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arXiv:1407.8183 · Physical Review A

Adiabatic quantum optimization in the presence of discrete noise: Reducing the problem dimensionality

Salvatore Mandrà, Gian Giacomo Guerreschi, Alán Aspuru‐Guzik

Published Dec 10, 2015• 17 citations
arXivHTMLPDF

Abstract

Adiabatic quantum optimization is a procedure to solve a vast class of optimization problems by slowly changing the Hamiltonian of a quantum system. The evolution time necessary for the algorithm to be successful scales inversely with the minimum energy gap encountered during the dynamics. Unfortunately, the direct calculation of the gap is strongly limited by the exponential growth in the dimensionality of the Hilbert space associated to the quantum system. Although many special-purpose methods have been devised to reduce the effective dimensionality, they are strongly limited to particular classes of problems with evident symmetries. Moreover, little is known about the computational power of adiabatic quantum optimizers in real-world conditions. Here we propose and implement a general purposes reduction method that does not rely on any explicit symmetry and which requires, under certain general conditions, only a polynomial amount of classical resources. Thanks to this method, we are able to analyze the performance of ``nonideal'' quantum adiabatic optimizers to solve the well-known Grover problem, namely the search of target entries in an unsorted database, in the presence of discrete local defects. In this case, we show that adiabatic quantum optimization, even if affected by random noise, is still potentially faster than any classical algorithm.

DOI
10.1103/physreva.92.062320

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Adiabatic quantum optimization is a procedure to solve a vast class of optimization problems by slowly changing the Hamiltonian of a quantum system. The evolution time necessary for the algorithm to be successful scales inversely with the minimum energy gap encountered during the dynamics. Unfortunately, the direct calculation of the gap is strongly limited by the exponential growth in the dimensionality of the Hilbert space associated to the quantum system. Although many special-purpose methods have been devised to reduce the effective dimensionality, they are strongly limited to particular classes of problems with evident symmetries. Moreover, little is known about the computational power of adiabatic quantum optimizers in real-world conditions. Here we propose and implement a general purposes reduction method that does not rely on any explicit symmetry and which requires, under certain general conditions, only a polynomial amount of classical resources. Thanks to this method, we are able to analyze the performance of ``nonideal'' quantum adiabatic optimizers to solve the well-known Grover problem, namely the search of target entries in an unsorted database, in the presence of discrete local defects. In this case, we show that adiabatic quantum optimization, even if affected by random noise, is still potentially faster than any classical algorithm.
In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on n -bits in time poly(n)2n/2 . In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of Ω(2n/2) has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.
Adiabatic Quantum Computing (AQC) is a powerful framework for designing quantum algorithms, yet its practical implementation is often hampered by the difficulty of engineering precise, continuous time-dependent Hamiltonians or the substantial computational overhead of time discretisation. This thesis develops alternative adiabatic frameworks based on dissipative and stochastic quantum evolution to circumvent these challenges.The core methodology introduces novel dynamics where discrete quantum operations, such as time-independent Hamiltonian evolution with Poisson-distributed phase randomisation or randomised unitary applications (qubitization), replace continuous evolution. This stochastic approach allows the system’s average behaviour to be rigorously analysed using new, generalised adiabatic theorems, including the development of novel quantitative bounds for unbounded Hamiltonians and specialised dissipative generators.Algorithmically, this work demonstrates that these discrete, stochastic procedures can achieve the optimal asymptotic time complexity of continuous AQC while mitigating discretisation costs. Specifically, adapted-schedule implementations are shown to yield optimal scaling for the Quantum Linear Systems Problem (QLSP), achieving complexity proportional to the condition number O(κ)O(\kappa)O(κ), and O(N/M)O(\sqrt{N/M})O(N/M​) scaling for unstructured search. By establishing highly efficient, implementable alternatives to continuous AQC, this research offers a significant pathway toward robust and scalable quantum computation. The study also investigates the general family of diagonal Hamiltonians for optimisation problems, revealing that finding the optimal scheduling required for peak performance is, in general, an NP-hard problem.
We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover's problem [L. K. Grover, Phys. Rev. Lett. 79, 325 (1997)] of finding a marked item inside of an unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters and is not exponentially small. This allows a significant reduction of the searching time, which is proportional to the number of qubits. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment, and we discuss alternate ways to solve NPNPNP-complete problems.
Abstract Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.
Analog models of quantum information processing, such as adiabatic quantum computation and analog quantum simulation, require the ability to subject a system to precisely specified Hamiltonians. Unfortunately, the hardware used to implement these Hamiltonians will be imperfect and limited in its precision. Even small perturbations and imprecisions can have profound effects on the nature of the ground state. Here we consider an imperfect implementation of adiabatic quantum optimization and show that, for a widely applicable random control noise model, quantum stabilizer encodings are able to reduce the effective noise magnitude and thus improve the likelihood of a successful computation or simulation. This reduction builds upon two design principles: summation of equivalent logical operators to increase the energy scale of the encoded optimization problem, and the inclusion of a penalty term comprising the sum of the code stabilizer elements. We illustrate our findings with an Ising ladder and show that classical repetition coding drastically increases the probability that the ground state of a perturbed model is decodable to that of the unperturbed model, while using only realistic two-body interaction. Finally, we note that the repetition encoding is a special case of quantum stabilizer encodings, and show that this in principle allows us to generalize our results to many types of analog quantum information processing, albeit at the expense of many-body interactions.
Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian.Such approaches have enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure.However, the most efficient of those procedures is still asymptotically suboptimal by a factor of log(κ).Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete-time evolutions.In combination with the qubitized quantum walk, our discrete adiabatic theorem gives a speed-up for all adiabatic algorithms.Here, we use this combination to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity.Our O[κ log(1/ )] complexity is also optimal in terms of the combined scaling in κ and the precision .Compared to existing suboptimal methods, our algorithm is simpler and easier to implement.Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.