Adiabatic Quantum Simulation Using Trotterization

Physical Review A, 2005

This paper explores several aspects of the adiabatic quantum computation model. We first show a way that directly maps any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. Specifically, we look for a smooth time-dependent Hamiltonian whose unique ground state slowly changes from the initial state of the circuit to its final state. Since this construction requires in general an n-local Hamiltonian, we will study whether approximation is possible using previous results on ground state entanglement and perturbation theory. Finally we will point out how the adiabatic model can be relaxed in various ways to allow for 2-local partially adiabatic algorithms as well as 2-local holonomic quantum algorithms.

SIAM Journal on Computing, 2007

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains.

Arguably one of the most important applications of quantum computers is the simulation of quantum systems. In the case where the Hamil- tonian consists of a sum of interaction terms between small subsystems, the simulation is thought to be exponentially more efficient than classical simula- tion. More generally, evolution under suitably specified sparse Hamiltonians may be efficiently simulated. In recent work we have shown that the complex- ity of simulating evolution under a Hamiltonian is very close to linear in the evolution time. In addition, we have shown that in the general case of a sparse Hamiltonian the complexity grows slowly with respect to the number of qubits. In this chapter we review these results.

Journal of Physics A: Mathematical and Theoretical, 2011

We develop an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian, and accounts for all computational resources. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed. This per-bit or per-qubit costing of oracle calls reveals hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. We generalize the requirement of sparse Hamiltonians to being a sum of sparse Hamiltonians in various bases for which the transformation to a sparse Hamiltonian may be performed efficiently.

Science, 2001

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.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2013

A universal quantum simulator would enable efficient simulation of quantum dynamics by implementing quantum-simulation algorithms on a quantum computer. Specifically the quantum simulator would efficiently generate qubit-string states that closely approximate physical states obtained from a broad class of dynamical evolutions. I provide an overview of theoretical research into universal quantum simulators and the strategies for minimizing computational space and time costs. Applications to simulating many-body quantum simulation and solving linear equations are discussed

We show that by a suitable choice of a time dependent Hamiltonian, Deutsch's algorithm can be implemented by an adiabatic quantum computer. We extend our analysis to the Deutsch-Jozsa problem and estimate the required running time for both global and local adiabatic evolutions. Quantum computation and quantum information theory has attracted a great deal of attention in recent times. Inherently quantum mechanical systems can in principle be used to implement a wide variety of computational algorithms wth enhanced efficiency [1–3]. The principle of superposition in quantum mechanics, according to which a system can be in a linearly superposed state of more than one eigenstate, is the key to this increased efficiency. One of the first algorithms that was first proposed in this context is Deutsch's algorithm [4]. In this, one would like to determine whether a function f : {0, 1} → {0, 1} is constant or balanced, i.e. whether f (0) = f (1) or f (0) = f (1) using a quantum computer. The four possible outcomes of f are: f (0) = f (1) = 0 (constant) f (0) = f (1) = 1 (constant) f (0) = 0 , f (1) = 1 (balanced) f (0) = 1 , f (1) = 0 (balanced) Ordinarily, one has to determine both f (0) and f (1) to infer the nature of the function, since the knowledge of one does not shed light on the value of the other. However, it was shown that by applying a certain sequence of unitary operators ('gates') on a given initial quantum mechanical state, and then making just one measurement on the final state, the nature of the function f can be determined [4]. Recently, a new framework of quantum computation has been proposed, in which the series of gates referred to above is entirely replaced by a Hamiltonian which changes continuously with time. The Hamiltonian is so chosen that the state of the system is its ground state at all times (although the ground state itself is time dependent), and the system slowly evolves to a desired final state [5]. Several applications of this have been considered [6]. Using this framework, it was shown that Grover's search algorithm can be efficiently implemented [7]. In this paper, we show that Deutsch's algorithm can be implemented as well, by choosing a suitable initial state and a Hamiltonian which evolves that state. Then a single measurement of the final state suffices to determine whether the function f is constant or balanced. Finally, we show that the results can be extended to the Deutsch-Jozsa algorithm involving n-qubits. Let us begin with a 2-level system, e.g. a spin 1/2 particle, with the basis kets {|0, |1}. We define the 'initial' and 'final' Hamiltonians H 0 , H 1 respectively as: H 0 = I − |ψ 0 ψ 0 | (1) H 1 = I − |ψ 1 ψ 1 | (2) (3) where the initial and final state vectors are given respectively by: |ψ 0 = 1 √ 2 (|0 + |1) (4) |ψ 1 = α|0 + β|1 (5) * Electronic address: saurya,randy,gabor@theory.uwinnipeg.ca

2011

In this paper we present a simulation environment enhanced with parallel processing which can be used on personal computers, based on a high-level user interface developed on Mathematica c which is connected to C++ code in order to make our platform capable of communicating with a Graphics Processing Unit. We introduce the reader to the behavior of our proposal by simulating a quantum adiabatic algorithm designed for solving hard instances of the 3-SAT problem. We show that our simulator is capable of significantly increasing the number of qubits that can be simulated using classical hardware. Finally, we present a review of currently available classical simulators of quantum systems together with some justifications, based on our willingness to further understand processing properties of Nature, for devoting resources to building more powerful simulators.

2008

Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.

Proceedings of the Australasian Computer Science Week Multiconference

We review existing quantum computational methods for solving the Hamiltonian cycle problem in different computational frameworks such as quantum circuits, quantum walks and adiabatic quantum computation. Then we present a QUBO (quadratic unconstrained binary optimization) formulation, which is suitable for the adiabatic quantum computation for a D-Wave architecture. Further, we derive a physical Hamiltonian from the QUBO formulation and discuss its adequateness in the adiabatic framework. Finally, we discuss the complexity of running the Hamiltonian cycle QUBO on a D-Wave quantum computer, and compare it with existing quantum computational methods.