmetry. Significance of the non-adiabatic factor in terms of transition probabilities is discussed.
The time evolution is studied for the Landau problem with a general time dependent electric field ${\bf E}(t)$ in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is derived with each factor having a clear physical interpretation. The factorization consists of a geometric operator (path-ordered magnetic translation), a dynamical operator generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic operator that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic transl…
Adiabatic quantum computing has recently been used to factor 56153 [Dattani & Bryans, arXiv:1411.6758] at room temperature, which is orders of magnitude larger than any number attempted yet using Shor's algorithm (circuit-based quantum computation). However, this number is still vastly smaller than RSA-768 which is the largest number factored thus far on a classical computer. We address a major issue arising in the scaling of adiabatic quantum factorization to much larger numbers. Namely, the existence of many 4-qubit, 3-qubit and 2-qubit interactions in the Hamiltonians. We showcase our metho…
We describe a theoretical scheme that allows for transfer of quantum states of atomic collective excitation between two macroscopic atomic ensembles localized in two spatially-separated domains. The conception is based on the occurrence of double-exciton dark states due to the collective destructive quantum interference of the emissions from the two atomic ensembles. With an adiabatically coherence manipulation for the atom-field couplings by stimulated Ramann scattering, the dark states will extrapolate from an exciton state of an ensemble to that of another. This realizes the transport of qu…
We report a quantum-classical hybrid scheme for factorization of bi-prime numbers (which are odd and square-free) using IBM's quantum processors. The hybrid scheme proposed here involves both classical optimization techniques and adiabatic quantum optimization techniques, and is build by extending a previous scheme of hybrid factorization [Pal et al., Pramana 92, 26 (2019) and Xu et al., Phys. Rev. Lett. 108, 130501 (2012)]. The quantum part of the scheme is very general in the sense that it can be implemented using any quantum computing architecture. Here, as an example, we experimentally imp…
We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two …
We show that Grover's algorithm defines a geodesic in quantum Hilbert space with the Fubini-Study metric. From statistical point of view Grover's algorithm is characterized by constant Fisher's function. Quantum algorithms changing complexity class as Shor's factorization does not preserve constant Fisher's information. An adiabatic quantum factorization algorithm in non polynomial time is presented to exemplify the result.
Trajectory-based mixed quantum-classical approaches to coupled electron-nuclear dynamics suffer from well-studied problems such as the lack of (or incorrect account for) decoherence in the trajectory surface hopping method and the inability of reproducing the spatial splitting of a nuclear wave packet in Ehrenfest-like dynamics. In the context of electronic non-adiabatic processes, these problems can result in wrong predictions for quantum populations and in unphysical outcomes for the nuclear dynamics. In this paper we propose a solution to these issues by approximating the coupled electronic…
In order to understand quantum decoherence of a quantum system due to its interaction with a large system behaving classically, we introduce the concept of adiabatic quantum entanglement based on the Born-Oppenhemeir approximation. In the adiabatic limit, it is shown that the wave function of the total system formed by the quantum system plus the large system can be factorized as an entangled state with correlation between adiabatic quantum states and quasi-classical motion configurations of the large system. In association with a novel viewpoint about quantum measurement, which has been direc…
Macroscopic cat states have been widely studied to illustrate fundamental principles of quantum physics as well as their application in quantum information processing. In this paper, we propose a quantum speedup method for adiabatic creation of cat states in a Kerr nonlinear resonator via gradient-descent optimal adiabatic control. By simultaneously adiabatic tuning the the cavity detuning and driving field strength, the width of minimum energy gap between the target trajectory and non-adiabatic trajectory can be widen, which allows us to speed up the evolution along the adiabatic path. Compar…
Farhi and others have introduced the notion of solving NP problems using adiabatic quantum com- puters. We discuss an application of this idea to the problem of integer factorization, together with a technique we call gluing which can be used to build adiabatic models of interesting problems. Although adiabatic quantum computers already exist, they are likely to be too small to directly tackle problems of interesting practical sizes for the foreseeable future. Therefore, we discuss techniques for decomposition of large problems, which permits us to fully exploit such hardware as may be availab…
Second law of thermodynamics in closed quantum many-body systems
RIKEN（理化学研究所）；Center for Quantum Computing, RIKEN（理科大学量子计算中心）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学所交叉理论与数学科学中心）；Institute for Photon Science and Technology, The University of Tokyo（东京大学光子科学技术研究所）；Center for Quantum Information and Quantum Biology, The University of Osaka（大阪大学量子信息与量子生物学中心）
We present a detailed derivation and numerical tests of a new mixed quantum-classical scheme to deal with non-adiabatic processes. The method is presented as the zero-th order approximation to the exact coupled dynamics of electrons and nuclei offered by the factorization of the electron-nuclear wave function [A. Abedi, N. T. Maitra and E. K. U. Gross, Phys. Rev. Lett., 105 (2010)]. Numerical results are presented for a model system for non-adiabatic charge transfer in order to test the performance of the method and to validate the underlying approximations.
