Dynamics of quantum adiabatic evolution algorithm for Number Partitioning
Vadim N. Smelyanskiy, Udo v. Toussaint, Dogan A. Timucin
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32 pages, 5 figures, 3 Appendices; List of additions compare to v.3: (i) numerical solution of the stationary Schroedinger equation for the adiabatic eigenstates and eigenvalues; (ii) connection between the scaling law of the minimum gap with the problem size and the shape of the coarse-grained distribution of the adiabatic eigenvalues at the avoided-crossing point
We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size $n$. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxilary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum excitation gap, $g_{\rm min}={\cal O}(n 2^{-n/2})$, corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simulteneous fipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimenssional quantum diffusion in the energy space. This effect provides a general limitation on the power of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.
The adiabatic theorem addresses the dynamics of a target instantaneous eigenstate of a time-dependent Hamiltonian. We use a Feshbach P-Q partitioning technique to derive a closed one-component integro-differential equation. The resultant equation properly traces the footprint of the target eigenstate. The physical significance of the derived dynamical equation is illustrated by both general analysis and concrete examples. Surprisingly, we find an anomalous phenomenon showing that a dephasing white noise can enhance and even induce adiabaticity. This new phenomenon may naturally occur in many p…
A major challenge facing adiabatic quantum computing is that algorithm design and error correction can be difficult for adiabatic quantum computing. Recent work has considered addressing his challenge by using coherently controlled adiabatic evolutions in the place of classically controlled evolution. An important question remains: what is the relative power of controlled adiabatic evolution to traditional adiabatic evolutions? We address this by showing that coherent control and measurement provides a way to average different adiabatic evolutions in ways that cause their diabatic errors to ca…
Adiabatic evolution is used in a variety of quantum information processing tasks. However, the elimination of errors is not as well-developed as it is for circuit model processing. Here, we present a strategy to accelerate a reliable quantum adiabatic process by adding Leakage Elimination Operators (LEO) to the evolution which are a sequence of pulse controls acting in an adiabatic subspace. Using the Feshbach $PQ$ partitioning technique, we obtain an analytical solution which traces the footprint of the target eigenstate. The effectiveness of the LEO is independent of the specific form of the…
Deutsch-Jozsa algorithm has been implemented via a quantum adiabatic evolution by S. Das et al. [Phys. Rev. A 65, 062310 (2002)]. This adiabatic algorithm gives rise to a quadratic speed up over classical algorithms. We show that a modified version of the adiabatic evolution in that paper can improve the performance to constant time.
Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final …
We decompose the quantum adiabatic evolution as the products of gauge invariant unitary operators and obtain the exact nonadiabatic correction in the adiabatic approximation. A necessary and sufficient condition that leads to adiabatic evolution with geometric phases is provided and we determine that in the adiabatic evolution, while the eigenstates are slowly varying, the eigenenergies and degeneracy of the Hamiltonian can change rapidly. We exemplify this result by the example of the adiabatic evolution driven by parametrized pulse sequences. For driving fields that are rotating slowly with …
Recently, some quantum algorithms have been implemented by quantum adiabatic evolutions. In this paper, we discuss the accurate relation between the running time and the distance of the initial state and the final state of a kind of quantum adiabatic evolutions. We show that this relation can be generalized to the case of mixed states.
The unitary operator corresponding to the classical canonical transformation that connects a general closed system to an open system under adiabatic conditions is found. The quantum invariant operator of the adiabatic open system is derived from the unitary transformation of the quantum Hamiltonian of the closed system. On the basis of these results, we investigate the evolution of the general quantum adiabatic system and construct a revised adiabatic theorem. The adiabatic theorem developed here exactly reduces to the well-known Berry adiabatic theorem when the control parameter
The geometric phase acquired by the vector states under an adiabatic evolution along a noncyclic path can be calculated correctly in any instantaneous basis of a Hamiltonian that varies in time due to a time-dependent classical field.
