The performance of the quantum adiabatic algorithm on spike Hamiltonians

Kong, Linghang; Crosson, Elizabeth

doi:10.1142/s0219749917500113

Cited by 13 publications

(16 citation statements)

References 32 publications

(40 reference statements)

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“…Without access to quantum hardware, comparison of SQA and QA is either limited to small system sizes where QA Hamiltonians can be exactly diagonalized ( 50 qubits), or to models for which analytical solutions of the quantum system are known (such as the spike problem we study here). We remark that the spike and related objective functions have the subject of recent analytic work [24,4], and that there have also been numerical studies of SQA [10,3], with findings that are consistent with our main result.…”

Section: Previous Worksupporting

confidence: 90%

“…Our proof does not bound the convergence time for SQA α + ζ > 1/2, although QA does work for some values of (α, ζ) in this range, such as when ζ = 0, α = O(1) [24] or α + 2ζ < 1 [4]. We conjecture that SQA will be efficient for these values as well (which is supported by numerical evidence), though this will require extensions of the present techniques.…”

Section: Discussionmentioning

confidence: 78%

“…for s < s * the ground state probability mass is concentrated on one side of the barrier, while for s > s * the opposite occurs, and for s = s * the probability mass on both sides of the barrier is O(1)). While a general understanding of what barriers admit such a tunneling description has not yet been found, there are many examples for which this is known to occur [16,24,27]. Assume that the QA spectral gap at s * is ∆ = 1/ poly(n), so that QA will be able to pass through the barrier efficiently.…”

Section: Discussionmentioning

confidence: 99%

“…. , n and k ∈ I S , because the ground state wave function is centered on the spike and the excited state wave functions have a greater spread, which can be seen from the explicit form of the spikeless eigenfunctions given in [24]. Now we define g i := t i − t i−1 and relabel the sum of t 1 , .…”

Section: Efficient Convergence Of Sqa For the Spike Cost Functionmentioning

confidence: 99%

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Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Crosson

Harrow

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA, also known as adiabatic optimization) finds the minimum of some cost functions exponentially more quickly than Simulated Annealing (SA), which is arguably a classical analogue of QA.One such cost function is the simple "Hamming weight with a spike" function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While this problem can be solved by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [16] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [28] to include barriers with width n ζ and height n α for ζ + α ≤ 1/2. This advantage could be explained in terms of quantum-mechanical "tunneling."Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling.Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general. * Caltech

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Section: Previous Worksupporting

confidence: 90%

Section: Discussionmentioning

confidence: 78%

Section: Discussionmentioning

confidence: 99%

Section: Efficient Convergence Of Sqa For the Spike Cost Functionmentioning

confidence: 99%

See 2 more Smart Citations

Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Crosson

Harrow

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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“…The ease of simulation does not imply that these are computationally easy problems without this extra information. Quantum search is extremely sensitive to the setting of the parameters [22,23], and may be more difficult to implement experimentally than other permutation symmetric problem Hamiltonians, such as the 'spike' problems discussed in [32][33][34][36][37][38]. While spike problems cannot yield a full quantum speedup, experimental implementations could provide a powerful tool for understanding the physics of large quantum superpositions in a computational setting.…”

Section: Discussionmentioning

confidence: 99%

Practical designs for permutation-symmetric problem Hamiltonians on hypercubes

et al. 2019

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We present a method to experimentally realize large-scale permutation-symmetric Hamiltonians for continuous-time quantum protocols such as quantum walk and adiabatic quantum computation. In particular, the method can be used to perform an encoded continuous-time quantum search on a hypercube graph with 2 n vertices encoded into 2n qubits. We provide details for a realistically achievable implementation in Rydberg atomic systems. Although the method is perturbative, the realization is always achieved at second order in perturbation theory, regardless of the size of the mapped system. This highly efficient mapping provides a natural set of problems which are tractable both numerically and analytically, thereby providing a powerful tool for benchmarking quantum hardware and experimentally investigating the physics of continuous-time quantum protocols.Quantum computing based on continuous time evolution rather than discrete gate operations offers a promising route for practical near-term quantum computing. This approach has a wide variety of natural applications including in finance [1-3], aerospace [4], machine learning [5][6][7], theoretical computer science [8], mathematics [9,10], decoding of communications [11] and computational biology [12]. Moreover, experimental quantum annealing has proven highly successful recently [13][14][15][16][17].While continuous-time quantum computing shows great promise, there are few known methods to experimentally implement test problems that can be used to prove the performance of hardware. For quantum computing based on discrete gates, solving unstructured search by Grover's algorithm [18] provides a quadratic speedup over any classical algorithm-the best possible speedup as proven by Bennett et al. [19]. There are continuous-time variants of quantum search algorithms which can obtain the same optimal speedup for both adiabatic quantum computation [20,21] and continuoustime quantum walk [22]. It has recently been shown that these are the two extremes of a continuum of protocols that all achieve the optimal quantum speedup [23].Continuous-time search algorithms are not easy to experimentally implement when encoded into qubits. In contrast, Grover's original algorithm can be efficiently decomposed into quantum gates [24]. A naive decomposition of the continuous-time search problem yields exponentially many terms coupling all possible subsets of qubits. To date, the largest qubit-encoded continuous-time quantum walks have been performed on two qubits [25,26]; neither implemented a search algorithm. Larger encoded discrete-time quantum walks and quantum searches have been experimentally realized [27,28], and alternative encodings have been explored in [29,30].Because of the difficulty of implementing qubitencoded continuous-time quantum search algorithms, this has been considered a toy problem: useful as a theoretical tool, but not practical experimentally. The search Hamiltonian can always be represented in a permutation symmetric basis, by transforming the marked state to either t...

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A double-slit proposal for quantum annealing

2019

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We formulate and analyze a double-slit proposal for quantum annealing, which involves observing the probability of finding a two-level system (TLS) undergoing evolution from a transverse to a longitudinal field in the ground state at the final time t f . We demonstrate that for annealing schedules involving two consecutive diabatic transitions, an interference effect is generated akin to a double-slit experiment. The observation of oscillations in the ground state probability as a function of t f (before the adiabatic limit sets in) then constitutes a sensitive test of coherence between energy eigenstates. This is further illustrated by analyzing the effect of coupling the TLS to a thermal bath: increasing either the bath temperature or the coupling strength results in a damping of these oscillations. The theoretical tools we introduce significantly simplify the analysis of the generalized Landau-Zener problem. Furthermore, our analysis connects quantum annealing algorithms exhibiting speedups via the mechanism of coherent diabatic transitions to near-term experiments with quantum annealing hardware.

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The performance of the quantum adiabatic algorithm on spike Hamiltonians

Cited by 13 publications

References 32 publications

Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Practical designs for permutation-symmetric problem Hamiltonians on hypercubes

A double-slit proposal for quantum annealing

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