The preparation of nonclassical states of mechanical motion conclusively proves that control over such motion has reached the quantum level. We investigate ways to achieve nonclassical states of macroscopic mechanical oscillators, particularly levitated nanoparticles. We analyze the possibility of the conditional squeezing of the levitated particle induced by the homodyne detection of light in a pulsed optomechanical setup within the resolved sideband regime. We focus on the regimes that are experimentally relevant for the levitated systems where the ground-state cooling is not achievable and …
In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, $γ=M/N$. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (versus only energy) is used, …
We study a system of two cavities each encapsulating a qubit and an oscillator degrees of freedom. An ultrastrong interaction strength between the qubit and the oscillator is assumed, and the photons are allowed to hop between the cavities. A partition of the time scale between the fast moving oscillator and the slow moving qubit allows us to set up an adiabatic approximation procedure where we employ the delocalized degrees of freedom to diagonalize the Hamiltonian. The time evolution of the $N00N$-type initial states now furnishes, for instance, the reduced density matrix of a bipartite syst…
We investigate the existence of the $macroscopic$ $quantum$ $phase$ in trapped ultracold quantum degenerate gases, such as Bose-Einstein condensate, in an asymmetrical two-dimensional magnetic lattice. We show the key to adiabatically control the tunneling in the new two-dimensional magnetic lattice by means of external magnetic bias fields. The macroscopic quantum phase signature is identified as a Rabi-like oscillation when solving the system of coupled time-dependent differential equations, described here by the Boson Josephson Junctions (BJJs). In solving the system of the BJJs we used an …
A surface-hopping algorithm recently derived from the exact factorization approach, SHXF, [Ha, Lee, Min, J. Phys. Chem. Lett. 9, 1097 (2018)] introduces an additional term in the electronic equation of surface-hopping, which couples electronic states through the quantum momentum. This term not only provides a first-principles description of decoherence but here we show it is crucial to accurately capture non-adiabatic dynamics when more than two states are occupied at any given time. Using a vibronic coupling model of the uracil cation, we show that the lack of this term in traditional surface…
Stimulated Raman adiabatic passage (STIRAP) describes adiabatic population transfer between two states coherently coupled via a mediating state that remains unoccupied. This renders STIRAP robust against loss in the mediating state, leading to profound applications in atomic- and molecular-beam research, trapped-ion physics, superconducting circuits, other solid-state systems, optics, in entanglement generation and qubit operations. STIRAP in optomechanics has been considered for optical frequency conversion where a mechanical mode provides the mediating state. Given the advances of optomechan…
The correspondence principle is important in quantum theory on both the fundamental and practical levels: it is needed to connect theory to experiment, and for calculations in the technologically important domain lying between the atomic and classical regimes. Moreover, a correspondence-principle part of the S-matrix is normally separated out in quantum electrodynamics in order to obtain a remainder that can be treated perturbatively. But this separation, as usually performed, causes an apparent breakdown of the correspondence principle and the associated pole-factorization property. This brea…
We show that no matter how slowly a quantum-to-classical symmetry breaking process is driven, the adiabatic limit can never be reached in a macroscopic body. Massive defect formation preempts an adiabatic quantum-classical crossover and triggers the appearance of a symmetric non-equilibrium state that recursively collapses into the classical state, breaking the symmetry at punctured times. The presence of this state allows the quantum-classical transition to be investigated and controlled in mesoscopic devices by supplying externally the proper dynamical symmetry breaking perturbation.
