Improved error bounds for the adiabatic approximation

Cheung, Donny; Hoyer, Peter Friedrich; Wiebe, Nathan

doi:10.1088/1751-8113/44/41/415302

Cited by 33 publications

(47 citation statements)

References 41 publications

(121 reference statements)

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“…1 for all s. It is shown in equations (30)-(32) of [16] that this observation leads us to the conclusion that equation (3) is, up to a constant multiple, an asymptotic upper bound for equation (6).…”

Section: Theorymentioning

confidence: 77%

“…We know from previous work that 15], and asymptotically tight expressions are known for E ν in the m = 0 case [8,16]. We therefore begin with this case to illustrate how our phase interference effect can be utilized.…”

Section: Theorymentioning

confidence: 99%

“…We also assume that the Hamiltonian is differentiable m + 2 times and that each derivative is bounded for all T . These last restrictions are put in place order to prevent issues that arise for Hamiltonians resembling that of the Marzlin-Sanders counterexample [11,16].…”

Section: Theorymentioning

confidence: 99%

“…Our technique naturally lends itself to Hamiltonians that couple the ground state to only one excited state, such as the search Hamiltonian given in equation (16). If the total error E is dominated by several transitions, this technique can still be adapted to approximately cancel multiple transitions simultaneously.…”

Section: Two-qubit Gatementioning

confidence: 99%

“…Final error amplitude |E| as a function of T for the search Hamiltonian(16) using N = 16 and equation(17).…”

mentioning

confidence: 99%

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Improved error-scaling for adiabatic quantum evolutions

Wiebe

Babcock

2012

New J. Phys.

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We present a new technique that improves the scaling of the error in the adiabatic approximation with respect to the evolution duration, thereby permitting faster transfer at a fixed error tolerance. Our method is conceptually different from previously proposed techniques: it exploits a commonly overlooked phase interference effect that occurs predictably at specific evolution times, suppressing transitions away from the adiabatically transferred eigenstate. Our method can be used in concert with existing adiabatic optimization techniques, such as local adiabatic evolutions or boundary cancelation methods. We perform a full error analysis of our phase interference method along with existing boundary cancelation techniques and show a tradeoff between error-scaling and experimental precision. We illustrate these findings using two examples, showing improved error-scaling for an adiabatic search algorithm and a tunable two-qubit quantum logic gate. Acknowledgments 15 References 15The adiabatic approximation underpins many important present-day and future applications, such as stimulated rapid adiabatic passage (STIRAP) [1,2], coherent control of chemical reactions [3] and quantum information processing (QIP) [4,5]. This approximation asserts that a system will remain in an instantaneous eigenstate of a time-varying Hamiltonian if the time-variation happens slowly enough. Errors in this approximation correspond to transitions away from the instantaneous ('adiabatically transferred') eigenstate. For highperformance applications, it is not always practical to minimize errors by slowing things down. Ambitious future technologies, such as quantum computing devices, will demand simultaneous maximization of both accuracy and speed. In this paper, we investigate a phase cancelation effect that appears during an adiabatic evolution and can be exploited to polynomially reduce the probability of a given transition at fixed maximum evolution time. This can lead to speed increases at fixed error probability. Unlike alternative methods that obtain improvements by modifying the adiabatic path [6,7], our technique chooses the evolution time so that destructive interference suppresses the transition. Furthermore, this phase cancelation effect can be exploited to improve existing adiabatic error reduction strategies such as local adiabatic evolutions or boundary cancelation methods. We provide an error analysis of our method and conclude that the accuracy improvements come at the price of increasingly precise knowledge of the time-dependent Hamiltonian; this implies that accuracy is an important and quantifiable resource for quantum protocols utilizing adiabatic passage. Adiabatic approximationThe adiabatic approximation states that if we consider the evolution of a quantum system under a time-dependent Hamiltonian that varies sufficiently slowly in time, then the time evolution operator approximately maps instantaneous eigenstates of the Hamiltonian at t = 0

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Section: Theorymentioning

confidence: 77%

Section: Theorymentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

Section: Two-qubit Gatementioning

confidence: 99%

“…Final error amplitude |E| as a function of T for the search Hamiltonian(16) using N = 16 and equation(17).…”

mentioning

confidence: 99%

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Improved error-scaling for adiabatic quantum evolutions

Wiebe

Babcock

2012

New J. Phys.

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Security Comparisons and Performance Analyses of Post-quantum Signature Algorithms

Raavi

Wuthier

Chandramouli

et al. 2021

Applied Cryptography and Network Security

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A study of heuristic guesses for adiabatic quantum computation

Perdomo-Ortiz

Venegas-Andraca

Aspuru‐Guzik

2010

Quantum Inf Process

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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.

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Improved error bounds for the adiabatic approximation

Cited by 33 publications

References 41 publications

Improved error-scaling for adiabatic quantum evolutions

Improved error-scaling for adiabatic quantum evolutions

Security Comparisons and Performance Analyses of Post-quantum Signature Algorithms

A study of heuristic guesses for adiabatic quantum computation

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