We use local adiabatic evolution to experimentally create and determine the ground state spin ordering of a fully-connected Ising model with up to 14 spins. Local adiabatic evolution -- in which the system evolution rate is a function of the instantaneous energy gap -- is found to maximize the ground state probability compared with other adiabatic methods while only requiring knowledge of the lowest $\sim N$ of the $2^N$ Hamiltonian eigenvalues. We also demonstrate that the ground state ordering can be experimentally identified as the most probable of all possible spin configurations, even whe…
Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution (H(epsilon t), epsilon ? 0), are insufficient to describe an evolution driven by the hamiltonian H(t) itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any hamiltonian H(t). As a corollary we demonstrate that the commonly used condition of a slow hamiltonian variation rate, compared to the spectral…
Quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving instances of NP-complete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one through the evolution. In this letter, we show a way of speedup in quantum adiabatic evolution algorithm, using the extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides the new point of view to reduce …
One of the difficulties in adiabatic quantum computation is the limit on the computation time. Here we propose two schemes to speed-up the adiabatic evolution. To apply this controlled adiabatic evolution to adiabatic quantum computation, we design one of the schemes without any prior knowledge of the instantaneous eigenstates of the final Hamiltonian. Whereas in another scheme, the control is constructed with the instantaneous eigenstate that is the target state of the control. As an illustration, we study a two-level system driven by a time-dependent magnetic field under the control. The phy…
Adiabatic processes in the quantum Ising model and the anisotropic Heisenberg model are discussed. The adiabatic processes are assumed to consist in the slow variation of the strength of the magnetic field that environs the spin-systems. These processes are of current interest in the treatment of cold atoms in optical lattices and in Adiabatic Quantum Computation. We determine the probability that, during an adiabatic passage starting from the ground state, states with higher energy are excited.
Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is m…
The smallness of the variation rate of the hamiltonian matrix elements compared to the (square of the) energy spectrum gap is usually believed to be the key parameter for a quantum adiabatic evolution. However it is only perturbatively valid for scaled timed hamiltonian and resonance processes as well as off resonance possible constructive Stückelberg interference effects violate this usual condition for general hamiltionian. More general adiabatic condition and exact bounds for adiabatic quantum evolution are derived and studied in the framework of a two-level system. The usual criterion is r…
In a recent Letter [Phys. Rev. Lett. {\bf 95}, 080502 (2005)], an interesting scheme was proposed to implement a type of conditional quantum phase gates with built-in fault-tolerant feature via adiabatic evolution of dark eigenstates. In this comment we elaborate the geometric nature of the gate scheme and clarify that it still belongs to a class of conventional geometric quantum computation.
We introduce a class of quantum adiabatic evolutions that we claim may be interpreted as the equivalents of the unitary gates of the quantum gate model. We argue that these gates form a universal set and may therefore be used as building blocks in the construction of arbitrary `adiabatic circuits', analogously to the manner in which gates are used in the circuit model. One implication of the above construction is that arbitrary classical boolean circuits as well as gate model circuits may be directly translated to adiabatic algorithms with no additional resources or complexities. We show that …
Accelerating quantum adiabatic evolution with $π$-pulse sequences
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University（清华大学低维量子物理国家重点实验室和物理系）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）
Precise and efficient control of quantum systems is essential to perform quantum information processing tasks. In terms of adiabatic speedup via leakage elimination operator approach, for a closed system, the ideal pulse control conditions have been theoretically derived by P-Q partitioning technique. However, it is a challenge to design the corresponding control pulses for an open system, which requires to tackle noisy environments. In this paper, we apply the stochastic search procedures to an open qutrit system and successfully obtain the optimal control pulses for significant adiabatic spe…
A cluster state cannot be a unique ground state of a two-body interacting Hamiltonian. Here, we propose the creation of a cluster state of logical qubits encoded in spin-1/2 particles by adiabatically weakening two-body interactions. The proposal is valid for any spatial dimensional cluster states. Errors induced by thermal fluctuations and adiabatic evolution within finite time can be eliminated ensuring fault-tolerant quantum computing schemes.
Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N. J. Cerf for Grover's search [ Phys. Rev. A. {\bf 65} 0423…
Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow external driving. According to the quantum adiabatic theorem, no transitions occur between non-degenerate instantaneous eigen-energy levels in such a dynamical evolution. However, this is true only when the driving rate is infinitesimally small. For a small nonzero driving rate, there are generally small transition probabilities between the energy levels. We develop a classical mechanics framework to address the small deviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians are …
We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from $\hat{H}_0$ to $\hat{H}_1$ in a total evolution-time $τ$, and our goal is to determine how the final state of the system depends on $τ$. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large $τ$, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not …
Quantum circuit evolutionary framework applied on set partitioning problem
QuIIN - Quantum Industrial Innovation, EMBRAPII CIMATEC Competence Center in Quantum Technologies, SENAI CIMATEC（塞纳伊-基马特克量子工业创新中心）；Latin America Quantum Computing Center, SENAI CIMATEC（塞纳伊-基马特克拉丁美洲量子计算中心）；Federal University of Latin-American Integration（拉丁美洲一体化联邦大学）
We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of non-degenerate quantum systems have to be taken into account to give the correct interference result in the calculation of physical quantities in states that are a superposition of instantaneous eigenstates of energy. To verify the contribution of those adiabatic phases in the interference phenomena, we consider the spin-1/2 model coupled to a precessing external magnetic field. In the model, the adiabatic phase increases in time up to reach the difference of the Berry's phases of the model whe…
We explain why quantum adiabatic evolution and simulated annealing perform similarly in certain examples of searching for the minimum of a cost function of n bits. In these examples each bit is treated symmetrically so the cost function depends only on the Hamming weight of the n bits. We also give two examples, closely related to these, where the similarity breaks down in that the quantum adiabatic algorithm succeeds in polynomial time whereas simulated annealing requires exponential time.
We aim to give more insights on adiabatic evolution concerning the occurrence of anti-crossings and their link to the spectral minimum gap $Δ_{min}$. We study in detail adiabatic quantum computation applied to a specific combinatorial problem called weighted max $k$-clique. A clear intuition of the parametrization introduced by V. Choi is given which explains why the characterization isn't general enough. We show that the instantaneous vectors involved in the anti-crossing vary brutally through it making the instantaneous ground-state hard to follow during the evolution. This result leads to a…
Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, …
We extend the concept of locality to enclose a situation where a tensor-product structure for the Hilbert space is not \textit {a priori} assumed; rather, this locality is related to a given matrix representation of the Hamiltonian associated to the system. As a result, we formulate a Lieb-Robinson-like bound for Hamiltonians local in a given basis. In particular, we employ this bound to obtain alternatively the adiabatic condition, where adiabaticity is naturally ensued from a locality in energy basis and a relatively small Lieb-Robinson bound.
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [Farhi et al., quant-ph/0001106]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order $\sqrt{N}$, where $N$ is the number…
A classical-kind phase-space formalism is developed to address the tiny intrinsic dynamical deviation from what is predicted by Wilczek-Zee theorem during quantum adiabatic evolution on degeneracy levels. In this formalism, the Hilbert space and the aggregate of degenerate eigenstates become the classical-kind phase-space and a high-dimensional subspace in the phase-space, respectively. Compared with the previous same study by a different method, the current result is qualitatively different in that the first-order deviation derived here is always perpendicular to the degeneracy subspace. A tr…
Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An n-vertex random graph has each edge included with probability 1/2, and a clique is a completely connected subgraph. There is no known classical algorithm that finds the largest clique in a random graph with high probability and runs in a time polynomial in n. For the small graphs we are able to investi…
The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with $N$-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this geometric phase captures the inherent geometric feature of the state evolution. Moreover, the geometric phase in the evolution of the eigenspace of an adiabatic action operator is also addressed, which is elaborated by a pullback U(N)-bundle. Several i…
We propose a method to produce fast transitionless dynamics for finite-dimensional quantum systems without requiring additional Hamiltonian components not included in the initial control setup, remaining close to the true adiabatic path at all times. The strategy is based on the introduction of an effective counterdiabatic scheme: a correcting Hamiltonian is constructed which approximatively cancels nonadiabatic effects, inducing an evolution tracking the adiabatic states closely. This can be absorbed into the initial Hamiltonian by adding a fast oscillation in the control parameters. We show …
Quantum adiabatic evolutions find a broad range of applications in quantum physics and quantum technologies. The traditional form of the quantum adiabatic theorem limits the speed of adiabatic evolution by the minimum energy gaps of the system Hamiltonian. Here, we experimentally show using a nitrogen-vacancy center in diamond that, even in the presence of vanishing energy gaps, quantum adiabatic evolution is possible. This verifies a recently derived necessary and sufficient quantum adiabatic theorem and offers paths to overcome the conventionally assumed constraints on adiabatic methods. By …
Starting from a simple mapping of a generator of local stochastic dynamics to a quantum Hamiltonian, we derive a condition, which allows us to use the quasi-adiabatic evolution and so relate gapped quantum phases with non-equilibrium's. This leads us to a study of invertible matrix product operators. Finally, we present an ansatz for constructing local stochastic dynamics for which the Perron-Frobenius vector has a Matrix Product Representation. Additionally, we get for free that the dynamics satisfy a generalized form of detailed balance.