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by means of classical computation. However, the order can be drastically reduced by converting the factorization problem to an optimization one and solving it using a quantum computer. Recent works involving both theoretical and experimental approaches use Shor's algorithm, adiabatic quantum computation and quantum annealing principles to factorize integers. H…
The classical behaviour of a macroscopic system consisting of a large number of microscopic systems is derived in the framework of the Bohmian interpretation of quantum mechanics. Under appropriate assumptions concerning the localization and factorization of the wavefunction it is shown explicitly that the center of mass motion of the system is determined by the classical equations of motion.
We study the stability of attractive atomic Bose-Einstein condensate and the macroscopic quantum many-body tunneling (MQT) in the anharmonic trap. We utilize correlated two-body basis function which keeps all possible two-body correlations. The anharmonic parameter ($λ$) is slowly tuned from harmonic to anharmonic. For each choice of $λ$ the many-body equation is solved adiabatically. The use of the van der Waals interaction gives realistic picture which substantially differs from the mean-field results. For weak anharmonicity, we observe that the attractive condensate gains stability with lar…
In the Graph Isomorphism problem two N-vertex graphs G and G' are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G into G'. If yes, then G and G' are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and can also determine all automorphisms of a given graph. We show how the GI problem can be c…
I argue that conventional estimates of the criterion for classical behavior of a macroscopic body are incorrect in most circumstances,because they do not take into account the locality of interactions, which characterizes the behavior of all systems described approximately by local quantum field theory. The deviations from classical behavior of a macroscopic body, except for those that can be described as classical uncertainties in the initial values of macroscopic variables,are {\it exponentially} small as a function of the volume of the macro-system in microscopic units. Conventional estimat…
We introduce two methods for speeding up adiabatic quantum computations by increasing the energy between the ground and first excited states. Our methods are even more general. They can be used to shift a Hamiltonian's density of states away from the ground state, so that fewer states occupy the low-lying energies near the minimum, hence allowing for faster adiabatic passages to find the ground state with less risk of getting caught in an undesired low-lying excited state during the passage. Even more generally, our methods can be used to transform a discrete optimization problem into a new on…
Despite the enormous technological interest in micro and nanolasers, surprisingly, no class-B quantum density-matrix model is available to date, capable of accurately describing coherence and photon correlations within a unified theory. In class-B lasers $-$applicable for most solid-state lasers at room temperature$-$, the macroscopic polarization decay rate is larger than the cavity damping rate which, in turn, exceeds the upper level population decay rate. Here we carry out a density-matrix theoretical approach for generic class-B lasers, and provide closed equations for the photonic and ato…
This work addresses quantum adiabatic decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system-environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when t…
We use a self-assembled two-dimensional Coulomb crystal of $\sim 70$ ions in the presence of an external transverse field to engineer a quantum simulator of the Dicke Hamiltonian. This Hamiltonian has spin and bosonic degrees of freedom which are encoded by two hyperfine states in each ion and the center of mass motional mode of the crystal, respectively. The Dicke model features a quantum critical point separating two distinct phases: the superradiant (ferromagnetic) and normal (paramagnetic) phases. We experimentally explore protocols that aim to adiabatically prepare the superradiant ground…
