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arXiv:quant-ph/0309150 · Physical Review A

Quantum adiabatic algorithms and large spin tunnelling

A. Boulatov, Vadim Smelyanskiy

Published Dec 24, 2003• 15 citations
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Abstract

We study the quantum adiabatic evolution algorithm with different evolution paths that correspond to different control Hamiltonians $H(\ensuremath{\tau})$ sampled from a random ensemble. The algorithm is applied to a random binary optimization problem where the cost function is symmetric with respect to the permutation of individual bits. We investigate the method proposed by Farhi et al. when different evolution paths preserve the bit symmetry of the control Hamiltonians $H(\ensuremath{\tau})$ and are generated by an ensemble of random $8\ifmmode\times\else\texttimes\fi{}8$ matrices. In this case, the algorithm dynamics is completely described in terms of the motion of a spin-n/2 system. We show that different evolution paths can be parametrized by a small number of independent parameters that are expansion coefficients of $H(\ensuremath{\tau})$ in a certain universal set of operators. One of these operators is responsible for avoiding the tunnelling for the spin-n/2 system and the corresponding coefficient determines the algorithm complexity for a given problem instance. We show that any problem instance can be solved in polynomial time by applying one of the two universal path modifications (due to the possible reflection symmetry of the problem). We show that a successful evolution path of the algorithm always corresponds to the trajectory of a classical spin $n/2$ and provide a complete characterization of such paths.

DOI
10.1103/physreva.68.062321

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We study the quantum adiabatic evolution algorithm with different evolution paths that correspond to different control Hamiltonians H(\ensuremathτ)H(\ensuremath{\tau})H(\ensuremathτ) sampled from a random ensemble. The algorithm is applied to a random binary optimization problem where the cost function is symmetric with respect to the permutation of individual bits. We investigate the method proposed by Farhi et al. when different evolution paths preserve the bit symmetry of the control Hamiltonians H(\ensuremathτ)H(\ensuremath{\tau})H(\ensuremathτ) and are generated by an ensemble of random 8\ifmmode×\else\texttimes\fi88\ifmmode\times\else\texttimes\fi{}88\ifmmode×\else\texttimes\fi8 matrices. In this case, the algorithm dynamics is completely described in terms of the motion of a spin-n/2 system. We show that different evolution paths can be parametrized by a small number of independent parameters that are expansion coefficients of H(\ensuremathτ)H(\ensuremath{\tau})H(\ensuremathτ) in a certain universal set of operators. One of these operators is responsible for avoiding the tunnelling for the spin-n/2 system and the corresponding coefficient determines the algorithm complexity for a given problem instance. We show that any problem instance can be solved in polynomial time by applying one of the two universal path modifications (due to the possible reflection symmetry of the problem). We show that a successful evolution path of the algorithm always corresponds to the trajectory of a classical spin n/2n/2n/2 and provide a complete characterization of such paths.
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 nnn. 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, gmin=O(n2−n/2)g_{\rm min}={\cal O}(n 2^{-n/2})gmin​=O(n2−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.
We apply a quantum adiabatic evolution algorithm to a combinatorial optimization problem where the cost function depends entirely on the of the number of unit bits in a n-bit string (Hamming weight). The solution of the optimization problem is encoded as a ground state of the problem Hamiltonian H_p for the z-projection of a total spin-n/2. We show that tunneling barriers for the total spin can be completely suppressed during the algorithm if the initial Hamiltonian has its ground state extended in the space of the z-projections of the spin. This suppression takes place even if the cost function has deep and well separated local minima. We provide an intuitive picture for this effect and show that it guarantees the polynomial complexity of the algorithm in a very broad class of cost functions. We suggest a simple example of the Hamiltonian for the adiabatic evolution: H(tau) = (1-tau) hat S_{x}^{2} + tau H_p, with parameter tau slowly varying in time between 0 and 1. We use WKB analysis for the large spin to estimate the minimum energy gap between the two lowest adiabatic eigenvalues of H(tau).
Quantum computation by adiabatic evolution, as described in quant-ph/0001106, will solve satisfiability problems if the running time is long enough. In certain special cases (that are classically easy) we know that the quantum algorithm requires a running time that grows as a polynomial in the number of bits. In this paper we present numerical results on randomly generated instances of an NP-complete problem and of a problem that can be solved classically in polynomial time. We simulate a quantum computer (of up to 16 qubits) by integrating the Schrodinger equation on a conventional computer. For both problems considered, for the set of instances studied, the required running time appears to grow slowly as a function of the number of bits.
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 tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing 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 analyze the power of the quantum adiabatic evolution algorithm (QAA) for solving random computationally hard optimization problems within a theoretical framework based on random matrix theory (RMT). We present two types of driven RMT models. In the first model, the driving Hamiltonian is represented by Brownian motion in the matrix space. We use the Brownian motion model to obtain a description of multiple avoided crossing phenomena. We show that nonadiabatic corrections in the QAA are due to the interaction of the ground state with the ``cloud'' formed by most of the excited states, confirming that in driven RMT models, the Landau-Zener scenario of pairwise level repulsions is not relevant for the description of nonadiabatic corrections. We show that the QAA has a finite probability of success in a certain range of parameters, implying a polynomial complexity of the algorithm. The second model corresponds to the standard QAA with the problem Hamiltonian taken from the RMT Gaussian unitary ensemble (GUE). We show that the level dynamics in this model can be mapped onto the dynamics in the Brownian motion model. For this reason, the driven GUE model can also lead to polynomial complexity of the QAA. The main contribution to the failure probability of the QAA comes from the nonadiabatic corrections to the eigenstates, which only depend on the absolute values of the transition amplitudes. Due to the mapping between the two models, these absolute values are the same in both cases. Our results indicate that this ``phase irrelevance'' is the leading effect that can make both the Markovian- and GUE-type QAAs successful.
Quantum computing seeks to use the powers of quantum mechanics to accomplish tasks that classical computers cannot easily accomplish. Adiabatic quantum computing is one flavor of quantum computing that slowly changes a system with time in such a way that the state remains close to the ground state for the entire evolution. An easy way to implement and study adiabatic computing is using stoquastic Hamiltonians with no-sign problem. These Hamiltonians can be readily simulated on classical computers using quantum Monte Carlo schemes; though, it is unknown whether such simulation captures the speed and power of the quantum computation.In this work, I study stoquastic Hamiltonians, focusing on quantum mechanical tunneling through classically inaccessible regions and barriers. I study multi-qubit tunneling problems to determine what properties of the barrier make this problem quantum mechanically or classically hard or inefficient. I compare quantum adiabatic computing with a path-integral Quantum Monte Carlo algorithm, which is a classical algorithm designed to simulate the quantum mechanical dynamics of the adiabatic evolution. Limited numerical data shows strong correlation between adiabatic and Monte Carlo runtimes, but due to computational limits, only small system sizes could be sampled.Additionally, I study the properties of the quantum adiabatic algorithm in both the asymptotic limit of a large number of qubits and long runtimes and more realistic finite system sizes and non-adiabatic conditions. I develop a modification of the existing Villain transformation that allows it to find the asymptotic running time of the quantum evolution for different barrier sizes. Furthermore, I compare this to finite system sizes and discover the extremely large numbers of qubits are needed for these asymptotics to be useful.When the quantum dynamics are run faster than adiabatically, new behavior arises that could potentially lead to enhancements in the quantum runtimes. I explore these enhancements, discovering when they do and do not occur. I also develop a broad framework for determining the nature of the near-adiabatic behavior of any quantum evolution, using only information about the energy gap between the ground state and the first excited state.