Quantum mixing of Markov chains for special distributions

Dunjko, Vedran; Briegel, Hans J.

doi:10.1088/1367-2630/17/7/073004

Cited by 12 publications

(11 citation statements)

References 21 publications

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“…Unitary k-designs capture the ability of a distribution to mimic the properties of the Haar measure in the sense that expectation values of polynomials of a certain order k are equal to those of the Haar measure. As pointed out above, they have a wide range of applications in quantum algorithm design [3,8,18], in quantum state and process tomography [19,20], and in notions of benchmarking [21] -basically as a powerful tool for partial derandomisation. Conceptually, they feature strongly in descriptions of equilibration, thermalisation and scrambling [2,9,29].…”

Section: Exact and Approximate Unitary Designsmentioning

confidence: 99%

“…Such random quantum processes, as they will be called in this work, again have applications in algorithms design, now quantum algorithms design [3,8,18]. They are used in quantum process tomography and low rank matrix recovery [19,20] and benchmarking [21], where they provide powerful tools to avoid significant overheads otherwise necessary with naive deterministic prescriptions.…”

Section: Introductionmentioning

confidence: 99%

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Mixing Properties of Stochastic Quantum Hamiltonians

Onorati

Buerschaper

Kliesch

et al. 2017

Commun. Math. Phys.

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Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.

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Section: Exact and Approximate Unitary Designsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixing Properties of Stochastic Quantum Hamiltonians

Onorati

Buerschaper

Kliesch

et al. 2017

Commun. Math. Phys.

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“…Markov chains (MCs) are central in computational approaches to physics [1], in computer science [2], and machine learning [3], and they form the crux of the ubiquitous Markov Chain Monte Carlo meth- should be possible [13] for the mixing problem. Such quadratic speed-ups have been demonstrated for various special cases of MCs [13][14][15][16][17][18], mostly relying on quantum walk [19,20] approaches. Quantum walks have also been utilized to speed-up simulated annealing [21][22][23], which often leads to the best runtimes in practice.…”

Section: Introductionmentioning

confidence: 99%

“…However, considering provable results for guaranteed mixing of general Markov chains, the best quantum algorithms achieve O √ δ −1 √ N , which falls short of the conjectured quadratic speedup, as it introduces the dependence on the system size N . Avoiding the O √ N dependence seems to be challenging, which further motivates investigating the settings with relaxed constraints, e.g., by restricting the MC family [13,18,24].…”

Section: Introductionmentioning

confidence: 99%

Faster quantum mixing for slowly evolving sequences of Markov chains

Orsucci

Briegel

Dunjko

2018

Quantum

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Markov chain methods are remarkably successful in computational physics, machine learning, and combinatorial optimization. The cost of such methods often reduces to the mixing time, i.e., the time required to reach the steady state of the Markov chain, which scales as δ −1 , the inverse of the spectral gap. It has long been conjectured that quantum computers offer nearly generic quadratic improvements for mixing problems. However, except in special cases, quantum algorithms achieve a run-time of O( √ δ −1 √ N ), which introduces a costly dependence on the Markov chain size N, not present in the classical case. Here, we re-address the problem of mixing of Markov chains when these form a slowly evolving sequence. This setting is akin to the simulated annealing setting and is commonly encountered in physics, material sciences and machine learning. We provide a quantum memory-efficient algorithm with a run-time of O( √ δ −1 4 √ N ), neglecting logarithmic terms, which is an important improvement for large state spaces. Moreover, our algorithms output quantum encodings of distributions, which has advantages over classical outputs. Finally, we discuss the run-time bounds of mixing algorithms and show that, under certain assumptions, our algorithms are optimal.Davide Orsucci: davide.orsucci@uibk.ac.at Hans J. Briegel: hans.briegel@uibk.ac.at Vedran Dunjko: v.dunjko@liacs.leidenuniv.nl ods [4]. In MC-based approaches the underlying objective is to produce samples from the steady state, i.e., the stationary distribution of a given MC. The MC is constructed so that this distribution encodes the solution of the problem at hand. The solution can then be reached by "mixing", i.e., by applying the MC transitions many times. For some problems, mixing processes constitute the fastest known classical solving algorithms, and play a vital role, e.g., in the Metropolis-Hastings methods [5], periodic Gibbs sampling [6], and Glauber dynamics [7].The fundamental parameter governing the time complexity of MC-based algorithms is thus the mixing time, that is, the number of steps required to attain stationarity. In most applications the MC is ergodic, i.e., has a unique stationary distribution, and timereversible, i.e., satisfies detailed balance [8,9]. The mixing time is tightly related to the spectral gap δ of the MC 1 and is bounded by Ω(δ −1 ) [10].Oftentimes direct mixing can be computationally prohibitive and thus heuristic methods, such as simulated annealing [11,12], are employed. Here one constructs a sequence of Markov chains which, for instance, encode the Gibbs (thermal) distributions at gradually decreasing values of the temperature, where the target distribution is specified by the final MC, i.e., the final temperature. Intuitively, this process increases efficiency by avoiding local minima, although the performance is typically not guaranteed. In simulated annealing, the neighbouring chains in the sequence are similar, in other words, the sequence is slowly evolving.The emergence of quantum computation offers a new possibil...

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“…We prove this general result by studying the mixing of quantum walks on Erdös-Renyi random networks: networks of n nodes with an edge existing between any two nodes with probability p independently, denoted as G(n, p) [22,23]. It is important to note that our problem differs from designing quantum algorithms for classical mixing: preparing a coherent encoding of the stationary state of a classical random walk [24][25][26]. Such problems involve running quantum algorithms for finding a marked node, known as quantum spatial search, in reverse.…”

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confidence: 99%

How Fast Do Quantum Walks Mix?

2020

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The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications, is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the quantum mixing time. However this quantity is only known for a handful of specific networks. In this letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e. the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example for dense random networks, where p is a constant, we show that the quantum mixing time is O(n 3/2+o(1) ). Besides opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.The quantum dynamics of any discrete system can be captured by a quantum walk on a network, which is a universal model for quantum computation [1]. Besides being a useful primitive to design quantum algorithms [2][3][4][5][6], quantum walks are a powerful tool to model transport in quantum systems such as the transfer of excitations in light-harvesting systems [7][8][9]. Studying the long-time dynamics of quantum walks on networks is crucial to the understanding of these diverse problems. As quantum evolutions are unitary and hence distancepreserving, quantum walks never converge to a limiting distribution, unlike their classical counterpart. However, given a network of n nodes, one can define the limiting distribution of quantum walk on the network as the long-time average probability distribution of finding the walker in each node [10]. Of particular interest is the quantum mixing time: starting from some initial state, the minimum time after which the underlying quantum walk remains close to its limiting distribution.The importance of the problem of mixing for quantum walks cannot be overstated: this is at the heart of quantum speedups for a number of quantum algorithms [2,11] and is also key to demonstrating the equivalence between the standard (circuit) and Hamiltonian-based models of quantum computation [12,13]. Unfortunately, no general result exists for quantum mixing time on networks: it has been estimated for a handful of specific graphs (graphs and networks are ...

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Quantum mixing of Markov chains for special distributions

Cited by 12 publications

References 21 publications

Mixing Properties of Stochastic Quantum Hamiltonians

Mixing Properties of Stochastic Quantum Hamiltonians

Faster quantum mixing for slowly evolving sequences of Markov chains

How Fast Do Quantum Walks Mix?

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