The adiabatic quantum algorithm has drawn intense interest as a potential approach to accelerating optimization tasks using quantum computation. The algorithm is most naturally realised in systems which support Hamiltonian evolution, rather than discrete gates. We explore an alternative approach in which slowly varying measurements are used to mimic adiabatic evolution. We show that for certain Hamiltonians, which remain frustration-free all along the adiabatic path, the necessary measurements can be implemented through the measurement of random terms from the Hamiltonian. This offers a new, a…
In [Phys. Rev. Lett. 95, 080502 (2005)], Zheng proposed a scheme for implementing a conditional phase shift via adiabatic passages. The author claims that the gate is "neither of dynamical nor geometric origin" on the grounds that the Hamiltonian does not follow a cyclic change. He further argues that "in comparison with the adiabatic geometric gates, the nontrivial cyclic loop is unnecessary, and thus the errors in obtaining the required solid angle are avoided, which makes this new kind of phase gates superior to the geometric gates." In this Comment, we point out that geometric operations, …
The adiabatic theorem is an important concept in quantum mechanics, it tells that a quantum system subjected to gradually changing external conditions remains to the same instantaneous eigenstate of its Hamiltonian as it initially in. In this paper, we study the another extreme circumstance where the external conditions vary rapidly such that the quantum system can not follow the change and remains in its initial state (or wavefunction). We call this type of evolution anit-adiabatic evolution. We examine the matter-wave pressure in this situation and derive the condition for such an evolution.…
A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We test one such algorithm by applying it to randomly generated, hard, instances of an NP-complete problem. For the small examples that we can simulate, the quantum adiabatic algorithm works well, and provides evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
We introduce an algorithm to perform an optimal adiabatic evolution that operates without an apriori knowledge of the system spectrum. By probing the system gap locally, the algorithm maximizes the evolution speed, thus minimizing the total evolution time. We test the algorithm on the Landau-Zener transition and then apply it on the quantum adiabatic computation of 3-SAT: The result is compatible with an exponential speed-up for up to twenty qubits with respect to classical algorithms. We finally study a possible algorithm improvement by combining it with the quantum Zeno effect.
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are un…
In quantum adiabatic evolution algorithms, the quantum computer follows the ground state of a slowly varying Hamiltonian. The ground state of the initial Hamiltonian is easy to construct; the ground state of the final Hamiltonian encodes the solution of the computational problem. These algorithms have generally been studied in the case where the "straight line" path from initial to final Hamiltonian is taken. But there is no reason not to try paths involving terms that are not linear combinations of the initial and final Hamiltonians. We give several proposals for randomly generating new paths…
A method of fundamental solutions has been used to investigate transitions in two energy level systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many level complex crossings as announced by Joye, Mileti and Pfister (Phys. Rev. {\bf A44} 4280 (1991)) have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.
In this paper, we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state. If the overlap between the initial state and final state of the quantum system is not equal to zero, both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity". But when the initial state has a zero overlap with the solution stat…
The use of the adiabatic approximation in practical applications, as in adiabatic quantum computation, demands an assessment of the errors made in finite-time evolutions. Aiming at such scenarios, we derive bounds relating error and evolution time in the adiabatic approximation that go beyond typical scaling relations. Using the Adiabatic Perturbation Theory, we obtain leading-order expressions valid for long evolution time $T$, while explicitly determining the shortest time $T$ and the largest error $\varepsilon$ for which they are valid. In this validity regime, we can make clear and precise…
Shortcuts to adiabaticity provide a general approach to mimic adiabatic quantum processes via arbitrarily fast evolutions in Hilbert space. For these counter-diabatic evolutions, higher speed comes at higher energy cost. Here, the counter-diabatic theory is employed as a minimal energy demanding scheme for speeding up adiabatic tasks. As a by-product, we show that this approach can be used to obtain infinite classes of transitionless models, including time-independent Hamiltonians under certain conditions over the eigenstates of the original Hamiltonian. We apply these results to investigate s…
We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computati…
